IMPROVEMENTS IN FINE-SCALE ESTIMATION
AND EVALUATION OF GEOGRAPHIC VARIABLES
USING CLIMATE DATA IN EAST AFRICA
By
Sarah L. Hession

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Geography
2011

ABSTRACT
IMPROVEMENTS IN FINE-SCALE ESTIMATION AND EVALUATION OF
GEOGRAPHIC VARIABLES USING CLIMATE DATA IN EAST AFRICA
By
Sarah L. Hession

Global environmental change has surfaced as a critical issue to both the
scientific community and the general public. One aspect of particular concern involves
climate change, which will exert impacts on ecosystems and economies, presenting
considerable challenge to human adaptation. In Africa, a continent that is vulnerable
due to multiple stressors and low adaptive capacity, climate change is expected to
significantly affect both people and ecosystems. Adaptation strategies are being
developed using information from studies that evaluate the impacts of climate variability
and climate change in Africa. Recommendations are made for local development of
adaptation strategies due to the heterogeneity of climate change and its effects on East
Africa’s climate. However, global climate change models are coarse in scale and mask
much of the local variation in regional climate, indicating the need for higher resolution
climate data. This dissertation addresses this need by comparing spatially explicit
statistical methods of interpolation and prediction, both theoretically and empirically;
expanding upon the method of universal kriging by incorporating complex feedback
relationships that may produce simultaneity between precipitation and its covariates;
and evaluating precipitation patterns over space in East Africa through a case study.
Mechanisms of precipitation have been considered in detail, expanding upon many
other spatially explicit applications of prediction methods to date. Further, spatially

explicit inferential regression models have been developed to better understand spatial
patterns and variability in East African precipitation. Predicted maps of precipitation,
generated at a resolution of 1 kilometer, accurately reflect the mesoscale influences of
topography and the presence of large water bodies (i.e., Lake Victoria) as well as the
seasonal influences of the passing of the intertropical convergence zone (ITCZ). In
terms of prediction, the spatially explicit methods considered herein clearly
outperformed a global data set (i.e., the CRU TS 3.1) in terms of error and ability to
reflect local variability. The method of local ordinary kriging generally outperformed the
multivariate kriging techniques, indicating that precipitation patterns in areas of high
topographic variability, such as East Africa, may be modeled as well or better using
local search neighborhoods in the kriging process rather than using complex
multivariate regression models. However, additional work to improve the multivariate
regression models and overall levels of correlation are expected to yield improved
prediction results. Furthermore, the case study successfully demonstrated that the
newly developed method of universal kriging with instrumental variables performs
similarly to other standard methods of estimation, and perhaps better in the presence of
significant measurable simultaneity.

Copyright by
SARAH L. HESSION
2011

DEDICATION

This dissertation is dedicated to my children, Maura Elizabeth Hession and
Patrick Joseph Hession III, who have endured this process with me and have made it all
worthwhile.

v

ACKNOWLEDGEMENTS

I would like to acknowledge my committee for their direction and insight into my
dissertation: Dr. Robert T. Walker (chair), Dr. Ashton Shortridge, Dr. Jennifer Olson, and
Dr. Jeff Andresen. I would also like to acknowledge the following for their invaluable input and
direction: Dr. David Campbell, Dr. Nathan Moore, Dr. Gopal Alagarswamy, and Dr. John Kern.
The work presented herein was supported by The Graduate School at Michigan State
University, the Climate Land Interaction Program (CLIP; An Integrated Analysis of Regional
Land-Climate Interactions, National Science Foundation Award Number BCS0308420) and East
Africa Climate Land Interaction Program – Savanna Ecosystems (EACLIPSE; Dynamic
Interactions Among People, Livestock, and Savanna Ecosystems under Climate Change,
National Science Foundation Award Number BCS0709671).

vi

TABLE OF CONTENTS
LIST OF TABLES ......................................................................................................... ix
LIST OF FIGURES ........................................................................................................ x
Chapter 1 Introduction ..................................................................................................1
1.1 Modeling a changing climate ..............................................................................1
1.2 In search of higher resolution climate data .........................................................9
1.2.1 Downscaling of GCM results..................................................................10
1.2.2 Use of statistical techniques with measured data ..................................11
1.3 Links and contributions to the Geographic traditions ........................................14
1.4 Research objectives .........................................................................................16
Chapter 2 Literature Review .......................................................................................19
2.1 Scales and mechanisms of precipitation in East Africa ....................................19
2.2 A review of statistical methods using in the analysis of precipitation
data ..................................................................................................................21
2.2.1 Kriging....................................................................................................21
2.2.2 Spatial regression ..................................................................................24
2.3 Historical use of methods to model climate patterns using
measured data..................................................................................................26
2.4 Gaps in the literature and contributions to address them .................................34
Chapter 3 Statistical Theory and Derivations .............................................................42
3.1 Background ......................................................................................................42
3.1.1 Kriging....................................................................................................42
3.1.2 Spatial regression ..................................................................................48
3.1.3 Summary of theoretical models and estimators .....................................56
3.2 Theoretical mapping between statistical approaches .......................................59
3.3 Modeling of feedback simultaneity in a spatial setting ......................................67
3.3.1 Simultaneous equations spatial regression model .................................68
3.3.2 An extension to universal kriging ...........................................................71
Chapter 4 Case Study ................................................................................................74
4.1 Study area description ......................................................................................74
4.2 Data ..................................................................................................................77
4.3 Methods ............................................................................................................83
4.3.1 Model selection ......................................................................................86
4.3.2 Consideration of other candidate OLS models ......................................91
4.3.3 Prediction of spatial patterns in average monthly precipitation ..............92
4.3.4 Comparison and evaluation of map accuracy ......................................102

vii 
 

Chapter 5 Hypothesis Testing ..................................................................................110
5.1 Introduction .....................................................................................................110
5.1.1 Study area .............................................................................................110
5.1.2 Identification of dependent variable and selection of representative
Data .......................................................................................................111
5.1.3 Identification of independent or descriptive variables ............................113
5.2 Interpretation and hypothesis testing ..............................................................116
5.2.1 Statement of hypotheses .......................................................................116
5.2.2 Results with interpretation .....................................................................120
5.3 Hausman test for simultaneity .........................................................................135
Chapter 6 Conclusions and Future Research ..........................................................137
6.1 Findings and conclusions ...............................................................................138
6.2 Gaps in the literature and contributions to address them ...............................140
6.2.1 Theoretical and methodological contributions ......................................141
6.2.2 Contributions to Regional Geography of Africa and
Geography as a whole .........................................................................143
6.3 Limitations ......................................................................................................143
6.3.1 Limitations in data in modeling .............................................................144
6.3.2 Limitations in statistical methodology ...................................................147
6.4 Future research ..............................................................................................148

APPENDIX Case Study Results .............................................................................151
BIBLIOGRAPHY .......................................................................................................172

viii 
 

LIST OF TABLES
Table 2-1

Summary of statistical methods for modeling climate pattern ................37

Table 3-1

Summary of common notation ...............................................................44

Table 3-2

Summary of theoretical models underlying kriging and spatial regression
approaches ............................................................................................57

Table 3-3

Summary of kriging and spatial regression estimators...........................58

Table 3-4

Summary of multipliers for each estimation type ...................................67

Table 4-1

Summary of independent variables used in OLS regression
analysis ..................................................................................................85

Table 4-2

Summary of OLS regression models ranked by AIC .............................89

Table 4-3

Summary of predicted precipitation amounts for each
prediction method ..................................................................................95

Table 4-4

Comparison of root mean squared errors ............................................103

Table 5-1

Summary of independent variables .....................................................115

Table 5-2

Summary of OLS regression models ranked by AIC ...........................123

Table 5-3

Summary of regression modeling results .............................................127

Table A-1

Summary of regression modeling results .............................................152

ix 
 

LIST OF FIGURES
Figure 1-1

Relationships between the Geographic traditions represented in this
research. ................................................................................................15

Figure 4-1

Location of the study area within East Africa (CLIP region)
and Africa...............................................................................................74

Figure 4-2

Meteorological station locations and study area boundary ....................75

Figure 4-3

Monthly precipitation (mm*10) averaged over all
meteorological stations within the study area ........................................78

Figure 4-4

Summary of the number of meteorological stations
measured by year with the study area ...................................................79

Figure 4-5

Monthly precipitation for the years 1980 through 1985 plotted
with long-term monthly averages ...........................................................80

Figure 4-6

Average monthly precipitation maps for April 1985 generated
using LOK, UK, UKIV, and regression techniques .................................97

Figure 4-7

Average monthly precipitation map for April 1985 generated using UK
without distance bands ..........................................................................98

Figure 4-8

Maps of cross validation residuals in April 1985 for LOK,
UK, UKIV, and regression residuals ....................................................105

Figure 4-9

Maps of CRU residuals in 1985 ...........................................................108

Figure 5-1

Location of the case study area within East Africa
(CLIP region) .......................................................................................111

Figure 5-2

Map of elevation (1 km resolution) with meteorological
station locations and study area boundary...........................................114

Figure 5-3

Monthly precipitation (mm*10) averaged over all
meteorological stations within the study area ......................................116

Figure 6-1

Comparison of meteorological station elevations and elevations
throughout the study area ....................................................................146

Figure A-1

January 1984 Average monthly precipitation maps .............................154

Figure A-2

April 1984 Average monthly precipitation maps ...................................155

x 
 

Figure A-3

August 1984 Average monthly precipitation maps ...............................156

Figure A-4

November 1984 Average monthly precipitation maps .........................157

Figure A-5

January 1985 Average monthly precipitation maps .............................158

Figure A-6

April 1985 Average monthly precipitation maps ...................................159

Figure A-7

August 1985 Average monthly precipitation maps ...............................160

Figure A-8

November 1985 Average monthly precipitation maps .........................161

Figure A-9

January 1984 Maps of significant cross validation residuals................162

Figure A-10 April 1984 Maps of significant cross validation residuals .....................163
Figure A-11 August 1984 Maps of significant cross validation residuals .................164
Figure A-12 November 1984 Maps of significant cross validation residuals ............165
Figure A-13 January 1985 Maps of significant cross validation residuals................166
Figure A-14 April 1985 Maps of significant cross validation residuals .....................167
Figure A-15 August 1985 Maps of significant cross validation residuals .................168
Figure A-16 November 1985 Maps of significant cross validation residuals ............169
Figure A-17 Maps of CRU residuals in 1984 ...........................................................170
Figure A-18 Maps of CRU residuals in 1985 ...........................................................171

xi 
 

Chapter 1
Introduction

1.1 Modeling a changing climate
Global environmental change has surfaced as a critical issue to both the
scientific community and the general public. One aspect of particular concern involves
climate change, which will exert impacts on ecosystems and economies, presenting
considerable challenge to human adaptation. Widespread impacts due to climate
change are expected in many regions of the world (Giorgi 2001; Hulme 1998; Lobell et
al. 2008).
In Africa, climate change is expected to result in warmer temperatures and
changes in precipitation patterns, significantly affecting both people and ecosystems
(Moore et al. 2005, 2006). According to the Fourth Assessment Report of the
Intergovernmental Panel on Climate Change (IPCC; Boko et al. 2007), “Africa is one of
the most vulnerable continents to climate change and climate variability, a situation
aggravated by the interaction of ‘multiple stresses’, occurring at various levels, and low
adaptive capacity.” Some of these stresses and vulnerabilities are described below, with
a focus on eastern Africa.
In East Africa, studies have projected warmer and wetter conditions (Hulme et al.
2001, Moore et al. in review) with possible decreases in precipitation during the northern
hemisphere summer months of June, July, and August (Giorgi 2001; Hulme 1998).
Warmer temperatures combined with higher potential evapotranspiration and variable

1

precipitation may already be causing decreased productivity of much of the East African
savanna (Moore et al. 2005).
Total rainfall in most areas has not significantly changed in the last 50 years,
although increases in interannual variability have been observed after the 1970s (Boko
et al. 2007, Fauchereau et al. 2003, Richard et al. 2001). Increased variability in
precipitation is due, in part, to changing climatic patterns. Two such climatic patterns
are the El-Nino Southern Oscillation (ENSO) and the Indian Ocean Dipole. ENSO has
experienced a marked decrease in frequency, concurrent with an increase in the
intensity and duration of this phenomenon (Camberlin et al. 2001, Mukabana and Pielke
1996). ENSO has a large impact on the variability of precipitation patterns; the impact
is different based on the phase of ENSO that is occurring. The two phases of ENSO,
the negative or warm phase (El Nino) and the positive or cool phase (La Nina),
generally influence the timing of seasonal precipitation in opposite ways (Majugu and
Magezi 1985). In La Nina years, the secondary rainy season generally occurs early in
the season (i.e., the rainy season begins in August or early September and ends in
October or early November). The end of this early rainy season is usually followed by a
marked decrease in rainfall, resulting in droughts throughout much of the region
(Majugu and Magezi 1985). Conversely, a typical El Nino year is characterized by a
late onset of the second rainy season (i.e., the rainy season begins in late September or
early October and ends in late November to early to mid December), with intense rains
and potential flooding in parts of the region (Majugu and Magezi 1985).
Distinct from ENSO, an intensifying dipole rainfall pattern has been identified that
occurs in a decadal pattern. The dipole is marked by increasing rainfall over the

2

northern sector and declining amounts over the southern sector (Boko et al. 2007,
Schreck and Semazzi 2004). This increased variability of precipitation over both time
and space greatly impacts water availability, crops and vegetation, farmers and others
who rely on these crops, livestock and wildlife that rely on vegetation and water
availability, as described below.
The eastern African region, like many regions within the tropics, experiences a
disproportionate share of climate extremes (Majugu and Magezi 1985). Changes in the
frequency and magnitude of extreme events, such as droughts and floods, have major
implications for numerous Africans. For many years, droughts caused “human
migration, cultural separation, population dislocation and the collapse of prehistoric and
early historic societies” (Pandey et al. 2003). Major impacts of drought include
decreased food production and shortages of potable water, which will be discussed
next.
The long-term relationship between climate and vegetation in the region has
been examined by Mworia-Maitima (1991, 1999) and colleagues, who show a clear
relationship between plant species composition and climate. Future climate change
may dramatically affect agricultural production across space and time; for example, the
length of the growing season in East Africa could increase in some areas and decrease
in others primarily as a result of altered precipitation amounts and timing (Jones and
Thornton 2003).
Scholes and Biggs (2004) refer to Sub-Saharan Africa as the “food crisis
epicenter of the world,” concluding that food security will worsen during the first half of
the twenty-first century due to projected changes in climate. Further challenges in

3

ensuring food security around the world, particularly in Africa, are expected due to the
impacts of climate change on agriculture (Thornton et al. 2009). In East Africa, much of
the population is largely dependent on rainfed cropping and pasture (approximately
80% of the population are agriculturalists); consequently, changes in productivity are
expected to have a profound effect (Thornton et al. 2009)]. Vulnerability mapping has
been used to identify areas that are presently vulnerable, both environmentally and
socially, that are expected to be severely affected by climate change (Thornton et al.
2009). Parts of eastern Africa, such as arid-semiarid rangelands and coastal regions,
are included as “hotspots” in these vulnerability maps (Thornton et al. 2009).
Climate change and variability are also expected to impose additional pressures
on water availability, water accessibility, and water demand in Africa (Boko et al. 2007).
Currently, about 25% of the African population experiences high water stress.
Furthermore, one-third of the people in Africa live in drought-prone areas and are
vulnerable to the effects of droughts (World Water Forum 2000). Although access to
freshwater was improved during the 1990s, access to improved water supplies was
available to only about 62% of the African population in 2000 (Boko et al. 2007,
WHO/UNICEF 2000). Women would be particularly affected by changes in the location,
quality and quantity of available domestic water as well as fuelwood, declining crop
yields, and changes in animal health and productivity (Wangui 2003, 2004).
For many Africans, adaptation to climate change is not an option but a necessity
(e.g., Thornton et al. 2006). Increasing numbers of studies are being carried out to
evaluate the impacts of climate variability and climate change in Africa, and to develop
adaptation strategies (Boko et al. 2007). Adaptation has been shown to be successful

4

and sustainable when conducted together with effective government input,
consideration of civil and political rights, and literacy (Brooks et al. 2005).
Thornton et al. (2009) suggest that adaptation strategies to climate change be
developed locally rather than for large, spatially contiguous regions, due to the
heterogeneity of climate change and its effects. Crop yield projections are highly
variable due, in part, to large variations in soils, topography, and current average
temperatures and corresponding variability in projected rainfall and temperatures
(Thornton et al. 2009). Consequently, adaptation strategies would best be informed by
higher-resolution climate data and developed through localized, community-based
efforts (Thornton et al. 2009).
Development of a range of adaptation strategies may improve the ability of local
populations to cope with climate change. Recent evidence from East Africa suggests
that diversification of coping mechanisms at the household level is greatest among the
poorest and wealthiest sections of society (Campbell 1999, Smucker and Wisner 2008).
Wealthy households have better access to non-farm activities and remittances that
reduce reliance on local resources during drought. The poorest increasingly rely on
wage labor or extractive activities (charcoal or fuelwood sales), subsistence agriculture
at the arable margins of the savanna, or food aid resulting in differential opportunities for
adaptation and new patterns of resource use.
Even if adaptation strategies are put in place, recent research suggests that, in
tropical regions, human responses to droughts and more gradual declines in vegetative
and livestock productivity, may lead to ‘tipping points’ in terms of the adaptive capacity
of households to manage with change. At some point, dependence on natural

5

resource-based livelihood options may not be able to satisfy households’ objectives and
needs (Moore et al. 2005, 2006).
Policy change or other higher level institutional responses could influence local
adaptation by either enabling or constraining those adaptations. Furthermore,
adaptation at the local scale may heighten or lessen the vulnerability of some societal
groups to future perturbations or stressors (Kates 2000, Wisner et al. 2004).
Under the near-term and future climate changes projected by the Climate-Land
Interaction Project (CLIP, funded by the NSF Biocomplexity Program, BE/CNH Award #
03088420), savanna ecosystems will be one of the most negatively impacted
ecosystems with large extents reaching ‘tipping points’ of dramatic changes in physical
conditions, functions, and services (Moore et al. 2005, 2006). Savanna areas are
dominated by pastoralism and, in some areas, wildlife. They are also at the expanding
edge of cropped agriculture and are the location of the most rapid in-migration and land
use change in the region over the past twenty years (Olson et al. 2004).
The savanna ecosystem responds to the highly variable rainfall with
disequilibrium, leading to heterogeneous vegetation that changes over space and
through time (Oba et al. 2003, Sankaran et al. 2005). This response will alter the
distribution of savanna grasses, thus affecting grazing patterns of wildlife and domestic
animals alike. Further degradation of pasture may occur due to the progressive growth
of bush that often results from increases in rainfall. Conflicts between communities and
ethnic groups over resource use have resulted from reductions in grazing lands (Oba et
al. 2000).

6

An increase in health stresses may also be a consequence of climate change.
Many contemporary African communities are affected by health stresses. Incidences of
malaria, including the recent resurgence in the highlands of East Africa, involve a range
of causal factors. Links to climate and other causal ‘drivers’ of change have recently
attracted a great deal of attention and debate (e.g., Hay et al. 2002, Pascual et al.
2006). Results from the “Mapping Malaria Risk in Africa” project show a possible
expansion and contraction, depending on location, of climatically suitable areas for
malaria by 2020, 2050, and 2080 (Thomas et al. 2004). However, new evidence
regarding micro-climate change due to land-use changes, such as swamp reclamation
for agricultural use and deforestation in the highlands of western Kenya, suggest that
conditions for larvae are being created and therefore the risk of malaria is increasing
(Boko et al. 2007, Munga et al. 2006).
Biodiversity in Africa is also under threat from climate variability and change and
other stress. Africa’s development is constrained by climate change, habitat loss, overharvesting of selected species, the spread of exotic species, and activities such as
hunting and deforestation, which threaten to undermine the integrity of the continent’s
rich but fragile ecosystems (Boko et al. 2007, UNEP/GRID-Arendal 2002).
Increased variation in rainfall related to effects of ENSO in East Africa also has
wide ranging socioeconomic impacts, especially in the agriculture and water resources
industries (Majugu and Magezi 1985). Economic losses from drought in the 1980’s
totaled several hundred million U.S. dollars (Boko et al. 2007, Tarhule and Lamb 2003).
Although future climate change seems to be marginally important when compared to
other development issues (Davidson et al. 2003), it is clear that climate change and

7

variability, and associated increased disaster risks, will seriously hamper future
development (Boko et al. 2007).
In efforts to plan for and mitigate these risks, powerful tools are being developed
that utilize traditional information on climate and an emerging ability to predict future
climatic events. These tools can be used to assist planning and management across all
socioeconomic activities and underpin sustainable development. Advance knowledge
of the probable climate extremes also plays a big role in mitigating the consequences of
climate hazards such as drought, floods, and tropical cyclones (Majugu and Magezi
1985). A central goal of global change science is to obtain more reliable assessments
of likely future climatic conditions and to assess the impacts on society, such as
poverty, food production, and the incidence of disease (McCarthy et al. 2001). Concern
with issues such as rising sea levels, increased frequency and intensity of extreme
climatic events, and variability in crop production is influencing policy discussions in
vulnerable countries.
To date, most research on climate change and its impacts has been global in
scale (e.g., Lobell et al. 2008, Lobell and Field 2006). The relatively coarse-scale data
simulated by general circulation models (GCMs) are useful for evaluating global trends
in climatic variables; however, the resolution of GCM simulated data is not sufficient for
regional evaluations of climatic patterns, particularly in the presence of high landscape
variability (Moore et al. in review, Thornton et al. 2009). In concurrence, the Fourth
Report of the Intergovernmental Panel on Climate Change (IPCC 4) stated, “Climate
scenarios developed from GCMs are very coarse and do not usually adequately capture
important regional variations in Africa’s climate. The need exists to further develop

8

regional climate models and sub-regional models at a scale that would be meaningful to
decision-makers.”
East Africa’s climate is highly variable on a local scale due, in part, to highly
variable topography. Substantial differences in local terrain occur across the region,
from Mount Kilimanjaro (5,895 m) and Mount Kenya (5,199 m), the two highest peaks in
Africa, and the Kenya Highlands, to the Great Rift Valley, Lake Victoria, and the Indian
Ocean coastline. East African climate is also influenced by multiple sources of
seasonality, such as the northerly and southerly migrations of the Intertropical
Convergence Zone (ITCZ) throughout the year which, give rise to bimodal precipitation
patterns at locations near the equator and unimodal patterns further north and south of
the equator (Stock 2004, Mutai and Ward 2000, Hastenrath et al. 1993), and the
complex seasonality resulting from Indian Ocean influences (Black et al. 2003). This
large variation in regional climate may be masked at the coarse scales of GCMs (Moore
et al. in review) and is difficult to detect from the sparse networks of climate
observations on the ground. Furthermore, climate data measured at East African
meteorological stations are sporadic over both time and space, and do not fully
represent landscape variability (Hession and Moore 2010), thus demonstrating the need
to develop higher resolution climate data.

1.2 In search of higher resolution climate data
Higher resolution climate information for impact assessment can be derived
using two basic approaches: (1) downscaling of output from GCMs, and (2) estimation
of conditions at unsampled locations using available measurements on the ground. The

9

former approach is more commonly employed; simulated data sets are more complete
over space and time, allowing for more straightforward evaluation of general climate
trends. Measured data are generally incomplete, but these direct measurements of the
variable of interest can also be used to model trends using state-of-the-art statistical
methods, the subject of this dissertation. Downscaling of simulated climate data is
described first. Statistical analysis techniques and their use in modeling climate trends
based on measured data are summarized in the following section.

1.2.1 Downscaling of GCM results
Downscaling methodologies developed over the last 30 years have begun to
bridge the gap between global climate modeling and regional applications.
Downscaling methods generally fall into one of two categories: statistical (empirical)
downscaling and dynamical downscaling. Statistical downscaling involves correlating
GCM simulated data with data from observed variables measured at specific locations
(e.g., temperature or precipitation) to downscale simulated GCM results (Rogers et al.
2003). A diversity of approaches are utilized in statistical downscaling (e.g, multiple
stepwise regression, logistic regression, canonical correlation analysis, and artificial
neural networks), deployed to predict localized conditions from large-scale atmospheric
parameters and GCM-derived aggregate data (e.g., Kahn et al. 2006, Salathe et al.
2007, von Storch et al. 1993). Dynamical downscaling forms the basis for regional
climate modeling. Regional climate modeling uses GCM simulated data to drive limitedarea, high-resolution regional climate model (RCM) simulations (Mearns et al. 2003;
Dickinson et al. 1989; Giorgi 1990, 2006) that are able to resolve smaller, regional scale

10

climatic forcings such as: complex topography, land-water interfaces, and vegetation or
land cover patterns. High-resolution climate impact assessments avail of these higher
resolution climate predictions to formulate regional adaptation strategies (e.g., Mearns
et al. 1999, 2001a, 2001b). Some major theoretical limitations impact both forms of
downscaling, however, including the propagation of errors from GCMs (Mearns et al.
2003). The research presented herein makes use of observed climate data, rather than
simulated data from GCMs or RCMs, and spatially explicit statistical techniques to
develop improved methods for prediction of data in a data-scarce environment, and to
apply these newly developed methods in East Africa. Furthermore, the accuracy of
results will be compared to a global interpolated data set generated at a spatial
resolution of 0.5 degrees by the Climatic Research Unit at the University of East Anglia
(Mitchell and Jones 2005).

1.2.2 Use of statistical techniques with measured data
Various statistical techniques have been used to generate higher resolution
climate data from observations on the ground. This dissertation focuses on techniques
from geostatistics, particularly kriging, and new approaches in regression analysis that
treat spatial problems to estimate higher resolution precipitation data. The methods of
greatest immediate relevance within these two broad categories are now briefly
discussed in order to motivate the dissertation work. The preferred techniques in this
dissertation are multivariate, and utilize data on variables expected to influence the
spatial distribution of precipitation (i.e., covariate data). Consequently, a description of
the scales and mechanisms of precipitation in East Africa is included in Chapter 2,

11

Literature Review. This knowledge informs the selection of appropriate covariates in
multivariate statistical analyses.
Uses of Kriging in the Estimation of Precipitation. Many climate studies have
evaluated the ability of interpolation methods to estimate climatic variables over space.
Historically, these studies have compared basic interpolation techniques (e.g., inverse
distance weighting (IDW), Theissen polygons, and nearest neighbor interpolation) to
various kriging techniques (e.g., Diodato 2005; Goovaerts 1999a, 1999b, 2000; PardoIguzquiza 1998). Basic interpolators provide estimates of a variable of interest at
unsampled locations; however, they do not assess uncertainty. Kriging techniques
have the added benefits of being spatially explicit (i.e., accounting for location and
configuration of samples as well as spatial autocorrelation in the data) and providing
estimates of uncertainty for the interpolated values. Multivariate kriging techniques (e.g,
universal kriging) incorporate covariate data to improve predictions and reduce
uncertainty (Diodato 2005; Goovaerts 1999a, 1999b, 2000; Kyriakidis et al. 2001;
Pardo-Iguzquiza 1998). In the development of multivariate kriging models, many
researchers have relied on establishing correlations between precipitation and elevation
to improve estimates of precipitation at unsampled locations (Diodato 2005; Goovaerts
1999a, 1999b, 2000; Kyriakidis et al. 2001; Pardo-Iguzquiza 1998). Although it is
certainly important, and linked closely to spatial distribution of precipitation in certain
parts of the world, elevation does not encompass the full range of mechanisms
impacting rainfall, particularly in East Africa.
Regression Analysis. Scientists studying rainfall mechanisms frequently use
multivariate statistical techniques in the interest of hypothesis testing. Many of these

12

researchers investigate climate on a more theoretical level with statistical inference and,
consequently, rely on regression models. In so doing, these researchers have
elaborated on relationships between precipitation and a wide variety of variables,
including elevation and its derivatives, vegetative cover, specific humidity, and
geographic descriptors such as distance to coastline, a variable of relevance to regions
such as East Africa which is heavily influenced by maritime conditions (Marquinez et al.
2003, Oettli and Camberlin 2005, Anders et al. 2006, Propastin et al. 2006, Ji and
Peters 2004). Such modeling has identified variables that are correlated with
precipitation and has thus improved the understanding of spatial patterns in rainfall, but
it is not designed explicitly to produce optimal estimates of values for sites out of
sample, the objective of many geostatistical exercises such as kriging. Moreover, given
the statistical estimation procedures generally used, these studies tend to neglect
issues of spatial autocorrelation and endogenous feedbacks between precipitation and
its covariates.
Modeling of Feedback Mechanisms. One widely noted feedback is the
relationship between precipitation and vegetation. Precipitation plays an obvious a role
in vegetation dynamics given the requirements of photosynthesis. But feedbacks to the
atmosphere also exist (Rodriguez-Iturbe and Porporato 2004, p. 2). A wide literature
covers many aspects of the relationship between vegetation and rainfall and a variety of
approaches have been used to characterize it (e.g., Lyon et al. 2008, Notaro et al. 2008,
Wang et al. 2008). To date, however, regression-based studies have tended to
overlook endogenous relations between precipitation and vegetation, despite advances

13

in spatial econometrics that treat both simultaneity and spatial autocorrelation (Kelejian
and Prucha 2004, Rey and Boarnet 2004).
To inform the selection of independent variables or covariates for multivariate
statistical analyses of spatial patterns in precipitation data, the scales and mechanisms
of precipitation will be considered. This knowledge will also inform hypothesis testing
that will be conducted as part of the multivariate statistical analyses. Scales and
mechanisms of precipitation are described in detail in Chapter 2.

Problem Statement: In sum, the predictive models of spatially explicit kriging
have not accounted adequately for the mechanisms of precipitation, and the inferential
models of regression addressing these mechanisms have not addressed spatial
relations affecting climate processes. Further, neither approach has considered
complex feedback relationships that may produce simultaneity between precipitation
and its covariates. This dissertation addresses these limitations in predictive modeling
by developing a kriging approach that accommodates simultaneous relations through
theoretical innovation. Each of these methods will be applied in a case study of climate
in East Africa, a data poor environment affected by climate change.

1.3 Links and contributions to the Geographic traditions
This dissertation spans and contributes to several strands of Geographic thought.
The context of the work presented herein links Regional Geography of Africa, Physical
Geography, and quantitative methods of spatial data analysis in an effort to answer

14

research questions that have been informed by Human-Environment researchers and,
in turn, to inform their research (Figure 1-1).

Physical
Geography

HumanEnvironment

Spatiotemporal
Methods of
Analysis

Regional
Geography of
Africa

Figure 1-1. Relationships between the Geographic traditions represented in this research.
For interpretation of the references to color in this and all other figures, the reader is
referred to the electronic version of this dissertation.

The results and findings of this dissertation add to the knowledge and
understanding of the spatial patterns of precipitation in East Africa. In addition, updated
and improved maps of spatial patterns in precipitation within the study domain are
generated, contributing to Physical Geography and Regional Geography of Africa.
Results of this dissertation contribute to work conducted by Human-Environment
researchers such as high-resolution climate impact assessments (Mearns et al.,1999,
2001a, 2001b) which may be used in the evaluation of coupled human natural systems
and formulation of associated policy. Furthermore, this dissertation provides new and
innovative techniques for spatiotemporal analysis of data which can be applied in any
context when the goal is to understand spatial patterns in any continuous data, to
predict data at unsampled locations, and/or to understand the relationship between a

15

variable of interest and other factors that may influence the spatial distribution of that
variable.
Lastly, this research is designed to complement ongoing NSF activities at
Michigan State University: the Climate Land Interaction Program (CLIP; An Integrated
Analysis of Regional Land-Climate Interactions, National Science Foundation Award
Number BCS0308420) and East Africa Climate Land Interaction Program – Savanna
Ecosystems (EACLIPSE; Dynamic Interactions Among People, Livestock, and Savanna
Ecosystems under Climate Change, National Science Foundation Award Number
BCS0709671). Results of this dissertation can be incorporated in climate change
studies such as these to inform adaptation strategies.

1.4 Research objectives
This dissertation research has four basic goals, which are to (1) contribute
theoretically and methodologically to the prediction of variables in data-scarce
environments utilizing improved kriging-based and spatial regression techniques; (2)
apply these techniques in a case study based in East Africa; (3) evaluate case study
results obtained using each technique and compare to those obtained from a gridded
global climate data set interpolated to a 0.5 degree resolution; and (4) conduct
hypothesis testing using selected spatial regression models to identify significant factors
that influence the spatial distribution of precipitation.

16

This dissertation is organized as follows:

Chapter 2 provides a review of literature that supports and expands the
introductory background discussion.

Chapter 3 presents the statistical modeling frameworks of kriging and spatial
regression, demonstrates the theoretical links between these frameworks, and details
the methodological innovations developed by this research.

Chapter 4 presents the case study. The case study area is described, including
details of the physical precipitation mechanisms in East Africa, to provide a foundation
for the selection of (1) independent variables for multivariate statistical analysis and (2)
variables for which feedbacks will be modeled. Precipitation estimates at unsampled
locations are generated using selected kriging method(s) and various formulations of
the spatial regression models described in Chapter 3 of this dissertation. Error
estimates from each analysis will be compared and evaluated for minimum error
variance and the presence of spatial patterns not captured by the model. Additionally,
error estimates from each analysis will be compared to error estimates generated by a
global climate data set interpolated to a 0.5 degree resolution (Mitchell and Jones 2005)
for comparison to the methods presented herein.

Chapter 5 focuses on understanding of spatial patterns in East African rainfall.
Initial hypotheses related to variables that influence the spatial patterns of precipitation

17

and vegetation are described, followed by hypothesis testing using the spatial
regression models developed in Chapter 4. This chapter follows the case study as an
extension to the case study. The primary objective of the case study is to develop and
compare prediction methods, some of which rely on multivariate regression models.
Chapter 5 demonstrates the added benefit of spatial regression models: that of gaining
an improved understanding of precipitation patterns in East Africa.

Chapter 6 presents findings, conclusions, contributions of the work presented
herein, and discusses limitations of this work and future research needs.

18

Chapter 2
Literature Review

This chapter will link climatic concepts with the use of statistics to better
understand the spatial patterns of precipitation and the scales over which they occur,
and to develop finer-scale estimates for climatic variables with sparse data
observations. The chapter begins with a discussion of the scales and mechanisms of
precipitation in East Africa in order to motivate the need for higher resolution data in
East Africa and the case study work in Chapter 4. Next, a review of statistical methods
historically used to evaluate climatic data for various purposes is presented. The
chapter ends with the identification of gaps in the literature and a discussion of how
these gaps will be addressed by this dissertation.

2.1 Scales and mechanisms of precipitation in East Africa
Mechanisms of precipitation occur over various geographic scales around the
world. Mesoscale processes take place over regions ranging from a few kilometers to
approximately one hundred kilometers in diameter (Ahrens 2007), including land/sea
breezes and orographic uplifting over mountainous terrain. Synoptic scale processes
impact on the spatial distribution of precipitation over areas ranging from hundreds to
thousands of square kilometers. These processes include high and low pressure areas
and associated weather fronts. In East Africa, the presence of low pressure in January
causes winds ranging from northeasterly (north of the Intertropical Convergence Zone
(ITCZ)) to southeasterly (south of the ITCZ), coming in from the Indian Ocean (Mutai

19

and Ward 2000). In July, the low pressure over Asia and India results in the summer
monsoon season there, and south to southwesterly winds in East Africa (Mutai and
Ward 2000). In the tropics, the position of the global-scale ITCZ also plays a role in
rainfall patterns. Its associated bands of rainfall move northward and southward over
the year, dominating the spatial pattern of precipitation on a global scale. The passage
of the ITCZ contributes to a generally bimodal seasonal pattern of rainfall near the
equator, and a unimodal pattern at the northern and southern extents of the study area.
Although the broad features of this climate system can be described, specific
questions remain unanswered, and constitute the empirical basis of this dissertation.
As demonstrated herein, rainfall processes at varying scales can be modeled through
the use of predictive or independent variables in a statistical correlation and regression
analysis. Patterns in precipitation can be better understood through consideration of
variables that are highly correlated with precipitation and for which data are more readily
available, such as topographic variables including elevation and its derivatives
(measures of mesoscale processes), geographic measures of location (proxies for
synoptic and global scale processes), and season-specific analysis (allowing for
seasonal variation in synoptic and global scale processes). Multivariate techniques that
utilize covariate information can make use of relationships between these variables and
precipitation to more accurately estimate precipitation patterns over space and time
(Diodato 2005; Goovaerts 1999a, 1999b, 2000; Kyriakidis et al. 2001; Pardo-Iguzquiza
1998). Furthermore, this dissertation will demonstrate that multivariate techniques able
to characterize endogenous relations between precipitation and vegetation provide a

20

new and exciting alternative to existing methodologies for producing maps of
precipitation over space.

2.2 A review of statistical methods used in the analysis of precipitation data
Various statistical techniques have been used to generate higher resolution
climate data using measured data, primarily at meteorological stations, rather than
simulated data from GCMs or RCMs. This dissertation focuses on techniques from
geostatistics, particularly kriging, and improved approaches in regression analysis that
treat spatial problems to estimate higher resolution precipitation data. The methods of
greatest immediate relevance within these two broad categories are summarized briefly
in this chapter, motivating the theoretical work presented in Chapter 3. Given the vast
literatures in question, the discussion in this chapter limits itself primarily to climaterelated applications.

2.2.1 Kriging
Kriging refers to a family of techniques developed in France by Matheron (1963)
based on the dissertation work of the South African mining engineer D. G. Krige (1951).
Since its origination, many forms of kriging have been developed to predict attribute
values at unsampled locations. Univariate kriging techniques consider only one
variable, the variable of interest (e.g., simple kriging, ordinary kriging). Multivariate
kriging techniques utilize secondary information in the form of independent variables or
covariate data (e.g., kriging with an external drift/regression kriging, cokriging). All
forms of kriging belong to a family of generalized least-squares regression algorithms,

21

four of which are described below. Each kriging method is described in detail in
Chapter 3. A fifth form of kriging, developed in Chapter 3, is also briefly described
below.
(1) Simple kriging is a univariate technique which is based on the assumption
that the mean value of the variable of interest is constant and known.
(2) Ordinary kriging is also a univariate technique which differs from simple
kriging in that the mean value, which is assumed constant, is unknown; to relax the
assumption that the mean is constant through the entire study area, local
neighborhoods can be established in which the mean is assumed constant.
(3) Kriging with a trend surface model is a multivariate technique in which a
trend surface model is used to estimate a varying mean across the study area; trends
surface modeling uses location coordinates and functions of these coordinates to
estimate the mean at each location with the assumption that the data point to be
predicted and the observed data are uncorrelated.
(4) Universal kriging (UK; Schabenberger and Gotway 2005), also known as
kriging with an external drift (KED; Goovaerts 1997), is another multivariate technique in
which the list of independent or explanatory variables is expanded to include other
variables which are: (a) correlated with the variable of interest, and (b) measured at
locations for which the variable of interest is to be estimated; estimators are developed
using the p explanatory variables assuming a general linear model that holds for both
the data and the unobservables, with the assumption that the data and the
unobservables are spatially correlated.

22

(5) Universal kriging with simultaneity (developed herein) expands upon
universal kriging by incorporating simultaneity between the variable of interest (i.e., the
dependent or endogenous variable) and an explanatory variable (i.e., the independent
or exogenous variable), such as that which occurs between precipitation and
vegetation; this is accomplished by deriving and including an instrumental variable
among the p explanatory variables in place of the variable that is simultaneously related
to the dependent variable (e.g., vegetation).

As the field of geostatistics has evolved, some inconsistencies have arisen with
regard to terminology (Hengl Heuvelink and Stein 2003) as well as an understanding of
mathematical similarities and distinctions between various kriging techniques. This may
be due in part to the fact that early developments in geostatistics generally occurred
outside of mainstream statistics (Christensen 1991). As succinctly stated by Hengl,
Hevelink, and Stein (2003),

“The most probable cause (for this confusion) is that similar applications
have been developed among different professions with different goals.
The second important cause of this confusion is that some authors, more
involved in the practice of kriging (‘geostatisticians’), consider these
techniques a special interpolation technique, while the other group
(‘statisticians’) consider kriging to be only a case of regression analysis
with spatially correlated data.”

In addition, more than one school of thought has evolved, resulting in
inconsistent naming conventions across kriging techniques. For example,
Schabenberger and Gotway (2005) describe ‘universal kriging’ as above. However, the

23

technique termed ‘universal kriging’ by Journel and Huijbregts (1978, pg. 313) utilizes
only polynomial functions of spatial coordinates, which is a method also known as
‘kriging with a trend model’ (Christensen 1991, Goovaerts 1997, Journel and Rossi
1989). Many authors agree that the term ‘universal kriging’ should be reserved for the
case when the only covariates used are polynomial functions of the spatial coordinates
(Hengl Heuvelink and Stein 2003). ‘Kriging with an external drift’ (e.g., Goovaerts
1997), is mathematically identical to ‘universal kriging’ as defined by Schabenberger
and Gotway (2005) and ‘regression kriging’ (Hengl Heuvelink and Stein 2003). In
mathematical terms it matters not which covariates are selected to model the mean
function,

; the mathematical derivation of the best linear unbiased predictor (BLUP) is

the same (Christensen 1991). This dissertation develops a system of classification
which recognizes this fact and seeks to clarify the links between forms of kriging as well
as spatial regression (described below).

2.2.2 Spatial regression
A line of generalized least-squares regression models have arisen for analysis of
spatial data: spatial regression models have been developed to explore causality using
multiple independent (exogenous) variables (e.g., Anselin and Bera 1998, Anselin 2006,
Caldas et al. 2007, Moore et al. 2007, Walker et al. 2000) and to account for
simultaneity between multiple dependent (endogenous) variables (Kelejian and Prucha
2004, Rey and Boarnet 2004). Spatial regression models can also be used to generate
estimates of dependent variables at unsampled locations taking into account
information on independent variables and simultaneity between multiple dependent

24

variables. This section first provides a brief summary of spatial regression models,
followed by a review of their current use and a discussion of potential future uses in
climate research.
Three forms of spatial regression models provide the basis for the modeling of
different forms of spatial autocorrelation:
(1) The spatial lag model, also known as a spatial autoregressive (SAR) model,
accounts for the presence of spatial autocorrelation in the dependent variable by
incorporating a spatial lag operator; spatial independence of the error terms is assumed.
(2) The spatial error model (SEM) allows for spatial dependence of the error
terms, which may occur through the omission of a spatially varying covariate.
(3) The general spatial model (SAC) can be employed when both forms of
spatial autocorrelation are present by incorporating both a spatial lag term and a
spatially correlated error structure.
If neither form of spatial autocorrelation is present, an aspatial OLS regression
model may be used. More detailed descriptions of these models are provided in
Chapter 3 and by Anselin (2006), Anselin and Bera (1998), LeSage (1998), and LeSage
and Pace (2009).
A decision process recommended by Anselin (2005) may be used to select
between OLS, SAR, and SEM models for analysis of the data. In summary, the process
involves completing an OLS analysis and calculating diagnostics for spatial
dependence. The OLS results may be relied upon only if the assumptions underlying
OLS are not violated. First, the lagrange multiplier (LM) statistics for both a spatial lag
and spatial error model are tested for significance. If neither LM is significant, the OLS

25

results may be used. If one of LM statistics is significant, the corresponding model
should be used to evaluate the data (e.g., if the LM statistic for only the spatial lag
model is significant, the spatial lag model should be used to evaluate the data). If both
of the LM statistics are significant, robust LM statistics should be tested for significance
and the appropriate model selected.
In the case that both robust LM statistics are significant, generalized spatial
modeling (SAC) may be used to account for this more complex form of spatial
autocorrelation (LeSage 1998).

2.3 Historical use of methods to model climate patterns using measured data
Recently-used methods for modeling and predicting patterns in precipitation and
vegetation based on measured data are summarized in
Table 2-1. Methods vary widely from simple OLS regression to spatially explicit
geostatistical methods, Laplacian smoothing splines, and process-based physical
models. The goals of many of these studies were to predict climatic variables over
space and to compare the ability of various interpolation methods. Other studies were
conducted with the goal of better understanding patterns in climatic variables and
factors that influence these patterns using regression models and formal hypothesis
testing. While some of these methods are univariate (i.e., they incorporate data from
only one variable, the variable of interest), many others are multivariate and incorporate
data from one or more independent or predictive variables (also known as covariates or
exogenous variables). This section first describes the independent variables that are
used in the studies below. Next, a review of past research is provided, with studies

26

grouped as follows: (1) studies to generate precipitation estimates at unsampled
locations, (2) studies to improve understanding of precipitation patterns, and (3) studies
to improve understanding of vegetation patterns.

Independent variables
In the development of multivariate models, many researchers have relied on
establishing a correlation between precipitation and elevation as a covariate to improve
estimates of precipitation at unsampled locations (Arora et al. 2006; Daly et al. 1994;
Diodato 2005; Goovaerts 1999a, 1999b, 2000; Hutchinson 1998b; Hutchinson and
Bischof 1983; KeifferWeisse and Bois 2001; Kyriakidis et al. 2004; Marquinez et al.
2003; Oettli and Camberlin 2005; Pardo-Iguzquiza 1998). Relatively high levels of
correlation have been identified; the relationship is generally an increasing one (Arora et
al. 2006; Spreen 1947): as elevation increases, precipitation increases. This is mainly
due to the “orographic effect” of the mountain terrain. However, precipitation can have
a very complex relationship with elevation (Arora et al. 2006), which can be complicated
by station distribution and other factors (Hulme and New 1997).
Many other factors have also been found to play a role in the distribution of
rainfall. Other researchers have expanded the list of potential predictive variables to
include derivatives of elevation and other variables summarizing geographic location
(Arora et al. 2006; Diodato 2005; Kieffer Weisse and Bois 2001; Kyriakidis et al. 2004;
Hutchinson 1998b; Hutchinson and Bischof 1983; Marquinez et al. 2003; Oettli and
Camberlin 2005; Spreen 1947). For example, Spreen (1947) found that distribution of
rainfall also depends on variables such as slope, exposure, and orientation. Similarly,

27

Keiffer Weisse and Bois (2001) found that topographic variables were correlated with
heavy rainfall events (e.g., 10- and 100-yr events), particularly when measured at short
time steps (i.e., less than three hours). Regional topographic variables (e.g., distance to
the Mediterranean, characterization of the general shape of the Alps, distance to
corresponding features of the Alps) were found to influence heavy rains, whereas local
measures of topography (e.g., altitude, slope, or azimuth) were less influential.
Marquinez et al. (2003) evaluated the relationships between precipitation distribution
and distance to coastline, distance to a location in the relative west, and elevation at two
geographic scales (i.e., local and sub-basin). Hutchinson (1998b) derived the east and
north components of the unit normal vector to represent slope and aspect for use as
independent variables.
In the vegetation studies described herein, normalized difference vegetation
index (NDVI; Rouse et al. 1974) was used to represent overall vegetation health.
Theoretically, NDVI ranges between -1 and +1, although values typically range between
0 for bare ground to 0.7 for lush dense vegetation. Independent variables which have
been considered to impact patterns in vegetation include precipitation (Propastin et al.
2006; Ji and Peters 2004), as well as potential evapotranspiration (PET), daily
maximum and minimum air temperatures, soil temperature, and solar irradiation (Ji and
Peters 2004). As shown by Ji and Peters (2004), precipitation and PET were the most
significant predictors of NDVI.

28

Prediction and mapping of precipitation
Numerous studies predicting and mapping precipitation patterns have compared
basic interpolation techniques (e.g., inverse distance weighting (IDW), proximal
interpolation via Theissen polygons, and nearest neighbor interpolation) to various
spatially-explicit kriging techniques that account for both location and configuration of
samples as well as spatial autocorrelation (e.g., Diodato 2005; Goovaerts 1999a,
1999b, 2000; Pardo-Iguzquiza 1998). In general, spatially explicit kriging techniques
outperformed basic interpolators. Further, multivariate kriging techniques incorporating
covariate data were generally shown to improve predictions and reduce uncertainty
(Diodato 2005; Goovaerts 1999a, 1999b, 2000; Kyriakidis et al. 2001; Pardo-Iguzquiza
1998). For example, Pardo-Iguzquiza (1998) found that kriging with an external drift (a
multivariate technique also known as universal kriging) performed better than Thiessen
polygons and ordinary kriging (univariate techniques). Goovaerts (1999a, 2000) found
that kriging with an external drift and cokriging (multivariate methods) outperformed the
basic interpolators (proximal interpolation with Thiessen polygons and IDW); however,
simple kriging with varying local means (a univariate technique) outperformed both
multivariate methods in this instance.
OLS regression has also been used to predict and map precipitation patterns
(Daly, et al. 1994; KeifferWeisse and Bois 2001). KeifferWeisse and Boise (2001) used
OLS regression to estimate heavy rainfall amounts at time steps ranging from 1 to 24
hours. Daly et al. (1994) stratified monthly and annual rainfall data over space into
individual “topographic facets” to account for differences between facets in an effort to
account for spatial autocorrelation. Independent regression analyses of precipitation

29

versus elevation for each facet were then conducted. It remains, however, that OLS
regression methods do not explicitly account for spatial autocorrelation.
Another group of methods has been used to estimate the spatial distribution of
rainfall: methods such as Laplacian smoothing splines and thin plate moving splines
have been used to smooth and interpolate precipitation data over space using location
coordinate information (Hutchinson 1998a; Hutchinson 1998b; Hutchinson and Bischof
1983). Rather than explicitly incorporating spatial autocorrelation into parameter
estimation, however, these studies utilized a simplified approach in which one sample
location was removed from close pairs of locations to avoid problems with short range
correlation over space. Hutchinson (1998a, 1998b) evaluated spatial patterns in one
day of rainfall data. Hutchinson and Bischof (1983) studied long-term mean seasonal
(i.e., precipitation data were stratified into seasons) and annual precipitation. Processbased physical models have also been used to predict the spatial distribution of
precipitation. For example, Barros and Lettenmaier (1993) modeled the advection of
moisture over topographic barriers utilizing a 4D Lagrangian model. This model
simulated orographically-induced precipitation at a scale sufficient to resolve dominant
topographic features. Although methods utilizing splines or process-based physical
models were used to interpolate rain data over space, they are not easily adapted for
formal hypothesis testing of relationships between precipitation and possible
explanatory variables.

30

Hypothesis testing
Hypothesis testing through regression analysis can be used to identify significant
predictors of precipitation patterns over space. Studies with this goal include those by
Arora et al. (2006), Marquinez et al. (2003) and Oettli and Camberlin (2005). These
studies generally aimed to model and understand patterns in the mean precipitation
function rather than generating estimates at unsampled locations, and they utilized
aspatial OLS regression techniques to accomplish these goals. If present, spatially
autocorrelated observations violate an underlying assumption of OLS regression, that of
independent observations (Haining 1990; Neter, Wasserman, and Kutner, 1990; Bailey
and Gatrell 1995; Schabenberger and Gotway 2005). Two major consequences of
violating this assumption are (Ji and Peters 2004):

1) underestimation of the regression coefficients’ standard errors, leading to the
identification of variables as significant when they are not; and
2) underestimation of the error variance term, yielding inflated coefficients of
determination (R2). Spatial regression techniques can account for spatial
autocorrelation, and thus are expected to provide better results in these
instances.

Relatively few efforts have been made to apply spatially explicit regression
techniques to climate data, probably because of the novelty of the methods. In
particular, the subject of spatial regression only enters the academic literature in the
80’s and 90’s. Early efforts in the analysis of spatial data include Ord (1975) and

31

Anselin (1988). More recently, use of spatial regression expanded into ecological work,
with applications to predicting the presence of species or species abundance (e.g.,
Augustin et al. 1996; Huffer and Wu 1998; Torbick, et al. 2010) and diseases (e.g.,
Gumpertz et al. 1997). An extensive recent review of spatial regression and
geostatistics as applied to ecological modeling is given by Miller, et al. (2007).
Use of spatially explicit regression is relatively new to the modeling of climate
variables and relationships with vegetation. Anders et al. (2006) utilize a form of
regression modeling that reflected spatial lags to estimate average annual precipitation
(as measured by the Tropical Rainfall Measuring Mission or TRMM) as a function of
surface saturation vapor pressure (Vp), slope (S), relative elevation (E), the product of
slope and Vp (Svp), and the product of slope and E (Se). This model incorporates two
sets of OLS coefficients combined additively, with the second set incorporating a spatial
lag. Significant predictors of precipitation are surface saturation vapor pressure (Vp)
and the product of Vp with slope (Svp). Ji and Peters (2004) model NDVI as a function
of precipitation, PET, maximum and minimum temperature, soil temperature, and solar
irradiation. They implement OLS with ridge regression for their initial model, then
integrate spatial variability using linear mixed models with a variogram component.
Significant predictors of NDVI are precipitation and PET, which account for relatively low
fractions of overall variability (46% for grassland areas, 24% for croplands). Propastin
et al. (2006) compare OLS regression and geographically weighted regression (GWR)
for modeling NDVI as a function of precipitation. GWR outperforms OLS, yielding
coefficients of determination (R2) of approximately 88%. Although the value of R2 is no
longer biased by the failure to account for spatial autocorrelation in the data, it is likely

32

that the reported values are overstated due to the handling of the precipitation data.
Gridded maps of precipitation were created by interpolating data from nine climate
stations using inverse distance weighting prior to performing the regression analysis.
The resulting smoothed data are likely to underestimate true variability and
overestimate the correlation between NDVI and precipitation. Further, GWR is a
spatially explicit regression technique that allows for spatially varying covariates, but
does not address the issue of spatially correlated error terms (Schabenberger and
Gotway 2005).
In sum, work has begun on modeling climate using spatially explicit regression
techniques; however, much remains to be done. Standard spatial models (e.g., SAR,
SEM) have not been widely applied, if at all. Further, the studies cited (Ji and Peters
2004; Anders et al. 2006; Propastin et al. 2006) involve modeling of climatic variables
for the purpose of hypothesis testing, which is somewhat limited given that spatial
regression also provides a potentially useful tool for generating predictions and
estimates of uncertainty at unsampled locations. Be this as it may, to date there are no
direct applications of spatial modeling as proposed in this context. Related efforts
include the mapping of Malaria incidence (Kazembe 2007) and species habitat
distributions (Gottschalk et al. 2007), but these applications do not utilize spatial
regression models in the form proposed here. Little used in the climate research
domain, spatial regression is a method that may afford researchers a value-added
approach to estimating climatic variables in data-sparse regions.
Finally, the ability to model multiple endogenous variables simultaneously in
spatial regression yields particularly useful and exciting possibilities in climate research

33

due to interactions among multiple climate variables. In the climate research domain,
an example of a feedback mechanism between climatic variables is that which occurs
between precipitation and vegetation. Precipitation clearly plays a role in vegetation
dynamics; however, as noted by Rodriguez-Iturbe and Porporato (2004), “vegetation
exerts important control on the entire water balance and is responsible for many
feedbacks to the atmosphere.” A wide literature covers many aspects of the feedback
between vegetation and precipitation (e.g., Brunsell 2006; Dekker et al. 2007; Kim and
Wang 2007; Mendez-Barroso et al. 2009; Notaro and Liu 2008; Roy 2009; Wang et al.
2008; Zeng and Yoon 2009). A variety of approaches have been used to detect and
model feedbacks, ranging from fully-coupled physical climate models to ensemble
simulations using climate models, simple correlation coefficient analysis, and aspatial
statistical feedback approaches developed initially for studies of ocean-atmosphere
interactions. However, no one has yet incorporated new statistical approaches to
simultaneity (e.g., Kelejian and Prucha 2004) to prediction, which is an objective of the
proposed dissertation.

2.4 Gaps in the literature and contributions to address them
A gap exists that is not completely filled by the variety of statistical techniques
most commonly used to evaluate climate data. The focus of the spatially explicit studies
described above is often to develop the most accurate estimates of precipitation without
necessarily understanding the role of various predictors or formally testing hypotheses
about them. Schabenberger and Gotway (2005) note that the use of predictive
variables is not the primary focus of many of these types of studies: predictive variables

34

are often used to “account for a spatially varying mean and to avoid bias.” In many
cases, there is no intention of interpreting the relationships between the predictive
variables and the dependent variable, or their significance.
When hypothesis testing and understanding are of primary interest,
Schabenberger and Gotway (2005) identify spatial regression as a form of data analysis
“where the focus is on modeling and understanding the mean function.” [Emphasis
added.] Understanding is gained when significant variables that play a role in
precipitation patterns are identified through hypothesis testing. However, as shown in
Table 2-1, recent studies designed to test hypotheses regarding multiple predictors and
their influence on rainfall rely heavily on OLS regression. Since OLS regression does
not account for spatial autocorrelation, results are biased in the presence of spatially
autocorrelated data.
More recently, spatially explicit regression techniques have begun to appear in
climate-related literature; however, each of the applications is somewhat limited, either
by the method used, the covariates selected, or the overall application (hypothesis
testing or estimation). Further, none of the approaches have characterized endogenous
relations, such as those between precipitation and vegetation. Accounting for these
feedback mechanisms is expected to produce improved predictions of precipitation over
space and refinements in the ability to separate sources of variability for the purpose of
hypothesis testing.
Spatial regression may thus provide a solution to multiple problems of interest in
the climate research domain: it allows for improved understanding through testing of
hypotheses related to multiple independent variables while accounting for spatial

35

autocorrelation and it provides a method for estimating climate data in data sparse
regions. This dissertation addresses these issues by experimenting with spatial
regression techniques, existing kriging algorithms, and by developing a kriging
approach that accommodates simultaneous relations through theoretical innovation.
From a theoretical perspective, much work has been done to mathematically
compare and categorize generalized least-squares approaches such as kriging and
spatial regression (e.g., Christensen 1991; Cressie 1993; Christensen 1996;
Schabenberger and Gotway 2005). However, this work has not been extended to
simultaneous equations spatial regression. Consequently, it is not known whether and
to what extent recent developments in spatial regression reach beyond kriging
capabilities.
Thus, the gaps in the literature are both empirical and theoretical. This
dissertation will address these gaps as described in Chapter 3.

36

Author(s)
Goovaerts
(1999a,
2000)

Goovaerts
(1999b)

Methods
OLS regression
Thiessen polygons
Inverse square
distance
OK, SKlm, KED, CK
OLS regression
SKlm, KED, CK

PardoIguzquiza
(1998)

Thiessen polygons
OK, CK, KED

Arora et
al. (2006)

OLS regression

Diodato
(2005)

IDW, OK, CK

Kyriakidis
et al.
(2004)

1) time series at
stations
2) spatial regression of
coeffs. with elevation
and atmospheric data
3) CK of residuals
4) reconstruction of
trend coefficients
5) conditional
stochastic simulation

Table 2-1
Application/Locat
Dependent
ion and Context
Predictors
variable
Methods relating precipitation to elevation
Estimation /
Elevation
Average monthly
The Algarve,
and annual
Southern Portugal,
precipitation
Atlantic Ocean to
south and west
Estimation /
Elevation (average of
Erosivity data
The Algarve,
values at 4 discrete points
averaged on a
Southern Portugal, in a 1 square kilometer cell) monthly and
annual basis
Estimation /
Elevation
Mean annual
Guadalhorce river
rainfall over 20
basin in southern
year period
Spain
Hypothesis testing Elevation
Seasonal and
/
Distance to lowest station
annual precip;
Chenab basin,
grouped by
western
mountain range,
Himalayas
windward and
leeward side
Estimation /
Elevation
Average
Benevento
Smoothed elevation (DEM)
seasonal and
Province, S. Italy,
Vegetation cover factor
annual
Mediterranean
Topographic index
precipitation
reion, variable
topography
Estimation /
Elevation
Daily
Northern California Large-scale specific
precipitation
coastal region,
humidity from NCEP/NCAR
characterized by
reassessment
complex
topography

37

Results
SKlm outperformed KED,
CK, and univariate methods
(cross-validation);
OK outperformed OLS
regression when r<0.75.
CK outperformed other
methods (cross-validation)

KED outperformed others
methods (cross-validation)

Different models generally
resulted for different
mountain ranges and for
windward/leeward sides

Highest correlation between
topographic index and
average annual precipitation
(r2=0.542)

Applicability of the method
was demonstrated; the
method was able to
reproduce the spatiotemporal
characteristics of observed
rainfall measurements

Table 2-1 (cont’d)
Author(s)

Methods

Marquinez
et al.
(2003)

OLS regression
(backwards stepwise
regression)

Daly, et al.
1994

OLS regression over
individual topographic
facets

Oettli and
Camberlin
(2005)

OLS regression
(forward stepwise
regression)
Cross-validation
Calculation of
estimates at gridpoints
Calculation of residuals
at gridpoints with
stations
Kriging of residuals
(cubic interpolation)
Regression model that
accounts for spatial
lags: utilizes OLS
coefficients for the
independent variables,
and OLS coefficients
for the spatial lags of
those variables

Anders et
al. (2006)

Dependent
Application
Predictors
variable
Methods relating precipitation to elevation
/
Distance from each station
Dry season
central area of the to coastline
Wet season
Cantabrian Coast, Distance from each station
Annual
northern Spain
to relative west
Elevation
Avg. elevation per subbasin
Average slope per subbasin
Estimation /
Elevation
Averaged on a
Willamette River
monthly and
Basin, Oregon
annual basis

Hypothesis testing
and estimation /
East Africa,
southern Kenya,
northeastern
Tanzania

Topo. principal components
For 35 different scaling
windows: average and
median elevation,
standard deviation,
amplitude, skewness, and
kurtosis, slope
Geographical locators (lat,
long, distance from Lake
Victoria)

Averaged on a
monthly and
annual basis

Hypothesis
Testing /
Himalayas

Surface saturation vapor
pressure (Vp), slope (S),
relative elevation (E),
product of slope and Sv
(Svp), product of slope and
relative elevation (Se); all
variables were standardized

Average annual
precipitation
generated from 4
yrs of TRMM
data (1998-2001)

38

Results
Adjusted correlation
coefficients ranged between
0.58 and 0.67

PRISM exhibited the lowest
cross-validation bias and
absolute error when
compared to kriging,
detrended kriging, and CK
See text

Best 1 parameter model
uses Vp; best 2 parameter
model uses Vp and Svp;
mean average error is not
substantially decreased by
adding more parameters

Table 2-1 (cont’d)
Author(s)

Methods

Dependent
Application
Predictors
variable
Methods relating precipitation to elevation
Estimation /
Regional variables such
Heavy rainfall
French Alps
as X and Y coordinates,
amounts (10-yr
distance to the
and 100-yr
Mediterranean; local
rainfall events)
variables such as
at time steps
elevation, smoothed
ranging from 1
elevation, exposure
hr to 24 hrs.
parameters, and slope
Estimation /
Latitude
Rainfall
Hunter Valley, Longitude
averaged on
New South
Elevation
seasonal and
Wales
annual basis

KeifferWeisse
and Bois,
2001

Multivariate OLS
regression(forward
stepwise regression) with
kriging of residuals

Hutchinson
and Bischof,
1983

Laplacian smoothing
splines

Hutchinson,
1998a

Thin plate smoothing
splines

Estimation /
Swiss Alps

X and Y coordinates

One day of
rainfall data

Hutchinson,
1998b

Thin plate smoothing
splines

Estimation /
Swiss Alps

X and Y coordinates
Elevation
E and N components of
the unit normal vector to
represent slope and
aspect

One day of
rainfall data

39

Results
Multivariate coefficients of
determination ranging from
0.77 for hourly data
decreasing to 0.57 for daily
100-yr data

Analysis is objective and
explicit; prior record
standardization not required;
each surface is valid for
entire catchment; surfaces
are consistent apart from
data points; method provides
percent predictive error
estimates
Estimates show good
agreement with withheld
data; short-range correlation
dealt with by removing one
point from close data pairs;
square-root transformation of
the data improved estimates;
higher order splines were
found to perform less well.
Analysis confirmed the
importance of incorporating
topographic variables

Table 2-1 (cont’d)
Author(s)
Barros and
Lettenmaier,
1993

Propastin, et
al. (2006)

Ji and Peters
(2004)

Dependent
Application
Predictors
variable
Results
Methods relating precipitation to elevation
Average areal precipitation
Process-based physical
Estimation
Seasonal and
was reproduced with errors
approach utilizing a 4D
annual runoff
in the range of 10-15%.
Lagrangian model
data
Point estimates
of monthly
precipitation
from snow
courses and lowelevation
precipitation
gauges
Spatially explicit statistical methods relating vegetation and climatic variables
OLS regression,
Hypothesis
Dependent variable: NDVI Summed 10-day OLS r2 ranged from 0.36 to
Geographically Weighted
testing /
Independent variable:
rainfall for each
0.67; GWR r2 averaged 0.88;
Regression (GWR)
Precipitation
of 17 years by
much reduced spatial
land cover class; autocorrelation in residuals
gridded maps
of GWR.
created from 9
climate stations
using inverse
distance
weighting and
resized to pixel
resolution of
NDVI
OLS model selection with
Hypothesis
Dependent var.: NDVI
Averaged NDVI
Precipitation and PET were
ridge regression,
testing /
Independent vars.:
within a 10 km
most significant variables;
integration of spatial
Northern and
precipitation, potential
weather station
stronger correlations for
variability using variogram central U.S.
evapotranspiration (PET), buffer, by land
grassland than cropland; r2
method with mixed linear
Great Plains
daily max and min air
cover class
of 46% for grassland, 24%
models (restricted
temperature, soil
for cropland.
maximum likelihood
temperature, solar
procedure)
irradiation
Methods

40

Table 2-1 (cont’d)
Dependent
Methods
Application
Predictors
variable
Results
Spatially explicit statistical methods relating precipitation, vegetation, and topography
Hession
Local OK, Universal
Hypothesis
Distance to Lake Victoria,
Total monthly
LOK performed best on the
(2011)
Kriging, Universal Kriging
testing and
distance to Indian Ocean,
precipitation
basis of RMSE, most
with Instrumental
estimation
elevation, northern and
representing 4
consistent significant
Variables, and Spatial
eastern components of
seasons for 2
predictor was distance to
Regression
the unit normal vector,
years
water body, followed by
surface curvature, NDVI
NDVI and elevation.
IDW-Inverse distance weighting; OK-Ordinary kriging; SKlm-Simple kriging with varying local means; KED-Kriging with an external drift, CKCokriging
Author(s)

41

Chapter 3
Statistical Theory and Derivations

This dissertation will incorporate both theoretical and empirical research. The
theoretical component is described here through the development of a theoretical
mapping between geostatistical kriging methods and spatial regression techniques, all
of which belong to the family of generalized least-squares regression algorithms. The
theoretical mapping is followed by the development of an extension to universal kriging
which accounts for simultaneity between two endogenous variables, such as that which
occurs between precipitation and vegetation previously as described in Chapter 2.

3.1 Background
Prior to mapping theoretical similarities and differences between kriging and
spatial regression techniques, each technique is presented in detail to provide a basis
for the theoretical work.

3.1.1 Kriging
Many forms of kriging have been developed to predict attribute values at
unsampled locations, including univariate techniques that consider only one variable
(i.e., the variable of interest) and multivariate techniques that utilize secondary
information in the form of independent variables or covariate data.

42

All forms of kriging belong to a family of generalized least-squares regression
algorithms and can be derived via the basic linear regression estimator Z*(u) defined as
(Goovaerts 1997):

∗

∑
(1)

where λα(u) is the weight assigned to the individual realization z(uα) of the random
variable (RV) Z(uα). The expected values of RVs Z(u) and Z(uα) are represented by
m(u) and m(uα). The number of locations used in the estimate at location u is
represented by n(u). Further, all varieties of kriging are derived with the goal of
minimizing the error variance

	
∗

(2)

with the constraint E{Z*(u) - Z(u)} = 0 (Goovaerts 1997). The model used to represent
m(u) distinguishes between the forms of kriging.
Several forms of kriging are summarized below using a common notation in
which m(u) is represented more generally by μ. A summary of the common notation
used throughout the remainder of this chapter is provided in Table 3-1. In general,
vectors and matrices are represented in bold type, whereas constant terms are not
bolded.

43

Table 3-1. Summary of common notation.

,

,
,

a vector of observations of the dependent variable at n locations
a constant, the mean value at a location, assumed to be known in simple
kriging; bolded, represents a vector of means at n locations
a vector of error terms at n locations, with mean 0 and covariance
the n x n covariance matrix of
a n x 1 vector of covariances between at a single location to be predicted
through kriging and the measured data at n locations
a n x 1 vector of 1’s
a n x (p+1) matrix of measured values for p covariates at n locations
preceded by a column of ones
a (p+1) x n vector of regression coefficients
a (p+1) x 1 vector of measured values for p covariates at a single locations to
be predicted preceded by an entry of one (1)
a (p+1) x n vector of regression coefficients derived through generalized least
squares
a constant representing the autoregressive coefficient under the spatial
autoregressive (SAR)
a n x n spatial weights matrix used in spatial regression
a n x 1 vector representing the spatially lagged dependent variable
for the SAR model, a n x 1 vector of independent and identically distributed
error terms at n locations, with a mean of 0 and variance σ ; for the spatial
error model (SEM), a n x 1 vector of spatially autocorrelated error terms
a (p+1) x n vector of regression coefficients for the SAR model
a n x n identity matrix
a constant representing the autoregressive coefficient for the error terms
under the SEM
a n x 1 vector of independent and identically distributed error terms at n
locations, with a mean of 0 and variance σ
a (p+1) x n vector of regression coefficients for the SEM model
a (p+1) x n vector of regression coefficients for the SAC model
the n x n covariance matrix of used in the derivation of simple kriging
estimators (same as )
a n x 1 vector of covariances between at a single location to be predicted
(same as )
the (n + p + 1) x (n + p + 1) augmented matrix used in the derivation of
universal kriging estimators
a (n + p + 1) x 1 augmented vector used in the derivation of universal kriging
estimators

44

1) The theoretical model underlying simple kriging is presented below in matrix
notation:

	

,

~

,

(3)

where y is a vector of observations measured at n locations, e is a vector of error terms
at n locations,

represents the covariance matrix of the vector e and

is known.

Furthermore, the mean μ is assumed to be constant and known.
The optimal linear predictor at an individual location developed under generalized
least squares is

(4)

and the simple kriging variance is

(5)

2

where σ = Var[y] at a single location is a constant and σ is a vector of covariances
between y at a single location to be predicted and the observed data at n locations
(from Schabenberger and Gotway 2005, pp. 223-224).

45

2) The theoretical model serving as the basis for ordinary kriging is stated
below.

,

where μ is constant but unknown and

~

,

(6)

is known.

The generalized least-squares form of the ordinary kriging predictor is

̂

̂

(7)

and the ordinary kriging variance is

1

where

2

(8)

= Var[y] (Schabenberger and Gotway 2005, pp. 226-227; from Cressie 1993,

p. 123). As previously noted, ordinary kriging estimators are developed with the
assumption that μ is constant but unknown; however, local neighborhoods can be
established throughout which μ is constant, relaxing the assumption that μ is constant
through the entire study area. This form of kriging is known as local ordinary kriging.

46

3) Universal kriging (UK; Schabenberger and Gotway 2005), also known as
kriging with an external drift (KED; Goovaerts 1997), is based on an underlying
theoretical model that allows for a varying mean throughout an area of interest. The
mean is modeled by a general linear regression model that holds for both the data and
the unobservables and incorporates p covariates or explanatory variables:

,

~

,

(9)
(10)

This model represents a linear mean function and a spatially correlated error process. It
is assumed that the data and the unobservables are spatially correlated with a variancecovariance matrix , Cov[y, y] =

, and Var[y] =

.

The regression coefficients β are estimated using generalized least squares as
follows:

,

(11)

resulting in the following best linear unbiased predictor of y:

(12)

47

with kriging variance

′

∙

′

∙

′

′
(13)

(from Schabenberger and Gotway 2005, pp. 241-242). A similar approach, regression
kriging, also known as kriging after detrending (Goovaerts, 1999b), is described by
Hengl, Heuvelink, and Stein (2003) and is compared to UK/KED.

3.1.2 Spatial regression
Three forms of spatial regression models provide the basis for the modeling of
different forms of spatial autocorrelation: 1) the spatial lag model, also known as a
spatial autoregressive (SAR) model; 2) the spatial error model (SEM); and 3) the
general spatial model (SAC). The SAR model accounts for the presence of spatial
autocorrelation in the dependent variable, but assumes spatial independence of the
error terms. The SEM allows for spatial dependence of the error terms, which may
occur through the omission of a spatially varying covariate. If neither form of spatial
autocorrelation is present, an aspatial OLS regression model may be used. If both
forms of spatial autocorrelation are present, the SAC model may be employed. Each of
these spatial regression models is presented below. More detailed descriptions are
provided by Anselin (2006), Anselin and Bera (1998), LeSage (1998), and LeSage and
Pace (2009).

48

1) SAR model: The SAR model incorporates a spatial lag operator and can be
theoretically represented as in (14) with its corresponding data generating process
(DGP; from LeSage and Pace 2009) shown in (15):

(14)
(15)

where y is a vector of dependent variable observations, ρ is the autoregressive
coefficient, W is a weights matrix, W y is a spatially lagged dependent variable, X is a
matrix of observations of the independent variables, β is a vector of coefficients for the
regression model, and

is a vector of independent and identically distributed error

terms.
2

As described by LeSage and Pace (2009), estimates for β, σ , and ρ can be
obtained through maximum likelihood estimation. A simplified approach is described in
2

which a log-likelihood function that is “concentrated” with respect to β and σ is
2

developed. First, closed-form solutions for β and σ are derived and substituted into
the log-likelihood function, yielding a log-likelihood that is concentrated with respect to β
2

and σ . This reduces the maximum likelihood to “a univariate optimization problem in
the parameter ρ” and obviates the need to simultaneously solve the first order
2

conditions for all three parameters. The estimators for β and σ obtained using the
concentrated log-likelihood function are identical to the form achieved by the full log-

49

likelihood function (Davidson and MacKinnon 1993, pgs. 267-269). The estimate for the
parameter ρ is not achievable in closed form and must be derived computationally.
The resulting estimator for β under the SAR model is:

′

′

.

(16)

Predicted values of y can be estimated completely at the n sampled locations: using
Haining’s (1990) terminology, the complete estimation referred to here can be
developed as the sum of the trend, signal, and noise (residuals) components, since the
values of the response variable are known at the sampled locations (Bivand 2009).
Thus, the complete estimator is:

(17)

obtained from the DGP or, replacing

with the residual term,

	

	

.

(18)

Predicted values of y at unsampled locations, however, can be obtained using
the trend component only of the SAR model:

(19)
50

Note that the dimensions of

and

in (19) reflect the number of locations at which the

response variable y will be estimated. Since no observations of the response variable
are available at unsampled locations, the signal component cannot fully reflect the
spatial variation (Bivand 2009). The following “feasible signal component” has been
suggested by Bivand 2009:

(20)

2) SEM model: The SEM allows for spatial autocorrelation in the residual term.
The theoretical form of the SEM follows (21) and (22), with a DGP of (23):

(21)
(22)
(23)

where in this case

is a vector of spatially autocorrelated error terms, λ is the

autoregressive coefficient for the error terms, and

u is a vector of independent and

identically distributed normal error terms.
2

Estimates for β, σ , and λ can be obtained through maximum likelihood
estimation, using the concentrated log-likelihood approach previously described

51

(LeSage and Pace 2009). Similar to the parameter ρ in the SAR model, λ is not
achievable in closed form and must be derived computationally.
The maximum likelihood estimator for β using the concentrated log-likelihood
under the SEM is

′

′

′

′

(24)

Estimation of y at the n sampled locations can be accomplished using (from Bivand
2009):

(25)

obtained from the DGP or, replacing

with the residual term,

	

(26)

At unsampled locations, Bivand (2009) describes estimation of y using the trend
component only of the SEM model:

(27)

52

The signal component is set to zero because the spatial smoothing process in the SEM
model is expressed only in terms of the error term u as shown in (25), for which the
expectation is zero. Again, the dimensions of

and

in (27) reflect the number of

locations at which the response variable y will be estimated.

3) SAC model: The SAC model incorporates both forms of spatial
autocorrelation through a spatial lag term and a spatially correlated error structure, and
is represented in (28) and (29). The DGP for this model is shown in (30)

(28)
(29)
(30)

where again

is a vector of spatially autocorrelated error terms and W1 and W2 are

weights matrices, ρ is the autoregressive coefficient for the dependent variable, λ is the
autoregressive coefficient for the error terms, and u is a vector of independent and
identically distributed normal error terms.
2

Estimates for β, σ , , and λ can be obtained through maximum likelihood
estimation, using the concentrated log-likelihood approach described above (LeSage
and Pace 2009). However, in the case of the SAC model, a bivariate optimization in

53

two parameters (

and λ) is required. Neither parameter is achievable in closed form;

therefore, both must be derived computationally.
The maximum likelihood estimator for β using the concentrated log-likelihood
under the SAC is:

′

′

′

′
(31)

Estimation of y at the n sampled locations can be accomplished using (from
Bivand 2009):

(32)

obtained from the DGP or, replacing

with the residual term,

(33)

At unsampled locations, estimation of y can be completed using the trend component
only of the SAC model:

(34)

54

The dimensions of

and

in (34) reflect the number of locations at which the

response variable y will be estimated. Similar to the SAR model, no observations of the
response variable are available at unsampled locations; consequently, the signal
component cannot fully reflect the spatial smoothing process. However, since the first
term in (33) is similar to that of (18), the corresponding formula for the SAR model, the
“feasible” signal component described by Bivand (2009) might also be considered:

(35)

A decision process recommended by Anselin (2005) may be used to select
between OLS, SAR, and SEM models for analysis of the data. In summary, the process
involves completing an aspatial OLS regression analysis and calculating diagnostics for
spatial dependence. The OLS results may be relied upon only if the assumptions
underlying OLS are not violated. First, the lagrange multiplier (LM) statistics for both a
spatial lag and spatial error model are tested for significance. If neither LM is
significant, the OLS results may be used. If one of LM statistics is significant, the
corresponding model should be used to evaluate the data (e.g., if the LM statistic for
only the spatial lag model is significant, the spatial lag model should be used to evaluate
the data). If both of the LM statistics are significant, robust LM statistics should be
tested for significance and the appropriate model selected.
In the case that both robust LM statistics are significant, generalized spatial
modeling (SAC) may be used to account for this more complex form of spatial
autocorrelation (LeSage, 1998).

55

3.1.3 Summary of theoretical models and estimators
Theoretical models are summarized in Table 3-2 for both kriging and spatial
regression models. Data generating processes (DGP) are also provided for the spatial
regression models (LeSage and Pace 2009). A summary of estimators for each
theoretical model follows (Table 3-3).
Note in Table 3-2 that the kriging models all incorporate spatially correlated error
terms. Conversely, the spatial regression methods incorporate spatial autocorrelation
directly in the model statement, resulting in uncorrelated error terms. Universal kriging
attempts to incorporate spatial autocorrelation into the modeling step through the use of
the generalized least squares estimate of β shown in (11). However, methods used to
estimate the variance-covariance matrix

, are not exact; consequently, it is expected

that the error terms retain some spatial autocorrelation. These observations illustrate
the following point made by Schabenberger and Gotway (2005):
“The fact that we consider such very different models for modeling (and
predicting) spatial data is due to the adage that, ‘one modeler’s fixed effect
(regressor variable) is another modeler’s random effect (spatial
dependency.’ Historically, estimation and prediction in models for spatial
data started at the two extremes: regression models with uncorrelated
errors (statistics) and correlated errors with a constant mean
(geostatistics).”
Fortunately, much progress has been made in both directions, resulting in an eversmaller gap between these approaches. This chapter will demonstrate the similarities
and differences between the geostatistical and spatial regression approaches at
present.

56

Table 3-2. Summary of theoretical models underlying kriging and spatial regression
approaches.
Theoretical
Model

Method

Distribution of
Error Terms

Geostatistical Kriging Methods

~

,

,

Simple Kriging

μ is assumed to be constant
and known

Ordinary Kriging

μ is assumed to be constant
and unknown

~
~

Universal Kriging

,

,

,

Spatial Regression Methods

SAR Model

SEM Model

SAC Model

~

DGP*:

57

,

~

DGP:

,

~

DGP:

,

Table 3-3. Summary of kriging and spatial regression estimators.
Method

Simple
Kriging

Ordinary
Kriging

Universal
Kriging

Estimators of y (y)
Estimation Method
Geostatistical Kriging Methods
Var[y] = , Cov[y, y] = :
populated using a variogram
model;
and : assumed known
Var[y] = , Cov[y, y] = :
populated using a variogram
̂
̂
model;
̂ : estimated by LS

′

Var[y] = , Cov[y, y] = :
populated using a variogram
model;
: estimated by GLS

′

Spatial Regression Methods

SAR Model

: concentrated ML;
: concentrated ML, numerical
estimation

∙
′

′
	

SEM Model

′

: concentrated ML;
: concentrated ML, numerical
estimation

′
∙
∙
∙

′

∙
SAC Model

′

∙
: concentrated ML;
, : concentrated ML, numerical
estimation

′
∙
∙ ′
∙

′

58

The spatial regression estimators for

are all shown in their complete form in Table 3-

3, which may be used when making predictions at previously sampled locations only.
When predicting at unsampled locations, only the trend components are estimated (i.e.,

,

,

). The signal (i.e., the spatial smoothing)

component is not reflected; however, each of the β terms incorporate spatial
autocorrelation terms:

(SAR),

(SEM),

and

(SAC). This is a limitation spatial

regression when interpolating at unsampled locations: since only the trend component
is estimated, the predictions are likely overly smooth.

3.2 Theoretical mapping between statistical approaches
Theoretical work presented in this section involves the development of a
classification system for kriging and spatial regression techniques, identifying
mathematical similarities and distinctions between them. An important objective of this
dissertation is to consider the theoretical correspondence between these two estimation
paradigms.
As previously noted, kriging and spatial regression algorithms both belong to the
family of generalized least-squares estimators. Generalized least-squares estimators
(GLSEs) can be presented in a general linear regression model, such as the following
linear regression model:

	

59

(36)

where

has a covariance structure of .
The Gauss-Markov estimator (GME) for β is of the form

′

′

and is the best linear unbiased estimator (BLUE) if
2004). Since
case, the GLSE
estimated

(37)

is known (Kariya and Kurata

is usually not known, calculation of the GME is not possible. In this
can be calculated as above using

in place of

.

The

must be positive definite. Additionally, many models use alternate

formulations for

, including

(38)

where M is a known square matrix (Kariya and Kurata 2004).
Note that the theoretical models for simple kriging, ordinary kriging, and universal
kriging, respectively, are similar. They are restated here for convenience.

,

~
,

′

~
,

~

60

,

(3)

,

(6)

,

(9)

Each equation comprises a term for the mean, which may or may not vary over
space, and a covariance structure for the error terms. GLSEs for all three forms of
kriging can also be presented using the following notation:

(39)

where y is a vector of the n observations of the variable of interest,
estimated kriging weights, and

is a vector of

is the kriging estimate at a given unsampled location.

The kriging weights are generally calculated using the inverse of the full covariance
matrix, which is augmented in the case of ordinary and universal kriging, multiplied by a
vector of covariances between observed locations and the location to be predicted. For
simple kriging, kriging weights are estimated as:	

(40)

where C is the covariance matrix between all pairs of sampled locations, and c is the
vector of covariances between each of the sampled locations and the location to be
estimated. The resulting vector (40) can be substituted in (39) to calculate the simple
kriging estimate.
The kriging weights for universal kriging, a multivariate form of kriging, are more
complicated and are calculated using the following formula and augmented matrices
(where matrix dimensions and row/column locations are shown in subscripts):

61

(41)
where

⋮
,

or

,

⋮

,

(42)

⋮
1

,

or

,

, and

(43)

⋮
						
						⋮
						
1		
		

⋮

⋯
⋯
⋯
⋯	
⋯
⋯
⋯

⋯
⋮ 				 ⋮
⋯
0 ⋯
					
0 ⋯
		⋮ 	⋮
			
		0 		⋯

			1
⋮ 			 			⋮
				1
1			 		0		
					
		0		
⋮ 					⋮
			
					0
	

⋮
0 				
0
⋮		
0		

(44)

which can also be written in a more condensed form as a partitioned matrix:

	

,

,
,

,

62

(45)

Note that cn,1 and Cn,n are the same as c and C in Equation (40).
,

To calculate

must first be inverted. As shown by Theil (1971,

pp. 17-19), a nonsingular matrix D is first defined as:

	

(46)

where

and

are non-singular symmetric matrices. Then

can be written as:

(47)
Since

= 0,

can be written as:

	

	
	

	

	

	
	

	

	

	

	

	

	

	

	
	

	

(48)

	

	

Substituting Equations (43) and (48) into Equation (41) yields:

	

	
	

	
	

	

	
	

	
	

	
	

	

	

	
	

	

	
	

∙
(49)

63

	
	

Since only

, from (42) is necessary for calculating the universal kriging estimate,
	

the top row of the augmented matrix multiplied by vector

	

	

	

	

	
	

	

The lower half of

	

	

	

can be rewritten as:

	

	

	

(50)

	

in (42) contains the Langragian multipliers necessary to

ensure that the results are unbiased and are not directly used in calculating the
universal kriging estimate.
After much matrix algebra, it can be shown that

	

	

	
	

	

	

(51)

	

and
	

	
	

	

	

	

(52)

	

Substituting (52) into (39) gives the universal kriging estimate as

	

	

	
	

	

	

(53)

	

which can also be written

	

	

′

	

′

′

	

64

′

	

′

	

(54)

and simplified as

(55)
or
(56)

	

In words, it can be shown that universal kriging reduces to simple kriging of the
residuals from a generalized least-squares analysis plus the generalized least-squares
estimate at the location of interest.
Estimators for the spatial regression models summarized in Equations (18), (26),
and (33) can also be written in the form of Equation (56). Recall that these complete
estimators are written for estimation at previously sampled locations, where information
on the response variable is available. For example, the SEM model can be written in
the form:

	

(57)

and the SAR model can be written as:

	

	

65

.

(58)

where

and

are constants, and

is a neighbor matrix. The estimated coefficients

for universal kriging, the SEM model, and the SAR model are:

′

′

′

	

′

(59)

	

′

′

′

′

.

(60)

(61)

Equations (56), (57), and (58) can be generalized as:

	

	

	

(62)

Furthermore, (59), (60), and (61) above can be generalized as follows:

	

′

	
	

′

	

	

(63)

Table 3-4 summarizes the contents of the matrices in (62) and (63) for universal kriging,
the SEM model, and the SAR model.

66

Table 3-4. Summary of multipliers for each estimation type.
Estimation
Type

M1

Universal
kriging

M2

M3

′

	

′∙

SEM model
SAR model

M4

	

′∙

	

3.3 Modeling of feedback simultaneity in a spatial setting
The spatial regression models presented in Section 3.1.2 have been expanded
upon in the field of spatial econometrics to allow for the modeling of feedback
simultaneity between two endogenous variables, such as precipitation and vegetation,
in systems of multiple equations (Rey and Boarnet 2004). The use of spatially-explicit
simultaneous equations to evaluate and understand feedback simultaneity is described
in the Section 3.3.1. Estimators using spatial two-stage least squares (S2SLS) are also
presented. In Section 3.3.2, concepts from the development of spatially-explicit
regression models with feedback simultaneity are merged with the universal kriging
technique (Section 3.1.1) to develop a new approach in multivariate kriging that
incorporates feedback simultaneity between two endogenous variables. This new
approach extends the ability to model and describe simultaneous feedback effects to
the prediction of attribute values at unsampled locations in a manner that is both
spatially explicit and reflective of feedback simultaneity between the dependent variable
and a predictive variable.

67

3.3.1 Simultaneous equations spatial regression model
Recent developments in the field of econometrics allow for an expansion into
systems of multiple, simultaneous equations (Kelejian and Prucha, 2004; Rey and
Boarnet, 2004). Rey and Boarnet (2004) present a taxonomy of spatial econometrics
models in simultaneous equations systems which allow for feedbacks between two
endogenous variables, spatial lags for one or both variables, and various forms of
cross-lags between both terms. The case which incorporates both feedback
simultaneity and spatial simultaneity is described in more detail as follows (corrections
made to equation 1 in Rey and Boarnet, 2004, p. 103):
(64)
(65)

where y1 and y2 are the n x 1 vectors of observations for each dependent variable, X is
an n x k matrix of observations for k independent variables associated with the k x 1
vectors β1 and β2 ,

and

provide for feedback simultaneity between the

dependent variables, W is an n x n spatial weights matrix, ρ11 and ρ22 are the spatial
autoregressive lag terms, ρ21 and ρ12 are spatial cross-regressive terms, and ε1 and ε2
are error terms with the following properties:

,

,

,

,

0,

,
,

,

0, for all i,

,

for all i ≠ j, and l = 1,2,

0, for all i ≠ j.

,

68

This system of equations can be expressed in matrix notation as follows:

(66)

where Y = [y1,y2] is a vector of endogenous (dependent) variables, X is a matrix of
exogenous or independent variables, B = (β1,β2), E = (ε1,ε2),

1
1

, and

.

Estimators for this system of equations have been proposed and initially
evaluated by Rey and Boarnet (2004). A spatial two stage estimator can be obtained
following the steps below:
1. Calculate the predicted values for the endogenous variable on the right hand side
(RHS) of the equation by running OLS regression on one or more of the
exogenous variables used in predicting the endogenous variable on the left hand
side (LHS) plus at least one exogenous variable not used in the prediction of the
LHS endogenous variable.
2. Calculate predicted values for the lagged endogenous values on the RHS (i.e,
Wy1 and/or Wy2) as above.

69

3. Replace each of the endogenous variables on the RHS with their predicted
values, then estimate the parameters of the equation using OLS regression.

An aspatial two stage estimator can be obtained by omitting the lagged terms and step
2 above.
For example, the S2SLS estimators of the parameter vector

,

(

,

) for the first equation in the system of equations incorporating a

feedback and two spatial lag terms (Rey and Boarnet 2004, Table 5.1, Model 13) can
be estimated in matrix terms as:

′

∗

where

,

,

,

∗

(67)

is the matrix of exogenous variables excluding the

additional exogenous variable(s) not used to estimate
variable),

,

, and

commonly known as the hat matrix). The matrix
variables, or the matrix

(i.e., the instrumental

∗

(which is also
is the full matrix of exogenous

above with the instrumental variable. The variables

and

are the instrumented variables or the instruments.
To test for the presence of feedback simultaneity, it is most straightforward to
start with an aspatial OLS model. First, fit an OLS model after calculating an instrument
for

(assuming that feedback simultaneity is present between

70

and

). For

example, the S2SLS estimators of the parameter vector

(

,

) can be

estimated in matrix terms as:

′

∗

where

,

	

∗

,

(68)

is the matrix of exogenous variables excluding the additional

,

exogenous variable(s) not used to estimate
The matrix

, and

.

is the full matrix of exogenous variables. Next, fit an OLS model for

treating the second endogenous variable

as an exogenous variable (assuming no

feedback simultaneity).

′

where

∗

.

′

(69)

The two models can be compared using the Hausman Test

(Hausman 1978).

3.3.2 An extension to universal kriging
This dissertation extends the multivariate kriging technique known as universal
kriging (UK; Schabenberger and Gotway 2005), also known as kriging with an external
drift (KED; Goovaerts 1997). Although UK can provide considerable improvement over
univariate forms of estimation, applications to date neglect critical relationships between

71

the kriged variables (e.g., precipitation) and selected covariates. In particular,
feedbacks, or simultaneity, between variables of interest are not considered by existing
kriging methods, a shortcoming in the kriging tool kit.
The presentation of universal kriging in Section 3.2 itself to extension through the
incorporation of simultaneity. The formulation of the universal kriging estimator shown
in (41) through (51) evidently assumes a lack of relationship between variables in the X
matrix, and the dependent variable, y. Given that the covariation structure is typically
taken to be purely spatial, possible co-variation introduced by simultaneity bias is
neglected in predicting . The research presented herein considers solutions to (56)
that account for endogenous relationships among the variables, specifically precipitation
and vegetative cover, by extending universal kriging through an instrumental variables
approach. Specifically, an instrument is included in the multivariate regression model
used to estimate the mean process at each location. For example, let

∗

be made up

of three independent variables (elevation (e), distance from the coast (d), and a
measure of vegetation (v)):

∗

1			

. The feedback or simultaneity

between the dependent variable precipitation and vegetation can be modeled using an
instrumental variable (e.g., a variable related to vegetation but less so to precipitation,
such as soil type (s)). Let

′

72

(70)

∗

where

. The instrumented variable

can be substituted for

in the

original matrix to correct for problems due to simultaneity, giving a new data matrix,
∗∗

			

.

∗∗

is then substituted for

in (70), and

is predicted

as before. This extended form of universal kriging is referred to as universal kriging with
instrumental variables (UKIV).
This dissertation implements UKIV in the case study (Chapter 5). As described
in Chapter 5, no simultaneity was identified based on testing using the OLS approach
described in Section 3.3.1 and the Hausman test. However, UKIV is retained in the
case study as a purely theoretical development.

73

Chapter 4
Case Study

4.1 Study area description
The case study area is located in eastern Africa (Figure 4-1), falling mainly in the
East African country of Kenya and overlapping into northern Tanzania. The geographic
coordinates of the study area range from 34.6° to 39.1° E longitude and 3.7° S to 1.7° N
latitude, falling in the center of the tropical belt, which ranges in latitude from the Tropic
of Cancer in the northern hemisphere at approximately 23.4° N to the Tropic of
Capricorn in the southern hemisphere at 23.4° S. Although the tropics are often thought
of as hot and humid, due to their position near the equator and the tropical rain belt,

Kilometers

Figure 4-1. Location of the case study area within East Africa (CLIP region) and Africa.

74

other features within the tropics (e.g., topography, continental and regional scale winds)
influence regional climates, resulting in conditions that range from arid to humid (Stock,
2004).
The study area contains Mount Kenya to the northeast, and much of the Kenya
Highlands to the west (Figure 4-2). Elevations in this study area range from 195 to
5,778 meters and average 1,291 meters with a standard deviation of 649 meters.
Substantial differences in local terrain occur across the region, from Mount Kenya, the
second highest peak in Africa, to the Great Rift Valley, which cuts through the center of
the region and Lake Victoria, bordering the western edge of the region. The size of the
study area was chosen to be large enough to observe spatial variability in precipitation
Rift Valley
West
Kenya
Highland
Mt. Kenya

Lake
Victoria

Mt. Kilimanjaro

Figure 4-2. Meteorological station locations and the study area boundary are shown above on a map of
elevation (1 km). Major topographic features in the study area are labeled.

75

and to allow for the evaluation of predictive variables at multiple scales. Furthermore,
the sample size of approximately 120 precipitation stations is adequate for statistical
analysis.
Equatorial East African rainfall seasonality is dominated by the “long rains”
(March through May) and “short rains” (October through December) associated with the
strong atmospheric convergence of the passing ITCZ (Hastenrath et al., 1993; Stock,
2004). This seasonal pattern in rainfall is illustrated on Figure 4-3. The long rains
“provide more rainfall than the ‘short rains’ and have a lower interannual variability”
(Camberlin and Okoola, 2003), but the short rains’ start is more predictable. Outside of
the long and short rains coincident with the passing of the ITCZ (i.e., January and
February, June through September), precipitation is typically localized convective
rainfall or, in the highland areas, stratiform rainfall during the cooler months (Ng’ang’a,
1992). Thus, the spatial scale of precipitation varies from large-scale during the long
rains and short rains to mesoscale during the drier months, driven by processes at their
respective scales. The months chosen for statistical analysis (i.e., January, April,
August, and November), therefore, represent the seasons in East Africa and the
corresponding forms of precipitation. Furthermore, independent variables related to
topography, such as elevation, were evaluated for predictive ability at two scales, 1 km
and 9 km, in an effort to capture the spatial scale over which precipitation occurs at
each season.
Higher rainfall occurs at higher elevations in part due to cooler temperatures and
also due to the mountains acting as barriers to moisture-bearing winds. The slopes of
the surfaces interact with elevation (and also the direction of the moisture-bearing

76

winds) such that steeper slopes extract more precipitation. Easterly flow from the Indian
Ocean is the dominant source of moisture for this region, particularly during the short
rains (Black et al., 2003). Since the relatively dry northeasterly and southeasterly
monsoonal winds are weaker at these transition times, onshore moisture transport is
stronger (Nicholson, 1996). Consequently, during the dry seasons generally wetter
conditions occur near Lake Victoria immediately west of the study area, and the Indian
Ocean coast to the east of the study area. While Lake Victoria acts a moisture source
for local convective rainfall in the surrounding highland areas, the coastal climate
differentiates itself from the highland climate due to small-scale diurnal convection from
the land/sea breeze (Camberlin and Planchon, 1997). Blocked by the Rift Valley
slopes, the western parts of East Africa receive moist westerly flows from the Congo
basin. Correspondingly, potential explanatory variables in the statistical analysis
included elevation (1 km and 9 km resolutions), a term that combined information from
slope and aspect (1 km and 9 km resolutions), distance from Lake Victoria and distance
from the Indian Ocean.

4.2 Data
Monthly precipitation is the dependent variable in this analysis. Monthly
precipitation data (mm) from approximately 120 meteorological stations obtained from
the Department of Meteorology, Government of Kenya were used in this analysis.
Meteorological station locations are shown on Figure 4-2. Meteorological stations are
generally located in areas of higher population; therefore, areas of high elevations (i.e.,
greater than 2,500 m) and low elevations (i.e., less than 1,000 m) are not well
represented.

77

Precipitation data for this analysis were chosen to represent the four seasons in
equatorial East Africa (i.e., the dry season in December, January, and February, when
the ITCA is in its southernmost position; the long rainy season in March, April, May, and
June, when the ITCZ is overhead; the cool dry season in July and August, when the
ITCA is in its northernmost position; and the short rainy season in September, October,
and November, when the ITCZ is again overhead) for two different years. The
individual months of January, April, August, and November were chosen as the most
representative month of each of the respective seasons (Figure 4-3).

Jan

Feb Mar

Apr

May Jun

Jul

Aug

Sep Oct

Nov Dec

Monthly precipitation (mm)
Figure 4-3. Long-term average monthly precipitation (mm) from 1926 to 1998 averaged over all
meteorological stations within the study area. The position of the intertropical convergence
zone (ITCZ) relative to the study area is labeled at various points in the year.

Precipitation data from 1984 and 1985 were chosen since these years represent
the overlap between years in which the largest numbers of meteorological stations were

78

measured (Figure 4-4) and the years in which remotely sensed vegetation data were
available (beginning in July 1982). Furthermore, the years to be evaluated were chosen
to represent a typical year (1985) and an atypical year (1984). Figure 4-5 illustrates
monthly precipitation for the years 1982 through 1985 plotted with long-term monthly
averages (dashed lines).

Figure 4-4. Summary of the number of meteorological stations measured by year within the
study area. The number of stations measured decreases after 1985 and continues to rapidly
decline through 2007. This decline is probably due to a combination changes in monitoring
locations or, more likely, lack of access to more recent meteorological station data due to
financial restrictions.

The year 1985 appears to be closest to the long-term average. The differences
between monthly averages for each month and the long term averages were quantified
by calculating average squared differences for each year as the sum of the differences
squared for that year divided by n (12). This value was lowest for 1985 (51.8). The
average sum of squared differences was highest for 1984 (288.4), with the largest

79

1980

1981

1982

J FMA MJJ AS OND

J FMA MJJ AS OND

J FMA MJJ AS OND

1983

1984

1985

J FMA MJJ AS OND

J FMA MJJ AS OND

Year shown

J FMA MJJ AS OND

1926 - 1998

Figure 4-5. Monthly precipitation for the years 1980 through 1985 plotted with long-term
monthly averages (dashed lines). The average sum of squared differences was lowest for 1985
(51.8). The average sum of squared differences was highest for 1984 (288.4), with the largest
numbers of months falling below the long-term average. Precipitation amounts in 1984 are
lower than average for most of the year.

numbers of months falling below the long-term average.
Independent variables included two distance measures: minimum distance of
each meteorological station from the Indian Ocean coastline and distance of each
station from the centroid of Lake Victoria. Since precipitation was not linearly related to
either of these distance measures, dummy variables for each measure were established
to represent “distance bands,” within which precipitation levels were similar. Distance
bands of 0 to 300 km, 300 to 450 km, 450 to 600 km, and greater than 600 km were
created as a measure for distance to Lake Victoria. Distance bands of 0 to 500 km and
greater than 500 km were used to represent distance to the Indian Ocean. All distance
bands were chosen based on inspection of precipitation versus distance plots; the cutoff

80

distances were generally observed to be indicative of a change in the relationship
between precipitation and distance from water body.
Other independent variables were a vegetation index, elevation, and derivatives
of elevation (measures of the degree to which a slope faces north and east, and the
curvature of the slope) at scales of 1 km and 9 km. The scale of 9 km was chosen to
evaluate the scale over which convective rainfall occurs. Furthermore, this scale is
consistent with the findings of Hession and Moore (2010) and Sharples et al. (2005);
Sharples et al. (2005) identify an optimal topographic scale of dependence of around 68 km. Sharples et al. also indicate that for scales of over 30 km, analyses incorporating
elevation yield results similar to analyses based on longitude and latitude alone.
Elevation at each station location was estimated using the SRTM 30 arc second
Digital Elevation Model (DEM) raster, shown on Figure 4-2. At locations near the
equator, 30 arc seconds is close to 1 km resolution. Average elevation at a 9 km
resolution was calculated for each raster cell using the nine cells centered on that cell
and the FOCALMEAN function in ArcGIS (ESRI, 2006). Elevation data within a large
buffer surrounding the study area were included to avoid edge effects or biased results
within the study area.
The eastern and northern components of the unit normal vector were used to
represent the effects of aspect and slope on precipitation (Hutchinson 1998b). These
components were calculated at both scales (1 km and 9 km) as follows:

p = cos(α) sin(θ)
q = sin(α) sin(θ)

81

where α is the angle of aspect in degrees and θ is the angle of the slope in degrees.
The values of p and q represent aspect scaled by the steepness of the slope. That is,
these values are largest in magnitude on the steepest slopes and approach zero in flat
areas. In addition, curvature was calculated in ArcGIS (ESRI, 2006) at the scales of 1
km and 9 km to evaluate whether curvature has an impact on precipitation patterns.
Vegetation was represented by one of the first vegetation indices created, the
normalized difference vegetation index (NDVI; Rouse et al 1974). NDVI is a ratio
between (Near infrared - Red) and (Near infrared + Red). Absorption patterns between
the red (0.55-0.68 μm) and near infrared (0.73-1.1 μm) portions of the spectrum provide
an indicator of vegetation amount and vigor. The ratio is sensitive to the difference
between near infrared and red, with increasing chlorophyll concentration and green leaf
vegetation density increasing NDVI value. Theoretically, NDVI ranges between -1 and
+1, although values typically range between 0 for bare ground to 0.7 for lush dense
vegetation. NDVI data were obtained from the Global Inventory Modeling and Mapping
Studies (GIMMS) dataset, a record of bimonthly NDVI beginning in July 1982. Average
monthly NDVI data was calculated from the two bimonthly measurements in each
month. A one-month lag was chosen as an optimal lag time to identify correlations
between precipitation and vegetation (Brunsell, 2006; Richard and Poccard, 1998;
Wang et al.,2003).
For the purpose of prediction using universal kriging with instrumental variables,
one independent variable expected to influence NDVI but not used to predict
precipitation patters was chosen. This variable was a measurement of soil pH,
specifically the pH in the water found in the interstitial spaces of the soil. These data

82

were obtained from the Harmonized World Soil Database (FAO/IIASA/ISRIC/ISSCAS/JRC 2009), developed in a collaborative effort between the Food and Agriculture
Organization of the United Nations (FAO), the International Institute for Applied Systems
Analysis (IIASA), the ISRIC-World Soil Information, the Institute of Soil Science –
Chinese Academy of Sciences (ISSCAS), and the Joint Research Centre of the
European Commission (JRC).
Data for each of the independent variables were extracted from the grid cells
overlapping meteorological station locations and combined with precipitation data at
each respective station using the Intersect Point Tool in Hawth’s Analysis Tools (Beyer,
2002), an add on to ArcGIS. Table 4-1 summarizes the independent variables
considered in the statistical analyses.
For comparison to results presented herein, the CRU TS 3.0 data generated by
the Climatic Research Unit of the University of East Anglia (CRU) were obtained. The
CRU TS 3.0 is a recent global data set, representing the years 1901 to 2006, based on
the methodology developed for CRU TS 2.1 (Mitchell and Jones 2005). The CRU TS
2.1 data were generated by interpolating climate data onto a regular 0.5 degree grid
following New et al (2000). The CRU TS 3.0 data were combined with the remaining
data by extracting from the 0.5 degree grid to sampled locations, as described above.

4.3 Methods
Four methods were used to generate estimated precipitation surfaces for the
case study area: local ordinary kriging (LOK), universal kriging (UK), a newly developed
extension to universal kriging that incorporates simultaneity between precipitation and
vegetation using an instrumental variable approach (UKIV), and spatial regression

83

modeling (SpReg). Each of these methods is described in detail in Chapter 3. LOK
was selected as the only univariate method for estimating precipitation (i.e., only
precipitation data were utilized in the analysis). The remaining methods are all
multivariate, and incorporate data from independent variables that are correlated with
precipitation in an effort to improve predictions at unsampled locations. For the
multivariate methods, initial model selection was conducted using ordinary least
squares (OLS) regression. The independent variables identified as predictors for
precipitation in the selected months were then incorporated in the multivariate kriging
analyses as well as the spatial regression analyses. The spatial regression techniques
were also used for the purpose of hypothesis testing (Chapter 5) in a demonstration of
their “added value”: spatial regression techniques are useful not only for prediction at
unsampled locations, but may also be used to improve understanding and test for
significant predictors of the spatial distribution of precipitation.

84

Table 4-1. Summary of independent variables used in OLS regression analysis.
Abbreviated names are shown as well.
Independent 
Variable 
Distance to Lake Victoria (km) 

Scale 

Abbreviated 
Name 
Notes 
dist2lv 
d2lv300 

= 1 if dist2lv <= 300 km,  
    0 otherwise 

d2lv450 

= 1 if dist2lv > 300 and <= 450 km,  
   0 otherwise 

d2lv600 

= 1 if dist2lv > 450 and <= 600 km,  
    0 otherwise 
if all 3 indicator variables are 0, then 
dist2lv > 600 km 

Distance to Indian Ocean (km) 

dist2coast 

Elevation (m) 

d2c500 
dem1km 
dem9km 

Combined measure of apsect 
and slope: measure of 
"northness" 
(unitless) 
Combined measure of apsect 
and slope: measure of 
"eastness" 
(unitless) 
Surface curvature 
(profile; unitless) 
NDVI 
(unitless) 

1 km 
9 km 

1 km 
9 km 

p1km 
p9km 

1 km 
9 km 
1 km 
9 km 
8 km 
8 km 
8 km 
8 km 
8 km 
8 km 
8 km 
8 km 

q1km 
q9km 
curv1km 
curv9km 
ndvi842 
ndvi845 
ndvi849 
ndvi8412 
ndvi852 
ndvi855 
ndvi859 
ndvi8512 

85

= 1 if dist2coast <= 500 km,  
    0 otherwise 

p = cos(α) sin(θ), where α is aspect 
angle and θ is slope angle  
p = cos(α) sin(θ) 

q = sin(α) sin(θ) 
q = sin(α) sin(θ) 

Average NDVI for February 1984 
Average NDVI for May 1984 
Average NDVI for September 1984 
Average NDVI for December 1984 
Average NDVI for February 1985 
Average NDVI for May 1985 
Average NDVI for September 1985 
Average NDVI for December 1985 

4.3.1 Model selection
Monthly precipitation data for four months (January, April, August, and
December) in 1984 and 1985 were evaluated in eight independent analyses over space.
Initial model selection was performed using OLS regression. Although automated
selection procedures such as stepwise regression are commonly used in model
selection, recent studies have identified many pitfalls in these approaches such as
model misspecification and inaccurate results due to multicollinearity and confounding
between independent variables. An alternative approach for developing and evaluating
a set of candidate models described by Burnham and Anderson (1998) was used.
Central to this approach is abandoning the use of automated variable selection methods
in favor of careful a priori specification of candidate models (hypotheses) of particular
interest. A number of candidate models were then compared directly and ranked based
on an information theoretic criterion, the Akaike Information Criterion (AIC; Akaike,
1974). This approach, therefore, provided a means to rank the relative strength of
models leading to an understanding of their predictive power and uncertainties.
Importantly, this methodology is not susceptible to the instabilities of variable selection
procedures previously mentioned.
Development of statistical models in this way requires careful thought and
engagement between discipline experts and statisticians. Variables identified as
possible predictors of spatial patterns in precipitation were selected in collaboration with
experts in East African climatology and vegetative land cover, then tested in several
regression model formulations.

86

As shown on Table 4-2, precipitation amounts for each month were evaluated
using four OLS regression model formulations, or candidate models. The candidate
models were developed to include the following sets of independent variables: (1)
average monthly NDVI for the subsequent month, the elevation term and its derivatives
(i.e., dem, p, q, and curv) at a 1 km scale, each indicator variable representing distance
to Lake Victoria and distance to the Indian Ocean; (2) average monthly NDVI for the
subsequent month, the elevation term and its derivatives (i.e., dem, p, q, and curv) at a
9 km scale, each indicator variable representing distance to Lake Victoria and distance
to the Indian Ocean; (3) average monthly NDVI for the subsequent month, the elevation
term and its derivatives (i.e., dem, p, q, and curv) at a 1 km scale, each indicator
variable representing distance to Lake Victoria and distance to the Indian Ocean and
interaction terms between distance to Lake Victoria (dist2lv) and the first two indicator
variables (i.e., d2lv300 and d2lv450); and (4) average monthly NDVI for the subsequent
month, each elevation term (dem, p, q, and curv) at a 9 km scale, each indicator
variable representing distance to Lake Victoria and distance to the Indian Ocean and
distance to the Indian Ocean and interaction terms between dist2lv and the first two
indicator variables (i.e., d2lv300 and d2lv450).
The AIC value is shown for each model formulation on Table 4-2. Lower AIC
values indicate better model fits. Each model is ranked based on AIC values; models
with a ranking of 1 were selected for subsequent use in each multivariate prediction
technique. Final models are summarized in Appendix 1.
Monthly precipitation values were transformed to normality using a normal score
transformation (Perttunen and Stuckman, 1990; Wu et al., 2006) prior to developing the

87

candidate models above. Transformations of the dependent variable are commonly
used in regression modeling to stabilize variance and improve normality (Gibbons,
1994; Hutchinson, 1998a; Hutchinson, 1998b; Neter et al., 1990; Schabenberger and
Gotway, 2005; Sharples et al., 2005), particularly for variables that exhibit asymmetrical
distributions, such as precipitation. Precipitation data are more often right-skewed in
distribution since the range of possible values is restricted by a lower bound of zero.
This issue may be exacerbated during dry months, when precipitation amounts are
lower and more often equal to zero.
The normal score transformation is a monotonic transformation that replaces the
observed value with a “typical” value from the standard normal distribution that
corresponds to the same order statistic, or ranked value. Thus, the normal score
transform ensures that the assumptions of normality and equal variances are met
(Perttunen and Stuckman, 1990). Back-transformation of the results, allowing for
interpretation in the original data scale, is relatively straightforward (Wu et al., 2006).
The normal score transformation was completed using an R function (Shortridge
2010), modified to allow for right skewed distributions such as precipitation. In
summary, this R function (and the normal score transform used for this application)
calculates normal scores for a vector of values such as precipitation at multiple
locations. First, the order statistics for the vector of values are determined

88

Table 4-2. Summary of OLS regression models ranked by AIC.
Model
1
2
3

Dep.
Variable
Jan 84
Precip
Nscores
Jan 84
Precip
Nscores
Jan 84
Precip
Nscores

Independent Variables
ndvi842, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500

AIC
293.9

Rank
(4)

ndvi842, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500

292.6

(3)

ndvi842, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

291.4

(2)

4

Jan 84
Precip
Nscores

ndvi842, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

288.4

(1)

1

Apr 84
Precip
Nscores
Apr 84
Precip
Nscores
Apr 84
Precip
Nscores

ndvi845, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500

291.3

(3)

ndvi845, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500

282.0

(1)

ndvi845, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

294.0

(4)

4

Apr 84
Precip
Nscores

ndvi845, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

284.3

(2)

1

Aug 84
Precip
Nscores
Aug 84
Precip
Nscores
Aug 84
Precip
Nscores

ndvi849, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500

229.1

(4)

ndvi849, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500

228.9

(3)

ndvi849, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

223.7

(1)

4

Aug 84
Precip
Nscores

ndvi849, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

224.3

(2)

1

Nov 84
Precip
Nscores
Nov 84
Precip
Nscores
Nov 84
Precip
Nscores

ndvi8412, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500

277.2

(2)

ndvi8412, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500

273.5

(1)

ndvi8412, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

281.2

(4)

Nov 84
Precip
Nscores

ndvi8412, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

277.4

(3)

2
3

2
3

2
3

4

89

Table 4-2 (continued). Summary of OLS regression models ranked by AIC.
1
Jan 85 Precip
ndvi852, dem1km, p1km, q1km, curv1km, d2lv300,
288.4
Nscores
d2lv450, d2lv600, d2c500
2
Jan 85 Precip
ndvi852, dem9km, p9km, q9km, curv9km, d2lv300,
296.6
Nscores
d2lv450, d2lv600, d2c500
3
Jan 85 Precip
ndvi852, dem1km, p1km, q1km, curv1km, d2lv300,
279.8
Nscores
d2lv450, d2lv600, d2c500, d2lv300*dist2lv,
d2lv450*dist2lv

(2)
(4)
(1)

4

Jan 85 Precip
Nscores

ndvi852, dem9km, p9km, q9km, curv9km,d2lv300,
d2lv450, d2lv600, d2c500, d2lv300*dist2lv,
d2lv450*dist2lv

291.0

(3)

1

Apr 85 Precip
Nscores
Apr 85 Precip
Nscores
Apr 85 Precip
Nscores

ndvi855, dem1km, p1km, q1km, curv1km, d2lv300,
d2lv450, d2lv600, d2c500
ndvi855, dem9km, p9km, q9km, curv9km, d2lv300,
d2lv450, d2lv600, d2c500
ndvi855, dem1km, p1km, q1km, curv1km, d2lv300,
d2lv450, d2lv600, d2c500, d2lv300*dist2lv,
d2lv450*dist2lv

311.7

(1)

315.5

(3)

312.6

(2)

4

Apr 85 Precip
Nscores

ndvi855, dem9km, p9km, q9km, curv9km, d2lv300,
d2lv450, d2lv600, d2c500, d2lv300*dist2lv,
d2lv450*dist2lv

317.1

(4)

1

Aug 85 Precip
Nscores
Aug 85 Precip
Nscores
Aug 85 Precip
Nscores

ndvi859, dem1km, p1km, q1km, curv1km, d2lv300,
d2lv450, d2lv600, d2c500
ndvi859, dem9km, p9km, q9km, curv9km, d2lv300,
d2lv450, d2lv600, d2c500
ndvi859, dem1km, p1km, q1km, curv1km, d2lv300,
d2lv450, d2lv600, d2c500, d2lv300*dist2lv,
d2lv450*dist2lv

218.9

(3)

214.7

(1)

220.7

(4)

4

Aug 85 Precip
Nscores

ndvi859, dem9km, p9km, q9km, curv9km, d2lv300,
d2lv450, d2lv600, d2c500, d2lv300*dist2lv,
d2lv450*dist2lv

216.6

(2)

1

Nov 85 Precip
Nscores
Nov 85 Precip
Nscores
Nov 85 Precip
Nscores

ndvi8512, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500
ndvi8512, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500
ndvi8512, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

268.2

(3)

264.4

(1)

271.5

(4)

Nov 85 Precip
Nscores

ndvi8512, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

268.0

(2)

2
3

2
3

2
3

4

Differences of less than 3 between AIC values are not considered
significant.

90

(i.e., the data are sorted and ranked from smallest to largest). The normal score for
each observation is then found as the “typical” value of the kth smallest number from a
standard normal distribution (i.e., a normal distribution with a mean of zero and a
standard deviation of one) (Perttunen and Stuckman, 1990). Results are backtransformed to the original data scale by linear interpolation between data. Values
above the highest observed datum are extrapolated: (1) the mean of the original data is
calculated and values above the mean identified, (2) the standard deviation of those
values is calculated, (e) if the mean score plus the score of the maximum value
multiplied by the standard deviation calculated in step 2 is less extreme than the actual
maximum, the actual maximum is used instead. No extrapolation was completed for
low values since the lower end of the output distribution was constrained to zero.

4.3.2 Consideration of other candidate OLS models
Many other models were considered prior to selecting the candidate models
summarized in Table 4-2. For example, models using the continuous distance
measures (dist2lv and dist2coast) were developed; however, AIC values were
consistently higher (worse) for these models. In addition, inconsistent results were
obtained due to multicollinearity between the continuous distance measures and other
variables such as NDVI.
Elevation cutoffs were developed to allow for potentially non-linear relationships
between precipitation and elevation and to account for the relatively small number of
precipitation stations at higher and lower elevations. Cutoffs of 1,000 meters and 2,500
meters were used. These variables were not significant in various modeling attempts

91

due, in large part, to the significance of the distance “bands” that were developed which
also indirectly stratified rain stations by elevation.
In addition, models were attempted without transforming the monthly precipitation
data. These models consistently yielded large numbers of negative predicted values for
precipitation and diagnostic plots that confirmed the presence of heteroskedasticity (i.e.,
unequal variances) and non-normal residuals, thus violating the underlying assumptions
of the regression techniques being used.

4.3.3 Prediction of spatial patterns in monthly precipitation
The following four methods were used to estimate precipitation surfaces for
precipitation in the months of January, April, August, and December, in 1984 and 1985:


Local ordinary kriging with nine nearest neighbors



Universal kriging



Universal kriging with an instrumental variable



Spatial regression (weights matrices developed using nine nearest
neighbors)

Local ordinary kriging and spatial regression both require specification of the
number of nearest neighbors to be incorporated in the analysis. A weighting scheme
incorporating nine nearest neighbors was selected, consistent with the work of New et
al (2000) and Piper and Stewart (1996). New et al (2000) used eight nearest neighbors
to interpolate climate data anomalies for mapping purposes. Piper and Stewart (1996)
used between five and ten neighboring stations to interpolate climate data.

92

Universal kriging with instrumental variables requires calculation of an instrument
for NDVI, the variable that is believed to be simultaneously related with precipitation. In
other words, a feedback simultaneity is believed to exist between precipitation and
vegetation (NDVI). An instrument representing NDVI, to be used in place of NDVI in the
UKIV analysis, was calculated using OLS regression, with the following independent
variables: the soil pH variable described in Section 4.2, Data, and a subset of
independent variables also used in the prediction of precipitation patterns, i.e., distance
to Lake Victoria, distance to the Indian Ocean coast, and elevation at a resolution of 9
km. Universal kriging with instrumental variables (UKIV) was included in the case study
as a purely theoretical development, however, since formal testing for simultaneity
between precipitation and the 1 month-lagged vegetation measure (NDVI) did not
identify simultaneity for any month in 1985, the year selected to represent a typical
precipitation year in East Africa. Formal testing was completed using the Hausman test
(Hausman 1978).
Maps for estimated precipitation in April 1985 are shown on Figure 4-6. The
remaining maps are provided in Appendix 1. Table 4-3 provides a summary of
predicted values generated by each method. For comparison purposes, a summary of
CRU TS 3.0 data is also provided on Table 4-3.
Predicted precipitation patterns are similar in all four maps in April 1985 (Figure
4-4), the first wet season of the year in the study area. The ITCZ is positioned directly
overhead the study area during this month. Peak estimated rains for the month
(approximately 600 mm) are depicted over Mount Kilimanjaro and, in particular, Mount
Kenya. Drier regions (< 50 mm) appear along the eastern side of the study area, and

93

dry regions also appear generally north/northwest of Mount Kenya, and southwest of
Mount Kenya. The LOK map illustrates streak- like features jetting out from the central
portion of the study area where rain stations are located; this is an effect of the
extrapolation tendencies of LOK and nearest neighbor configuration.
The effects of the distance bands, which were significant for all three cutoffs (i.e.,
300 km, 450 km, and 600 km from Lake Victoria), are evident in the maps created using
universal kriging, universal kriging with simultaneity, and spatial regression. The
distance bands allowed for decreasing precipitation at locations further from Lake
Victoria in a stepwise fashion. It was hoped that the continuous distance measure
would result in a smoother surface; however, distance bands were necessary due to the
complicated, non-linear relationship between distance from Lake Victoria and amount of
precipitation and the confounding between the continuous distance measure and other
independent variables (e.g., elevation and NDVI). In addition, elevation at a resolution
of 1 km was a significant predictor of April 1985 precipitation, which is evident in the
increased rainfall amounts occurring at higher elevations (e.g., Mount Kenya, Mount
Kilimanjaro, and the highlands of western Kenya). Figure 4-7 shows a map derived
using universal kriging but excluding the dummy variables used to create the distance
band; no “bulls-eye” effect is evident, demonstrating that this effect is due to the use of
the distance bands.

94

Table 4-3. Summary of predicted precipitation amounts for each prediction method.
Month
Jan-84

Apr-84

Aug-84

Nov-84

Jan-85

Method
Observed
OK
UK
UKIV
SpReg
CRU
Observed
OK
UK
UKIV
SpReg
CRU
Observed
OK
UK
UKIV
SpReg
CRU
Observed
OK
UK
UKIV
SpReg
CRU
Observed
OK
UK
UKIV
SpReg
CRU

Min.
0
0.8118
0
0
0
2.82
3
5.936
3
3
3
9.87
0
0.117
0
0
0
0.2
0
49.8
0
0
0
0.29
0
1.262
0
0
0
0.43

1st Qu. Median
1.45
7.8
7.268
15.89
0
0
0
1.156
0
0
6.69
8.49
67.4
96.7
69.34
119.8
37.31
57.35
44.65
67.57
58.46
71.46
16.51
19.77
2.613
20.81
4.475
13.05
0
0
0
0
0
0
3.1
5.99
87.45
131
99.63
162.8
55.5
87.28
55.5
82.15
73.02
95.24
1.017
1.85
3.7
13.3
10.68
15.72
1.411
4.5
3.403
5.011
3.035
4.902
1.03
1.385

95

Mean
3rd Qu.
20.2
31.8
20.7
30.06
8.464
3.818
9.487
4.1
9.587
3.5
9.518
11.58
138.2
199
130.2
181
64.69
75.31
79.56
86.43
70.12
86.59
20.28
22.54
54.3
78.34
25.36
26.06
17.23
10.87
11.18
5.86
17.74
14.02
7.256
12.55
170.4
202.3
183.4
245.9
88.27
116.2
84.96
108.6
94.19
119.2
2.477
3.71
27.55
44.37
18.01
19.78
8.378
10.02
9.124
11.3
8.857
10.41
2.17
2.89

Max.
129.8
74.45
143.3
149.8
158.5
19.49
821.1
728.4
821.1
821.1
821.1
43.53
283.7
226.5
283.7
283.7
283.7
16.44
767.2
529
767.2
767.2
767.2
8.6
147.3
86.83
147.3
147.3
147.3
11.79

Table 4-3 (continued). Summary of predicted precipitation amounts for each prediction
method.
Month
Apr-85

Aug-85

Nov-85

Method
Observed
OK
UK
UKIV
SpReg
CRU
Observed
OK
UK
UKIV
SpReg
CRU
Observed
OK
UK
UKIV
SpReg
CRU

Min.
1st Qu. Median
3.2
175.4
258.9
12.79
147.3
187.5
3.2
55.15
175
3.2
87.16
196.1
3.2
149.8
189.7
3.82
8.385
10.81
0
2.71
29.03
0.1317
7.015
20.24
0
2.02
4.764
0
5.2
10.8
0
2.566
5.613
0
2.91
11.3
21.3
67.45
102.1
28.23
87.27
112.4
21.3
45.29
63
21.3
49.29
76.03
21.3
44.82
61.55
4.21
7.08
8.96

Mean
3rd Qu.
251.9
335.8
191.9
233.5
157.1
241.2
195.3
267.3
170.9
221.9
13.26
15
63.14
109.1
38.65
62.5
28.68
37.18
39.21
72.73
22.17
25.44
8.353
11.69
121.1
140.9
118.9
144.1
70.33
91.12
76.5
94.53
63.61
82.81
14.47
21.05

Note:  UK, UKIV, and SpReg all used normal score transformations of average
              monthly precipitation values.  Values were transformed back to the
              original scale using a back transform function.

96

Max.
543
455.4
543
543
543
56.31
246.1
189.2
246.1
246.1
246.1
14.68
561.1
426.9
561.1
561.1
561.1
44.6

Ordinary Kriging (nn=9)

 

Universal Kriging

Universal Kriging with IV

Spatial Regression

Figure 4-6. Precipitation maps for April 1985 generated using LOK (top left), UK (top
right), UKIV (bottom left), and regression techniques (either ordinary least squares or
spatial lag models, as indicated; bottom right).

97

Universal Kriging

Figure 4-7. Precipitation map for April 1985 generated using UK without distance bands.

Estimated precipitation maps for all of the months evaluated in this dissertation
are provided in Appendix 1. Based on these maps, the following observations were
made.
January is the month during which the ITCZ is at its most extreme southerly
position; consequently, it is a dry month near the equator. All four of the January 1984
precipitation maps show the highest levels of precipitation on the western edge of the
study area, nearest Lake Victoria. Peaks shown in the eastern central and southern
portions of the OK map, corresponding with high elevation locations (i.e., Mount Kenya
and Mount Kilimanjaro), are also evident on the UK map. These spatial patterns in
precipitation are consistent with the expectation that rains occurring in the dry seasons
are mainly localized convective rainfall near Lake Victoria or stratiform rainfall in the
highland areas (Ng’ang’a 1992). Maps for UK, UKIV, and OLS regression (Figure 1,

98

Appendix 1) appear very similar. These three models were all based on the same OLS
model which incorporated categories for distance from Lake Victoria with interaction
terms for the < 300 km and 300 to 450 km categories. The scales for these maps are
also similar, extending to approximately 150 mm.
The range of precipitation amounts is much higher in April 1984, extending to
728 mm in the OK map, and to approximately 820 mm in the remaining maps. This is
expected since the ITCZ is directly over the study area at this time of year. The OK
map also depicts precipitation across the study area, although in the remaining maps it
appears that the rainfall is concentrated mostly in the west, near Lake Victoria. This is
partly the result of the categorization scheme used in these map scales, since the
lowest category extends to approximately 55 mm. The OK map shows a strong peak at
the location of Mount Kilimanjaro. This peak also occurs on the UK and UKIV maps.
Precipitation maps for August 1984 once again show a generally lower range of
precipitation amounts, ranging up to 280 mm. The ITCZ is at its northernmost position
in August, causing the second dry season of the year in the study area. Similar to
January 1984, higher levels of precipitation in August 1984 occur mainly in the western
portion of the study area near Lake Victoria and in the Western Highlands. Higher
precipitation levels are also seen over the peaks of Mounts Kenya and Kilimanjaro. The
rings shown in the UK, UKIV, and OLS regression maps illustrate the interaction
between elevation effects within distance band nearest Lake Victoria. The second
distance band, from 300 to 450 km from Lake Victoria, illustrates the effect of location
within that band, which is decreased precipitation. A decreasing trend with elevation is
evident within the second distance band. The overall spatial pattern of precipitation in

99

August 1984 is also consistent with the expectation that dry season rains are mainly
localized convective rainfall near Lake Victoria or stratiform rainfall in the highland areas
(Ng’ang’a 1992).
In November 1984, the ITCZ is over the study area again, providing the second
rainy season, also known as the “short rainy season” in the region. Precipitation
amounts range up to approximately 770 mm in the UK, UKIV, and spatial regression
maps, and up to 530 mm in the OK map. Precipitation is generally present in elevated
amounts across the study region, with the highest amounts in the eastern central
portion of the area. The spatial patterns in the November 1984 precipitation maps,
illustrating more widespread areas of higher precipitation, are consistent with the largerscale effects of the passing ITCZ. The map based on spatial regression appears
somewhat smoother than the kriging-based maps.
The precipitation maps for January 1985 appear similar to the January and
August 1984 maps than the January 1984 maps; the range of precipitation is closer to
that of January 1984 (i.e., maximum predicted rainfall amounts reach approximately 150
mm). As it has been shown, precipitation is predominantly in the western portions of the
study area in the dry seasons, consistent with the mesoscale effects of Lake Victoria
and of increased elevation. Although not observed in January 1985, the interaction
between the distance to Lake Victoria terms in the multivariate models and elevation are
evident, similar to August 1984.
Similar to April 1984, precipitation maps for April 1985 depict the higher rainfall
levels that occur during the long rainy season ranging up to approximately 550 mm.
Increased levels of rainfall are evident throughout most of the study area, consistent

100

with the large-scale effects of the passing of the ITCZ. The highest levels of rainfall are
observed over Mounts Kenya and Kilimanjaro; this pattern is most evident in the OK,
UK, and UKIV maps. The spatial regression map appears smoother in spatial
distribution than the kriging-based maps, with less differentiation between areas of peak
rainfall and areas of lower rainfall amounts.
Precipitation amounts predicted for August 1985 are similar in spatial
distribution. All four maps depict higher rainfall levels in the northwest of the study
region, ranging up to approximately 250 mm. All three of the multivariate prediction
methods (UK, UKIV, and spatial regression) show more detail in the central and south
central portions of the study area, where precipitation amounts appear higher than in
the OK map. These spatial patterns in precipitation are consistent with the mesoscale
effects of Lake Victoria and of higher elevations expected during the dry seasons in
East Africa. The streaking features common in maps generated by LOK are clearly
present.
November 1985 precipitation patterns are similar in distribution to the November
1984 rainfall amounts, although generally lower overall. Although 1985 was observed
to be a more typical year, 1984 was atypical in that rainfall amounts were lower than
usual in the early part of the year and higher than most years in the later part of the
year. Precipitation amounts range to approximately 430 mm (LOK map) to
approximately 560 mm (UK, UKIV, and OLS regression maps). The highest
precipitation amounts are shown in the southeastern portion of the study area over
Mount Kilimanjaro, extending northward towards Mount Kenya. The effect of Lake
Victoria is also evident in all four maps.

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The bimodal nature of precipitation throughout the year along the equator in East
Africa is also evident in Table 4-3, which provides a summary of predicted precipitation
amounts by prediction method. Summary statistics of predicted precipitation amounts
for all four prediction methods illustrate two wet seasons and two dry seasons each
year. There is a tendency of the multivariate methods (i.e., UK, UKIV, and regression
methods) to predict larger proportions of zeroes, particularly in the dry months and
higher maximum predictions, resulting in more highly right-skewed rainfall amounts.
LOK generally yields a smaller range of precipitation amounts: the lower percentiles are
generally higher for LOK, and the maximum values are lower.
CRU precipitation estimates are consistently lower than estimates generated
herein (Table 4-3). This is expected since global data sets are often too smooth and do
not reflect the more extreme values. In some months, CRU results correspond to lower
percentiles of the distribution more closely (e.g., January 1984, April 1984, August
1984, January 1985, and August 1985). Months with particularly low CRU values
include April and November of 1984, and April and November of 1985, representing the
wet seasons in the study area. It appears that the CRU data greatly underestimate
precipitation in wet East African seasons.

4.3.4 Comparison and evaluation of map accuracy
Accuracy of the precipitation maps was evaluated by comparing root mean
square errors (RMSEs) for each the four maps generated at each time interval. RMSEs
are summarized in Table 4-4.

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Table 4-4. Comparison of root mean squared errors

Month
Jan-84
Apr-84
Aug-84
Nov-84
Jan-85
Apr-85
Aug-85
Nov-85

OK
rmse
20.162
74.707
34.903
101.027
23.669
89.125
34.798
73.881

UK
UKIV Sp. Reg.
rmse
rmse
rmse
24.368 20.840 20.529
166.342 99.637 103.903
82.620 39.360 38.558
175.267 122.138 119.139
37.768 26.888 24.781
270.690 90.644 96.990
89.018 39.504 35.705
130.970 72.749 67.992

CRU
rmse
26.461
161.403
79.831
212.218
40.682
267.046
84.766
130.234

Root mean square errors are generally lowest for LOK, the univariate method of
estimation. Spatial regression provided the lowest RMSE for November 1985
precipitation estimates. In general, RMSE values are closest in dry seasons,
particularly January 1984, January 1985, August 1985. Spatial regression yielded lower
RMSEs than either UK method in four of the eight months.
RMSE values are consistently the highest for the CRU data, particularly during
the wet seasons. In some cases, the CRU RMSE is twice as high as the others,
indicating the high amount of error from corresponding observations collected at
meteorological stations in the study area.
Since RMSEs represent overall map accuracy with a single number, precipitation
map accuracy was also evaluated using maps of standardized residuals to evaluate for
spatial patterns in error terms. Large positive residuals (i.e., zscore > 2) are indicated
with red dots and large negative residuals (i.e., zscore < -2) are shown with blue dots.
A map of standardized residuals is shown in Figure 4-7 for April 1985.

103

In April 1985, two areas of significant residuals are generally evident: in the
northwest of the study area in the western Kenya highlands, and in the center of the
study area near Mount Kenya. In most cases, both high and low significant residuals
are observed in the western Kenya highlands, indicating that there is a large amount of
variability in precipitation in this region. This may be due, in part, to the highly variable
topography in the area. The UK model underestimated precipitation in a few locations
in the center of the study area, while the SpReg model overestimated precipitation in
this region. The UKIV model has the fewest significant standardized residuals,
indicating a generally improved model fit. The RMSE for the UKIV model (29.79) is very
close to the smallest RMSE (29.71) achieved by the OK model.
Standardized residual maps for all of the months evaluated in this dissertation
are provided in Appendix 1. Based on these maps, the following observations were
made.
Patterns of significantly high or low residuals are somewhat similar across
prediction methods and months being evaluated. In January 1984, several significant
negative residuals are shown for the OLS method. A cluster of low residuals is present
in the northwest of the maps for all four methods, although a high residual is shown very
near to at least one low residual in all cases, indicating highly variable rainfall amounts
in the region. Large positive residuals are indicative of the model underestimating
actual precipitation; large negative residuals occur when the model overestimates actual
precipitation.

104

Ordinary Kriging (nn=9)

 

Universal Kriging

Universal Kriging with IV

Spatial Regression

Figure 4-8. Maps of Significant Cross Validation residuals (|z-score|>2) in April 1985 for
LOK (top left), UK (top right), UKIV (bottom left), and regression residuals (for
regression model; bottom right). Significantly high residuals are shown in red, and
significantly low residuals are shown in blue.
Significant residuals are present in the central/eastern regions of the study area
in April 1984. UK residuals depict a cluster of high residuals east of Mount Kenya, and
another cluster of high residuals in northern Tanzania, indicating that the UK model
overestimated precipitation in these locations.

105

In general, there are more significantly low residuals than high residuals in
August 1984. UK and UKIV residual maps show four significantly low residuals in the
center of the study region; however, in most cases they are surrounded by nonsignificant residuals. Another area in which significant residuals is present is the in
along the western edge of the study area. Both positive and negative residuals appear
in this region, indicating high amounts of rainfall variability over space.
UK and UKIV residual patterns are similar to each other in November 1984, with
negative residuals in the center of the study area (indicating that the models
overestimate in these locations) and positive residuals present in the northwest and
southeast, where the model underestimates precipitation. Significant residuals extend
to the northwest and southeast regions of the study area. Significant residuals are
mainly negative on the LOK and SpReg maps, indicating that the models generally
overestimate precipitation in these areas.
There are fewer significant residuals in January 1985 than in January 1984.
LOK predicted values significantly underestimate actual precipitation amounts in several
locations. UK and UKIV maps illustrate a general scattering of significant positive and
negative residuals, with the negative residuals falling in more central areas.
In August 1985, negative residuals are shown generally in the center of the
study region, with the exception of LOK, for which two high residuals are shown in the
central study area. This indicates that UK, UKIV, and SpReg overestimate precipitation
at a cluster of locations in the center of the study region, while LOK is more likely to
underestimate precipitation is this region.

106

In November 1985, there are generally more negative than positive residuals;
however, in this month the positive residuals are generally interspersed with the
negative residuals in the central portion of the study area, indicating higher precipitation
variability in November 1985.
Maps of CRU residuals for 1985 (the “typical” year) were also plotted to evaluate
for spatial patterns in the residuals (Figure 4-8). From these plots, it can be seen that
there are large numbers of significantly high residuals (|standardized residual or zscore|
> 2), generally in clusters. High residuals indicate locations where the CRU modeled
values significantly underestimate precipitation observed at the corresponding
meteorological station. Again, this is expected due to the overly-smoothed nature of
global climate model estimates of climatic variables. The significant residuals are
generally clustered, in large part, because the global model estimates do not take into
account local features that influence precipitation on a local scale, such as topography
and surface water bodies. Because these features are correlated over space, error
terms would also be expected to be correlated over space.

107

April 1985
April 1985
1
0
-1
-3

-2

CRUprecip854$Y

0
-1
-2
-3

CRUprecip851$Y

1

January 1985
January 1985

35

36

37

38

35

36

CRUprecip851$X

38

CRUprecip854$X

0
-1
-2
-3

-3

-2

-1

0

CRUprecip8511$Y

1

November 1985
November1985

1

August 1985
August1985

CRUprecip858$Y

37

35

36

37

38

35

CRUprecip858$X

36

37

38

CRUprecip8511$X

Figure 4-9. Maps of CRU residuals in 1985. Significantly high residuals (zscore > 2)
are shown in red, and significantly low residuals (zscore < -2) are shown in blue.
In identifying an optimal model, LOK performed the best on the basis of minimum
RMSE, a single map-wide measure. This is result is consistent with the findings of
Goovaerts (1999a, 2000), in which a local form of simple kriging (simple kriging with
varying local means) outperformed multivariate techniques when applied to the
mountainous terrain in The Algarve, Southern Portugal.

108

The multivariate techniques were close runners up in some months, and had the
lowest RMSE in November 1985, in which the spatial regression RMSE was the lowest.
As shown on Figure 4-5, the least typical year in terms of monthly rainfall amounts was
1984 compared to all other years from 1980 through 1985, with droughts occurring in
the early part of the year, and surplus rain in the later months of the year. It could be
surmised from these results that multivariate methods of interpolation perform better in
typical years than in atypical years, in which other drivers of precipitation may contribute
to monthly precipitation patterns.
Although many studies have identified the importance of including topographic
variables such as elevation and its derivatives in predicting precipitation patterns over
space (Arora et al. 2006; Daly et al. 1994; Diodato 2005; Goovaerts 1999a, 1999b,
2000; Hutchinson 1998b; Hutchinson and Bischof 1983; KeifferWeisse and Bois 2001;
Kyriakidis et al. 2004; Marquinez et al. 2003; Oettli and Camberlin 2005; PardoIguzquiza 1998), it can also be hypothesized that local forms of univariate kriging
perform better than multivariate techniques in areas of highly variable terrain due to the
difficulty in modeling the complexity of precipitation patterns with multivariate models in
these regions.

109

Chapter 5
Hypothesis Testing

5.1 Introduction
This chapter makes use of the regression modeling results described in Chapter
4 for the purpose of hypothesis testing in order to gain an improved understanding of
the significant predictors of monthly precipitation patterns over space within the case
study area. Significant variables in predicting the spatial patterns of precipitation are
identified through a combination of a regression model selection process, in which
competing models with varying selections of independent variables are compared and
selected on the basis of the Akaike Information Criterion (AIC, Akaike 1974), and formal
testing of the regression coefficients within each selected model. One additional test,
the Hausman Test, is presented and used to identify whether significant simultaneity
can be documented between precipitation and vegetation (Hausman 1978).

5.1.1 Study area
The position of the study area in East Africa, as well as its proximity to Lake
Victoria and the Indian Ocean are illustrated in Figure 5-1. Understanding of these
relative locations is importance since explanatory variables for describing variation in
monthly rainfall include distance to Lake Victoria and distance to the Indian Ocean.

110

Lake
Victoria

Indian
Ocean

Kilometers

Figure 5-1. Location of the case study area within East Africa (CLIP region). Meteorological stations
(green diamonds) are shown in proximity to Lake Victoria and the Indian Ocean.

5.1.2 Identification of dependent variable and selection of representative data
Monthly precipitation (mm) represents the dependent variable in this analysis.
Precipitation data from roughly 120 meteorological stations obtained from the
Department of Meteorology, Government of Kenya were used. Meteorological station
locations are shown on Figure 5-1.
Monthly precipitation was evaluated through eight different regression analyses;
the data for these analyses were chosen to represent the four seasons (i.e., the dry
season in December, January, and February; the long rainy season in March, April,
May, and June; the cool dry season in July and August; and the short rainy season in
September, October, and November) in eastern Africa for two different years. The
individual months of January, April, August, and November were chosen as the most

111

representative month of each of the seasons described above within the case study
area (Figure 5-2).

Jan

Feb Mar

Apr

May Jun

Jul

Aug

Sep Oct

Nov Dec

Monthly precipitation (mm)
Figure 5-2. Long-term average monthly precipitation (mm) collected from 1926 to 1998 averaged over all
meteorological stations within the study area. The position of the intertropical convergence zone (ITCZ)
relative to the study area is labeled at various points in the year.

The years 1984 and 1985 were chosen for this analysis since these years
represent the overlap between years in which the largest numbers of meteorological
stations were measured (approximately 120 stations) and the years in which remotely
sensed vegetation data were available (beginning in July 1982). Furthermore, the years
to be evaluated were chosen to represent a typical year (1985) and an atypical year
(1984). Additional detail regarding selection of years for analysis is provided in
Chapter 4.

112

5.1.3 Identification of independent or descriptive variables
Distance from major water bodies (i.e., Lake Victoria and the Indian Ocean) was
used to evaluate monthly precipitation in East Africa. As described in Chapter 4,
categorical “distance bands” were used in lieu of the linear distance measures since
precipitation was not linearly related to either of these measures. Dummy variables
were established to identify areas within which precipitation levels were similar.
Distance bands of 0 to 300 km, 300 to 450 km, 450 to 600 km, and greater than 600 km
were created as a measure for distance to Lake Victoria. Distance bands of 0 to 500
km and greater than 500 km were used to represent distance to the Indian Ocean.
Other potential explanatory variables included NDVI (a vegetation index; Rouse
et al. 1974), elevation, and derivatives of elevation (i.e., measures of the degree to
which a slope faces north and east, and the curvature of the slope) at scales of 1 km
and 9 km. The scale of 9 km was chosen to evaluate the scale over which convective
rainfall occurs. Furthermore, this scale is consistent with the findings of Hession and
Moore (2010) and Sharples et al. (2005); Sharples et al. (2005) identify an optimal
topographic scale of dependence of around 6-8 km.
Elevation at each station location was estimated using the SRTM 30 arc second
Digital Elevation Model (DEM) raster, shown on Figure 5-3. A measure of 30 arc
seconds is approximately 1 km near the equator. Average elevation at a 9 km
resolution was also calculated using elevation at surrounding pixels, as described in
Chapter 4.
The influence of aspect and slope on precipitation was evaluated using the
eastern and northern components of the unit normal vector (Hutchinson 1998). These

113

values characterize aspect scaled by the steepness of the slope. Results are largest in
magnitude on the steepest slopes and approach zero in flat areas. In addition,
curvature was calculated in ArcGIS (ESRI, 2006) to evaluate whether curvature has an
impact on precipitation patterns. A summary of all potential explanatory variables used
in this study is provided in Table 5-1.

Rift Valley
West
Kenya
Highland
Mt. Kenya

Lake
Victoria

Mt. Kilimanjaro

Figure 5-3. Map of elevation (1 km resolution) with meteorological station locations and study area
boundary are shown above. Topographic features within the study area are labeled.

114

Table 5-1. Summary of independent variables used in regression analysis and
hypthosis testing. Abbreviated names are shown as well.
Independent 
Variable 
Distance to Lake Victoria (km) 

Scale

Abbreviated
Name
Notes
dist2lv
d2lv300

= 1 if dist2lv <= 300 km,  
    0 otherwise

d2lv450

= 1 if dist2lv > 300 and <= 450 km,  
   0 otherwise

d2lv600

= 1 if dist2lv > 450 and <= 600 km,  
    0 otherwise
if all 3 indicator variables are 0, then 
dist2lv > 600 km

Distance to Indian Ocean (km) 

dist2coast

Elevation (m) 

d2c500
dem1km
dem9km

Combined measure of apsect 
and slope : measure of 
"northness" 
(unitless) 
Combined measure of apsect 
and slope: measure of 
"eastness" 
(unitless) 
Surface curvature 
(profile; unitless) 
NDVI 
(unitless) 

1 km
9 km

1 km
9 km

p1km
p9km

1 km
9 km
1 km
9 km
8 km
8 km
8 km
8 km
8 km
8 km
8 km
8 km

q1km
q9km
curv1km
curv9km
ndvi842
ndvi845
ndvi849
ndvi8412
ndvi852
ndvi855
ndvi859
ndvi8512

115

= 1 if dist2coast <= 500 km,  
    0 otherwise

p = cos(α) sin(θ), where α is aspect 
angle and θ is slope angle  
p = cos(α) sin(θ)

q = sin(α) sin(θ)
q = sin(α) sin(θ)

Average NDVI for February 1984 
Average NDVI for May 1984 
Average NDVI for September 1984
Average NDVI for December 1984
Average NDVI for February 1985 
Average NDVI for May 1985 
Average NDVI for September 1985
Average NDVI for December 1985

5.2 Regression model selection and testing
Initial regression model selection for precipitation in each month (i.e., January,
April, August, and November) in 1984 and 1985 was conducted using OLS regression
and the recommended approach of Burnham and Anderson (1998). OLS regression
was used in this step since it is likely to identify more significant variables than spatially
explicit regression models: in the presence of spatial autocorrelation, precision of the
OLS estimates tends to be overstated, resulting in elevated estimates of the OLS
coefficients and values such as the coefficient of determination (R2).
All regression modeling was completed using a normal score transformation of
the precipitation data (Chapter 4). Once an OLS model formulation was selected for
each month in consideration, these models were further scrutinized to determine if they
could be improved through spatial regression modeling (Anselin, 2006; LeSage 1998,
LeSage and Pace 2009). This process is described in more detail in Chapter 2 of this
dissertation. In Chapter 4, the final selections of regression models were utilized in all
three of the multivariate methods used to generate predicted precipitation maps.
In this chapter, the final selection of regression models was further evaluated
through hypothesis testing. The hypotheses to be tested are specified in Section 5.2.1
along with the basis for the hypotheses. Results of hypothesis and model interpretation
are presented in Section 5.2.2.

5.2.1 Statement of hypotheses
A summary of the independent variables tested through regression modeling is
provided in Table 5-1. This section provides a brief description of the independent

116

variables and their expected relationships with precipitation followed by a formal
statement of hypothesis.
Distance measures (i.e., distance to Lake Victoria and distance to the Indian
Ocean) were used to model synoptic and global scale processes (Figure 5-1). The
distance variables, which were categorized to represent “distance bands,” are expected
to represent a complex interaction of different local, regional, and mesoscale processes
that include the lake/sea breeze near Lake Victoria in the western portion of the study
area and orographic effects in the highland areas (Camberlin and Planchon 1997,
Ng’ang’a 1992). In general, higher levels of precipitation are expected to occur closer to
the large water bodies, particularly in dry seasons when mesoscale effects dominate. It
is also expected that the role of these factors will vary according to season.

Hypothesis 1
H10:

Precipitation amounts do not vary with distance from water bodies.

H11:

Precipitation amounts decrease as distance from water bodies increases.

Elevation (Figure 5-3) and its derivatives are expected to reflect mesoscale
processes related to orographic precipitation. Increasing precipitation is expected to
occur with increasing elevation (Arora et al., 2006; Spreen, 1947). This hypothesis was
evaluated at the scales of 1 and 9 km.

117

Hypothesis 2
H20:

Precipitation amounts do not vary with elevation.

H21:

Precipitation amounts increase as elevation increases.

The eastern and northern components of the unit normal vector were used to
represent the effects of aspect and slope on precipitation (Hutchinson, 1998). The
largest values of p and q occur on the steepest slopes; the lowest values occur on
peaks, valley floors, or generally flat areas such as savannas. Since p and q
incorporate both slope and aspect, the effects of both can be evaluated. The direction
of these effects can also be evaluated without reference to the prevailing wind field
(Hutchinson, 1998). Steeper slopes provide stronger orographic lifting; thus, increasing
values of p and q are expected to be associated with higher rainfall (locally, at least;
Buytaert et al., 2006).

Hypothesis 3
H30:

Precipitation amounts do not vary with the eastern and northern
components of the unit normal vector, representing slope and aspect.

H31:

Precipitation amounts increase with increasing values of the
eastern/northern components of the unit normal vector.

The above relationship, or the significance of this relationship, is expected to vary
by season. It is also anticipated that the spatial scale at which these processes occur

118

will vary by season due to different synoptic forcings, wind patterns, and other seasonal
variations (e.g., long rains and short rains) (Ng’ang’a, 1992).

Curvature provides a measure of the degree to which a surface is convex or
concave. It is calculated as the slope of the slope (the second derivative of the
surface). High values of curvature represent upwardly convex surfaces, such as a
hilltop. Conversely, low values of curvature correspond with upwardly concave
surfaces, such as a valley bottom. Since higher curvature would generally occur at the
relative higher elevations (e.g., at peaks), higher precipitation is expected to correspond
with increased curvature. Furthermore, curvature was found to be less correlated with
other independent variables than elevation; consequently, it is less likely to be
confounded with other variables and may be a more effective predictor of precipitation.

Hypothesis 4
H40:

Precipitation amounts do not vary with curvature.

H41:

Precipitation amounts increase with increasing curvature.

Increased NDVI values (measured in the subsequent month) are expected to
correlate with increased precipitation in the month being evaluated. This would be
expected since increased precipitation causes increased greenness in most natural
landscapes (Rodriguez-Iturbe and Porporato, 2004).

119

Hypothesis 5
H50:

Precipitation amounts do not vary with NDVI.

H51:

Precipitation amounts increase with increasing NDVI.

As previously noted, significance of the above variables and their corresponding
hypotheses is expected to vary from month to month according to seasonal variability in
the factors that affect spatial patterns in precipitation, such as position of the ITCZ, the
relative importance of elevation and orographic precipitation in dry seasons versus rainy
seasons, and the relative importance of constant sources of moisture, such as Lake
Victoria.
Furthermore, the two years evaluated in this dissertation, 1984 and 1985, were
chosen to represent atypical (1984) and typical (1985) in terms of monthly rainfall
averages compared to long-term monthly averages. In particular, early months in 1984
were characterized by drought conditions; later months experienced higher precipitation
than long term averages. The impact of this distinction on causal factors of precipitation
patterns is unknown; therefore, no hypothesis is stated. Similarities and differences
between these years were noted for future hypothesis development.

5.2.2 Results and interpretation
Initial model selection was performed using OLS regression, as described in
Chapter 4. OLS regression was used in this step since it is likely to identify more
significant variables than spatially explicit regression models: in the presence of spatial
autocorrelation, precision of the OLS estimates tends to be overstated, resulting in

120

elevated estimates of the OLS coefficients and values such as the coefficient of
determination (R2).
A number of candidate models were compared and ranked based on an
information theoretic criterion, the Akaike Information Criterion (AIC; Akaike, 1974).
This approach provided a means to rank the relative strength of the OLS models.
Regression models with the lowest AIC value were identified with a rank of (1) and
selected for further hypothesis testing. Through this model selection process, some
independent variables were excluded from further consideration.
As illustrated in Table 5-2, precipitation amounts for each month were evaluated
using four OLS regression model formulations, or candidate models. The candidate
models were developed to include the following sets of independent variables: (1)
average monthly NDVI for the subsequent month, the elevation term and its derivatives
(i.e., dem, p, q, and curv) at a 1 km scale, each indicator variable representing distance
to Lake Victoria and distance to the Indian Ocean; (2) average monthly NDVI for the
subsequent month, the elevation term and its derivatives (i.e., dem, p, q, and curv) at a
9 km scale, each indicator variable representing distance to Lake Victoria and distance
to the Indian Ocean; (3) average monthly NDVI for the subsequent month, the elevation
term and its derivatives (i.e., dem, p, q, and curv) at a 1 km scale, each indicator
variable representing distance to Lake Victoria and distance to the Indian Ocean and
interaction terms between distance to Lake Victoria (dist2lv) and the first two indicator
variables (i.e., d2lv300 and d2lv450); and (4) average monthly NDVI for the subsequent
month, each elevation term (dem, p, q, and curv) at a 9 km scale, each indicator
variable representing distance to Lake Victoria and distance to the Indian Ocean and

121

distance to the Indian Ocean and interaction terms between dist2lv and the first two
indicator variables (i.e., d2lv300 and d2lv450).
Results of initial model selection are summarized in Table 5-2. The AIC value is
shown for each model formulation: lower AIC values indicate better model fits. Each
model is ranked based on AIC values; models with a ranking of (1) were selected for
subsequent use in each multivariate prediction technique.
The initial model formulations selected for each month and year
were further tested using spatially explicit methods to determine if use of the OLS is
appropriate, or if a spatially explicit model is required. The three spatially explicit
regression models considered (SEM, SAR, and SAC) are described in detail in Chapter
3 of this dissertation, as well as the decision process for selecting the appropriate model
type.

122

Table 5-2. Summary of OLS regression models ranked by AIC.
Model
1

2

3

Dep.
Variable
Jan 84
Precip
Nscores
Jan 84
Precip
Nscores
Jan 84
Precip
Nscores

Independent Variables
ndvi842, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500

AIC
293.9

Rank
(4)

ndvi842, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500

292.6

(3)

ndvi842, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

291.4

(2)

4

Jan 84
Precip
Nscores

ndvi842, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

288.4

(1)

1

Apr 84
Precip
Nscores
Apr 84
Precip
Nscores
Apr 84
Precip
Nscores

ndvi845, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500

291.3

(3)

ndvi845, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500

282.0

(1)

ndvi845, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

294.0

(4)

4

Apr 84
Precip
Nscores

ndvi845, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

284.3

(2)

1

Aug 84
Precip
Nscores
Aug 84
Precip
Nscores
Aug 84
Precip
Nscores

ndvi849, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500

229.1

(4)

ndvi849, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500

228.9

(3)

ndvi849, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

223.7

(1)

4

Aug 84
Precip
Nscores

ndvi849, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

224.3

(2)

1

Nov 84
Precip
Nscores
Nov 84
Precip
Nscores
Nov 84
Precip
Nscores

ndvi8412, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500

277.2

(2)

ndvi8412, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500

273.5

(1)

ndvi8412, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

281.2

(4)

Nov 84
Precip
Nscores

ndvi8412, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

277.4

(3)

2

3

2

3

2

3

4

123

Table 5-2 (continued). Summary of OLS regression models ranked by AIC.
1
Jan 85 Precip
ndvi852, dem1km, p1km, q1km, curv1km, d2lv300,
288.4
Nscores
d2lv450, d2lv600, d2c500
2
Jan 85 Precip
ndvi852, dem9km, p9km, q9km, curv9km, d2lv300,
296.6
Nscores
d2lv450, d2lv600, d2c500
3
Jan 85 Precip
ndvi852, dem1km, p1km, q1km, curv1km, d2lv300,
279.8
Nscores
d2lv450, d2lv600, d2c500, d2lv300*dist2lv,
d2lv450*dist2lv

(2)
(4)
(1)

4

Jan 85 Precip
Nscores

ndvi852, dem9km, p9km, q9km, curv9km,d2lv300,
d2lv450, d2lv600, d2c500, d2lv300*dist2lv,
d2lv450*dist2lv

291.0

(3)

1

Apr 85 Precip
Nscores
Apr 85 Precip
Nscores
Apr 85 Precip
Nscores

ndvi855, dem1km, p1km, q1km, curv1km, d2lv300,
d2lv450, d2lv600, d2c500
ndvi855, dem9km, p9km, q9km, curv9km, d2lv300,
d2lv450, d2lv600, d2c500
ndvi855, dem1km, p1km, q1km, curv1km, d2lv300,
d2lv450, d2lv600, d2c500, d2lv300*dist2lv,
d2lv450*dist2lv

311.7

(1)

315.5

(3)

312.6

(2)

4

Apr 85 Precip
Nscores

ndvi855, dem9km, p9km, q9km, curv9km, d2lv300,
d2lv450, d2lv600, d2c500, d2lv300*dist2lv,
d2lv450*dist2lv

317.1

(4)

1

Aug 85 Precip
Nscores
Aug 85 Precip
Nscores
Aug 85 Precip
Nscores

ndvi859, dem1km, p1km, q1km, curv1km, d2lv300,
d2lv450, d2lv600, d2c500
ndvi859, dem9km, p9km, q9km, curv9km, d2lv300,
d2lv450, d2lv600, d2c500
ndvi859, dem1km, p1km, q1km, curv1km, d2lv300,
d2lv450, d2lv600, d2c500, d2lv300*dist2lv,
d2lv450*dist2lv

218.9

(3)

214.7

(1)

220.7

(4)

4

Aug 85 Precip
Nscores

ndvi859, dem9km, p9km, q9km, curv9km, d2lv300,
d2lv450, d2lv600, d2c500, d2lv300*dist2lv,
d2lv450*dist2lv

216.6

(2)

1

Nov 85 Precip
Nscores
Nov 85 Precip
Nscores
Nov 85 Precip
Nscores

ndvi8512, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500
ndvi8512, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500
ndvi8512, dem1km, p1km, q1km, curv1km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

268.2

(3)

264.4

(1)

271.5

(4)

Nov 85 Precip
Nscores

ndvi8512, dem9km, p9km, q9km, curv9km,
d2lv300, d2lv450, d2lv600, d2c500,
d2lv300*dist2lv, d2lv450*dist2lv

268.0

(2)

2
3

2
3

2
3

4

Differences of less than 3 between AIC values are not considered
significant.

124

The regression models selected for each month/year are summarized in
Table 5-3. For each model, the regression technique used is given as well as the list of
independent variables tested, their estimated coefficients, and measures of statistical
significance. Since regression modeling was conducted using standardized
precipitation data based on the n-score transformation (see Chapter 4 for additional
detail), estimated coefficients are interpreted as the average change in standardized
precipitation for each unit change in the independent variable. Significance at the levels
of 0.1%, 1%, 5%, and 10% are indicated. Further discussion of significant variables is
based on a 10% level of significance.
In January 1984, significant predictors of precipitation included NDVI (p=0.0259),
distance to Lake Victoria (distances less than 300 km [p=0.0618] and between 300 and
450 km [p=0.0198]) and distance to the Indian Ocean (distances less than 500 km
[p=0.0002]). As expected, locations closer to Lake Victoria were found to experience
increased rainfall. Regression coefficients for the indicator variables are interpreted as
the increase in mean expected rainfall (standardized) over that expected in the greatest
distance category (i.e., greater than 600 km) when located within the given interval. For
example, locations within 300 km of Lake Victoria were expected to receive
approximately 10 additional standardized rainfall units (SRU) than locations greater than
600 km away. In this way, the greatest distance category is used as a baseline for
evaluating the remaining distance categories. Locations between 300 and 450 km from
Lake Victoria were expected to receive almost 4 more standardized rainfall units than
locations more than 600 km away. Since the interaction term between d2lv450
(indicating a location of between 300 and 450 km from Lake Victoria) and the
125

continuous distance measure dist2lv was significant (p=0.0221) with a negative
coefficient, rainfall was shown to decrease with increasing distance in this interval, as
would be expected. Locations within 500 km of the Indian Ocean were expected to
receive 1.2 additional SRU than further locations.
In January 1984, NDVI was negatively associated with precipitation (p=0.0259),
indicating that increased NDVI corresponded with decreased precipitation. This result
was not as expected. A possible explanation is the relatively low coefficient of
determination (R2=0.3) obtained for the January 1984 OLS regression model, indicating
that only 30% of the variation in rainfall over space was explained by the model.
Addition of one or more other explanatory variables may improve model interpretations.

126

Table 5-3. Summary of regression modeling results.
Month
Jan-84

Method
OLS

Apr-84

SARlag

Aug-84

OLS

Nov-84

SARlag

Independent Variable
(Intercept)
ndvi842
dem9km
p9km
q9km
curv9km
d2lv300
d2lv450
d2lv600
d2c500
I(d2lv300 * dist2lv)
I(d2lv450 * dist2lv)
(Intercept)
ndvi845
dem9km
p9km
q9km
curv9km
d2lv300
d2lv450
d2lv600
d2c500
Rho
(Intercept)
ndvi849
dem1km
p1km
q1km
curv1km
d2lv300
d2lv450
d2lv600
d2c500
I(d2lv300 * dist2lv)
I(d2lv450 * dist2lv)
(Intercept)
ndvi8412
dem9km
p9km
q9km
curv9km
d2lv300
d2lv450
d2lv600
d2c500
Rho

Estimate
-1.3530
-1.5500
0.0003
-0.2401
0.0468
-50.6300
10.2200
3.7850
0.6514
1.1610
-0.0271
-0.0081
-1.9160
0.7846
0.0001
-0.0479
0.2024
-136.4866
1.4931
0.9401
1.0393
0.3932
0.6040
-2.1604
0.7530
0.0006
-0.0632
0.1014
-3.9075
-1.5345
3.3980
-0.1001
0.4681
0.0116
-0.0076
-1.4962
-0.1582
0.0001
-0.3114
-0.0247
66.7779
1.4101
0.7143
1.2727
0.7044
0.2500

127

St. Error
0.6059
0.6857
0.0003
0.1573
0.1590
93.3400
5.4080
1.5980
0.5481
0.2960
0.0194
0.0035
0.4924
0.4462
0.0002
0.1241
0.1256
72.5866
0.4961
0.4808
0.4441
0.1891
0.1017
0.4720
0.5121
0.0002
0.1242
0.1205
6.7899
4.1207
1.2118
0.4035
0.2371
0.0148
0.0026
0.6446
0.6878
0.0002
0.1415
0.1390
81.0492
0.5406
0.5338
0.4849
0.2504
0.1629

t value
-2.2330
-2.2610
1.1930
-1.5260
0.2950
-0.5420
1.8890
2.3690
1.1890
3.9210
-1.3990
-2.3260
-3.8910
1.7584
0.5272
-0.3864
1.6123
-1.8803
3.0094
1.9552
2.3404
2.0787
5.9358
-4.5770
1.4710
3.2120
-0.5090
0.8420
-0.5750
-0.3720
2.8040
-0.2480
1.9740
0.7880
-2.8860
-2.3211
-0.2300
0.4865
-2.2008
-0.1776
0.8239
2.6082
1.3382
2.6246
2.8133
1.5348

Pr(>|t|)
0.0278 *
0.0259 *
0.2358
0.1302
0.7690
0.5887
0.0618 .
0.0198 *
0.2375
0.0002 ***
0.1649
0.0221 *
0.0001 ***
0.0787 .
0.5981
0.6992
0.1069
0.0601 .
0.0026 **
0.0506 .
0.0193 *
0.0376 *
0.0000 ***
0.0000 ***
0.1446
0.0018 **
0.6119
0.4020
0.5663
0.7104
0.0061 **
0.8046
0.0511 .
0.4327
0.0048 **
0.0203 *
0.8181
0.6266
0.0278 *
0.8591
0.4100
0.0091 **
0.1808
0.0087 **
0.0049 **
0.1248

Table 5-3. Summary of regression modeling results.
Month
Jan-85

Method
SARlag

Independent Variable Estimate
(Intercept)
-0.7867
ndvi852
0.5605
dem1km
-0.0002
p1km
0.0418
q1km
0.4667
curv1km
12.2468
d2lv300
-4.8020
d2lv450
3.3218
d2lv600
0.2569
d2c500
0.3788
I(d2lv300 * dist2lv)
0.0218
I(d2lv450 * dist2lv)
-0.0072
Rho
0.2970
Apr-85 SARlag (Intercept)
-2.0913
ndvi855
-0.2445
dem1km
0.0005
p1km
0.1716
q1km
0.2458
curv1km
-6.1177
d2lv300
1.5083
d2lv450
1.0736
d2lv600
1.1885
d2c500
0.2507
Rho
0.6349
Aug-85 SARlag (Intercept)
-1.0773
ndvi859
0.9904
dem9km
0.0005
p9km
0.0794
q9km
0.2018
curv9km
-9.1431
d2lv300
0.1276
d2lv450
-0.3907
d2lv600
-0.4021
d2c500
-0.0420
Rho
0.4170
Nov-85
OLS
(Intercept)
-2.7490
ndvi8512
2.2120
dem9km
-0.0004
p9km
-0.0934
q9km
-0.0210
curv9km
177.2000
d2lv300
2.2380
d2lv450
1.6200
d2lv600
1.5390
d2c500
0.9699
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1
Bolded variables are significant.

128

St. Error
0.6318
0.5799
0.0002
0.1515
0.1441
8.7368
4.5302
1.5306
0.5711
0.2664
0.0161
0.0033
0.1613
0.6588
0.7459
0.0002
0.1474
0.1510
8.3879
0.5984
0.5685
0.5218
0.2290
0.1050
0.4340
0.4516
0.0002
0.1078
0.1082
64.7750
0.4247
0.3993
0.3687
0.2040
0.1329
0.6493
0.5926
0.0002
0.1640
0.1629
94.1500
0.6147
0.6146
0.5507
0.2356

t value
-1.2450
0.9665
-0.7656
0.2757
3.2398
1.4018
-1.0600
2.1702
0.4498
1.4220
1.3515
-2.1659
1.8406
-3.1744
-0.3277
2.5010
1.1637
1.6278
-0.7294
2.5206
1.8883
2.2777
1.0947
6.0476
-2.4820
2.1929
3.0242
0.7373
1.8647
-0.1412
0.3005
-0.9787
-1.0904
-0.2057
3.1382
-4.2330
3.7330
-1.4950
-0.5690
-0.1290
1.8820
3.6410
2.6360
2.7940
4.1160

Pr(>|t|)
0.2131
0.3338
0.4439
0.7828
0.0012 **
0.1610
0.2891
0.0300 *
0.6529
0.1550
0.1765
0.0303 *
0.0657 .
0.0015 **
0.7431
0.0124 *
0.2446
0.1036
0.4658
0.0117 *
0.0590 .
0.0227 *
0.2736
0.0000 ***
0.0131 *
0.0283 *
0.0025 **
0.4610
0.0622 .
0.8878
0.7638
0.3277
0.2755
0.8370
0.0017 **
0.0001 ***
0.0003 ***
0.1383
0.5705
0.8976
0.0629 .
0.0004 ***
0.0098 **
0.0063 **
0.0001 ***

April 1984 precipitation was evaluated using a spatial lag model with the same
independent variables as those in the January 1984 model except for the interaction
terms. A model without interaction terms was found to provide a better fit to the April
1984 precipitation amounts during initial model selection, indicating that the interaction
terms did not improve model fit. All of the additive indicator variables for distance were
significant and positive, indicating increased rainfall compared to the furthest distance
category. Locations closest to Lake Victoria are expected to receive approximately 1.5
more than locations more than 600 km from Lake Victoria (p=0.0026) in terms of SRU;
locations between 300 and 450 km from Lake Victoria are expected to receive 0.94
additional SRU (p=0.0506), and locations between 450 and 600 km should receive 1.04
SRU more than the furthest distance category (p=0.0376). These results indicate that
distance from Lake Victoria plays a significant role in precipitation amounts even during
the long rainy season when the ITCZ is overhead. Profile curvature at a scale of 9 km
was marginally significant (p=0.0601), with a negative coefficient. This finding does not
support the hypothesis that increasing precipitation would occur with increasing
curvature for this particular month.
Elevation at a scale of 1 km was found to be a significant predictor of August
1984 precipitation (p=0.0018), with a positive coefficient. This indicates that
precipitation increases with increasing elevation at the estimated rate of
0.0006 SRU per meter, or 6 mm per kilometer. Significance of the elevation term in the
short dry season confirms the hypothesis that local scale, orographic precipitation
occurs during this season. Precipitation was also found to be approximately 3.4 SRU
greater in locations between 300 and 450 km from Lake Victoria (p=0.0061) than the
129

most distant locations (greater than 600 km). The interaction term was also significant
(p=0.0048) and negative, indicating that precipitation amounts decrease as distance
from Lake Victoria increases within this distance band, as expected. Locations within
500 km of the Indian Ocean were expected to receive 0.47 SRU additional rainfall on
average than more distant locations (p=0.0511).
In November 1984, the eastern component of the unit vector was a significant
predictor of precipitation, indicating that east facing slopes were expected to receive
0.31 SRU less rainfall than west facing slopes (p=0.0278). The ITCZ was overhead
during this time. Furthermore, generally higher precipitation was observed in the
eastern portion of the study area. This would indicate that increased precipitation
should occur on the east-facing slopes. However, the grid of the eastern component of
the unit vector at a spatial resolution of 9 km appears highly variable and pixilated in this
region. It is possible that some meteorological stations in this region are on west-facing
slopes; however, this is not clear. In addition, higher rainfall amounts were shown to
coincide with distances of less than 300 km from Lake Victoria (1.4 SRU, p=0.0091),
between 450 and 600 km from Lake Victoria (1.3 SRU, p=0.0087), and less than 500
km from the Indian Ocean (0.7 SRU, 0.0049) compared to greater distances. Again,
this illustrates that distance from surface water bodies has a significant impact on
precipitation amounts during rainy seasons (in this case, the short rainy season).
Distances from Lake Victoria between 300 and 450 km were shown to receive
3.3 SRU more in January 1985 than locations greater than 600 km, on average
(p=0.0300). These amounts were shown to decrease as distance from Lake Victoria
increased in this distance band (p=0.0303), as expected. Furthermore, north-facing
130

slopes were shown to receive more rainfall than south-facing slopes (p=0.0012).
Inspection of the measured data for this month indicated that the greatest amount of
precipitation fell in the northwest of the study area, on the north- and west-facing slopes
of the highland mountains. Elevation at a spatial resolution of 1 kilometer was not a
significant predictor of precipitation in the study area during this dry season (p=0.7828);
it is possible that orographic precipitation occurred at scales other than that considered
herein, or that other causal factors were involved.
Similar to April 1984, all three components of the distance to Lake Victoria
measure were significant predictors of precipitation in April 1985 (p=0.0117, p=0,0590,
and 0.0227, respectively, for each distance band). All of the coefficients were positive,
indicating that the distance bands closer to Lake Victoria received more rainfall than the
furthest distance category (greater than 600 km), as expected. In addition, elevation at
a scale of 1 km was a significant predictor of precipitation, with an additional 0.0005
SRU per meter (or 5 SRU per km) of elevation expected.
In August 1985, NDVI was identified as a significant predictor of precipitation
(p=0.0283): higher NDVI values corresponded with increased rainfall, as expected.
Elevation at a scale of 9 km was also significantly (p=0.0025) and positively related to
precipitation, with an additional 0.0005 SRU of precipitation per meter of elevation
expected. The northern component of the unit vector was also significant in August
1985 (p=0.0622) with a positive coefficient. This indicates that higher precipitation
amounts were expected on north-facing slopes. Inspection of the measured
precipitation data indicated that higher rainfall amounts were observed in the
northwestern portion of the study area, similar to January 1985 in which the northern
131

component was also significant and positive. Furthermore, the ITCZ was in its
northernmost position during this month.
All of the distance bands for Lake Victoria and the Indian Ocean were significant
and positive in November 1985. Precipitation within 300 km of Lake Victoria was
expected to be 2.2 SRU higher on average than precipitation more than 600 km from
the lake (p=0.0004). Locations between 300 and 450 km from the lake were expected
to receive 1.6 SRU more (p=0.0098), while locations between 450 and 600 km were
expected to receive 1.5 SRU more than locations beyond 600 km from Lake Victoria
(p=0.0063). Locations within 500 km of the Indian Ocean were expected to receive
almost 1 SRU above more distant locations, on average (p=0.0001). NDVI was also a
significant predictor of precipitation in November 1985 (p=0.0003): precipitation was
shown to increase with increasing NDVI, as expected. Curvature at the 9 km scale was
positively related to precipitation (p=0.0629), indicating that increased precipitation
corresponded with upwardly convex surfaces (hilltops, representative of higher
elevation), as expected.
In general, distance measures were the most common significant predictors of
precipitation in the study area during both rainy and dry seasons. The significant
influence of distance measures or general geographic locators (e.g., latitude and
longitude) was also identified by Arora et al. (2006), Marquinez et al. (2003), Oettli and
Camberlin (2005), KeifferWeisse and Bois (2001); Hutchinson and Bischof (1983), and
Hutchinson (1998a, 1998b). In some cases, an interaction term between the indicator
variables and the continuous distance measure was able to refine the predictions,
allowing for decreasing precipitation as distance from Lake Victoria increased within a
132

distance band. It is interesting to note that this decreasing trend occurred only within
the 300 to 450 km distance band, and only during dry seasons. The only exception is
August 1985, in which no distance terms were significant.
NDVI was a significant predictor in four of the eight months considered (i.e.,
January 1984, April 1984, August 1985, and November 1985). The relationship was
positive in all but one month. These results are difficult to interpret and are not well
understood, since there was not a clear seasonal pattern to the significance of NDVI.
Elevation and its derivatives were significant only sporadically. Elevation at a 1
km resolution was significant in August 1984 and April 1985. Elevation at a 9 km
resolution was significant in August 1985. However, significance of the an elevation
term in the months of August (i.e., the second dry season) supports the hypothesis that
orographic rainfall is a significant contributor to precipitation in this dry season. Oettli
and Camberlin (2005) found similarly that mean elevation was not highly correlated to
precipitation amounts except, perhaps, in relatively smaller study areas in East Africa
due to the “considerable inconsistency in the elevation–rainfall relationship, which only
applies to either very small or very large space scales.”
The eastern component of the unit vector was significant in November 1984 only,
whereas the northern component was significant in both January and August 1985.
Significance of this term in January and August 1985 can be understood based on
inspection of the observed precipitation data: higher rainfall amounts were observed in
the northern/northwestern portions of the study area in these months. Furthermore, the
ITCZ was in its northernmost position in August 1985.

133

Curvature was significant but negative in April 1984, which was not expected.
However, curvature was significant and positive in November 1985, indicating that
higher precipitation amounts occurred at hilltops (or local areas of higher elevation), as
expected.
With regard to model type, OLS models were adequate for the modeling of
precipitation in January 1984, August 1984, and November 1985. Spatial lag models
provided better fits to precipitation data in the remaining months. No obvious
seasonality was evident in model selection based on these findings.
Comparison of results for 1984 and 1985 indicates similarities in the significance
of distance variables: increased precipitation was observed in the 300 to 450 km
distance band in January 1984 and 1985 (the first dry season), with decreasing
precipitation in this band as distance from Lake Victoria increases. This pattern was
also observed in August 1984 (the second dry season), although it was not detected in
August 1985. In addition, nearly all of the distance categories were significant in all
rainy seasons (i.e., April 1984, August 1984, April 1985, and August 1985). This pattern
indicates that significantly increased precipitation is most likely observed in the range of
300 to 450 km from Lake Victoria during the dry seasons. Significance of all distance
terms in the rainy seasons indicates that rainfall is generally widespread at this time,
with the exception of the most distant locations from water bodies (e.g., greater than
600 km from Lake Victoria.
Elevation terms were significant predictors of variability of rainfall over space in
both August 1984 and August 1985, the second dry season of each year, supporting the

134

likelihood of the occurrence of orographic precipitation during these dry seasons.
However, elevation was not significant in either January 1984 or January 1985.
Precipitation in January 1984 was lower than the long-term average for this month.
January 1985 was also slightly lower than the long term average; this could explain
reduced precipitation of all forms, including orographic, in these months.
Observed similarities between significant predictors of 1984 and 1985 monthly
precipitation over space would suggest that causal factors do not vary significantly
between years that are typical and atypical in terms of monthly precipitation amounts.
However, additional study is required to test this hypothesis.

5.3 Hausman test for simultaneity
Formal testing for simultaneity between monthly precipitation and the
1 month-lagged vegetation measure (NDVI) was conducted for each month in 1985, the
year selected to represent a typical precipitation year in East Africa.
Formal testing was completed using the Hausman test (Hausman 1978). The
Hausman test compares two regression models, a model that incorporates potential
simultaneity through the use of an instrumental variable and the same model without the
instrumental variable, to test the following hypothesis:
Hypothesis 6
H60:

Simultaneity does not exist between monthly precipitation and vegetation
(represented by a 1-month lagged NDVI).

H61:

Simultaneity does exist between monthly precipitation and vegetation.
135

Essentially, the Hausman identifies whether a regression model formulated under
the alternative hypothesis is significantly better than the regression model under the null
hypothesis. If so, then significant simultaneity is concluded to be present.
Formal testing of the above hypothesis failed to identify the presence of
significant simultaneity for any month in 1985. This result may be due, in part, to a
limitation in available data and an over-simplification of the lag between increased
precipitation and increased NDVI. A one-month lag was used to model the relationship
between precipitation and NDVI based on citations in the literature. However, due to
the highly variable landscapes and land cover types of East Africa, it is likely that the lag
between precipitation and vegetation varies over space according to vegetation type
and other factors. Improved results could be obtained by allowing for a variable lag time
according to land cover type.

136

Chapter 6
Conclusions and Future Research

In the context of understanding the impacts of global climate change and the
development of effective adaptation strategies, the need for higher resolution climate
data is clear. Although the relatively coarse-scale data simulated by global models can
be useful for evaluating global climate trends, they are not sufficient for regional
evaluations of climatic patterns, particularly in the presence of high landscape variability
such as that which occurs in eastern Africa. This concern has been identified by
individual researchers as well as Intergovernmental Panel on Climate Change (Boko et
al. 2007), who state that finer-scale climate data are necessary for decision-makers.
Furthermore, development of successful adaptation strategies requires higherresolution climate data and the combined efforts of localized, community-based efforts
(Brooks et al. 2006, Thornton et al. 2009).
This dissertation is devoted to gaining a more complete understanding of existing
techniques for developing finer-scale data by theoretically mapping similarities and
differences between two statistical paradigms: kriging and spatial regression;
developing improvements to those methods by expanding the capabilities of universal
kriging to incorporate feedback simultaneity that may occur between the dependent
variable of interest, rainfall in this study, and a variable such as vegetation; and applying
these techniques to a case study set in East Africa in hopes that it will be useful for
improved understanding of climate change and its impacts, and for further development
of adaptation strategies to climate change in East Africa.

137

6.1 Findings and conclusions
Precipitation maps were plotted using ordinary kriging (LOK), universal kriging
(UK), universal kriging with an instrumental variable (UKIV) and a selected spatial
regression model for each month evaluated as part of this dissertation. The maps
primarily illustrated the effects of elevation, with maximum values of monthly
precipitation generally occurring on or near Mts. Kenya and Kilimanjaro, and distance
from Lake Victoria, particularly in the dry seasons. In the dry seasons represented by
January and August, the precipitation maps illustrated that the highest levels of
precipitation occur on the western edge of the study area, nearest Lake Victoria, and in
areas of high elevation (i.e., Mount Kenya and Mount Kilimanjaro). These spatial
patterns in precipitation are consistent with the expectation that rains occurring in the
dry seasons are mainly localized convective rainfall near Lake Victoria or stratiform
rainfall in the highland areas (Ng’ang’a 1992). Precipitation maps for the months
representing the wet seasons, April and November, higher levels of precipitation were
observed to occur more uniformly over the study area. This is expected since the ITCZ
is directly over the study area during these months, influencing precipitation on a large
scale.
A comparison of map accuracies indicated that root mean squared errors
(RMSEs) were consistently lower for LOK, with the exception of November 1985 in
which the UK estimate had the lowest RMSE. However, there were many close runners
up which included one or more of the other model types (UK, UKIV, and/or spatial
regression) depending on the month. Maps of standardized error terms were also
reviewed for overall number of significant error terms and spatial patterns in the error

138

terms. Clusters of standardized residuals were common in the west Kenya highlands
and in the vicinity of Mt. Kenya. However, the clusters in the northwest often included
high and low values, indicating the presence of highly variable precipitation amounts
over space.
In April 1985, the standard error map for UKIV (the newly derived method)
indicated that the number and configuration of significant standard errors were better
than that for LOK. This confirms that UKIV can perform similarly to other standard
methods of estimation, and perhaps better in the presence of significant measurable
simultaneity.
Two prominent approaches to generating higher resolution climate data were
identified: downscaling modeled values from generalized circulation models (GCMs)
and statistical techniques that utilize measured climate data. Although the former is
common, due to the availability and completeness of modeled data sets, the latter has
been shown to provide more accurate regional predictions in East Africa. Data from the
CRU TS 3.1 global data set were mapped in the East African study area and compared
to the maps described above. The CRU TS 3.1 data were found to substantially
underestimate precipitation amounts in the region. Furthermore, large contiguous areas
of significantly low estimates illustrated spatial patterns in the error terms of the CRU
data set, indicating its shortcomings in reflecting local geographic features.
Hypothesis testing was conducted to better understand the system of
precipitation in the study region by identifying significant explanatory variables for
describing the spatial patterns in precipitation. Spatially explicit regression models were
used where diagnostic testing indicated that OLS models were inappropriate. In

139

general, distance measures were the most common significant predictors of
precipitation in the study area during both rainy and dry seasons. In some cases, an
interaction term between the indicator variables and the continuous distance measure
was able to refine the predictions, allowing for decreasing precipitation as distance from
Lake Victoria increased within a distance band; this decreasing trend occurred only
within the 300 to 450 km distance band, and only during dry seasons, however. OLS
models were adequate for the modeling of precipitation in January 1984, August 1984,
and November 1985. Spatial lag models provided better fits to precipitation data in the
remaining months. Consequently, no obvious seasonality was evident in model
selection based on these findings.
Comparison of results for 1984 (an atypical year with drought conditions in the
early part of the year) and 1985 (a typical year) indicates similarities in the significance
of distance variables and elevation, in particular. Observed similarities between
significant predictors of 1984 and 1985 monthly precipitation suggest that causal factors
do not vary significantly between atypical and typical years in terms of monthly
precipitation amounts. However, additional study is required to test this hypothesis.

6.2 Gaps in the literature and contributions to address them
A gap exists that is not completely filled by the variety of statistical techniques
historically used to evaluate climate data. The focus of the many spatially explicit
precipitation studies has often been to develop the ‘best’ estimates of precipitation, not
to understand the rainfall system in the region under study. To better understand
systems of precipitation, aspatial ordinary least squares regression techniques have
140

often been used, although results are biased when spatial patterns in the data are not
considered. Spatial regression consists of several regression techniques that are not
only spatially explicit, but can also be used to improve understanding of variables that
influence precipitation patterns. Spatially explicit regression techniques have begun to
appear in climate-related literature; however, each of the applications is somewhat
limited, either by the method used, the covariates selected, or the overall application
(hypothesis testing or estimation). Further, none of the approaches have characterized
endogenous relations, such as those between precipitation and vegetation. This
dissertation addresses this gap by 1) including spatial regression as a potential
technique for estimation of precipitation at unsampled locations, 2) developing an
extended version of universal kriging that explicitly represents and incorporates
endogenous relations between vegetation and precipitation into estimations at
unsampled locations, and 3) conducts hypothesis testing of significant factors in
explaining precipitation patterns in a spatially explicit manner.

6.2.1 Theoretical and methodological contributions
This dissertation has provides a theoretical summary of kriging and spatial
regression techniques, ultimately demonstrating that one set of equations can be used
to represent both kriging and spatial regression methods. The differences between the
set of equations for universal kriging, the SEM spatial regression model, and the SAR
spatial regression model are summarized in Table 3-2; differences between these
approaches are restricted to the spatial weights matrices used to derive the regression
coefficients, and the spatial weights applied to the model residuals. Thus, this

141

dissertation has successfully identified the similarities between kriging approaches and
spatial regression techniques. Furthermore, the differences between these
approaches, contained within the spatial weights matrices, have been specified.
The theoretical portion of this dissertation also included the development of a
modified universal kriging algorithm that accounts for complex feedbacks between
precipitation and a covariate, in this case vegetation, by incorporating the simultaneity
between these variables. This was accomplished through an instrumental variables
approach. First, an instrumental variable was created by regressing vegetation (i.e.,
NDVI) against a subset of the variables included in the precipitation model, plus one
variable that was correlated with NDVI but not precipitation. This variable was soil pH.
Predicted values for NDVI were generated then substituted for the observed NDVI data
in the universal kriging model used to predict precipitation. Formal testing for
simultaneity between precipitation and vegetation did not confirm the occurrence of
simultaneity feedback; however, the new method of universal kriging with instrumental
variables was retained in the case study to further develop the application of this
method.
The new and innovative technique for spatial analysis of data developed in this
dissertation can be applied in any context when the goal is to understand spatial
patterns in any continuous data, to predict data at unsampled locations, and/or to
understand the relationship between a variable of interest and other factors that may
influence the spatial distribution of that variable. This technique lends itself to
evaluation of many types of data, including data used in ecological evaluations,
environmental contamination studies, deforestation studies, geological and mining

142

studies, and econometric studies. Furthermore, these techniques are applicable to
evaluating data collected anywhere in the world, potentially improving the ability to study
regional patterns using these multivariate techniques.

6.2.2 Contributions to Regional Geography of Africa and Geography as a whole
The results and findings of this dissertation add to the knowledge and
understanding of the physical mechanisms of precipitation in East Africa as well as
spatial patterns of precipitation within the study area. In addition, once more recent
precipitation data are available, updated and improved maps of spatial patterns in
precipitation within the study domain can be generated, contributing to Physical
Geography and Regional Geography of Africa.
Results of this dissertation will contribute to work conducted by HumanEnvironment researchers such as high-resolution climate impact assessments (Mearns
et al.,1999, 2001a, 2001b) which may be used in the evaluation of coupled human
natural systems and formulation of associated policy.

6.3 Limitations
This dissertation has achieved many of the goals established at the outset;
however, looking to the future, improvements can be made. This section describes
some of the limitations of the work presented herein and suggestions for future
improvements.

143

6.3.1 Limitations in data and modeling
As noted in Chapter 4, meteorological stations are generally located in areas of
higher population, and do not fully represent the range of elevations in the study area.
The representativeness of station elevations and the impact that this has on the
representation of precipitation in the region is considered here. Figure 6-1a
summarizes elevation for the entire study area; for comparison Figure 6-1b summarizes
elevation for the meteorological stations only. The maximum elevation of a
meteorological station of 2,773 m falls well below the maximum elevation for the study
area of 5,778 m; the minimum elevation for a meteorological station of 454 m is
somewhat closer to the minimum elevation in the region of 195 m. Inspection of the
histograms also indicates that the locations of the meteorological stations overrepresent elevations in the range of roughly 1,300 m to 2,500 m. The most notable
under-representation occurs in the elevations below 1,300 m; more than 50% of the
study area falls below 1,300 m. The statistical approaches considered herein are based
on an assumption that the data used in the analysis are representative of the variable of
interested (i.e., precipitation). To accurately represent precipitation in the study area,
some form of randomization would be necessary in the placement of meteorological
stations. This is clearly not the case, resulting in a bias in the results of this dissertation.
While it is unlikely that a randomized and fully representative distribution of
meteorological stations will be available in the near future, solutions to this problem can
be addressed by better stratifying the study area based on elevation, or possible
through the use of declustering techniques.

144

A one-month lag was used to model the relationship between precipitation and
NDVI based on citations in the literature. It is likely that the lag between precipitation
and vegetation varies over space according to vegetation type and other factors. A
surface of spatially varying lag times could be developed in which the lag time depends
on factors such as land cover type; however, limitations in the temporal resolution of
vegetation data would have to be considered. For example, NDVI data are available in
biweekly time steps only from the Global Inventory Modeling and Mapping Studies
(GIMMS).

145

(a)

(b)

Figure 6-1. Comparison of elevations throughout the study area (a). Histogram of
elevations at meteorological station locations (b). Elevations were obtained from the
SRTM 30 arc second (approximately equivalent to 1 km at the equator) Digital Elevation
Model.

146

Elevation measurements were not available for every meteorological station.
Consequently, elevation was estimated for each station using a 1 km DEM. This
increases the uncertainty in the effect of elevation on precipitation. In addition,
elevation was considered at spatial resolutions of 1 km and 9 km based on previous
work cited in the literature. While this scale may be the best for representation of
localized effects on precipitation due to the presence of large water bodies and variable
topography, it is my opinion that elevation should be considered in other scales or
represented using other factors. For example, variables such as aspect and curvature
likely impact precipitation patterns at scales much larger than 9 km (e.g., when
identifying the windward and leeward side of a mountain). Also, the distance bands
used in categorizing distance from Lake Victoria were highly correlated with a
categorical elevation variable that was initially developed and not used due to
multicollinearity. This suggests that the distance bands were acting as a proxy for
average elevation. This observation is supported by the fact that the first distance band
generally lines up with the West Kenya Highlands, the second distance band generally
corresponds to the Rift Valley, and the third corresponds to higher elevations to the east
including Mounts Kenya and Kilimanjaro.

6.3.2 Limitations in statistical methodology
Use of the spatial regression techniques to estimate precipitation at unsampled
locations is currently limited: when predicting at unsampled locations, only the trend
components were included. The signal (i.e., the spatial smoothing) component is not
estimated, even though each of the β terms incorporate spatial autocorrelation terms.

147

Potential estimators for the signal component have been suggested, but not yet
evaluated for performance. This limitation results in overly smoothed predictions using
this method.

6.4 Future research
Future work is needed to evaluate the effect of modifications to the methods
used in this dissertation. For example, rain station locations were generally clustered in
the vicinity of higher population areas; consequently, the observed data may not have
adequately represented less populated (or unpopulated) areas at higher elevations or in
the savanna regions. Use of stratification or declustering techniques may be useful to
provide a better representation of precipitation over space.
Analysis of more recent data is also important for generating current finer-scale
precipitation estimates, and for identifying changes in significant predictive variables
through hypothesis testing. Consequently, efforts should be made to obtain more
recent, spatially complete data from meteorological stations in East Africa.
Additional work can be done to improve the results of the UKIV model.
Incorporating a spatially varying lag term may help in identifying any feedback
simultaneity that occurs between precipitation and vegetation. Improved regression
results through consideration of additional variables for use in predicting patterns in
NDVI (i.e., in generating the instrumented variable) may improve overall predictive
ability of the UKIV method. Further, additional research in the mechanisms of
precipitation may identify additional variables for inclusion in regression models,
potentially providing improved results.

148

Lastly, additional research is necessary to expand the methods presented herein
to evaluate trends in both space and time.
The statistical methods used in this dissertation are broadly applicable to many
types of data, including environmental, economic, epidemiological, among others. The
ability to incorporate simultaneity between multiple endogenous variables when creating
predicted surfaces of key variables will be of practical benefit. For example, many
contaminants of concern in environmental investigations are be related endogenously
(e.g., concentrations of various metals vary together naturally or due to similar
mechanisms of environmental release). In some cases, however, one or more
variables may be more costly to measure, such as dioxin concentrations in soil or blood
levels. The ability to utilize information on this endogeneity may reduce sampling and
analytical costs, improve study accuracy, and lead to more efficient and effective
environmental remediation.
In the context of climate studies, the methods used herein are broadly applicable
to other regions of the world. Understanding and predicting spatial patterns in East
African precipitation is highly complex due to changing seasonal patterns and widely
varying topography in this region of the world. The univariate statistical method of LOK
proved to be most effective for mapping precipitation patterns in East Africa due to the
complexity of causal factors and their influence on precipitation patterns in the region.
However, multivariate, spatially explicit statistical methods are expected to prove more
effective in other regions of the world with less variable seasonality and topography over
space, such as the North America plain states, or larger, more spatially contiguous
moisture sources, such as the Congo Basin in Central Africa. Furthermore, the

149

combined approaches of spatially explicit regression modeling and interpolation
methods yield not only improved prediction surfaces, but also better understanding of
the factors that influence precipitation patterns over space.

150

`

Appendix

151

APPENDIX Case Study Results
Table A-1. Summary of regression modeling results.
Month
Jan-84

Method
OLS

Apr-84

SARlag

Aug-84

OLS

Nov-84

SARlag

Independent Variable
(Intercept)
ndvi842
dem9km
p9km
q9km
curv9km
d2lv300
d2lv450
d2lv600
d2c500
I(d2lv300 * dist2lv)
I(d2lv450 * dist2lv)
(Intercept)
ndvi845
dem9km
p9km
q9km
curv9km
d2lv300
d2lv450
d2lv600
d2c500
Rho
(Intercept)
ndvi849
dem1km
p1km
q1km
curv1km
d2lv300
d2lv450
d2lv600
d2c500
I(d2lv300 * dist2lv)
I(d2lv450 * dist2lv)
(Intercept)
ndvi8412
dem9km
p9km
q9km
curv9km
d2lv300
d2lv450
d2lv600
d2c500
Rho

Estimate
-1.3530
-1.5500
0.0003
-0.2401
0.0468
-50.6300
10.2200
3.7850
0.6514
1.1610
-0.0271
-0.0081
-1.9160
0.7846
0.0001
-0.0479
0.2024
-136.4866
1.4931
0.9401
1.0393
0.3932
0.6040
-2.1604
0.7530
0.0006
-0.0632
0.1014
-3.9075
-1.5345
3.3980
-0.1001
0.4681
0.0116
-0.0076
-1.4962
-0.1582
0.0001
-0.3114
-0.0247
66.7779
1.4101
0.7143
1.2727
0.7044
0.2500

152

St. Error
0.6059
0.6857
0.0003
0.1573
0.1590
93.3400
5.4080
1.5980
0.5481
0.2960
0.0194
0.0035
0.4924
0.4462
0.0002
0.1241
0.1256
72.5866
0.4961
0.4808
0.4441
0.1891
0.1017
0.4720
0.5121
0.0002
0.1242
0.1205
6.7899
4.1207
1.2118
0.4035
0.2371
0.0148
0.0026
0.6446
0.6878
0.0002
0.1415
0.1390
81.0492
0.5406
0.5338
0.4849
0.2504
0.1629

t value
-2.2330
-2.2610
1.1930
-1.5260
0.2950
-0.5420
1.8890
2.3690
1.1890
3.9210
-1.3990
-2.3260
-3.8910
1.7584
0.5272
-0.3864
1.6123
-1.8803
3.0094
1.9552
2.3404
2.0787
5.9358
-4.5770
1.4710
3.2120
-0.5090
0.8420
-0.5750
-0.3720
2.8040
-0.2480
1.9740
0.7880
-2.8860
-2.3211
-0.2300
0.4865
-2.2008
-0.1776
0.8239
2.6082
1.3382
2.6246
2.8133
1.5348

Pr(>|t|)
0.0278 *
0.0259 *
0.2358
0.1302
0.7690
0.5887
0.0618 .
0.0198 *
0.2375
0.0002 ***
0.1649
0.0221 *
0.0001 ***
0.0787 .
0.5981
0.6992
0.1069
0.0601 .
0.0026 **
0.0506 .
0.0193 *
0.0376 *
0.0000 ***
0.0000 ***
0.1446
0.0018 **
0.6119
0.4020
0.5663
0.7104
0.0061 **
0.8046
0.0511 .
0.4327
0.0048 **
0.0203 *
0.8181
0.6266
0.0278 *
0.8591
0.4100
0.0091 **
0.1808
0.0087 **
0.0049 **
0.1248

Table A-1. Summary of regression modeling results (Continued).
Month
Jan-85

Method
SARlag

Independent Variable Estimate
(Intercept)
-0.7867
ndvi852
0.5605
dem1km
-0.0002
p1km
0.0418
q1km
0.4667
curv1km
12.2468
d2lv300
-4.8020
d2lv450
3.3218
d2lv600
0.2569
d2c500
0.3788
I(d2lv300 * dist2lv)
0.0218
I(d2lv450 * dist2lv)
-0.0072
Rho
0.2970
Apr-85 SARlag (Intercept)
-2.0913
ndvi855
-0.2445
dem1km
0.0005
p1km
0.1716
q1km
0.2458
curv1km
-6.1177
d2lv300
1.5083
d2lv450
1.0736
d2lv600
1.1885
d2c500
0.2507
Rho
0.6349
Aug-85 SARlag (Intercept)
-1.0773
ndvi859
0.9904
dem9km
0.0005
p9km
0.0794
q9km
0.2018
curv9km
-9.1431
d2lv300
0.1276
d2lv450
-0.3907
d2lv600
-0.4021
d2c500
-0.0420
Rho
0.4170
Nov-85
OLS
(Intercept)
-2.7490
ndvi8512
2.2120
dem9km
-0.0004
p9km
-0.0934
q9km
-0.0210
curv9km
177.2000
d2lv300
2.2380
d2lv450
1.6200
d2lv600
1.5390
d2c500
0.9699
Significance codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1
Bolded variables are significant.

153

St. Error
0.6318
0.5799
0.0002
0.1515
0.1441
8.7368
4.5302
1.5306
0.5711
0.2664
0.0161
0.0033
0.1613
0.6588
0.7459
0.0002
0.1474
0.1510
8.3879
0.5984
0.5685
0.5218
0.2290
0.1050
0.4340
0.4516
0.0002
0.1078
0.1082
64.7750
0.4247
0.3993
0.3687
0.2040
0.1329
0.6493
0.5926
0.0002
0.1640
0.1629
94.1500
0.6147
0.6146
0.5507
0.2356

t value
-1.2450
0.9665
-0.7656
0.2757
3.2398
1.4018
-1.0600
2.1702
0.4498
1.4220
1.3515
-2.1659
1.8406
-3.1744
-0.3277
2.5010
1.1637
1.6278
-0.7294
2.5206
1.8883
2.2777
1.0947
6.0476
-2.4820
2.1929
3.0242
0.7373
1.8647
-0.1412
0.3005
-0.9787
-1.0904
-0.2057
3.1382
-4.2330
3.7330
-1.4950
-0.5690
-0.1290
1.8820
3.6410
2.6360
2.7940
4.1160

Pr(>|t|)
0.2131
0.3338
0.4439
0.7828
0.0012 **
0.1610
0.2891
0.0300 *
0.6529
0.1550
0.1765
0.0303 *
0.0657 .
0.0015 **
0.7431
0.0124 *
0.2446
0.1036
0.4658
0.0117 *
0.0590 .
0.0227 *
0.2736
0.0000 ***
0.0131 *
0.0283 *
0.0025 **
0.4610
0.0622 .
0.8878
0.7638
0.3277
0.2755
0.8370
0.0017 **
0.0001 ***
0.0003 ***
0.1383
0.5705
0.8976
0.0629 .
0.0004 ***
0.0098 **
0.0063 **
0.0001 ***

Universal Kriging

Ordinary Kriging (nn=9)

 

Spatial Regression

Universal Kriging with IV

Figure A-1. January 1984 Average monthly precipitation maps generated using LOK
(top left), UK (top right), UKIV (bottom left), and regression techniques (either ordinary
least squares or spatial lag models, as indicated; bottom right).

154

Universal Kriging

Ordinary Kriging (nn=9)

 

Spatial Regression

Universal Kriging with IV

Figure A-2. April 1984 Average monthly precipitation maps generated using LOK (top
left), UK (top right), UKIV (bottom left), and regression techniques (either ordinary least
squares or spatial lag models, as indicated; bottom right).

155

Ordinary Kriging (nn=9)

 

Universal Kriging

Universal Kriging with IV

Spatial Regression

Figure A-3. August 1984 Average monthly precipitation maps generated using LOK
(top left), UK (top right), UKIV (bottom left), and regression techniques (either ordinary
least squares or spatial lag models, as indicated; bottom right).

156

Universal Kriging

Ordinary Kriging (nn=9)

 

Spatial Regression

Universal Kriging with IV

Figure A-4. November 1984 Average monthly precipitation maps generated using
LOK (top left), UK (top right), UKIV (bottom left), and regression techniques (either
ordinary least squares or spatial lag models, as indicated; bottom right).

157

Universal Kriging

Ordinary Kriging (nn=9)

 

Spatial Regression

Universal Kriging with IV

Figure A-5. January 1985 Average monthly precipitation maps generated using LOK
(top left), UK (top right), UKIV (bottom left), and regression techniques (either ordinary
least squares or spatial lag models, as indicated; bottom right).

158

Universal Kriging

Ordinary Kriging (nn=9)

 

Spatial Regression

Universal Kriging with IV

Figure A-6. April 1985 Average monthly precipitation maps generated using LOK (top
left), UK (top right), UKIV (bottom left), and regression techniques (either ordinary least
squares or spatial lag models, as indicated; bottom right).

159

Universal Kriging

Ordinary Kriging (nn=9)

 

Spatial Regression

Universal Kriging with IV

Figure A-7. August 1985 Average monthly precipitation maps generated using LOK
(top left), UK (top right), UKIV (bottom left), and regression techniques (either ordinary
least squares or spatial lag models, as indicated; bottom right).

160

Universal Kriging

Ordinary Kriging (nn=9)

 

Spatial Regression

Universal Kriging with IV

Figure A-8. November 1985 Average monthly precipitation maps generated using
LOK (top left), UK (top right), UKIV (bottom left), and regression techniques (either
ordinary least squares or spatial lag models, as indicated; bottom right).

161

Ordinary Kriging (nn=9)

Universal Kriging

Universal Kriging with IV

Spatial Regression

Figure A-9. January 1984 Maps of significant cross validation residuals
(|z-score|>2) for LOK (top left), UK (top right), UKIV (bottom left), and regression
residuals (for regression model; bottom right).

162

Ordinary Kriging (nn=9)

Universal Kriging

Spatial Regression

Universal Kriging with IV

Figure A-10. April 1984 Maps of significant cross validation residuals
(|z-score|>2) for LOK (top left), UK (top right), UKIV (bottom left), and regression
residuals (for regression model; bottom right).

163

Ordinary Kriging (nn=9)

Universal Kriging

Spatial Regression

Universal Kriging with IV

Figure A-11. August 1984 Maps of significant cross validation residuals
(|z-score|>2) for LOK (top left), UK (top right), UKIV (bottom left), and regression
residuals (for regression model; bottom right).

164

Ordinary Kriging (nn=9)

Universal Kriging

Universal Kriging with IV

Spatial Regression

Figure A-12. November 1984 Maps of significant cross validation residuals
(|z-score|>2) for LOK (top left), UK (top right), UKIV (bottom left), and regression
residuals (for regression model; bottom right).

165

Universal Kriging

Ordinary Kriging (nn=9)

Universal Kriging with IV

Spatial Regression

Figure A-13. January 1985 Maps of significant cross validation residuals
(|z-score|>2) for LOK (top left), UK (top right), UKIV (bottom left), and regression
residuals (for regression model; bottom right).

166

Ordinary Kriging (nn=9)

Universal Kriging

Universal Kriging with IV

Spatial Regression

Figure A-14. April 1985 Maps of significant cross validation residuals
(|z-score|>2) for LOK (top left), UK (top right), UKIV (bottom left), and regression
residuals (for regression model; bottom right).

167

Ordinary Kriging (nn=9)

Universal Kriging

Spatial Regression

Universal Kriging with IV

Figure A-15. August 1985 Maps of significant cross validation residuals
(|z-score|>2) for LOK (top left), UK (top right), UKIV (bottom left), and regression
residuals (for regression model; bottom right).

168

Ordinary Kriging (nn=9)

Universal Kriging

Universal Kriging with IV

Spatial Regression

Figure A-16. November 1985 Maps of significant cross validation residuals
(|z-score|>2) for LOK (top left), UK (top right), UKIV (bottom left), and regression
residuals (for regression model; bottom right).

169

April 1984

January 1984

August 1984

November 1984

Figure A-17. Maps of CRU residuals in 1984. Significantly high residuals (zscore > 2)
are shown in red, and significantly low residuals (zscore < -2) are shown in blue.

170

April 1985

January 1985

August 1985

November 1985

Figure A-18. Maps of CRU residuals in 1985. Significantly high residuals (zscore > 2)
are shown in red, and significantly low residuals (zscore < -2) are shown in blue.

171

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