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THEORY AND APPLICATIONS OF SENSITIVITY ANALYSIS
TO ENZYME KINETICS

By

Thomas Henry Pierce III

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1981

Copyright by
Thomas Henry Pierce III
1981

 

 

ABSTRACT

THEORY AND APPLICATIONS OF SENSITIVITY ANALYSIS
TO ENZYME KINETICS

By

Thomas Henry Pierce III

The theory of Sensitivity Analysis and its applications
to Enzyme Kinetics are examined. The Walsh Sensitivity
Analysis Procedure, WASP, is develOped and shown to be a
powerful probe of the theory of Sensitivity Analysis as well
as the preferred method for discrete models. The Fourier
Analysis Sensitivity Test, FAST, method is reviewed and
shown to be the preferred method of Sensitivity Analysis for
continuous models. The linear Taylor series approach to
Sensitivity Analysis is given as an aid in interpreting

Sensitivity Analysis results.

The theory of Walsh function Sensitivity Analysis is
derived and its advantages are investigated. The Walsh
technique is shown to be an exact technique for discrete
model output functions. For continuous model output
functions the Walsh method yields an averaged finite
difference Taylor series with respect to the parameters.
Walsh Analysis and 2-point discrete Fourier Analysis are

shown to be identical. Since Walsh Analysis is easily

Thomas Henry Pierce III

related to both the Fourier method and to the linear Taylor
series method, it is a valuable tool for further development

of Sensitivity Analysis.

The applications of the mass action laws of chemical
kinetics are used to develop models which are analyzed with
respect to their parameters. Enzyme Kinetics models for
hysteresis and allosterism are investigated by the
techniques of Sensitivity Analysis. The mechanism of
hysteresis in the Frieden Model and the Ainslie model is
clearly shown to be an effect of the rates of isomerization
of the inactive enzyme-substrate complex to the active
enzyme-substrate complex. The Ainslie model is dynamically
equivalent to the simpler Frieden model for a large set of
rate constants. The Frieden model also displays apparent
allosterism if the "correct" set of rate constants and
initial condititons are used. Therefore the Frieden model
is the simplest one-site enzyme kinetic model which displays
both burst and lag hysteresis as well as both positive and

negative cooperativity (allosterism).

Fourier Sensitivity Analysis was applied to a pH
Tryptophanase model where the parameters and their
variations were obtained from experimental data. This type
of analysis gave insight to the design of future
experiments. Over the experimentally accessible range of

pH, the lower pH region is shown to contain the most

 

 

 

 

Thomas Henry Pierce III

information on the parameters which can be examined in

future experiments.

A computer program is given which is used for routine
application of Sensitivity Analysis, both Fourier and Walsh,
to other models. This program has been extensively revised
to clarify its logic and to simplify its use. Any
mathematical model which can be simulated on a computer may

be directly inserted into this program.

Suggestions for future work are discussed. Research in
the connections between Statistics and Sensitivity Analysis
should lead to insight into both areas. The investigation
of "approximate" Walsh Sensitivity Analysis may lead to

faster algorithms for Sensitivity Analysis.

 

ACKNOWLEDGEMENTS

A great many people assisted me in this endeavor
and I am especially grateful to those mentioned here.
I would like to thank Rex Kenner, Bill Waller, Barb
Kennedy and Aristides Mavridis for their help in getting
me started; Jim Mehaffey and Mark Ondrias for keeping me
sane. My parents and Grandmother Pierce I thank for
their encouragement. I am grateful to Cathy Stewart
for her conviction and care during these difficult
times. I appreciate the everpresent support and
guidance of Professors Robert Cukier and James Dye
and acknowledge Dr. Robert Ball (Dr. Bob) for his
penetrating discussions, computer programs (GETNUM,
PLOT) and especially his friendship without which
this work would have been impossible.

I am also indebted to Michigan State University,
Chemistry Department, for an assistantship as well as

support from NSF Grant No. PCM 78—15750.

ii

TABLE OF CONTENTS

Chapter Page
LIST or TABLES . . . . . . . . . . . . . . . g . . . V
LIST OF FIGURES. . . . . . . . . . . . . . . . . . . vii

CHAPTER I. SENSITIVITY ANALYSIS --

AN OVERVIEW. . . . . . . . . . . . . . . 1
CHAPTER II. FOURIER SENSITIVITY ANALYSIS. . . . . . 9
CHAPTER III. WALSH SENSITIVITY ANALYSIS . . . . . . 28
CHAPTER IV. EXAMPLES OF SENSITIVITY ANALYSIS. . . . 52

CHAPTER V. SENSITIVITY ANALYSIS OF SIMPLE

ENZYME KINETICS MODELS . . . . . . . . . 128
Michaelis-Menten Model . . . . . . . . 128
Reversible Michaelis-Menten Model. . . . . . . . 1A0
Models with Slow Conformational
Changes. . . . . . . . . . . . . . . . . . . . . 1A1
Model of Ainslie, Shill and Neet . . . . . . . . lu2
Frieden Model. . . . . . . . . . . . . . . . . . 157
Summary. . . . . . . . . . . . . . . . . . . . . 167

CHAPTER VI. SENSITIVITY ANALYSIS OF A

TRYPTOPHANASE KINETIC MODEL . . . . . . 170
CHAPTER VII. FUTURE WORK AND DEVELOPMENT. . . . . . 189
APPENDIX 1 — Relationships of Fourier Co-

efficients to Taylor Series
Coefficients. . . . . . . . . . . 19A

iii

 

 

 

 

Chapter

APPENDIX

APPENDIX
APPENDIX

APPENDIX

APPENDIX

APPENDIX

APPENDIX

APPENDIX

REFERENCES .

9

Histograms of Fourier Trans-
formation Functions

Fast Walsh Transform.

Parseval's Equation for Walsh
Functions

The Calculation of walsh Partial

Variances . . . . . . .

Derivation of walsh Coupled
Partial Varianc Formulas.

Relationship Between Linearly
Dependent Equations and their
Fourier Coefficients. . .

Sensitivity Analysis Fortran
Programs. . . . . . .

Approximate walsh Sequency Sets

iv

Pag

(D

198

202

20“

209

213

216

218
283
284

LIST OF TABLES

Table Page

3.1 Multiplication Table for the p = 2 A
Group of Walsh Functions . . . . . . . . . . 32

5.1 Parameter Values for the Ir—

reversible Michaelis-Menten
Model. . . . . . . . . . . . . . . . . . . . 130
5.2 Parameter Values for the Re—

versible Michaelis-Menten

Model. . . . . . . . . . . . . . . . . . . . 130
5.3 Parameter Values for the Ainslie

Model. . . . . . . . . ... . . . . . . . . . 1A6
5.” Summary of SAM and Computer Data . . . . . . 1A?
5.5 Parameter Values for the Frieden

Model. . . . . . . . . . . . . . . . . . . . 159
5.6 Rate Constants for Allosteric

Test . . . . . . . . . . . . . . . . . . . . 168
6-1 Parameters for Tryptophanase

Model . . . . . . . . . . . . . . . . . . . 172

 

 

Table

6.3

Page
Best-fit Parameters Based Upon
Scheme 1 and Their Marginal
Standard Deviation Estimates . . . . . . . . 173
Legend for Figures 6.5—6.11. . . . . . . . . 173

Vi

 

 

LIST OF FIGURES
Figure
Figure 3.1 The four Walsh functions defined by
two binary digits. . . . . . . . . . . .

Figure 3. 2 Plot of the sampling points in the
multidimensional u—space.. . . . . .

Figure 4.1 The linear sensitivity coefficients
from the Linear Model. . .

Figure 4.2 The averaged value of the Linear Model
from a Walsh Sensitivity Analysis. . . . . . .

Figure 4.3 The standard deviation of the
simulations sampled in the Walsh Analysis of the
Linear Model.. . . . . . . . . . . . . . .

Figure 4.4 The Welsh expansion coefficients from
the Linear Model. . . . . . . .

Figure 4.5 Walsh partial variances from the
Linear Model. . . . . .

Figure 4.6 Fourier expansion coefficients from
the Linear Model.

Figure 4.7 Fourier partial variances from the
Linear Model.

Figure 4.8 The standard deviation of the sampled
simulations from the Linear Model using Fourier
Analysis.. ................

Figure 4.9 Linear sensitivity coefficients from
the Exponential Model. . . .

Figure 4.10 The averaged value of the Exponential
Model using Walsh Sensitivity Analysis with 10%
variation in the parameters. . . . . . .

Figure 4.11 Walsh eXpansion coefficients from the
Exponential Model (10% variation). .

Figure 4.12 Walsh partial variances from the
jExponential Model (10% variation). . . . .

IFigure 4.13 The standard deviation from the Walsh

vii

Page

33

A8

56

57

58

6O

63

65

66

67

7O

71

72

7A

Analysis of

the Exponential Model (10% variation).

Figure 4.14 The relative deviation from the Walsh

Analysis of

Figure 4.15
analysis of

the Exponential Model (10% variation).

The averaged value from the Walsh
the Exponential Model with the

parameters varied by 60%. -

Figure 4.16
Exponential

Figure 4.17
Exponential

Figure 4.18
Exponential

Figure 4.19
Exponential

Figure 4.20
Exponential

Figure 4.21
Exponential

Figure 4.22
Exponential

Figure 4.23
Exponential

Figure 4.24
Exponential

Figure 4.25
Exponential

Figure 4.26

the Exponential Model (10% variation).

Figure 4.27
Exponential

Figure 4.28

Walsh expansion coefficients from the
Model (60% variation).

The Walsh partial variances from the
Model (60% variation)“ . . . .

The Walsh standard deviation from the
Model (60% variation)“

The Walsh relative deviation from the
Model (60% variation). . .

The Walsh averaged value from the
Model with 100% parameter variation.

Walsh expansion coefficients from the
Model (100% variation)..

Walsh partial variances from the
Model (100% variation)..

The Walsh standard deviation from the
Model (100% variation)..

The Walsh relative deviation from the
Model (100% variation)..

The Fourier averaged value from the
Model (10% variation).

Fourier expansion coefficients from

Fourier partial variances from the
Model (10% variation). . . .

The Fourier standard deviation from

the Exponential Model (10% variation).

Figure 4.29

The Fourier relative deviation from

the Exponential Model (10% variation).

viii

Page

75

77

78

79

. 8O

81

82

. 8A

85

86

87

88

9O

91

. 92

. 93

9A

Figure 4.30 The Fourier averaged value from the
Exponential Model (100% variation).

Figure 4.31 Fourier expansion coefficients from
the Exponential Model (100% variation).

Figure 4. 32 Fourier partial variances from the
Exponential Model (100% variation).. .

Figure 4.33 The Fourier standard deviation from
the Exponential Model (100% variation).. .

Figure 4.34 The Fourier relative deviation from

the Exponential Model (100% variation).

Figure 4.35 The averaged concentrations from the
Unimolecular Model. . - . . . . . . . . . .

Fi re 4 36 Linear sensitivity coefficients for
[B in the Unimolecular Model. - . . .

Figure 4 37 Fourier partial variances for [B] in

the Unimolecular Model.

Fi re 4.38 Linear sensitivity coefficients for
fiuin the Unimolecular Model.

Figure 4.39 Fourier partial variances for [C] in
the Unimolecular Model.

Fi re 4.40 The Fourier standard deviation for
iuin the Unimolecular Model. . . . . .

Fi re 4.41 The Fourier relative deviation for
[C in the Unimolecular Model.

Figure 4.42 The averaged concentrations in the
Branched Unimolecular Model. . . . .

Figure 4.43 Fourier partial variances for [B] in

the Branched Unimolecular Model.

Figure 4.44 Fourier partial variances for [C]. in

the Branched Unimolecular Model.

Figure 4.45 Fourier partial variances for [D] in
the Branched Unimolecular Model. . . . . .

Figure 4.46 The averaged Substrate concentration
in the Michaelis-Menten Model (1% variation).

ix

Page

96

97

98

99

100

103

105

106

107

108

110

111

113

11A

116

117

120

Figure

Figure A.A7 The relative deviation of Substrate
in the Michaelis-Menten Model (1% variation).

Figure A.A8 Partial variances of the rate
constants for Substrate in the Michaelis-Menten
Model (1% variation).

Figure A.A9 Partial variances of the rate
constants for Velocity in the Michaelis—Menten
Model (1% variation).

Figure “.50 Averaged Substrate concentration in
the Michaelis-Menten Model (80% variation).

Figure A.51 Partial variances for Substrate in
the Michaelis-Menten Model (80% variation).

Figure 4.52 Partial variances for Velocity in the
Michaelis-Menten Model (80% variation).

Figure 5.1. Average concentrations and standard
deviations of the concentrations Michaelis-
Menten models. The symbols represent: 4), S;

O, P; A, E; +, ES. . . . . . . . . . . .

Figure 5.2 Partial variance plots for the
Michaelis—Menten models. A number represents
the partial variance for that rate constant.
Coupled partial variances are represented as
follows: in (a) by *, 81,3; in (b) by *,
3%,3; +9 81,25 X, 82,3; in (C) by *9 81,33 +9

,3. o o o o o o . o o . o o . o o . . o o 0
Figure 5.3 Average concentrations and the
standard deviations of the concentrations for
the Ainslie models. The s mbols represent:
0, S; A, E; +, E*S;¢, P; , ES: )1, E*P..

Figure 5.A Partial variance plots for the
Ainslie lag model. A number represents the
partial variance for the rate constant. Other
partial variances are represented by: B, 8113
D, S13; F, 815. Coupled partial variances are
represented as follows: in (b) by +, 85, ; *,
87,113 in (c) by +, 81;,13; *, 81,15; in (d)
by , 85,7; in (e) by , 13,15.. . . . . . .

Figure 5.5 Partial variance plots for the
Ainslie burst model. A number represents

the partial variance for that rate constant.
Other partial variances are represented by:

B. 811; D. 813; F, S15. Coupled partial vari—
ances are represented as follows: in (a) by
*3 811,7; in (b) by *, 81.15; +, 813,15. The

X

Page

121

122

123

125

126

127

133

136

1A8

151

Figure

symbol "s" represents the sum of the displayed
partial variances.

Figure 5.6 Average concentrations and the standard
deviations of the concentrations of the Frieden
models. The symbols represent: 0, S; A, E; +,
E*S;<>, P;X, E*;)(, ES. ... . . . .

Figure 5.7 Partial variance plots for the Frieden
lag model. A number represents the partial vari-
ance for that rate constant. Coupled partial vari-
ances are represented as follows: in (b) by +,

. ' . 9(-
85,9, in (C) by +, 85,9, , 81,7. . . . . . . . .

Figure 5.8 Partial variance plots for the Frieden
burst model. A number represents the partial vari-
ance for that rate constant. Coupled partial vari-
ances are represented as follows: in (a) by *,

85,9; in (b) by *, 81,7.

Figure 6.1 Averaged value of De1(ABS) from the pH
drop Tryptophanase Model (standard deviation
variation). . . . . . . . . . . . . . . . .

Figure 6.2 Averaged value of De1(ABS) from the pH
jump Tryptophanase Model (standard deviation
variation). . . . . . . . . . . . . . . . .

Figure 6.3 Averaged value of k' from the
Tryptophanase Model (standard deviation
variation). . . . . . . . . . . . . . . . .

Figure 6. A Partial variances of De1(ABS) from the
pH jump Tryptophanase Model (standard deviation
variation). . . . . . . . . .

Figure 6.5 Partial variances of De1(ABS) from the
Tryptophanase Model (10% variation).. . . . . .

Figure 6. 6 Partial variances of k' from the pH
jump Tryptophanase Model (standard deviation
variation). . . . . . . . . . . . . . . .

Figure 6.7 Partial variances of k' from the pH
jump Tryptophanase Model (10% variation).

Figure 6. 8 Partial variances of De1(ABS) from the
pH drop Tryptophanase Model (standard deviation
variation). . . . . . . . . . . . . . . . . .

Figure 6. 9 Partial variances of De1(ABS) from the
pH drop Tryptophanase Model (10% variation). .

xi

Page

161

16“

166

175

175

178

179

180

182

183

18A

185

 

 

 

Figure A.1 Histogram of log—uniform distribu-
tion function

Figure A.2
function.

Histogram of uniform distribution

Figure A.3 Histogram of Gaussian—type distribu—
tion function . . . . . . . .

Figure A.A
function.

Histogram plot of sin—transformation

xii

Page

198

199

200

201

I. SENSITIVITY ANALYSIS — AN OVERVIEW
INTRODUCTION

Mathematical models have exerted tremendous influ—
ence in science. Many people have commented on the "seem-
ingly exact" way that mathematical equations model nature
(Benacerraf, gt al. 196“). It is this ability to 'describe'
physical processes that makes mathematics so useful in
science.

Mathematical models are composed of four parts; inde-
pendent variables through which the model evolves, dependent
variables which change as a function of the independent
variables, parameters which are constant during a Simula-
tion but may change from one simulation to another, and
constants which never change, such as the velocity of light
in a vacuum.

Once a mathematical model is proposed, if it is 'cor-
rect', we can use it to predict the future behavior of a
physical system. It may be used to explain previous be-
havior of the physical system. To do this many models
require the 'adjustment' of parameters. It is this ability
of these parameters to describe different physical systems

by changing their values which gives great generality to

mathematical models and causes confusion as to whether
or not the model is 'right'. Often two or more models
give the correct results using only different parameters.
Mathematical models depend on their parameters.

Sensitivity analysis describes precisely how the mathe—
matical model depends on its parameters. An intuitive
sensitivity analysis method would be to vary a parameter
over two values and observe how the model changes. If a
particular value of the model output increases as the
parameter increases, we say that the output is positively
affected. This can be generalized by asking for the quanti-
tative sensitivity of a particular output function to a
parameter. The most popular method is to take the deriva-
tive of the output function with respect to the parameter

'p', evaluated at a nominal value DO-

—"I (-—) (1.1)

Many models have more than one parameter. A collection
of the derivatives with respect to the parameters permits
observations to be made about the model. For example, we
can order the parameters according to their significance.
The most significant parameter is the one which has the
largest effect on the value of the model output function.
This leads to an ordering of the parameters with respect

to their effect on the model, from most significant to

 

 

least significant.

Often models have parameters with at least one in-

dependent variable. An example is the temperature de-

Ipendence of a rate constant:

-Ea/RT
k= Ae (1.2)

Here A and Ea'are parameters, R is a constant, and tempera-

ture, T, is the independent variable. The model output

function is the rate constant k, the dependent variable.
In such cases the sensitivity of the output function

depends not only on the parameters but also on the inde-

pendent variable. The model output function is the rate

constant k, the dependent variable. In such cases the

sensitivity of the output function depends not only on the

parameters but also on the independent variable. When

measurements are repeated at different values of the in—
dependent variable, sequences of parameter sensitivity

Values can be collected over a temperature range of interest
A sequence of parameter sensitivity values may also

give other useful information. From it we may be able to

1dentify regions of sensitivity. Using the previous ex-

aJTlple, there may be temperature ranges where the sensi-

tILVity ordering of parameters changes. In one region the

parameter 'A' may be the most important, while in a dif-

ferent region the parameter 'Ea' may be the most important.

Therefore, to measure 'A', we should measure it in the
first region where the model is most sensitive to 'A'.

For a more complex model it may happen that in the
zregion of interest the model has pg significant sensi—
tsivity to one or more of the parameters. In this case
at;he model may be reduced to a simpler model by formally

1?:ixing the value of insensitive parameter to zero (or
.c)118): For example, it may be possible to remove the step
corresponding to the parameter in question from a mechanism.

Application of sensitivity analysis can also help to

\rzailidate a model. Knowing the rank-order of the parameters'
sseallsitivities and the region of maximum sensitivity permits
title design of experiments to probe the test model over
tzklese regions. A fit of the model to the new data, gives
fPLlrther evidence that the model is correct.

As described above, sensitivity analysis can lead to
IDeetter understanding of the model. Since many models are
<3<Jmp1ex an intuitive understanding of model behavior may
1363 difficult to obtain. It is useful to have quantita-
‘tiive mathematical tools which help to crack a complex model
Zirlto separate parts, which can then be independently ana-
lyzed and understood.

Classical linear sensitivity analysis is a useful
fTLrst approach which is straightforward. One examines the
Ckuange in the model output function caused by a unit

change in a parameter, gg (Beck 1977). However, this

method is only rigorously correct for linear models; those
models whose output functions are linear in their param-
eters, since the first derivative is the only variable
taffect attributable to a parameter.
With nonlinear models, higher order effects may be
important. The presence of higher order terms can be
x/werified by generalizing classical senSitivity analysis

t3<3 a Taylor series with respect to the parameters.

n
— it;
f(p1,p2..-.pn) - f(p_)| = + 121 (3101)] = Api
P. E, p -0
n n 2
+ 1/2 2 5: 13—99-11 An Ap

+ . . . (1.3)

In linear models higher derivatives, f", f"', etc,
Etre zero so that changes caused by parameters are weighted
anly'by f' evaluated at the nominal value, p=po. For non—
lainear models, the first derivative approximation is good
1d? all higher derivatives are small or if the region of

Vtiriation is so small that (Ap)2 3 0.

Both of these conditions are very restrictive. We

would like to vary a parameter over its entire valid range,
which is often many orders of magnitude. In some cases
higher derivatives are as large or larger than first deri-
vatives. With these problems in mind the idea of alternate
:representations of the model output function in terms of
{she parameters is a natural step.

In 1973 Cukier 22 El, derived a technique which repre-
sseents the model function in terms of a Fourier series.
Ublley related the sensitivity of each parameter to a separate
Fourier coefficient. Since any expansion of a well-
t>€ehaved function is identical to another expansion which
r1213 been rearranged, it can be seen (Appendix 1) that
tskdese Fourier coefficients are functions of all the higher
Clearivatives of the model function with respect to the
IDEIrameters. This is the ideal relationship required for
rnanlinear functions. It allows sensitivity measures where
‘Clae parameters are varied over orders of magnitude with
,Eyg restrictions on the model output function.

The implementation of the Fourier method on a com-
Fnlter requires approximations as explained in Chapter
UTwo. The approximations limit the method through an ac-
clmmulation.of approximation error. However the sources
<>f"the error have been described by Cukier §t_§l, (1975).
‘TIlese errors are controllable and they can be bounded by

a. maximum.error estimate.

Other expansions are possible and in Chapter Three we

will investigate Walsh series expansions (Walsh 1923,
Fine 19U9). It will be shown that for a discrete model
we incur pg errors in analyzing the sensitivity of a
Inodel. However for continuous models it will be shown
tshat a Walsh sensitivity expansion is identical to a
1Finite-difference Taylor series..

In Chapter Four the three approaches are compared,
Eilfld we can see that they give similar results when the

gpazrameter variation approaches zero. In the case of a
global analysis of a strongly nonlinear model only
tzrie Fourier method gives the correct results for the
sensitivities.

In Chapter Five we apply the Fourier technique to
Esc>me steady-state enzyme kinetics models. Here we Show
Tillat two apparently different models are dynamically id-
GBIltical over a rather extensive range of parameter varia-
‘tzion. Also the sensitivity analysis of progress curves
Sdaows that some parameters may 'accumulate' sensitivity
2111 time. At short times they are relatively insensitive,
bnat as the reaction proceeds they become the most important
parameters in the model.

In Chapter Six we apply the Fourier technique to a
rwecently-studied transient-state enzyme model (June 33 El;
1979, 1980).

The future work and development of sensitivity analysis

texzhniques is discussed in Chapter Seven. It is suggested

that the study of approximate Walsh sensitivity analysis
and the frequency sets used in approximate analysis will
extend both methods. Also the relationship of sensitivity
analysis and statistics is discussed. It is our belief
that these questions will direct research into fruitful
gareas which will advance the usefulness of sensitivity
lgznalysis techniques with extremely complex models.
Appendix 8 contains the various programs and their
c>1>eration instructions for the application of both Fourier
ELITd welsh Sensitivity Analysis. The programs are model

j.r1dependent so that any type of numerical model may be

‘ulssed. It is my hope that sufficient theory and examples

axlre given here to encourage others to use these powerful

t:eechniques.

II. FOURIER SENSITIVITY ANALYSIS

In this chapter we will discuss the Fourier method

caf sensitivity analysis. Only an overview of the theory

veill be given as this technique was extensively reviewed
E337 Cukier et a1. (1978). Here we will examine the details

<31? the implementation and review the approximations and

21.jJnitationS of this method. The particular model chosen

era this implementation is derived from the laws of mass

EL<2tion kinetics.
Chemical rate equations as derived from postulated

‘nleechanisms can be described by sets of first order in time,

C:c>up1ed ordinary differential equations of the form,

iii. = Fitgucg) (2.1)
dt
1 = l,2,3,...,m
“nith prescribed initial conditions,
0 (2.2)

Ci(t=0) = C1

In Equation (2.1) Ci(t) is the concentration of the

ifiih species at time t, g = (C(1),C(2),...,C(m)), is a

 

10

vector of all the species concentrations and k = (k(1),
k(2),...,k(n)) is a vector of all the rate constants
(parameters). The function Fi symbolizes the rate law

for the species 1. We shall assume that these rate equa-
tions can be solved for C(t), given the initial conditions,
the values of the rate constants, and the Specific func-
tional form of the rate laws. If this cannot be done
analytically, it can almost always be done numerically.

We require the sensitivities of the concentration
(31(t) to uncertainties in the values of kg, the rate co-
eefTflcients. The uncertainties in the rate coefficients
aaxre, in this method, represented statistically. That is,
vve: assign a probability density, pZ (k£)dk£ as the prob-
ability that the z'th rate coefficient lies between kg
and kl + dkz. These probability densities reflect our
lcrlc1wledge of the possible values of the rate coefficients
i.r1 a given elementary chemical reaction. If one has ac-
<3111rate data, then the probability density can be chosen
t3<> be narrow to reflect this information. However, if
CiEifbaare sparse or not reliable, the uncertainty can be
C3I1<33en as large as desired.

The joint distribution function may then be written

1‘).
P(_Ig) = I} p5, (kg) (2.3)

 

 

 

11

as we require the rate constants' probability distributions

to be linearly independent, but not necessarily identically

distributed. Once the joint distribution is known we may
"'construct moments of the multidimensional function by multi—

dimensional integration. The first moment, the average

value, is written

<C(_k_)> = f dk C(k)P(k) (2A)

tvhere dk = dkldkz...dkn is the multidimensional volume
eelement in the rate coefficient Space. In the present
eeJcample <C(k)> represents the average concentration of a
g;1;ven species as calculated from the rate laws and the
IPEite constants, where the rate constants are varied over
1:11eir entire set of possible values.

Similarly we may construct multidimensional variances

C>f the function, by calculating
(o2)i = <ci(k)2> - <Ci(k)>2 (2.5)

UDInis would represent the expected spread of concentrations
C>‘ver values accessible to species 1 because of uncertainties
len the rate constants. Similarly partial variances, the

‘réariance along only one parameter dimension, say the first

 

l2
parameter can be computed as
2 - * 2 9(- 2
(Ol)i ‘ ((Ci(kl)) > ‘ (Ci(kl)> (2.6)

where G:(kl) is the function averaged over 5 = (k2...kn)

and the integration is over kl. This gives the spread

of concentrations caused by uncertainties in kl. This

idea may be extended to coupled partial variances, variances
over more than one parameter at a time.

These variances would be very informative. We would
be able to characterize the extent that the model de-
pended on the parameters. This also would tell us which
parameters were most important (those whose variances
were largest). If a parameter's variance were small then
the effect of a parameter changing over its entire range
is negligible to the behavior of the model. This means
that the model may be simplified by excluding parameters
'whose variances are small.

The coupled partial variances would tell us how the
Ibarameters interact. If these coupled variances are large
11hen the model also depends on the relationship of coupled
parameters. The effect of one parameter acts in concert
unith.another parameter. These coupled variances may be
eoctended to arbitrary number of parameters coupled together
(but less than n).

The only requirement here is the construction of the

 

13

joint probability distribution and its multidimensional
integration. Presumably this could be done by numerical
quadrature. We would sample each parameter over its
domain, then solve the model for each different combina-
tion of the parameters. The solutions would be numerically
integrated against the joint probability distribution to
give the desired moments.

The required amount of calculation to accomplish this

 

is enormous. If we chose 10 different values for each
parameter, and there were only five parameters in the
model, we would have to solve the model 510 times, ap-
proximately 10 million times. Even so, if the ranges of
the parameters were large or the model highly structured,
we would need still more sampled points to accurately carry
out these variance calculations by such a brute force
method. To calculate the variances in finite time we need
a different approach. We need a way to compute the multi-
dimensional integrals without exhaustive sampling of the
output function.

In 1938 Hermann Weyl (Weyl, 1938) derived an integral
identity which, under certain conditions, reduces a multi-
dimensional integral to a single path integral. To apply
his theorem we must return to our definition of parameters
and transform them into periodic functions.

The rate constants, considered as random variables,

may be related to a generating function,

 

 

1A

k2 = 82, (119.) (2.7)

where g2 is the generating function and u2 is the inde—

pendent variable. As ul is varied from -w to m, k, is

varied over all its possible values. Consequently, uz

also has a probability distribution. Since the kz's
I l
A s, the u, s are also

independent.o We may then write the total joint probability

are independent functions of the u

density function in u-Space as

) (2.8)

It is convenient to let u2 be related to 91 through
the transformation function

such that as e, traverses -m to m, u, also goes from

-w to w, so no information has been lost. Now we further
write 9, = (wls) with -m <s<m. The new parameter, m2,

is called the A'th frequency. This procedure relates
each parameter to a frequency, ”2’ so that by varying S

over its range all the parameters vary simultaneously,

 

at different frequencies, over their ranges.

 

 

15

_The probability distributions in u-space are easily
related to a probability distribution in e-space. It is
desirable for all unit lengths in 9-space to be equi-
probable, 14g;, we want to insure that P(u£)du2 = P(e£)dez.
Note that we are using the same symbol for the probabili-
ties even though we have transformed the independent vari-
able uz to 6,. This is done to simplify notation.

The chain rule may be used to derive the equation

relating these two probabilities.

P(8£)d8£ = P(u2)du£

13(6),) = 1901,) fig;- (2.10)
dx _ _ 2 1/2
35 - cose - (l-x )

P(6) = for the half interval

aha

Writing ”A = 02 we obtain

 

 

 

l6

dGz

2)l/2.___
dx

% = p(G2)(1 - x (2.11)

This is a first order differential equation whose solution

is the transformation function G To uniquely solve this

2.
equation we need an initial condition.

k0 exp(u£), and note that when

A A
s = 0, x = sin(wls) = O, we see that ul = G (sin(w£s)) = 0

If we define k

which implies that 0(0) = O. This is the required initial
condition. This means that u, is restricted to a poly-
nomial in Sin(w£s) with no additive constants. Some pos-
sible transformation functions are given in Appendix 2
along with the distributions that they generate.

Now that the rate constants are related to the search

variable, 5, we can apply Weyl's theorem.

1m 11.;IT F(s)ds (2.12)

_ L
(5;) fdgr(g) - T,” 2T

Weyl showed that this integral identity would be exact
if

“sz

6, = wzs and calm2 # O (2.13)

1

(:1

1n

 

 

 

 

 

17

for all possible integer values of cg. In this case the
frequency set, {“2}, is called incommensurate.

An incommensurate frequency set is easily constructed
if we use irrational numbers for the frequencies. How-
ever, since we will be using a computer for solving the
model, irrational numbers are not feasible. What is done

is to define an order of accuracy 'M' such that

E elm, # O for i Iail i M + 1 (2.1M)

or more concretely

Z a w = 0 for min [Ila l = M + 2]
2 E Z a Z 2

Once we have defined an order of accuracy it can be
seen that irrational numbers for the frequency set are no
longer required. Now a frequency set will be associated
with its value of M. In fact, we may use integer fre-
quencies as it simplifies further calculations.

The finding of arbitrarily accurate frequency sets is
apparently quite difficult. For the special case of

Ath order accurate sets, i.e., M = A, we may exploit the

18

idea of sums and differences of two frequencies to find

the sets. Schailbly 23 al. have tabulated Ath order ac-
curate frequency sets for parameter sets of up to fifty

parameters. These sets are stored in the program in Ap-
pendix 8.

Since the parameters are proportional to Sines, it is
prudent to use only odd frequencies in order to exploit
the periodicity of the sine of an odd frequency (see
Cukier 1978). This helps in the search for frequencies
by eliminating half of the integers.

Given the frequency set, we can approximate the multi-
dimensional e-Space integral in s-space. But the s-space
integral has other valuable properties. We have related
each parameter to a function of a sins of a frequency.
Since Sines of different frequencies are part of an ortho-
normal family, the Fourier series, the effect of each
parameter may be easily projected out of the s-Space
integral.

Expanding the output function in a Fourier series

(Zygmund 1959), given

A on
01(g(s)) = 39 + E. [Ajcosfljs) + Bjsin(js)] (2.15)

j 1
A} = % £2, Ci(s)cos(js)ds
1 1 n i .
Bj = F Lm' C (S)sin(js)ds

19

or equivalently in exponential format,

“'Ci(k(s)) = E0 cJ exp(i2sz)
where
03-: % t2. 01(k(s))exp(12njs)ds (2.16)

A + B
Note that CJ = 41.5.41 separates the output function

into its frequency components. ("1" denotes the ith
concentration). Since the th parameter is associated
with the 2th frequency, the magnitude of the Fourier co-
efficients of this frequency and its harmonics measure
the sensitivity of the 2th parameter.

We can illustrate this with a simple example. Con-
sider a reaction scheme with three rate coefficients
associated with three frequencies, Wl’ WB, and W3. Since
we vary the rate coefficients as Sin(w£s) the ith concen—
tration as a function of s will consist of sums and products
of these sin(w£s) factors. When strings of these Sines
are multiplied together the result is Sines and cosines
of sums of the w 3 factors. If we assume that the fre-

A

quencies in the sums of the w 5 factors are incommensurate,

A
no linear combination of the frequencies can be formed

which sums to zero. That is, there are always three

20

independent components in these sums. If the Fourier
decomposition of C(s) is performed the Fourier coefficients
of indices wl, 2wl, 3wl,... can only be due to the rate
coefficient kl since only kl varies with W1 and no com-
bination of w2 and W3 can add up to a multiple of W1 to
cause an interference. Thus, the Fourier analysis enables
us to isolate the effect on the ith concentration of un-
certainty 1n_the th rate coefficient.

The above definitions of A3 and B3 are exact only if
we are able to analytically integrate the equations.

Since we will numerically integrate the model equations

for 01(3) only concentration-time points will be available.
We must then use a discrete Fourier transform instead of

a continuous one. The concentration points may be numeri-
cally integrated into discrete Fourier coefficients.

There are two kinds of error involved when this ap-
proach is used. We obtain the largest error by exchang-
ing integers for irrational numbers in Weyl's integral
identity (Cukier 1975). By choosing a value of M and its
frequency set we postpone addition errors, the sum of the
harmonics equalling zero, beyond the combination of M
frequencies or harmonics. Since Fourier coefficients
decrease by at least (l/n), error from the combination
of harmonics can be maintained at a low level.

This form of error also depends on the model. If

there is no combination of M parameters multiplied together

 

 

 

 

21

in the model then the possible error from different fre-

quencies adding to the same frequency, breaking a type

of linear independence, is eliminated. Also if the out-

put function of the model does not have parameters raised

klO, then the frequencies

to high powers, such as k5 or
will not add in the high harmonics to give error. In

any case one may always increase the value of M to de-
crease this type of error, provided the frequency set is
known.

The second source of error is from the use of a finite
discrete transform instead of an infinite continuous one.
This error comes about by sampling the function at equally
spaced points, Egl. This equal spacing gives an aliasing

error, frequencies which oscillate faster than the sampling

rate fold their effects into lower frequencies (Cukier

1975)
acalc = A + A + I (A + A )
k k 2N—k m=0 2Nm-k 2Nm+k
bcalC = B + B + I (B + B 1
k k 2N—k m=0 2Nm-k 2Nm+k’ (2.17)

Aliasing sets a limit on the maximum frequency that

one may compute from a sampled function. This maximum

22

frequency is determined from the Shannon Sampling Theorem
(A. J. Jerri 1977) and is called the Nyquist Frequency.
One can reduce this type of error by sampling more points.
In the literature (Cukier 1975, Jerri 1977) the usual
number of samples taken to insure accuracy is uwmax’

where wmax is the largest frequency desired.

2wmax < N (2.18)

This sampling rate tells us the minimum number of
samples that are required to compute a particular fre-
quency.

Having examined the error terms we find that the
number of points chosen is important. However adding a
Single point implies that we will do an additional Simula-
tion with a new parameter vector. The cost in computation
time in each simulation can be high. If the model is com-
posed of ordinary differential equations then the computa-
tion of the required simulations accounts for about 90%
of the total required computation time involved in sensi-
tivity analysis. We would, therefore, like to minimize
the number of simulations necessary to calculate the
Fourier coefficients. If odd frequencies are chosen,
the number of simulations necessary is reduced by one-
half. It was shown (Cukier et a1. 1978) that the symmetry

relations for Sines of odd frequencies

 

 

 

 

23

MW - s) = f(S)

f(s - n) = f(-s)

1‘(s + 121) = f(s - 35-)

'f(s - 121) = f(-s - 321) (2.19)

allow us to sample the output function in the range
[-w/2, w/2] and then reflect these values into [0,2w].
In this way we get twice as many Fourier coefficients as
sampled points.

Given a finite number of simulations we can construct
a finite Fourier series approximation to the function in
s-space. The coefficients in this approximation may be
evaluated in two possible ways. The direct application
of the transformation, a brute force approach, would re-
quire N2 multiplications on the computer. This can be
seen by examining the equations for the Fourier coefficients

given below.

C(sj)exp(i2NjP/N) (2.20)

O
'0
II
II M Z

1

OI‘

2“

N
a3 = % 3:1 C(sj)cos(g%%£)
2 N 21.11:
bJ = N E C(sj)sin( N’ )

N an odd integer

Alternatively we could use the Fast Fourier Transform
algorithm, FFT, of Cooley and Tukey (Cooley, Tukey 1965).
This method uses approximately Nlog(N) multiplications.
Usually this algorithm is applied when the number of
samples is a power of two. In this case the algorithm is
at its most efficient. Since we want to minimize the
number of required samples, sampling the function a power
of two times is too strict a requirement. This usual
restriction in the number of samples is pg; a requirement
for use of the algorithm. In fact, as long as the number

of samples taken lg not a prime number the FFT algorithm

 

is a much faster and more accurate technique than the
direct method. If the number of samples, N, is factored
into its prime factors, n1n2n3...n2 = N, then the number
of operations that this generalized FFT algorithm takes
is a§,mg§§ (nlN + n2N + ... nzN) (Dahlquist 197A).

By using the FFT method on R points in s-space, we

extract 2R coefficients. Some of these coefficients are

 

 

 

 

25

ambiguously related to the parameters. This ambiguity
depends on the order of accuracy M. The ambiguity implies
that more than one linear combination of the wfset fre-
quencies adds to the same frequency. We donlt, then, know
to which linear combination to assign this frequency.
Hence it is considered as an error term and only used

in the calculation of the total variance.

Obviously for a Mth-accurate frequency set, only those
combinations of w-frequencies whose a-set sum to less than
flEg’may be unambiguously defined. That is, we know, to
Mth-accuracy, what combination of parameters add to these
frequencies. These Fourier coefficients may be easily
combined into the desired variances as was earlier pro-
posed.

Parseval's formula for Fourier Series (Zygmund 1959)

gives us the total variance of the model output function

N-l a 2
2 _ 2 2 0
0total ' 121 (31 + bi) + (7?)

(2.21)

where N = 2R, and R is the number of simulations.

The contribution of the ith parameter to this total
variance is contained in the coefficients of the ith
frequency and their harmonics (Cukier gt_§1, 1978)

K 2 2

2
o = Z a + b (2.22)
2’ p=l pwg, prL

26
where K = M/2, for a Mth accurate frequency set.

From this we can construct the reduced partial variances

S, =-—§———- (2.23)

The contribution of the coupling between parameters
is contained in the coefficients of the combination fre-

quencies, jwk + pw2(Cukier gt a; 1978).

a, 82
a: k = 2 z a2 +. + b2 +. (2.2m)
: 1=_Bl J=‘B2 JWQ’ 1Wk JWQ, ka
113 H 0

Where maxEBl + 82] = (M+1) - M/2, thereby preventing double
counting of frequencies which may be included in single
partial variances. This does allow for the possibility
of some double counting in the coupled partial variances.
However, these high harmonics of the fundamental frequencies
are usually attenuated so the error is very slight.

From this we can construct the reduced coupled partial

variances,

27

82.,k = T— (2.25)

Cukier (1975) has related the EW WA coefficient to the
linear sensitivity coefficient (— c) in the limit of small

pt
parameter variation.

Bw£= <(ac,(U))(% p)> + 0(p2 ) (2.26)

This equation shows a direct relationship between the
averaged linear sensitivity coefficient and the Fourier
coefficient, Bwl’ which is used to construct the w,
partial variance.

The Fourier method of sensitivity analysis is imple-
mented in a program given in Appendix 8. This program
is not restricted to models involving ordinary differential
equations. Rather any type of mathematical model may be
inserted in the program through use of the subroutine
MODEL. Hence the global parameter sensitivities of any

type of model are easily obtained.

III. HADAMARD-ORDERED WALSH FUNCTIONS

Cukier gt a1, (1978) proposed that alternate orthogonal
expansions also may be possible, leading to other types
of sensitivity analysis. This approach was investigated
and an alternate expansion was found. It was discovered
that a Walsh function expansion (walsh 1923) could be used
for sensitivity analysis. If we consider a model whose
parameters take on a finite set of values, then we may
use welsh Sensitivity Analysis. Here the model output
function has a finite set of discrete responses dependent
on the parameters. For these discrete model functions
the use of walsh functions eliminates the approximations
inherent in the Fourier method. With continuous model
output functions (those functions whose parameters vary
continuously over a domain) the Walsh method is closely
related to Taylor Series Sensitivity Analysis. In this
chapter we develop the theory of the Walsh method.

A welsh function (Ahmed and Rao 1975) may be defined
as a function of two arguments, a time variable and a
sequency variable, Similar to frequency in Fourier analysis.
Walsh functions form a complete orthonormal set of step
functions. Here the Hadamard definition (Ahmed and Rao

1975) is used to represent Walsh functions.

28

 

29

t
._ 11
WALH(w,t) = (-1)l'l (3.1)

Where t=(tl,t2,...,tp) is the binary representation of t

and (wl,w2,...wp) is the binary representation of w.

walsh functions are defined over the time range
[0,1], 14g;,the time variable is a real number less than
1. However, the sequency variable is an integer less than
2p. This means that the binary point for the time vari-
able is placed to the left of t1 and the binary point for
the sequency variable is placed to the right of wp. Note
that the indexing chosen here labels the most significant
digit of the variable first.

With this definition of walsh functions, the time
variable is defined with respect to the sequency variable
in order to cancel dimensions. It should be noted that
the time referred to here i§_ngt_"model time". Model time
is defined as the independent variable in a model such as
the model of a chemical reaction which evolves in time.

To clarify the evaluation of a walsh function, let
us consider the walsh function, WALH(2, .75). Only two
binary digits are necessary to represent these arguments,
therefore let p=2. writing out the binary expansions of

the arguments we obtain

30

W = 210 = (100)2 (302)
t = 07510 = (oll)2

Substituting these binary representations into the defin-

ing Equation 3.1 we obtain

w,tl+w2t2 1

l.l+O.l=(_l) = -1

WALH(2, .75)=(-1) =(-1)

(3.3)

walsh functions are constrained by the number of binary
digits required to represent w and t. For this reason
we get different groups of functions for each choice of
p, the number of binary digits used in the representation.
Each group is closed with respect to ordinary multiplica-
tion. If we multiply one Walsh function by another walsh

function from the same group, we obtain a third member

 

of the group. This means that multiplication of a p—digit
walsh function with a k—digit walsh function is not de-
fined.

This group property can be illustrated by examining a
walsh function multiplication table for p-2. Here we have
a group of Walsh functions with only four members,

WALH(O,t), WALH(l,t), WALH(2,t), and WALH(3,t). By using

31

Equation 3.1 we can generate Table 1.

Figure 1 gives the plots of these four Walsh func-
tions. They are piecewise continuous functions. These
functions may be integrated but they may not be differen-
tiated without the introduction of distributions which
include delta functions.

walsh functions have some useful properties. They

are invariant to an exchange of arguments, i.e.,

WALH(w,t) = WALH(t,w) (3.1))
Proof:
Ewiti itiwi
WALH(w,t) = (-1) = (-l) = WALH(t,w) (3.5)

As has been claimed, the Walsh functions are orthog-

onal (Walsh, 1923),

A} WALH(n,t) WALH(w,t)dt = 2pan (3.6)

W

Proof:

32

 

 

 

Table 3.1. Multiplication Table for the p=2 Group of
walsh Functions.

* WALH(O,t) WALH(l,t) WALH(2,t) WALH(3,t)
WALH(O,t) WALH(O,t) WALH(l,t) WALH(2,t) WALH(3,t)
WALH(l,t) WALH(l,t) WALH(O,t) WALH(3,t) WALH(2,t)
WALH(2,t) WALH(2,t) WALH(3,t) WALH(O,t) WALH(l,t)
WALH(3,t) WALH(3,t) WALH(2,t) WALH(l,t) WALH(0,t)

 

 

33

.muflmflc apmnwn 03p hp cosflmmc msoepocze gnaw: L309 one _.n epzmflm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

c8:
00... and 00.0 0N0 00.0
_ . _ _
_ . _ _ _
00. P 05.0 on .0 0N0 00.0
Q3050 N I Q

mcotoczbl aback

 

 

3A

{3 WALH(m,t)WALH(w,t)dt

p D
l l l i:lniti iggiti
= 2 z z (-1) {-1)
ti=0 t2=0 t =0
p
l l 1E£ni+w1)ti
= z 2 (-l)
tl=0 tp=0
p
+
l 1 nlwl 122(91 w1)t1
= z ... Z (1 4- (-l) ) {-1)
t=0 t=0
2
p
p - p
:2 H6 -2(S
1:1- ni’wi nlw

If we divide each Walsh function by two we obtain an
orthonomal set of functions. Completeness of Walsh func-
tions was shown by Walsh (1923).

Utilizing the orthonormality and completeness proper-
ties of Walsh functions we know that any continuous func-
tion, f(t), can be expanded into an infinite Walsh series

with O i t < 1. This exact infinite expansion of an

35

arbitrary function may be approximated by a finite expan-

sion:

N—l
f(t) = z CnWALH(n,t) (3.8)
n=0

Here we restrict to to a discrete set of N points (ti)

where,
Nt = t1 (3.9)

Note that O 1 ti < N-l and ti is an integer. The error

incurred by this approximation is

error = 2 CnWALH(n,t) (3.10)

Since the coefficients of a Walsh expansion decrease
in magnitude by (l/n) as shown by Fine (1955), we can
approximate the major structure of any arbitrary function
by using a finite expansion of Walsh functions.

Walsh coefficients are a linear transformation of
the function sampled at each ti. We can compute the co-

efficients by exploiting the orthogonality of Walsh functions.

36

By writing the function in its finite Walsh expansion,
Equation 3.8, then multiplying both sides by WALH(m,t)
and integrating over "t", we will project out the Cm

coefficient of the Walsh expansion.

_1_ fl f(t)WALH(m,t)dt

O

= 2 Cm 3; fl WALH(n,t)WALH(m,t)dt
n=0 2p 0

(s = C (3.11)

II
M
O

This procedure for calculation of the N coefficients
of the Walsh expansion from the N sampled values of the
function requires N2 multiplications. Using matrix factor-
ization techniques an algorithm may be developed where
this transformation only requires Nlog(N) multiplications.
This is known as the Fash Walsh Transform (Ahmed and Rao
1975, Andrews and Caspari 1970) (See Appendix 3).

To use Walsh functions in sensitivity analysis we
must be able to calculate multidimensional moments, which
are averages of the output function over all its param-

eters. To calculate these moments we must express the

37

function in terms of its parameters. If we relate each

parameter, ui, to a generating function in ti’
1.11 = 0'o(ti) (3.12)

we may expand each parameter dimension in a Walsh series.
Thereby we obtain a multi-dimensional Walsh series expansion.

For example, let f(ul,u2) be an output function with
two parameters, ul and u2. If we relate the parameter ul

to a generating function, gl(tl), we may write f(q) as
f(ul,u2) = f(gl(tl),u2) = f(tlau2) (3-13)

Note again that the same symbol for the function, f, is
retained. This will be done throughout this chapter when
the meaning is obvious.

This function may be formally expanded in a Walsh

series in t1 with u2 treated as a parametric constant.
f(tl,u2) = g Cj(u2)WALH(J,tl) (3.1M)

Since the coefficients are now functions of u2 we
may relate u2 to a generating function in t2 and expand

the coefficients in a Walsh series in t2.

38

f(tl,u2) = f(tl,g2(t2)) = g CJkWALH(j,tl)WALH(k,t2)

z
k
(3.15)

This yields a two-dimensional WAlsh series expansion for
f(ul,u2).

. The key to the utility of Walsh functions in sensi-
tivity analysis is the multiplication identify. For an
expansion to be efficient, products of the orthogonal basis
set must reduce easily. In the case of Walsh functions, the
product of two Walsh functions in the same group is a third

Walsh function, given by
WALH(n,t)WALH(w,t) = WALH(n + w,t) (3.16)

where + is binary addition without carry. That is,
O + O = O, O + l = 1, 1 + O = 1, and l + 1 = 0.

Proof:

2 W t

Z n.t i i i
(-l)

i 1 i
WALH(n,t)WALH(w,t) = (-l)

2

(n +w )t. Z (“1 + W1)t1
(-l)1 i i 1

= (-l)1

WALH(n + w,t) (3.17)

39

We can apply this multiplication property to reduce
the multidimensional integral for the multidimensional
Walsh series coefficients to a one dimensional integral.
This will allow us to connect the multidimensional param-
eter space to a single dimensional line.

In summary, if f(u) is a multidimensional function in
u_with u = (u1,u2,...up) we may expand the function in a
finite multidimensional Walsh series. Each dimension of
f(g) will be related to a dimension ti which will be ex-
panded in a single dimensional Walsh series. We may

write a generalization of Equation 3.15 as a Cartesian

product.
m -1 m -l m -1
l o
f(u) = f(t) = z z z n
_ _ . _ C WALH(w t )
wl-O w2-O wp-O _ i=1 ’ i
(3.18)
The coefficients may be expanded as finite sums
1 ml-l mp-l ( ) p
C = C 2 . 2 f t, N WALH(w t )
w wlw2 .wp N t1=0 tp=0 i=1 i’ i
(3.19)
m +m -m
N _ 2 l 2 p

Let us specialize to the case where each parameter

“O

has only two distinct values. In this case each Walsh
sequency expansion of a parameter will consist of only
two sequencies. Since there are only two distinct points
in each parameter dimension only two samples will be taken
from each dimension, i;g;, m1 = 2. In this case we will
need only one binary digit to represent a particular
parameter dimension (Kunz 1979).

In the multidimensional parameter space, the function
is then defined only on the binary hypercube. Each sepa-
rate walsh series requires only a one digit representation.

This one digit Walsh function is written

1

kiti Z kiti k t

= {-1)i=1 = (-1) l 1

MT)

WALH(k,t) = (4)1:l
(3.20)

When the coefficient equation is rewritten with 1-

digit Walsh functions the result is

p w t
n {-1) i
=1

l l

l
2 2 000 X i
p:

l
w

C =—
— 2p t

l=0 t2 = O t O i

(3.21)

By applying lexicographic ordering (Kunz 1979) to

W1 and ti we may define W and T as

L11

20 (3.22)

Therefore, T and W are p-digit binary numbers which range
from O to 2p—1 as ti and W1 take on their allowed values
of O and 1.

Upon substitution of W and T the coefficient equation

becomes

p
Z w t

i i
f(T><-1>1=1

3

2p (3.23)

2H4
2
u

N-l
C =
W T O

Z

Note that the sum over T encounters all of the two-term
sums in Equation 3.21.
We may now associate the finite multidimensional ex-

pansion with a finite single dimensional expansion.

N-l
f(T) = 2 C WALH(W,T) (3.2M)

w=o w

From Equation 3.29 we may derive the required

M2

relationship for varying the parameters. Each parameter
is associated with a separate dimension. Each dimension
may be expanded in a Walsh sequency expansion as in Equa-
tion 3.18. Since these sequencies belong to different
parameter dimensions we have the requirement that the
binary sum of the sequencies be unique for any combina-

tion of sequencies.

1WALHCw1,t) WALd(wl+w +...+wp,t +t2+t3+...+tp)

2 l

WALH(W,T) (3.25)

This means that the binary sum of the w2's must never add
to the same W-value for different wi's. Analogous to the
Fourier method, this restriction is called "binary incom-
mensurate". Note that this procedure involves the con-
version of p one-digit Walsh functions to one p-digit
Walsh function.

Since, for exact analysis, we must sample from each
dimension so that we never repeat the search curve, we
then must assign a unique sequency to each parameter where
the sequencies are binary incommensurate. The simplest
set of such sequencies is the set of powers of two,
0,21,22,

2 etc. (3.26)

“3

In this way, for a model with 'p' parameters, parameter
ul would be associated with the least significant binary
digit in a p-digit number, u2 with the next most significant
digit, etc.

Each parameter can take on two values given by the

generating function gi(ti),

/

0 1-1
3

gi(ti) = ui = ui + AWALH(w t 21-1) (3.27)

i

where u? is the average value of ui and A determines the
range. With this generating function, when ti takes on
its values of O and 1, 111 oscillates between u? + A and
u: - A at the 21.1 sequency.

By defining the parameters of a model to be functions
of different sequency Walsh functions, as in the Fourier
method, we can expand the model output function in an
infinite walsh series. By truncating the expansion to a
finite Walsh series we incur no error. This can be i1-

1ustrated by a two dimensional example. If we set u1 =

WALHQm,t) and u2 = WALH(k,t), we may then write

f(ul,u2) = f(WALH(m,t), WALH(k,t)) (3.28)

uu

By expanding Equation 3.28 in a two dimensional walsh
series, we will obtain a finite set of terms. Using the
multiplication identity we can see that there are only

four possible terms, a O-sequency term, a k-sequency term,
a m-sequency term, and a (k + m)-sequency term. No other
terms are possible regardless of the nature of the function
f.

In the calculation of the Walsh coefficients with the
Fast Walsh Transform, we use 2p equally spaced samples.
This enables us to calculate 2p different sequency. Co-
efficients, CO’Cl""’Cn-l' Therefore, we will have, as
a subset of these coefficients, all four of the required
sequencies. Of the 2p computed coefficients only these
four will be non-zero. In summary, for any Walsh driven
function, there will be no error incurred in approximating
the function by using finite Walsh expansions.

From Equation 3.2“ we can derive the total variance

of the model output function (See Appendix H). The var-

iance is
o = E C2 (3 29)
T = 1 °

This is the same formula as in the Fourior method. In

an analogous manner, the single parameter variance is

“5

constructed by calculation of the variance with respect
to only one parameter, say the first, using the model
output function which has been averaged over all the other

parameters (See Appendix 5).

2
01 = Cw (3.30)

Nowever, in the Walsh case we get only one term! There
is no infinite series to truncate as there is in the Fourier
method. In the walsh expansion the reduced partial vari-

ance is given by

y 02 c:
l _ l
T 2 Ci
i=1

which is exact.

In a similar vein we can construct coupled partial
variances. The coupled partial variances are the Walsh
coefficients whose sequency is that of the desired single

sequencies added together by binary addition without
carry (See Appendix 6). Then we divide by the total

variance to get the reduced coupled partial variances.

 

 

M6

(3.32)

For the partial variances to be rigorously correct,
$42,, to contain no error terms, we must sample the entire
parameter space. This will only be true if we are using
a discrete model whose parameters can only take on two
values.

When Walsh Sensitivity Analysis is applied to a con-
tinuous model, an approximation is made. This a-proxima-
tion is that the influence on the output function of the
range of parameter variation in a continuous model may be
approximated by using only two values of a parameter chosen
from a continuous range of possible values. A good choice,
when comparing the Walsh method with a continuous method,
would be the extremes of the interval over which the
parameter is varied in the continuous analysis.

To illustrate this we will examine a one-parameter
model. First we write down the one dimensional Walsh
expansion in terms of the parameter, ul, where u1 =

WALH(l,t).

l t
f(ul) = 2 CW <-1)w (3.33)

47

From this expansion we can calculate two terms. The
CO term and the Cl term. The CO term is the sum of all

the sampled function values

c0 = § (f(ul=l) + f(u1='1>) (3.3u)

This may be interpreted as the average value of the func-
tion at f(ul=0), where we note that when t=O, f(WALH(1,0))
= f(l) and when t = 1, f(WALH(l,l)) = f(-l).

To calculate Cl we subtract the two function values:

1
cl = 5 (f(l) - r(-1)) (3.35)

If we were to do a Walsh sensitivity analysis on a
one parameter model, the minimal sequency with which to

vary that parameter is wl = 1. So the C coefficient would

1
be proportional to the "sensitivity" of the model to its
parameter. In linear analysis, if we used a central dif-
ference formula for the first derivative of the function
with respect to a parameter, we would get the same equa-

tion for the sensitivity of the parameter.

With a two dimensional model the possible Walsh

AB

coefficients are 00’ C1, C2, C3. We can write out the
walsh transform, from Equation 3.23, for the Walsh co-

efficients as

= —2' (3.36)

 

 

 

 

 

 

The sampled parameter sets can be plotted in the two-
dimensional parameter space (Figure 2). This identifies

the sampled function values in the parameter space.

 

ulf
p/‘o ° 0
. 2 xfo
u2->
“6 0
f3 ‘f1

 

Figure 2. Plot of the sampling poinfisin the multidimen-
sional u—space.

“9

The equation for C may be rewritten as

l

f —f f -f
- 1 l 3

 

 

which is the average of the two central difference approxi-
mations to the first derivative with respect to the first
parameter (See Figure 2).

Similarly, for C2

 

 

+ 2 ) (3.38)

which is the average of the two central difference approxi-
mations of the first derivative with respect to the second

parameter.

Finally, the C3 coefficient is exactly as expected,

a central difference approximation to A2f/AulAu2.

c3 = % (rO-rl—r2+r3) . (3.39)

If we examine the general expression for the Walsh

coefficient of the kth parameter, wk-l, we note that for

50

an P-dimensional system the coefficient is always the
central difference approximation average to the derivative
with respect to the kth dimension (without loss of gen-

erality we may set R = 1).

 

p
1_[ l l - iElwltl
c = c = ... z ... z f(t t ...t )(-l) ’ J
p
1 l l f(0t2...tp)-f(1t2...tp)
Clo .0 = ‘EII z . z { 2 }
2 t2=0 tp=o
... <é£> (3.u0)
Aui

Similarly for all coupled Walsh coefficients (21"1 +

Zz-l-sequency) we get the approximation to the average

of the mixed derivative (See Abramowitz and Stegun pp—

889).

 

C 1 l l ( ){ )t1+t2}
=‘—— z ... 2 .f t ...t (-1
110...O 2p t =0 t =0 1 p
l p
= 31 g % {f(OOtl..tD)-f(01t3..tp)-f(10t3..tp)+f(llt3..tp)}
-7;:§ _ .. =
2 t3—O tp o u
= _—.é2_f_> (3.}41)

51

This implies that a walsh function expansion to a
continuous function is the first 2p terms of a Taylor

series using average derivatives.

 

f(3) = f(’_c_)
p t p p 2 t +t
= <f> + Z <ޣ>(-1) i + Z z <AuA Af (-1) 1
i=1 1 i=1 J i 1 u°
p p p 3 t +t.+t
+ z z 2 Au fiquu (-1) i J k + ... (3.u2)
1 < j < k i J k

with the restriction that each term be at most of degree

one in any parameter.

IV. EXAMPLES IN SENSITIVITY ANALYSIS

This chapter will consider the application of the dif-
ferent types of sensitivity analysis to some simple models.
To understand these models does not require rigorous
sensitivity analysis. In fact, many of the results are
intuitively obvious. However, the application of the dif-
ferent sensitivity analysis techniques to these models will
help to distinguish the domains of applicability of these
techniques. The analyses will also assist in the inter-
pretation of the results of sensitivity analyses of more
complex models.

The simplest mathematical basis for sensitivity
analysis (Beck 1977) is a Taylor series of the model
output function in terms of the parameters of the function.

0

Exapnding the output function around a nominal value, g ,

we can write the Taylor series as

f(gow) = f(go) + f'QgOM + 31,— f"(§0)A2 +

(A.1)

'Classical' sensitivity analysis is concerned with

the first derivative term in this expansion, usually

52

53

multiplied by a scaling factor to remove the dimension of

the parameter. The sensitivity coefficient of k

l,
Xk , is written,
1
Xk = k? (La?) o (14.2)
1 1 sac.

With such first derivatives, classical sensitivity
analysis attempts to explain the effects on the model
function from changing the parameters' values. In order
for this to work, the higher order terms must be small with

respect to the first derivative term, which can be

0

i' In this case all higher

guaranteed if (ki - kg) << k
order terms are multiplied by a number close to zero.

The requirement that higher order terms be small
restricts the domain of classical sensitivity analysis to
regions localized around the nominal value, k0. However,
if the function is linear in the parameters all higher
derivatives in the expansion are zero. This special case
has been developed into a widely-used practical method
(Beck 1972). From this viewpoint of ranges of parameter
variations, the different sensitivity analysis techniques
may be segregated. This can be seen by examining models

where different ranges of parameter variation are used.

The first model is a straight line with two parameters,

5A

the slope, m, and the intercept, b.

y

mt + b (11.3)

This model has a Taylor series expansion

y = f(m,t,b)

f(mO,t,bO) + t(m-m0) + b - bO

mt + b (A.A)

As expected, the expansion is exact for the linear problem.

The scaled sensitivity coefficients are

Xm = tmO; Xb = b0 (A.S)

A sensitivity coefficient is large when a change in
the parameter changes the value of the output function to
a large degree. In this case the value of the output
function depends critically on the value of the parameter.

In the linear model Xm is large when t is large. Hence,

55

for accurate 'm' estimation, measurements should be taken
at large values of t where the output function is most
'sensitive' to the value of ’m'. The b-sensitivity co-
efficient shows that measurements at t = 0 or small t
values, where sensitivity to m is small, will allow an
accurate estimation of b. These coefficients are plotted
in Figure A.1. This confirms previous knowledge (Acton .
1966).

Applying walsh sensitivity analysis to the linear
model results in the same sensitivity coefficients as
the Taylor series approach. This will happen since the
average finite-difference expansion calculates exact
derivatives in the linear case (Lanczos 1955).

A Walsh sensitivity analysis of the linear model
requires a set of nominal values, and a range of variation
for the parameters. The nominal values m = 0.0, b = 0.0
along with a range of variation of :10 for each parameter
were chosen. In order to vary two parameters over this
range, four simulations were required (N = 2p = 22).

Figure 4.2 is a plot of the average value of y, averaged
over the four simulations. This plot shows 'typical'
values of the output function over the selected parameter
space. Note that for this simple case the average and
nominal values are the same. Figure u.3 shows the
'standard deviation of the four simulations (square root

of the total variance) from the average value. If the

 

 

 

 

 

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59

standard deviation is small then the range of parameter
space examined has little effect on the output function.
Figure A.A is a plot of the Walsh expansion coefficients.
These coefficients are related to the linear sensitivity
coefficients shown in Figure “.1. A Walsh expansion co-
efficient.may be thought of as an averaged derivative of
the model output function as shown in Chapter Three. How-
ever, the equation depends on the particular transforma-
tion function used in the analysis. In Chapter Three the

2-1

transformation function u2 = WALH(2 ,Tg) was used. In

this case, by using the chain rule, it can be shown that
du = dt. Similarly, the conversion of “2 to t2 must be
accounted for when a different transformation function is

used.

In the linear model the transformation function used

0

was the arithematic transformation function uz = uz +

AWALH(22‘l,t£). Applying the chain rule to the equation for

a Walsh expansion coefficient, 3.A0, results in

BfCuz) 31.1
C = <-—————-. t

2
_ > (4.6)
2% l auz 3 2

 

which in the case of the arithematic Walsh transformation

function yields

60

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0.0P 0.0 0.0 0.ml 0.0—..1
OOPII _ _ _ _ _ _ . OOPII

 

o
m
I
1
l
O
”‘1’

I

 

l
O
swapgyeog uogsuodxg

 

 

 

 

a a — q 00 —.
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61

8 0 2—1
5?; {u£ = u2 + AWALH(2 ,t£)}

Bug _

§E_ - A (4.7)
2

therefore
BfCul)

(322-1 " A<-—3-E;—> (”.8)

From this equation the relationship to the linear sensi-

tivity coefficient is easily shown.

0 ”(‘12) “1?
x11: u, (151—) = (T) 021-1 (11.9)

Therefore the Walsh expansion coefficients, using an
arithmetic transformation function, are equal to linear
sensitivity coefficients scaled by a scale factor which is
the nominal value, ug, divided by the range of parameter
variation, A. In the linear model the scale factors for

the parameters are (mo/Am) and bO/Ab) which are l and 0,

62

respectively.

Alternatively the reduced partial variances may be
plotted, (henceforth the reduced partial variances will
be called partial variances with the assumption that they
are divided by the total variance). These partial variances
are shown in Figure 9.5. Using the notation developed in
Chapter Two, the partial variance of the first parameter,
m, is written 81' Similarly the partial variance of the
second parameter, b, is written 82. The point at which
the curves intersect, here at t = i1, depends on the range
of parameter variation and the nominal values chosen for
the Walsh Sensitivity analysis. Here both parameters were
varied over [-10.,10.].

In the linear model the interpretation is simple.

To estimate the b-parameter, measurements should be taken
near t = 0 where 82 is largest, near t = 0. Measurements
far from t = 0 are highly sensitive to the value of m as
shown by 81' These measurements taken from this region
would be best for accurate estimation of m.

Treating the linear model with the Fourier method
gives similar results. The nominal values and parameter
variations were the same as in the walsh analysis. How-
ever, more simulations were required to estimate the Fourier
coefficients. The frequency set used was the 6th-order
accurate set, [3,5], which requires, at a minimum, 11

simulations (21 simulations were used). The expansion

63

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coefficients versus t are plotted in Figure A.6. These
expansion coefficients are proportional to the first
derivatives in the Taylor series.

The partial variances shown in Figure “.7 are nearly
identical to those derived from walsh analysis, Figure
“.5. The minor differences are a result of the approxi-
mate nature of the Fourier method, since a Fourier expansion
of an angled line requires an infinite number of terms.
However, the partial variances shown capture more than 99%
of the total variance.

The standard deviation curve given in Figure 4.8 also
shows nearly the same range of variation in the output
function as was examined by the Walsh method. However,
the standard deviation curve weights simulations far from
the average simulation more than those close to the average,
causing the Fourier standard deviation curve to be smaller
in magnitude than its Walsh counterpart. This happens
as the parameter vectors are chosen throughout the param-
eter variation interval in the Fourier analysis while the
parameter vectors are chosen at the extremes in the Walsh
method.

Now let us examine a simple nonlinear model. Here
nonlinear means that the Taylor series expansion of the
output function with respect to the parameters is composed
of terms containing second or higher derivatives of the

output function. Perhaps the most commonly used nonlinear

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model in chemistry is the single exponential.

f = k2e 1 (1.10)

This function can be expanded in its Taylor series as

o o
_ O O kit 0 0 klt O
f(kl,k2,t) - f(kl,k2,t) + e (k2-k2) + k2te (kl-kl)
k0 t o o
l _ -
+ k0t2ek%.t(k -k0)2 + (A ll)
2 l l 0 O O 0

Although the expansion continues for an infinite num-
ber of terms, a linear sensitivity analysis would only

examine the first derivative terms.

gteklt (14.12)

69

These sensitivity coefficients are plotted in Figure
A.9 using k = —O.25 seconds and k = 1000.0 as the nominal
values.

From this sensitivity analysis we can say that the

best measurements for k2 are at small t, and the best

measurements for kl are at t A seconds, the maximum of
the curve. However, if higher order terms are considered
note that they may be large and could affect the value of
the output function.

To use Walsh sensitivity analysis on this model a
range of parameter variation is needed. First, let us
examine 'local behavior'; behavior of the model when the
parameters are varied only slightly. In this case, for
small variations in the parameters, the walsh coefficients
should be equivalent to the results of classical sensi-
tivity analysis. But for large variations in the value of
the parameters the Walsh method will give different results
as the higher derivatives become significant.

Again a parameter set must be chosen. Figure 4.10
shows the plot of the averaged value for the four simula-
tions of the exponential model with k2 = 1000 t 100 and
k1 = -0.25 i 0.025 seconds, 202;: 10% variation. Since
there are two parameters in this model, the curve in
Figure “.10 is the average of four different simulations

where each simulation has a unique combination of parameters.

The expansion coefficients are plotted in Figure A.11.

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73

The C1 coefficient is that Walsh expansion coefficient

which, in the case of a continuous model, is an averaged

 

finite-difference measure of the first derivative of the
output function with respect to the first parameter, kl.
Similarly, the C2 coefficient is a finite-difference ap-
proximation to the first derivative of the output function
with respect to the second parameter, k2. Comparing
Figure “.11 with Figure “.9 we see that they are identi-
cal curves to within a constant scaling factor.

To display the sensitivities of the parameters it is
mOre convenient to examine the partial variances shown in
Figure “.12. As before, these plots also show the most
sensitivity to the second parameter at short times, and
to the first parameter at long times. This figure also
shows that there is very little coupling in the sensi-
tivity between the two parameters. This can be observed
by noting that the sum of the two partial variances (Sl
+ S2) is nearly 1.0. This means that almost all of the
variance in the output function is assigned to S1 or 82.

The standard deviation curve for this analysis is
given in Figure “.13. This curve, which looks like an
exponential decay, reflects the decay of the output func-
tion. It is tempting to claim that since the standard
deviation is only 5% of its maximum at 20 seconds that
statements about the sensitivity of parameters at these

long times are meaningless. However, upon examining

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76

the dimensionless plot in Figure “.1“ we see that the

relative deviation, which is the standard deviation

 

divided by the average value, actually increases in time.
This means that as the magnitude of the output function
decreases the relative variation grows. Therefore, the
sensitivities of the parameters at long times can be
significant if the relative deviation, rather than the
absolute deviation, is nearly constant.

One advantage that the Walsh method has over linear
analysis is that it explicitly uses a range of variation
for the parameters. If this range of variation is increased
(from 10% to 60%) and the mathematical model reanalyzed
it can be seen from Figure “.15 that slightly different
behavior results. In Figure “.15 the average value does
not decay away as fast as the earlier analysis. Figure
“.16 shows that the coefficient curves have shifted the
maximum sensitivity of the decay constant, kl’ to longer
times, 5 seconds, reflecting the effect of the nonlinear
behavior of the model. The sensitivity of the pre-exponen—
tial parameter, k2, also decays away more slowly. The
partial variances in Figure “.17 also show the effect of
a larger range of variation by shifting the crossover
point from “ seconds to 6 seconds. Note that the nonlinear
effect of the model is to delay the sensitivity to k1
into longer times. However, the standard deviation curve,

Figure “.18, and the relative deviation curve, Figure “.19,

77

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83

have the same behavior as for the corresponding cases of
a small range of the parameters although the magnitudes
have changed.

Choosing an even larger range of variation for the

parameters of the model (k2 = 1000 i 1000, k = 00,25 i

l
0.25 (seconds)7¥) results in the curves in Figures “.20-
“.2“. In this analysis the average value does not even‘
decay away to zero! This is not typical behavior as

shown in the previous two analyses. This behavior is
caused by the particular sets of rate constants used in
this analysis. At 2.5 seconds two simulations have reached
their final values, and at 12 seconds the other two simu-
lations have reached completion. This is the danger en-
countered when the analysis uses only the extremes of the
parameter variation intervals. If the intervals are large
enough the behavior of the model at the extremes of the
parameter intervals may be completely different than its
behavior closer to the nominal value.

With the Walsh method we can examine the onset of
nonlinear behavior by expanding the range of analysis from
the nominal value. This is important, especially for
models which are numerically solved so that the degree
of nonlinearity in the model solution is unknown.

When the analysis was repeated using the Fourier
method with the same set of nominal parameters and over
the same small range of variation, k = -0.25 t 0.025

(seconds)-l, k2 = 1000 t 100, the same results as were

— __ __

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as with linear and Walsh techniques (Figure H.25L The
expansion coefficients, Figure H.26, have different mag-
nitudes but the behavior is the same. The maximum of
the C coefficient occurs at the same time point for both
the Walsh and Fourier methods.

The Fourier partial variances, Figure “.27, are also
identical to those of the small variation Walsh analysis.
There is one slight difference in the two partial vari-
ances at very early times. This is caused by the slightly
different parameter ranges used. The Fourier method used
a log-uniform transformation function which varied the
parameters over [-0.221, -O.227] and [1395, 905]. How-
ever, within one second, the Walsh partial variances and
the Fourier partial variances, both normalized by their
respective total variances which are different (compare
Figure 4.13 with Figure H.28), reach identical values.
Consequently the Fourier partial variances have the same
interpretation as did the previous Walsh partial variances.

For comparison purposes, the relative deviation curve
for the small variation Fourier analysis is plotted in
Figure h.29. Note that it is always smaller than the cor-
responding Walsh curve, Figure 3.1“. This reflects both
the slightly restricted range of parameter variation and
the increased number of simulations used in the Fourier

method.

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Figures u.3o—u.3u give the results of Fourier analysis
of the exponential model when a large range of variation of
the parameters is used, (k1 = -O.25 i .25 (seconds)-l,
k2 = 1000 t 1000). This analysis reveals the nonlinear
aspects of the model. Figure “.30 shows a slower averaged
decay of the output function than for the small range case.
The expansion coefficients for the case of large variations,
Figure 4.31, have nearly the same behavior as those ob-
tained with the small variations (Figure u.26). However,
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times, similar to the behavior of the Walsh Cl coefficient
shown in Figure “.16, although the shift is not as great.
The large variation C2 coefficient doesn't decay away as
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H.32. The partial variance of kl, 81’ reaches a maximum
at about 13 seconds and then slowly decays away. The
maximum is important since it selects a time region which
is optimal for the measurement of that parameter. Also
over this larger range of parameter space there is sig-
nificant coupling between the sensitivities of the two
parameters. This is shown in Figure H.32 by the coupled
partial variance 81,2.

Coupled partial variances indicate the degree of linear
dependence between pairs of parameters. When a coupled

partial variance is large it is difficult to separate the

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101

effect of one parameter from that of the other. Note
that because the partial variances and coupled partial
variances are relative measures of sensitivity, they must
be used in conjunction with the total variance to provide
an understanding of the sensitivity. In particular,
if the total variance is very small, there is little point
in carefully examining its components since they are Just
a partitioning of this very small total variance into the
individual contributions.
In examining the exponential model it can be seen
that the walsh method is equivalent to the linear analysis
for small variations in the parameters. Its advantage
over linear analysis is that as the range of parameter
variation increases it also picks up the nonlinear effects
in the model. The Fourier method is also similar to linear
analysis, in the limit of small variations of the param-
eters. However, for large variations in the parameters,
since it samples the whole of parameter space, it gives
correct results while the walsh and linear methods fail.
The Fourier method requires more simulations to achieve
its results than does the walsh method, for models with a
small number of parameters; i;g;, fewer than seven. The
number of simulations required in Fourier analysis is
heavily dependent on the order of accuracy of the fre-
quency set. For a 6th—order accurate set for 10 parameters

the Fourier method requires, at a minimum, 28M3 simulations,

102

whereas a Nth-order set requires only #11 simulations.
It should be noted that the walsh method is exact for
discrete models and for 10 parameters requires only 102”
simulations.

By applying the Fourier method to the kinetics of
simple chemical reactions we can further improve our
understanding of the interpretation of partial variances.
One of the simplest reaction schemes in chemical kinetics
is the unimolecular first-order decay of species A to

species B.

A + B (11.13)

However, the mathematical model for this reaction is the
exponential model which we have already examined. A
slightly more complicated model reaction has two coupled
first-order reactions, which may be written

k k
A -} B -3 c (n.1u)

Choosing k = 0.1 i 0.01 (seconds).1 and k = 0.01 i

l 2
0.001 (seconds)-1 with A = 10000, B = c = 0 we are able
k
to simulate a reaction with a ’bottleneck' step, (B 4? C),
since kl >> k2.
Figure “.35 displays the averaged concentrations for

this reaction. Application of linear sensitivity analysis

103

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to this model to examine the sensitivities of the B-
concentration would result in sensitivity coefficients
similar to the expansion coefficients shown in Figure “.36.
Since the curves are not normalized they are difficult to
interpret. One might be tempted to say that since Cl

is at a maximum at 20 seconds measurements in this time
region are Optimal for the determination of kl. Similarly
C2 has its most effect on the concentration of B at 90
seconds. Therefore measurements of B near 90 seconds would
pin down the k2 rate constant.

If we examine the partial variances for the B-concentra-
tion we clearly get different results. From Figure “.37
we see that measurements of the B—concentration before 10
seconds have elapsed and after “5 seconds will give ac-
curate estimates of kl and k2 respectively.

The sensitivity of the C-concentration in linear
analysis gives curves shown in Figure “.38. Here we see
that, since the k2 step is a bottleneck, we will have a
difficult time measuring kl because the effect on C from
k2 is so large. Only at short times are the sensitivities
of k1 significant.

Figure “.39 is even more revealing. This plot of the
partial variances of the C-concentration clearly demon-
strates that k2 is the most important parameter in the
model. It also shows that kl contributes to the C-concen—

tration only at short times. Therefore to estimate kl

105

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109

from measurements of C we should take the measurements
at very short times. To estimate k2, measurements after
50 seconds are adequate.

Figure “.40 and “.41 show the standard deviation and
relative deviation curves, respectively, of the C-concen—
tration. From the relative deviation curve we see that we
are varying the C-concentration only slightly. Therefore,
the Fourier expansion coefficients are equivalent to the
linear sensitivity coefficients.

The basic difference in the ease of interpretation
of the partial variances over the expansion coefficients
is that the partial variances explicitly account for the
range of parameter variation by being normalized by the
total variance. The linear sensitivity coefficients or
their equivalent, the expansion coefficients, do not ac-
count for the amount of variability introduced by varying
the parameters. This is a great weakness in linear analysis.

A common feature in chemical kinetics models is the
occurrence of a competing reaction. This is a reaction in
which two steps compete for the same reactant. Which
step predominates is dependent on the rate constants for
the two steps. A simple scheme for this problem is

written

k
B +3 D (14.15)

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112

Here the final products are a mixture of C and D. The
ratio of C to D depends on the two rate constants k2

and k3. Since this is a reaction problem often found in
chemical kinetics models a sensitivity analysis on this
scheme was done in order to determine the nature of the
partial variances.

Fourier Sensitivity Analysis was applied to this
scheme with k1 = 0.1 i 0.01 seconds, k2 = 0.01 t 0.001,
seconds, and k3 = 0.003 1 0.00003 seconds with a log-uni-
form transformation function to vary the rate constants
uniformly in log-space. A 6th-order frequency set was
used, [9, 15, 19], with 37 simulations.

The averaged concentrations of the four chemical
species are shown in Figure M.U2. From this figure it
can be seen that the time range chosen covers virtually
all of the reaction. Since the concentration of species
B both grows and decays it is the most active. Inspection
of this model shows that k2 and k3 cannot be separated by
measuring only the concentration of B. In fact, only the
sum k2 + k3 could be determined. As expected, Figure M.U3
shows that the sensitivity of B to the rate constants is
very similar to that of the coupled first-order model,
Figure “.37, since B does not "know" which path it will
take and both paths are treated as a single sink. The
only difference between this model and the coupled reaction

model, with respect to B, is that 10% of the sensitivity

113

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115

to the 'sink' parameter k2 is given to k3. This small

sensitivity to k which is Just the ratio of the nominal

3
rate constants means that any measurements of B over the
whole time range even in conjunction with measurements of
C or D would be of little help in estimating k3 Since the
sensitivity of B to k2 is so large.

The partial variances of the C concentration, Figure
a.uu, are as expected for a competing reaction. Here k2
is the most important rate constant for C. This is, of
course, expected as k2 controls the only path for the
production of the C concentration. Note however that k3
'accumulates' sensitivity over the time course of the
reaction. This is interpreted as showing how k3 controls,
to a lesser extent than k2, the amount of C produced by
the end of the reaction. Hence to estimate k3 in this
model measurements of C near the end of the reaction are
required.

The partial variances of the D concentration, Figure
a.us, are subject to similar interpretations. During the
first 50 seconds of the reaction, the formation and initial
decay of the B concentration, the partial variances of D
exhibit the same structures as those of the coupled scheme.
Here k3 controls the concentration of D, and the sensi-
tivity to kl quickly decays away as B is created faster
than destroyed. In this case the sensitivity to the rate

constant from the competing reaction, k2, accumulates

116

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faster and to a larger degree than k3 did in the partial
variance of C. This is because k2 controls a more rapid
step which removes B from the pool available to k3, thus
preventing the conversion of a larger amount of B into D.
It is then the nominal value of k2, which is greater than
the nominal value of k3, which causes the sensitivity of k2
to accumulate faster over the same time range. Note that
this model is simple enough to permit conclusions of this
type to be made by inspection. However, the verification
of these conclusions by sensitivity analysis lends confi-
dence to the treatment of more complex models.

In enzyme kinetics the most commonly used model is

the Michaelis-Menten model. This model may be written

k} k3

E + S < >ES -* E + P (H.16)

k2

Often one starts with an excess of Substrate, S, to enzyme,
E, in order to make a steady-state assumption on the con-

centration of ES. This results in a simplified rate equa-
tion for the change in substrate in time, often called the

'velocity' of the reaction (Fersht 1977).

 

S(t=0) = 30 (A.1?)

119

Where Vmax = k3EO and Km = (k2 + k3)/kl. This equation
may be integrated to obtain progress curves of substrate
versus time. If substrate is measured as a function of
time either the integrated equation or the expression for
the velocity could be used to determine Km and Vmax'

To determine which equation, the integrated form or
the differential form, would be better suited for esti-

mating Km and Vfia a Fourier sensitivity analysis was per-

x
formed. Figure U.U6 shows the averaged values of substrate
for this analysis with Km = 11000 t 110 and Vmax = 50 t 5
and SO = 11000 i 110. The relative deviation curve given
in Figure u.u7 shows that the substrate was only varied
over a small range by using these parameters.

Figure a.ua shows the partial variances of the sub-

strate. The first parameter, V is the most important

max’
parameter over this entire time range with the second
parameter, Km, being much less important and the third
parameter, SO, even less important.

For the velocity equation we get different results
as shown in Figure M.U9. Here at long times the Km
parameter is twice as important as it was in the inte-
grated equation. Therefore to measure Km we should take
velocity data at long times and fit to the velocity equa-

tion. If estimation of Vha is our only concern then

X

use of the integrated equation with measurements during

the initial phase of the reaction is sufficient. Note

120

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also that the use of the Michaelis-Menten model to estimate
the initial concentration of substrate, SO, requires that

the other two parameters, Km and Vm be previously known

ax’
since the model is more sensitive to these parameters.
Repeating the analysis over a larger parameter range
(80% of the nominal value) resulted in essentially the
same observations, Figures h.50-H.52. S0 is still the
least sensitivity parameter as shown by the partial vari-
ances of the substrate, Figure “.51. The velocity equa-
tion appears to be the more sensitive formulation of the
Michaelis-Menten model. This is dramatically shown by the
partial variances in Figure H.52. Vmax is the most im-
portant parameter in this section of parameter space.
Even its couplings with Km and Vmax are more important than

Km or Vmax in the region where the reaction has gone to

u0% completion.

125

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V. SENSITIVITY ANALYSIS OF SIMPLE
ENZYME KINETICS MODELS

Many enzyme kinetics models are composed of inter-
locked Michaelis—Menten models (Segel 1975).' These models
are proposed to explain kinetic behavior patterns which
the single Michaelis-Menten model alone is unable to do
(Segel 1975). This behavior is even defined as "non-
Michaelis-Menten" (Whitehead 1970). It was our desire to
examine models which presumably exhibited non-Michaelis-
Menten behavior patterns and were composed of linked
Michaelis-Menten models. In this chapter we investigate
four enzyme kinetics models, the irreversible Michaelis-
Menten model, the reversible Michaelis-Menten model
(Michaelis gt_al.1913), the Ho-Frieden model (Ho 1976,
Bates & Frieden 1973), and the Ainslie, Shill, and Neet
model (Ainslie gt a; 1972). Of course, to fully understand
the linked Michaelis-Menten models we must first examine

the Michaelis-Menten model itself.

Michaelis—Menten Model

The simplest model used in enzyme kinetics is the ir-

reversible Michaelis-Menten model:

128

129

kl k
3
E+S=‘-‘-‘-‘ES—-—)E+P. (5.1)
k
2

In order to illustrate the utility of the partial variances
to evaluate the sensitivities of concentrations to the

rate constants, we applied the Fourier sensitivity analysis
method, FSAM, to this simple model. Although the range

of rate constants used and the substrate and enzyme concen-
trations selected insure that steady-state conditions are
established very rapidly, the model was solved numeric-
ally without including any steady-state or equilibrium
assumptions. Of course, in this situation, if one could
observe only the substrate or product concentrations, it
would be possible to determine only k3 and (k2 + k3)/kl
since the steady-state assumption yields [S] and [P]

in terms of these two "constants". We use here the time-
development of (E), (8), (ES), and (P) in terms of k1,

k2, and k3 in order to illustrate the method and permit
comparison with more complex models.

Examination of the range of values of kl, k2, and k3
tabulated (Fersht, 1977) for a variety of enzyme reactions
which follow Michaelis-Menten kinetics shows that most
lie within an interval of four orders of magnitude
centered on the nominal values listed in Table 1. Because

FSAM was designed to apply to situations with arbitrarily

130

Table 5.1. Parameter Values for the Irreversible
Michaelis-Menten Model.

 

 

 

 

(0)
k1 Aki
o o
k +k
kl 1.0 (uM sec)-1 10:2 nominal K&0)= 2 3==llOOO uM
ko
1
k2 10” sec‘1 10i2
k 103 sec‘1 10.452 1.1 M : KVI < 1.1 *103 ”M
3 1 -
s0 = 11,000 uM
(Assay Conditions)
E0 = 0.05 uM

 

 

Table 5.2. Parameter Values for the Reversible Michaelis-
Menten Model.

 

 

 

 

0
k1 Aki
kl 1.0 (uM sec).l 10:2 0
k k0 7
k 1014 sec-1 10:2 10‘9 < KO: 1 3= o 1 < 10
2 - T ko ko —
k3 103 sec"1 10:2 2 A
k4 1.0 (uM sec)"1 10:2 S0 = 11’000 “M

m
o
u
C)
C)
U1
t
3

 

 

131

large ranges in the rate constants, it was possible to
explore the sensitivities of the concentrations to the three
rate constants while each was allowed to vary independently
by up to four orders of magnitude. The rate constant rang
ranges and initial conditions given in Table l were used

in these simulations. The initial concentrations correspond
to "assay conditions" (SO >>130). It is important to note

that the equilibrium constant K = kl/k2 was not held

1
constant when the rate constants were varied. This per-
mitted exploration of the overall sensitivity of the

model to a range of maximum velocities which covered four
orders of magnitude and a range of Michaelis constants
which spanned eight orders of magnitude. It would also

be possible to test a more restricted model by fixing the
equilibrium constant as is done later for more complex
models.

In Figures 5.1a and 5.1b we have plotted the average
concentrations, which are the concentrations summed over
all the different rate constant sets divided by the number
of simulations, and the standard deviations [square root
of the total variance defined in Equation (2.21) for the
irreversible Michaelis-Menten Modell All these curves
are scaled to the percent of the total enzyme concentra-
tion E0 for enzymatic species and to the percent of the

initial substrate concentration, 8 for the product and

O,
substrate. Two concentrations of the four are linearly

132

Figure 5.1. Average concentrations and standard devia-
tions of the concentrations Michaelis—Menten
Models. The symbols represent: 0, S; 0, P;
A, E; +, ES.

 

 

133

 

 

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related by the mass balance equations. Hence only two
standard deviations, those of P and ES, are shown in
Figure 5.lb.

Figure 5.1a shows that on the average the concentra-

 

tion of substrate is decreased by only 57% in 300 seconds.
However, the rapid growth of the standard deviation curve
for substrate, shown in Figure 5.1b, indicates that a large
spread of calculated concentrations would be rapidly
attained for this range of rate constants. In fact, the
wide range of concentrations of substrate becomes so pro—
nounced that a substantial number of simulations go to
completion after 20 seconds and another group after 110
seconds. This returns the enzyme concentration to its
initial value for these simulations, and results in ER?
parent breaks in the curves of the total standard devia-
tion for E and ES. The origin of this effect will be more
fully discussed in connection with the partial variances.
In Figure 5.2a we have plotted the sensitivity of
the product concentration to the uncertainties in the
three rate constants as a function of time. The values of
the partial variances 81’ 82, S3 indicate that for this
range of rate constants, the product concentration depends
most strongly on the value of k3, the rate constant for
the formation of product from the ES-complex. Next in
importance is the reverse step with rate constant k2.

Thus the value of k3 is the most important in determining

135

Figure 5.2. Partial variance plots for the Michaelis—
Menten Models. A number represents the
partial variance for that rate constant.
Coupled partial variances are represented

1,3; in (b) by

1’2; X, 82,3; in (c) by *, 81,3;

as follows: in (a) by *, S
*’Sl ;+S

3
+, 82,3.

13AR1’IAL. VAFLIABHZES

PARTIAL VARIANCES

 

136

 

 

 

 

 

MlCHAELlS-MENTEN
Irreversible Reversible
PRODUCT PRODUCT
I‘ P
3 .... .. 11:1 1 (c)
1 “ ~ ‘ ‘s
3 S S 3 1 S 'S"
‘1 U 7
1 Z
‘1 < ‘-
-: c: 1
y <
J 3 3 3.3-3 3 > j
-3w3 1
3 ya 4. j
1 3 < 4 3‘ 3 3
< H .3 3 3 3 3
1. -__ 1 1 11~1 1 1 < 4 . . 1 1 1 1 1 1
‘ ’. ‘ 0‘ :11 a
' - . .. . 1 5 -2 .2... ..E..L-,..§...',.
2 2 2 2 2 4. 2" 2-; Zu‘ B: 9“ ‘ ¢ «+-
0? 2 2. 2 r2 -2 r *fi 0‘2 1
O 3 O 3
ES- COMPLEX ES-COMPLEX
I, I}
1
(b) 3 (d)
u
2
< i
. H 1
CE 1
1 < 1
3 '3 3 > 41
1 _. 1 ,
11 a 3 “ 1
3 H . : 3
. . H 12 2 . -7lax
' 31 3? m 1 2 c - ~ 4 2
.,H‘ "* *-r-+2+:z-+2 . 4. q q ‘*'
o' *‘3 . x“ *1 0w - a
O 3 O 3

TIME (sec) 1116-2

Figure 5.2

137

the product concentration. This is not surprising since
k3 controls how fast substrate is converted into product,
while the binding step yields a steady state ES concentra-
tion essentially instantaneously on the time scale of

Figure 5.2a. As time increases, the specific k3 value

chosen has an even greater effect on the accumulated

 

product concentration so the relative importance of k3
increases with time.

Figure 5.2a also indicates that the product concentra-
tion is sensitive to the coupling between kl and k3,
especially at early times. This indicates that the ac-
curate determination of k3, for example, from the early
portion of a single progress curve would be hampered by
coupling to kl’ resulting in significantly larger mar-
ginal deviations of the rate constant than would be ob-
tained by using the entire progress curve. It must be
emphasized that the entire analysis described here applies
to full time-course behavior rather than only initial rate
behavior. If one wished to study the sensitivity of
initial rates to substrate concentration for example,

a different procedure would be used.

The analysis shows that if the concentration of ES
were measurable, it would be most sensitive to RI at short
times as indicated by Figure 5.2b. This reflects the
fact that the formation of ES is the dominant initial

step. However, the sensitivity to k3 grows rapidly as

138

substrate is depleted. There is also strong coupling
between kl and k3, implying that the errors in determin-
ing these rate constants from the time-dependence of the
concentration of ES would be strongly correlated. As
was found for the product sensitivities, ES is not very
sensitive to the value of k2 over the range examined.
Figure 5.2b shows apparent discontinuities in the
sensitivity of the ES concentration to the rate constants.
One might be tempted to attribute this to numerical errors
or instabilities in the calculation, but this is not the
case. When the ranges over which the rate constants can
vary are drastically reduced the discontinuities disappear.
The discontinuous curves (obtained with these time steps)
originate because of the large ranges available to the
three rate constants. Within a narrow time span an ap-
preciable number of simulation runs reach completion.
When this occurs the ES complex disappears and the concen-
tration of E builds up for these simulations. This com-
pletely eliminates the sensitivity of ES to the rate
constants for this subset of simulations. The result
is a series of apparent breaks in the partial variances.
It is possible to eliminate these breaks in either of two
ways. As indicated above, the ranges of the rate con-
stants can be restricted so that the number of simulations
which reach completion is insignificant. Alternatively,

the initial substrate to enzyme ratio can be made so

139

large that the simulations do not reach completion even
for the most favorable combination of rate constants.
Neither of these alternatives is entirely satisfactory
since they both impose restrictions on the model.

In order to be certain that the origin of the great-
est sensitivity of product concentration to k3 and least
sensitivity to k2 is not Just the large ranges permitted
for the rate constants, sensitivity analyses were per-
formed over several reduced ranges about the same nominal
values. The general results were the same except that,
as noted above, the apparent breaks in the sensitivity of
E8 to the rate constants disappeared.

Under the assay conditions modeled here, the concentra-
tion of the enzyme-substrate complex, ES, assumes a steady-
state value at very short times, as shown in Figure 5.1a.
However, even under steady-state conditions the concentra-
tion of the complex changes with time as substrate is
used up. This is reflected in sensitivities which also
change with time. The growing sensitivity to k3 means
that in a full time-course analysis, this rate constant
would be relatively more accurately determined by following
the progress curve for an extended period of time than
could the other rate constants or combinations of these

rate constants.

1&0

Reversible Michaelis-Menten Model

Only slightly more complex than the irreversible
Michaelis—Menten model is the reversible model in which
equilibrium is ultimately reached. In testing this model,
the same nominal rate constants and ranges were used as

for the irreversible case but a reverse step was added:

E+S==—=’ES===-E+P. (5.2)
k
u

Table 5.2 gives the nominal rate constants, their ranges,
the initial conditions, and the frequency set used. Fig-
ures 5.lc and 5.1d show the average concentrations and
standard deviations for the reversible case.

As shown in Figure 5.20, the first MO seconds gives
approximately the same product sensitivities as the ir-
reversible model. The reverse step ku only begins to
become important at later times as the concentration of
product becomes large enough to bind to the enzyme.

Initially the sum of the product partial variances
and the higher partial variance which couples kl and k3
accounts for 93% of the total variance. During approxi-
mately the first 100 seconds, this sum decays to 85%
and remains constant. The other 15% of the sensitivity

is spread over couplings among the parameters but no

1A1

individual coupling is large enough to appear on the
graph.

The major difference between the irreversible model
and the reversible model is seen in the sensitivity of
ES to the rate constants. In the irreversible case the
sensitivity to k3 grows, while in the reversible model
k1 remains most important. Apparently the reverse step
(kn) can serve to stabilize the concentration of BS as
the reaction approaches equilibrium. Since substrate is
present in excess, the concentration of the complex
continues to be dominated by sensitivity to kl. As with
the irreversible case, apparent breaks in the sensitivity
of ES to the rate constants are observed (see Figure 5.2d).
These are again caused by a subset of the simulations

which in this case reach their equilibrium values.

Models with Slow Conformational Changes

Except for a (usually undetectable) lag in product
formation caused by storage of substrate as the ES com-
plex, the Michaelis-Menten model is not capable of des-
cribing bursts or lags. Nor can it lead to allosteric
behavior since the phenomenon of allosterism as defined
(Segel, 1975; Fersht, 1977) in terms of deviations of
the reaction velocity from the predictions of the

Michaelis—Menten model. In order to examine these

192

phenomena it is necessary to devise more complex models.

The most common interpretations (Monod 33 al., 1965;
Koshland gt al., 1966) of allosteric behavior involve multi-
subunit enzymes in which interactions among the subunits
make the addition of another substrate molecule easier or
more difficult than those which were previously bound.
These models are intrinsically thermodynamic in nature
since they refer to interactions which affect binding
constants. It was suggested some time ago (Whitehead,
1970) that allosterism could arise without subunit inter-
actions as a natural consequence of kinetic models which
involved slow steps such as conformational changes. Such
models have also been proposed to describe bursts and lags
in product production.

In this section we apply sensitivity analysis to a
model first examined by Ainslie, Shill and Neet (1972)
using steady-state methods. Our sensitivity analysis
of the model showed that similar behavior can be obtained
with less complex models. Therefore, these simpler models

are also examined in some detail.

Model of Ainslie, Shill and Neet

In 1972, Ainslie gt 1. proposed an enzyme model

which they studied by using steady-state techniques
coupled to slow conformational changes. They showed

that appropriate choices of the 16 rate constants could

1M3

be made so that the model displayed either bursts or lags
in product production. They also showed that the varia-
tion of the final steady-state velocity with substrate
concentration could be made to exhibit allosterism, leading
to behavior similar to either positive or negative co-
operativity depending upon the choice of rate constants.
Because of the wide variations in behavior exhibited by.
this model, brought about merely by changing the values
of the rate constants, we felt that this model would pro-
vide an excellent test of the methods of sensitivity
analysis.

This model, which we refer to as the Ainslie model,

is described by the following scheme:

1 16 in
E + S:;=3 ES ;==3 EP;;=£E E + P
2 15 13
10++9 3++u ++
11 5 7
3* + 3 ==:: E*S :Efi E*P ==e E* + P (5.3)
12 8

The numbering sequence for the 16 rate constants is also
given in scheme (1A) above. Equilibrium constants,
K1, K3, K5... are defined as kl/kZ’ k3/ku, k5,k6, etc.

With its 16 rate constants, this model is complex

luu

enough to defy intuitive understanding of its detailed
dynamic behavior. By applying SAM to this model, with
the nominal rate constants and ranges already tested by
Ainslie, et al., we can determine the important pathways
which lead to bursts and lags. The analysis also showed
that the model need not be this complex to yield the same
general behavior. .

Ainslie, 33 al. separated the rate constants into two
sets: those which gave lags in product growth and those
which gave bursts. Each of these sets was also divided
into two groups which showed allosteric behavior similar
to positive cooperativity and negative cooperativity,
respectively. Hill plots (Hill, 1925; Segel, 1975;
Fersht, 1977) were used to classify the cooperativity.
Negative cooperativity gives Hill coefficients less than
one while positive cooperativity leads to Hill coefficients
greater than one.

In this sensitivity analysis it was only necessary
to use two groups of rate constant ranges corresponding
respectively to bursts and lags in order to cover the
entire range studied by Ainslie, gt al. In order to
decrease the complexity of the problem, simplify the
interpretation, and include the thermodynamic constraints
demanded by the presence of mechanistic loops, we main-
tained all of the equilibrium constants at fixed values

in each of the two sets studied. This corresponds

1H5

approximately to the choices made by Ainslie, e2 _1.
who used constant values for most of the equilibrium
constants_while varying the rate constants. This simpli-
fication reduces the number of independent parameters to
eight but does not alter the general behavior of the model.
The eight differential equations which describe the
time-dependence of the concentrations of the eight species
in this scheme can be reduced to six coupled non-linear
differential equations by using the two algebraic equa-
tions of mass balance. The nominal values of the rate
constants, the values of the equilibrium constants used,
and the initial conditions are given for the lag and
burst sets in Table 5.3, while the frequency sets and
computer data are given in Table 5.“. The ranges allowed
for each rate constant were 10:1 times.the nominal value.
In Figure 5.3 the average concentrations and the
standard deviations of the two sensitivity analysis runs
are displayed. In the lag set the product growth is
initially slow but it rapidly increases reaching 27% of
its equilibrium value after 120 seconds. In contrast, the
product growth of the burst set starts out fast and then
slows down, reaching only 11% of its equilibrium value
after 120 seconds. This leads to a large range of con-
centrations in the lag set, but to a restricted set in
the burst case. In both cases the less active free

enzyme, E, is a minor species.

1U6

Table 5.3. Parameter Valuesa for the Ainslie Model.

 

 

 

 

 

(Lag Set) (Burst Set)
0 (i) O (i)
1 k1 Keq ki Keq
1 10 (0143)”1 10‘2(uM)‘l 10. (uMs)‘l 0.1(uM)'1
3 10'2 s"1 3.0 10‘3 s‘1 10‘2
5 10” s‘1 3.0 105 5‘1 1.0
7 103 5'1 30.0 (UM)_1 103 3'1 100.0 (11M)-1
9 10"1 3‘1 10 10‘2 a-1 10.0
11 10.0 (uMs)‘l 0.3 (uM)‘1 10 (uMsYl 10"2 mm)“1
13 1.0 (ms?1 10‘3 uM 10 (uMs)_1 10"3 pM
15 10 s‘1 112.7 103 s"1 100
_ b _ _ _
EO - 0.5 uM sO — u000.0 0M E0 - 0.05 uM sO - u000.0uM
[P] [P]
K = 27 =-——431 K = 1.0 = eq
T [Sjeq T [Sleq
aThe range of the rate constants was 10:1 times the nominal
value, k3.

bInitial distribution: 90% E, 10% 3*.

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Figure 5.3. Average concentrations and the standard

deviations of the concentrations for the
Ainslie models. The symbols represent:

0, S;A, E: +, 39631-0, sz, ES: 11’: E*P.

1H9

Careful examination of the partial variances shown
in Figure 5.” shows that the lag mechanism operates by
shuttling the ES complex to the E*S complex which then
rapidly forms product. The top (E+S + E+P) cycle has
slow turnover relative to the bottom (E* + S + E* + P)
cycle and the important bridge between them is the iso-
merization step of the enzyme-substrate complex. The
small amount of E* present initially starts turning over
substrate so that the substrate concentration is most
sensitive to k7, the product formation rate constant in
the bottom cycle. As the reaction proceeds the total
concentration of enzyme in the bottom cycle is increased
by the conversion of ES to E*S; this increases the sensi-
tivity to k3 and to the binding step E* + S + E*S (kll).
This shift to the bottom cycle is verified by the rapid
decay of substrate sensitivity to le.

The coupled partial variances of Figure 5.Ab rein-
force the above conclusions. The rapid decay in time of
the coupled partial variance 85,7 is consistent with the
growing importance of the depletion of the substrate
concentration via the bottom cycle. Furthermore, the
isomerization step which increases the total active enzyme
concentration and thus increases the importance of the kll
step results in the rapid growth in time of the coupled
partial variance 87,11.

Turning to the sensitivity of the enzyme concentration,

Figure 5.“.

150

I

Partial variance plots for the Ainslie lag
model. A number represents the partial
variance for the rate constant. Other par-

tial variances are represented by: B, S

11’
D, 813; F, 815. Coupled partial variances
are represented as follows: in (b) by +,

. «)6 . ' o
85.7: ’ 37,11, 1“ (C) by +’ 313.15: *’
81,15; in (d) by *, 85,7; in (e) by *,

S13.15'

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152

displayed in Figure 5.Uc, we note that there is negli-
gible sensitivity to kl, since the binding step reaches
equilibrium so rapidly on the time scale of this display
that variations in kl cannot change the concentration of E.

(Recall that the equilibrium constant K is fixed.) On

1
the other hand, the relative amounts of enzyme present

as E, ES and EP strongly depend on the values of the

other rate constants k13 and le in the top cycle at

early times when the product concentration is low. The
large initial values of the partial variances 813 and 815
and the coupled partial variance 813,15 support this asser-
tion.

The sensitivity to the top cycle rate constants is
then lost to k3 as the inactive enzyme isomerizes to
E*S. Since 90% of the enzyme is initially in the inactive
form, this transfer to E*S changes the E concentration
significantly. All the other complexes also display this
feature of a rapid rise to a large sensitivity to the
isomerization rate k3.

The sensitivity plots for E*S and E*P in Figures 5.He
and 5.ur respectively, are consistent with the interpre-
tation of the other sensitivity plots. Once again the
sensitivity to the binding step (kll) for the bottom
cycle is negligible due to the rapidity of this step.

The steps with rate constants k5 and k7 are important

initially but the k3 isomerization step grows to major

153

importance. The similarity of the E*S and E*P sensitivi-
ties suggests that the inclusion of both intermediate com—
plexes may not be necessary in the formulation of a
mechanism that leads to lag behavior.

The burst mechanism operates by forming product
initially via the fast bottom cycle. The rate constants
are such that, as time progresses, enzyme is shunted from
the lower cycle to the upper cycle primarily through the
enzyme-substrate complex isomerization step. Since the
top cycle is relatively slow, the turnover of substrate
slows after the initial period, hence the burst behavior.
The bottom E* cycle remains the major route for substrate
turnover as shown by the large sensitivity to k7 in Figure

5.5a. Though the partial variance S does drop from 88%

7
to 60% of the total variance at 115 seconds while 815
grows somewhat, it is k7 that dominates the substrate
sensitivity even more than in the lag case.

In notable contrast to the lag analysis, the enzyme
sensitivity displayed in Figure 5.5b does involve kl.
However, examination of the total variance, that is the
sum of all the partial and coupled partial variances,
reveals that it is very small. Since the partial var-
iances are all defined relative to this total variance

we obtain non-negligible values for the partial variances

even though, as just noted, the total variance is

154

AINSLIE BURST

 

 

 

 

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Figure 5.5.

TIME (sec) x tO’z

Partial variance plots for the Ainslie burst
model. A number represents the partial vari-
ance for that rate constant. Other partial
variances are represented by: B, 811; D,
813; F, 51 . Coupled partial variances are
represente as follows: in (a) by *, 811 7;
in (b) by *, 311 The symbo

"s" represents the sum 0%3 the displayed
partial variances.

155

negligible. Thus we may conclude that the enzyme concen-
tration in the burst region of parameter space is not sig-
nificantly affected by the rate constants. Of course, if
the equilibrium constants were allowed to vary the results
could be greatly altered.

The sensitivity plots of the intermediates, ES, E*S,
- and E*P, shown in Figures 5.5c, 5.5d and 5.5e, respec-
tively, are dominated by the sensitivity to the isomeriza-
tion of E8 to E*S. At very short times the top and bottom
cycle rates have some sensitivity, but the total variance
is very small here.

From the above detailed analysis a simple rationale
of the operation of this model with regard to burst and
lag behavior can be formulated. The initial relative
concentrations of E and E* are determined by K9. It is
the relatively slow transformation of E*S to ES and 1132:
ngga brought about by substrate binding which gives
rise to the bursts and lags. The formation of E*S and
ES from E* and E, respectively, is rapid compared with
the rate of interconversion of these forms. Because of
the importance of the step with rate constants k3 and ku
it is not surprising that the sensitivity to k3 (fixing k3
also determines ku through the constant value of the
equilibrium constant, K3) provides an important clue to
the behavior of the model. If we wish to focus on bursts

and lags in product production, then the most informative

156

sensitivities will be those which relate product (or sub-
strate) concentration to the rate constants.

With this information we can now formulate a reduced
model which exhibits essentially the same sensitivities

as the complete Ainslie model:

E+SZESZEPIE+P
I (5.3)

E* + S 1 E*S I E* + P

 

Of course to obtain the proper very long time behavior it
would be necessary to incorporate the E 2 E* step in the
model to return E* to E. However, the E z E* step plays
no significant role in the behavior of the model in the
region of parameter space and the time range explored
here. As long as the rate constant sets are chosen such
that a pool of enzyme is bound up in the ES intermediate
which is then slowly converted to E*S, the lag behavior
will result. For burst behavior one needs more E*
present initially to cycle through the bottom than the
isomerization ES 1 E*S would yield at equilibrium. Since
the reduced model can exhibit these features, bursts and
lags will result from this model.

Furthermore, the mechanistic steps by which EP is

157

created and destroyed are of secondary importance over
this region of rate constant space. This suggests the

possibility of still further reduction of the model.

Frieden Model

A model proposed by Bates and Frieden (1970) to account
for time-lags in enzyme reactions and also studied by Ho

(1976) may be represented by the scheme:

1 7
E + S ES E + P
2 8
111112 311. 11
5 9
E* + S E*S E* + P. (5.A)
6 10

In order to permit comparison of this model with that
of Ainslie gt_al. (1972) the same rate constants were used
for equivalent steps. To evaluate k7, k8, kg, and klo,

a steady-state approximation was applied to dEEPJ/dt and
d[E*P]/dt. This related the product release rate and
equilibrium constants of the Frieden model to those of
the Ainslie model by the following equations, in which

the primed rate constants refer to the Frieden model,

158

kk kk
16 1H, 1 13
k'=—-—-—, k'=r—L—
7 k1u+k15 8 1u+k15
kk kk
k.__._§_Y_ k' 2.2.51, (5.5)

KO

These rate constants, the other nominal rate constants, the
equilibrium constants, and the initial conditions for the
burst and lag runs of the Frieden model are given in Table
5.5, while the frequency sets and computer data are given
in Table 5.4. As with the Ainslie model, the equilibrium
constants were fixed and the rate constant ranges were
10:1 times the nominal values.

Figure 5.6 shows the average concentrations and the
standard deviations of the Frieden model. Both the lag
and burst cases are similar to those of the Ainslie model.
It is interesting to note that there is more "effective"
enzyme in the Frieden model since there are fewer inter-
mediate complexes.

The substrate sensitivity shown in Figure 5.7a is
largest for k9, the rate constant for release of product
from the active form. Initially, the corresponding upper
cycle rate constant k7 for the inactive form contributes
about 20% to the rate of product formation but this

decreases to less than 5% as the isomerization step

159

Table 5.5. Parameter Values for the Frieden Model.

 

 

a
LAG Burst

i k0 k kiO)

1 eq eq

 

1 10 (um s)“1 10"2 (uM)-l 10.0 (uM s)“1 10‘1 (01"1)‘l

 

 

3 10"2 s‘1 1.0 3 x 10'“ s"1 10‘2
5 10 (uM s)“1 10‘1 uM 10 (uM s)‘1 10"2
7 30 s.1 3.1 x 103 pM 10 3'1 10.0 uM
9 750 s‘1 3.1xx 102 um 9.9 x 103 3‘1 100 um
11 10'2 s‘1 10‘1 10‘3 s‘1 10"1
_ b
ET - 0.05 uM
S0 = AOOO uM
[P] [P]
_ _ e __ _ eg
KT - 31 - mg: KT — 1.0 - [5180

+
aThe range of rate constants was 10"1 times the nominal

value k1.

bInitial distribution: 90% E, 10% E*.

160

Figure 5.6. Average concentrations and the standard
deviations of the concentrations of the
Frieden models. The symbols represent:

G). S; A, 1513+, E*S;®. PsX. E*;)1. ES-

161

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162

proceeds and the isomerization rate constant, k3, becomes
more important.

The partial variances shown in Figure 5.7b are par-
ticularly revealing. The coupling between k3 and k9
grows and decays during the time range of the isomerica-
tion of ES to E*S. As this occurs the rate constant, k5,
for the binding of substrate to active enzyme grows in
importance as does its coupling with k9. By examining
these time-developments, one can gain a rather clear pic-
ture of the lag behavior as product production shifts
from the upper((slow) cycle to the lower (fast) cycle.

These dynamic effects are also mirrored in the sensi-
tivities to the various enzyme forms. For example, the
enzyme sensitivity shown in Figures 5.7c and 5.7d shifts
with time from the E-cycle to the E* cycle. This shift
is responsible for the lag behavior. Note the rapid
growth and decay of the sensitivity of E to the coupling
between kl and k7, the slightly slower growth and decay
of its sensitivity to k3, and the slower growth in its
sensitivity to k5 and kg and to the coupling between k5
and k9. These plots show how the dependence of the
concentration of free less active enzyme on the various.
rate constants changes with time.

The major route for the formation of E*S is the iso-
merization step ES 2 E*S. This is shown in Figure 5.70

by the dominance of the sensitivity of the E*S concentration

Figure 5.7.

163

Partial variance plots for the Frieden lag
model. A number represents the partial
variance for that rate constant. Coupled
partial variances are represented as fol-
lows: in (b) by +, 85,9; in (c) by +,

85,9; *, 81,7.

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165

to the isomerization rate constant k3. The growth in
sensitivity to R3 is accompanied by a rapid decrease in
the sensitivity to k5. At longer times, some sensitivity
to k5 and k9 accumulates.

The burst set of rate constants gives a reversal of
this behavior pattern as shown in Figure 5.8. Again, the'
E* cycle initially controls the rate of product production.
For this range of rate and equilibrium constants, however,
E*S is converted to ES which is less active with the result
that the overall rate of product production is decreased.

These simulations clearly show that the simpler Frieden
scheme can give both bursts and lags. In fact, the in-
sensitivity to kll and kl2 shows that an even simpler
model without the E 2 E* step would also describe the time-
behavior of these systems, provided one started with an
equilibrium distribution ofELandggfi. This is because in
the models studied here, the direct interconversion of E
and E* is slow enough that it cannot compete with the
ES 2 E*S isomerization.

Since one of the "bonuses" of the Ainslie model was
its ability to describe allosteric behavior without the
need for cooperative subunits it was of interest to see
whether the simpler Frieden model could also give ap-
parent cooperativity. In order to do this, two simula-
tions were performed with the Frieden model which were

designed to yield "initial velocities" at various substrate

166

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167

levels under steady-state conditions. In these simula-
tions, the rate constants k8 and klO’ which yield overall
reversibility in the Frieden model, were set equal to zero.
After steady-state had been achieved, the reaction velocity
dP/dt was evaluated as a function of substrate concentra—
tion. This is equivalent to the evaluation of initial
steady-state velocities appropriate to separate assays.

It was possible in this way to find sets of rate
constants and concentrations which gave Hill coefficients
which vary from 0.125 (rate constant set 1 in Table 5.6)
to 2.645 (rate constant set 2). These results mimic the
behavior usually attributed to negative and positive co-
operativity, respectively, and show that even a model as
simple as that of Frieden can be made, with a suitable
choice of rate constants and initial conditions, to exhibit

allosteric behavior.

Summary

Model reduction is an important goal of sensitivity
analysis. By applying sensitivity analysis to complex
mechanistic schemes, one is able to determine which steps
in a reaction are essential to the behavior being examined
and which are not; perhaps permitting a simple model to
be formed as a subset of the more complex scheme.

Another major aim of sensitivity analysis is to

determine which rate constants are most likely to be

168

Table 5.6. Rate Constants for Allosteric Test.

 

 

 

Set #1 Set #2
_l _l
kl = 100 (HM s) kl = 10 (UM s)
_ -l _ -1
k2 - 1000 s k2 - 1.0 3
k3 = 10‘"I s‘l k3 = 3 x 10'” s"1
2 -l -l
k = =
5 10 3 k5 10 5
k6 = 10” s‘1 k6 = 10‘1 s'1
-1 -l
= k =
k7 10 s 7 10 5
k8 = 0 k8 = 0
k9 = 10“ s‘1 k9 = 9900 s"1
k10 = 0 klO = 0
_ -2 ' -l _ -3 -1
kll — 10 (uM s) kll _ 10 (uM s)
_ -l -l _ -2 -1
S0 = 15000 UM S0 = M000 UM
E0 = 0.05 UM E0 = 0.05 uM
time range of test
(1200 sec to 1750 sec) (85 sec to 1&5 sec)
substrate decay over this time range
(5550 uM to 3150 uM) (zuuo to 2160) ”M

Hill coefficient=0.125 Hill coefficient=2.6u5

correlation coefficient correlation coefficient
0 = .999 o = 0.997

 

 

169

measurable and which have so little effect on the concen-
tration-time curves that they cannot be accurately de-
termined. For example, by following the full time course
of product growth for a system which follows the Frieden
mechanism and exhibits a lag one might expect, if the
equilibrium constants are known, to be able to determine
k9, k7; k3 and k5. However, k3 and k9 would be strongly
coupled as would k5 and kg. The rate constant k7 would
be measurable primarily from the behavior at short times
but this time period contains essentially no information
about k3 and k5.

Our primary motivation for implementing these tech-
niques was to ultimately apply them to the study of
transients in enzyme kinetics. Under conditions where
one can follow the production and disappearance of inter-
mediates, it should be much easier to distinguish among
various mechanisms. The application of sensitivity analysis
should provide very useful information about the sensitivity
of the various concentrations to the rate constants so that
the latter can be arranged in order of their accessibility

of measurement.

VI. SENSITIVITY ANALYSIS OF A
TRYPTOPHANASE KINETIC MODEL

Recently the mechanism of TryptOphanase catalysis
has been under investigation in our laboratories. Of
particular interest is the variation of the ultra-violet-
visible absorbance spectrum of Tryptophanase with pH.
As the pH is changed the enzyme apparently changes its
conformation which results in changes in the spectral
shape (June gt a1. 1979). The model proposed (June et a1.

1980) for this is

BY BY BY’ BY
EYH+ KEH EY + H+
de ++ kY6
E5
Scheme 1

170

171

which is composed of three interconvertible manifolds
designated 8, y, and 5. During 1979-1980, two types of
incremental pH change experiments were done to probe the
absorbance changes which accompany a change in pH. Incre-
mental pH Jump experiments were done to examine the con-
version of the low pH form of the enzyme to the high pH
form. The reverse reaction was also tested with incre-
mental pH drop experiments. The results of the rapid
changes (t < 10 seconds) were analyzed in terms of a re-
duced model using only the 3 and y manifolds since the
growth or decay of form 6 is slow. This simplification,
along with the restriction that the protonation-deprotona-
tion reactions occur within the mixing time of the stOpped-
flow experiment allowed the mechanism to be reduced to an
apparent first order scheme with the apparent first order

rate constant, k', and a model output function AAO The

bs'
equations used to fit the data (in a least squares sense)
are given in TableéSLL The program KINFITA (Dye, Nicely,
1971) was used (June, Dye, Suelter 1980) to obtain the
parameters and their standard deviations, shown in Table
6.2. To clarify these results and to propose further
experiments a sensitivity analysis of the model was under-
taken.

The sensitivity analysis of this model can be separated
into two regions, the investigation of the general sensi-

tivity of the model with respect to the parameters, and

172

Table 5-1- Parameters for Tryptophanase Model

 

 

= __ v
Aobs Aco + (AA)exp( k t)

+
AA” + AAO-Ka/(H )

 

 

 

 

 

 

AA =
obs l + Ka/(H+)
(e 8t W)[K = K 1
AA” = KB 8H
[1 + —Y———][: + (H+) J
KBHKBY 8H 0
AA = (eBBt)(H+ )oEKBH/KyH — lJKBY
o +
[1 + KBYJEKBH + CH )0]
K = KBHEl + KBYJ
a 1 + KBHKBY/KYH
H + H
k; = kBY KBH + k BY<H ) + kBY + kBY K gH/(H+ )
KBHKBYEl + (H )/KYH] 1 + KBH/(H )
. H _
Thermodynamic constraints require that KBY - KBHKBY/KYH
= . H a H H
and, of course kYB kBY/KBY’ and kYB k8 YB/K BY

 

 

173

Table 6.2. Best-fit Parameters Based Upon Scheme 1 and
Their Marginal Standard Deviation Estimates.

 

 

Standard Deviation

 

 

 

Parameters i Value Value Percent
O
(SBBt)Jump 0.080 cm“1 0.001 1.u
1
o -1
(asst)drop 0.0175 cm 0.0008 u.6
KBH 2 2.0::10“lo M 0.le10‘10 20
KBY 3 39 8 20
KYH u 1.7::10"7 M 0.33:10'7 18
-1
KBY 5 8.3 sec 1.6 18
KEY 6 0.0297 sec-1 0.0045 15
Ka 7.7x10‘9 M 0.ux10‘9 5
-1
. 6
KYB 0.212 sec 0 012
HEY 0.0u5 sec‘1 0.010 20
H -1
kYB 0.66 sec 0.0“ 6

 

 

Table 6.3. Legend for Figures 6.5-6.11.

 

 

KYH =0 kBY =

 

 

17”

the investigation of the actual "fitted" parameters along
with their estimated standard deviations. This procedure
should permit one to see in which regions the measurements
might provide better estimates of the parameters. The
investigation of the general sensitivity of the model was
done by varying each parameter over a fixed relative range.
A 10% variation, iLg;,-within 10% of the nominal values,
was chosen for this technique. Since each parameter is
varied over an equal range the sensitivity of the model to
its parameters enables us to rank order them in terms

of their relative effects on the output function.

A second sensitivity analysis with the parameters
varied only over the estimated standard deviations (as
determined by KINFITH) was performed. This type of
analysis indicates which regions should be studied in
order to refine the estimates of the parameters.

Sensitivity analysis for each different experiment
was done, one for the pH drop and one for the pH jump.
Each sensitivity analysis had two output functions, k'
and AAobs' Six parameters were varied in each analysis
using 99 simulations with a fourth-order accurate Fourier
frequency set. The value of [H+]O was set equal to the
initial H+ concentration used in each experiment. For
the pH drop [H+]O = 10-8'7 M and for the pH Jump [H+]O =
10‘7'0 M.

Figure 6.1 shows that the average value of AAobs for

175

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176

the pH drop analysis grows as the pH decreases from 8.7
to 6.U. Both the 10% parameter variation and the standard

deviation variation give the same averaged nominal value
for AAobs' Similarly, Figure 6.2 shows the average value

of AAOb for the pH Jump analysis. Since the range of

3
pH covered in the pH Jump analysis is more symmetric about
the apparent pKa value of 8.1 than is the pH drop analysis,
Figure 6.}.is more like a complete titration curve than

is Figure 6.2.

Figure 6.3 displays the averaged value of k' as a

function of pH. The same averaged values were obtained

for both the pH drOp and the pH Jump analysis as well as

for both sets of parameter variations. This is, of course,

expected from the functional form of k' which contains
only rate constants, equilibrium constants and [H+]o

The partial variances of AAobs for the standard devia-

tion (std. dev.) pH Jump sensitivity analysis are shown
in Figure 6.R. This output function is only sensitive

to €882, KBH’ and KB Note that there is no large varia-

Y.
tion of sensitivity in the pH region 7.0-8.5. The sensi-

tivities only differ at the ends of the pH region.

Figure 6.5 gives the partial variances for AAobs where

the range of variation was 10%. This plot shows that the
most important parameter is 683:. Note that the sensitivity

to e B: is much lower in Figure 6.¥, where the standard

8
deviation variation was used. Since the model is so

177

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sensitive to this parameter June 33 a1. (1980) were able
to fit it to less than 5% error. Both Figure 6.9 and
Figure 6.5 show that the measurement of 888: can be more

accurately made at high pH values in pH Jump experiments.

The essentially equal sensitivities to KBH and KBY reflect

the fact that the amplitudes are most sensitive to
Ka a K8H KBY as indicated by the computed standard devia-
tion of Ka in Table 6.2.

Figure 6.6 is the standard deviation analysis for the

apparent first order rate constant k'. Here at low pH,
KBY is the most important parameter. At higher pH, say
9, kBY and K8H are the most important parameters. The

partial variances for the 10% parameter variation, Figure
6.3, are not much different than those derived from the
fitted standard deviation analysis. The overall shapes
of the sensitivity curves are the same but the magnitude
differ slightly. The maximum sensitivities of the param-

eters are grouped in two regions, KBH and kBY are large

at a pH of 9, while the other parameters reach their peak

in the pH range of 7.0-7.9. The sensitivity to ng in

the pH Jump analysis is significant only at low pH values

and neither the amplitude nor the rate constants in the

ij Jump analysis show appreciable sensitivity to KYH'

Figures 6.8 and 6.9. are the partial variances of the

‘parameters in the pH drop model. The amplitude param-

eter 888: is the most sensitive in the 10% deviation

182

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analysis, but since it was accurately measured, the stan-
dard deviation analysis shows that the equilibrium constant
KBY is the largest. There are only two different regions
of sensitivity in these analysis, a high pH set (KBY
> 888: > KBH) and a low pH set CKBY > K8H >> 888:).
Figures 6.10 and 6.11 show the partial variances of
the k' output function in the pH drop analysis. Here,
.for the first time, some sensitivity to KYH appears at low

pH values.

From the partial variance plots for the Tryptophanase

model we see that the parameters can be grouped according
to the pH dependence of their effect on the output func-

tions. KBH’ KBY’ and e 8: determine the value of AA

8 obs

with reasonably uniform sensitivities at pH values below
9 when the standard deviations are used. Since 288%

is by far the most important parameter, allowing the same
relative deviation for it as for the other parameters
causes it to take most of the partial variance, from u0%
at low pH values to over 95% at higher pH values. By
restricting the ranges to the standard deviation values,

KBH and KB become the dominant parameters except at high

Y
pH values.

Partial variances obtained when k' is the output func-

tion show that K dominates at pH values around 7 but

BY
becomes third in importance above pH = 8.2. The parameters

k and K

BY 8H become the most important at high pH values

187

and maintain substantial sensitivity down to pH values

of about 7.“. Only at pH values below about 7.6 do the
sensitivities to kgy and KYH begin to become important.
This reflects the fact that these parameters refer to a

low pH pathway for the interconversion of the B and y mani-
folds. To determine kgy and KYH with greater precision,
the measurements should be extended to lower pH values if
possible.

The sensitivities of k' to its parameters does not
change much when the parameters are allowed to vary over
equal relative intervals instead of over the estimated
standard deviations. However, examination of Table 6.2
shows that the relative standard deviations for the most
important parameters are not very different. Therefore,
a change from 110% relative deviation to to is nearly the

same as a change from :10% on all parameters to :20% on

all parameters so that we would not expect much difference.

L):

The sensitivity analysis applie here suggests what
experiments should be done to further refine the parameters.
Incremental pH Jump experiments to higher pH values than
the limit of 9.3 used to date would probably result in

better estimates Of kBY and KB? while pH drop experiments

5
I
V

to lower final pH values than “.7 would greatly improve

A

U .
the determination of ng and .YH,

This application of sensitivity analysis to the

Tryptophanase model was made a ter the model had been

188

developed and the parameters fit to the data. The marginal
standard deviation estimates given by KINFITA for all of
the parameters gave us an indication of their reliability.
However, sensitivity analysis, not only confirmed these
ideas, but also clearly delineated the regions of pH in
which the absorbance changes and rate constants are most
affected by particular parameters. Thus the maJor goals

of sensitivity analysis, to rank the parameters in order

of their importance to the output functions, and to assist

in the design of future experiments, were both realized

in this example.

VII. FUTURE WORK AND DEVELOPMENT

The previous chapters examined the theory and applica-
tions of Sensitivity Analysis. This chapter reviews those
areas which should be profitable fields of research for
further development of sensitivity analysis.

The most useful theoretical development would be in
the relationship between the Walsh and Fourier methods.
Christenson (1952) has laid the groundwork for this problem.
He noted that Walsh functions may be generalized to sets of
orthogonal functions with more than two values. This is
done by relating the Walsh function to powers of (exp(2iH/N),
where the two-value Walsh functions are obtained by letting
N = 2, thereby giving powers of (-l). The generalized
Walsh function may then take on N different values.

This relationship suggests that the N-point discrete
Fourier transform may be totally developed from a dis-
crete algebraic viewpoint without recourse to the continuous
Fourier transform. If this were done, a clearer understand-
ing of the errors involved in aliasing and choice of fre-
quency sets should result. This would also lead to a more
direct relationship between the linear sensitivity co-
efficients (Taylor series) and the Fourier expansion co-

efficients.

189

190

.Choice of the frequency sets for sensitivity analysis
has always been a limitation. In the Fourier method we
may do sensitivity analysis with up to 50 parameters since
their Uth order accurate frequency sets are known. How-
ever 6th order or higher accurate frequency sets are not
known for an arbitrary number of parameters. It appears
to be a difficult number - theoretic problem to even find
a higher-order accurate set. However, finding higher-
order accurate sets for arbitrary number of parameters
would enable the computation of more accurate Fourier
sensitivity analyses.

In Walsh analysis an arbitrary number of parameters
may be evaluated. All the frequencies required for exact
analysis are known (21). Unfortunately, the largest re-
quired frequency for a p-parameter set is 2p-l. This
requires 2p simulations to compute the 2p-l coefficient.
For large values of p this becomes impractical. Analogous
to the Fourier method we can develop approximate Walsh
frequency sets to a required order of accuracy. Appendix
9 has an approximate Walsh frequency set which is Ath-
order accurate. With this set of frequencies we can do
approximate walsh sensitivity analysis with up to 21

12 21

parameters using only 2 simulations instead of 2

which would be required for exact analysis.
This technique will work for any set of approximate

frequencies, and with the apparent relationship of Fourier

191

and Walsh expansions we should be able to connect the ap-
proximate frequency sets from the two methods with each
other. Unfortunately, an algorithm for finding approximate
Walsh frequency sets has not been discovered, although

it is easier to invent approximate Walsh sets than it is

to invent approximate Fourier sets. The set given in Ap-
pendix 9 was chosen in an intuitive fashion. Obviously
more work is required to deve10p a systematic method of
finding approximate Walsh frequency sets for any desired
accuracy. This should also clear up the problem of find-
ing approximate Fourier frequency sets of arbitrary accuracy.

Another useful area of research is the connection of
statistics and sensitivity analysis. Sensitivity analysis
measures the effect on the output function of variations in
the parameters. Statistics deals with the reverse problem,
the effect on the parameters caused by variations, or er-
rors, in the output function. Research in the relation-
ships between sensitivity analysis and statistics would
unite the more theoretical aspects of sensitivity analysis
with the real world measurements used in statistics.

One direct approach is to "feed" the "answers" obtained
from a least squares analysis of data directly into the
sensitivity analysis programs. The least squares program
delivers "best" estimates of the parameters and standard
deviation estimates for each parameter. By using these

values as the nominal parameters along with the standard

192

deviation as the range of variation a sensitivity analyis
may be done on this model. From the partial variance curves
obtained in this way one may determine whether the output
function is sensitive to that particular parameter space.
If there are maxima in the partial variance curves then one
should make more measurements in that region to pin down the
"best" value for the parameter in a least squares sense.
Such an approach should be useful in both model reduction
and experimental design.

The computer programs are well-designed. However,
by examining the timing data printed by the programs it
seems likely that improvement in the matrix transpose
algorithm (SUBROUTINE TRANP) would decrease the amount
of required computer time. Other than this, there are no
new, faster algorithms (that I know of) which should be
substituted for the ones presently used. However the
programs were written to facilitate the replacement of
sub—programs if better ones are developed.

OnecWher place that the programs could be modified
is in SUBROUTINE MODEL. It may cause a significant de-
crease in computer time if models written in terms of
differential equations are recast into an integral equa-
tion form. Integral equations are usually more stable
numerically than differential equations. This type of
change could result in a decrease of orders of magnitude

in the computer time spent computing the required simulations.

193

Applications of both the Fourier and Walsh sensitivity
analysis should be straightforward. Interpretation of
the results will, of course, depend on the problem. It
is hoped that the applications and interpretations pre-
sented here are sufficiently detailed to enable interested
researchers to perform sensitivity analysis on their own
models. The insight available from sensitivity analysis

is only realized after the model has been analyzed.

APPENDICES

APPENDIX 1

RELATIONSHIP OF FOURIER COEFFICIENTS TO
TAYLOR SERIES COEFFICIENTS

If a function can be expanded in a Taylor series over
an interval, it may also be expanded in terms of orthogonal
polynomials over an equivalent interval. This may be

written

 

where PJ(x) is an arbitrary orthogonal polynomial and

(J)
(x0)
evaluated at x = x0.

f is the Jth derivative of f(x) with respect to 'x'
By exploiting the orthogonality of the PJ(x) polynomials
we can relate the aJ expansion coefficients with the Taylor

series coefficients, i.e.,

 

oo 00 f
ak = jgo fw(x) 13.0.) Pk(x)dx = 3E0 rm.) ——%0—)— (x-xO)JPk(x)dx
' (J)
°° f(xo)
_ 320 J' ka

19A

195

where
wjk = fw(x) (x-xO)JPk(x)dx

From this equation we see that an orthogonal polynomial
coefficient, ak, is a weighted sum of all derivatives of the
function evaluated at the nominal value, x0. This implies
that an orthogonal expansion coefficient is composed of the
'effects' of all the derivatives of the function.

We can specialize this result to the orthogonal series
of Sines and cosines. Expanding f(x) in terms of frequencies

we obtain

a

f(X) = Z {a. cos(Jx) + b sin(Jx)} + -g
3:1 J J 2
where
l H
ak = F f—w f(x) cos(kx) dx
_ 1 Ir
bk - n f_1T f(x) sin(kx) dx

Substituting in the Taylor series eXpansion for f(x) we

obtain

bk

196

= % £3, (f(xo) + f'(xO)(x-x0) + ...)cos(kx) dx
= %.{:T (f(xo) + f'(xO)(x-x0) + ...)sin(kx) dx

Let y = x-xO, then

Using

=l|I—‘

H+XO

if

1 2
1T -n+x (f(xo) + f'(xo)y + 5 f"(x0)y +...)cos(y+xo)dy

O

(f(xo) + f'(xo)y+-% f"(xo)y2)sin(y+xo)dy

the expansion for sin(a+B) and cos (0+8) we obtain

1 Tr+x0 .

F f_fi+x0[f(xo)cos(y)cos(x0) - f(xo)sin(y)51n(xo)]dx

w+x0 .
.LW+XO[1”(Xo)(y)(cos(y)COS(XO))-f'6%9y(sin(y)31n(xo))]dx+...
l w+x
2;.Lfl+xo[IWXO)sin(y)COS(XO) + f(xO)cos(y)sin(x0]dx

H+X

.LW+X [f”(xo)y sin<y>cos(xo)+f'(xo)y cos(y)sin(xO]dx+...

197

Setting the nominal value, x to zero, we may reduce

0,
equations as follows
- °° 1 .. (2.1) 2.1
ak - J20 W [Jr f(O) y cos(ky)dy

.+ .
% f” f(ZJ l) y2J+l

_w (0) sin (ky) dy

<<y23><fE§%)><cos<ky>>> + <y23+le81+l)Sin(ky)>

IIM 8

0

Similarly bk may be reduced to

2j+lf(2J+1)

(0) sin(ky)> - <y23f(23)

bk = E <y (0) cos(ky)>

This clearly shows that the Fourier coefficients are com-

posed of all derivatives of the expansion function.

198

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

APPENDIX 2
8 L I I I_
In. _ .
r~
>- _
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2:
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C3“3 m
w I-I
a: H
U. '—
‘—
m-l Fh_.— —
N I“mam-In
‘00 0.75 .I.5 2.25 3.0
PRRRMETER

Figure A.1. Histogram of Log—uniform Distribution Func-
tion.

This function is given by:

Parameter - nominal *exp(A% sin-1(sin(sq))

where here
1/2

nominal (PHI/PLO)

1/2 £n(PHIPLO).

199

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o l L l
m D'-
ID.. -
V
>—
L)
2
DJ
23cm. r
em
DJ
0:
U4
9?." "
I
°-1.0 -0.5 0.0 0.5 1.0
PRRRMETER

Figure A.2. Histogram of uniform distribution function.
This function is given by:

Parameter = nominal + Asin_l(sin(sq)).

where here

)

nominal %(P

HI+PLO

1
2(PHI‘PL0)

200

 

FREQUENCY
Sp 7? 100
trim.
‘17:
EI’A
izr"

S
x
I

I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘13:o ";h.5 015"' 1.0

0.0
PRRRMETER

Figure A.3. Histogram of Gaussian-type distribution func-
tion.

This function is given by:

 

1 + sin(sq)J

2
Parameter — a 10S [1 _ sin(sq)

where

_ 100
PHI'PLO

such that 90% of the samples are between PHI and PLO'

201

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

O 1 L l
as -
as ~
)—
LJ
2:
DJ
ZDCL. L
GCD
UJ
I: I
U.
.. w E -
I. .
0.1-0 77"0 05 M CCCCC 1005 100

7 0.0
PRRRMETER
Figure A.A. Histogram plot of sin-transformation function.

This function is given by
Parameter = nominal + Asin(sq)

where here

_H_I___LQ

nominal

p _
A: HI LO

202

APPENDIX 3

This procedure follows the original Fast Fourier
Transform, the Cooley-Tukey algorithm. In fact, some Fast
Fourier Transform programs may be converted directly into
Walsh transforms by simply setting all the trignometric
values to i1 and deleting the complex part of the trans—
formation (since the Walsh transform is real).

The factorization of the transform relies on the
lexicographic ordering of the sampled function values.
Writing out the transform using binary representation for
the time, t = (tlt2,...,tp), and for the sequency, m =

(ml,m2,...,mp)

N—l
C = C(m m ...m ) = l 2 f(tn)WALH(n3tn)
m l 2 p N _
n-0
p
Z t.m
= )3 X Z f(t t ..t ) (—1) (A-1)
ml 0 m2 0 mp_0

The calculation of the transform is carried out in a
series of stages. There is one stage for each power of
two in the number of points, 2p = N. The first stage is
to derive a partial transformation series, X1, from the
input series, f(t), by expanding the first sum in the equa-

tion (ignoring the scaling factor for now).

Xl(tlt2... p—l p t -0

. m
0) + (-1) pf(tl...t 1)

l...tp_l p-1

Now we pass through the data, either adding or subtracting

adjacent function values. The second stage is constructed

from the first by expanding the second sum. Then
1
X t m
2( lt2 tp_2mp_l p) t 23:0 xl(tl tp_lmp)
p—l

This procedure is continued until all P-stages have
been computed. The values of the last stage are the de-

sired Walsh coefficients.
C = C(mlm2...mp) = Xp(mlm2...mp)

This is an extremely fast transform on a computer as only

additions and subtractions are required.

20A

APPENDIX A
PARSEVAL'S FORMULA FOR WALSH FUNCTIONS

The total variance of a function may be expressed as
the sum of its squared Walsh expansion coefficients. This
may be easily seen by computing the variance for an arbi-

trary function. The defining equation for variance is

2

OTotal = <(f(x))2> - <f(x)>2

where <f(x)> is defined as the average of the multidimen-

sional function f(x) with x = (xl,x2,...x <(f(x)2>

D).

-

is then the average of the square of the function f(x).
Expanding f(x) in a finite multidimensional Walsh

series we obtain the following series:

( > 1 l l p <
f 5 = z z z c. —. WALH k x )
k =0 k =0 k =0 ‘1k2°'°kp E 1’ i
1 2 p 1-1
p
= z c n WALH (k.,x )
E K i=1 1 i

205

To compute the average of f(x) we need only compute

the average of the series expansion

C

X WALH (k X ) dX

quo

<f(x)> = f
x 1

This equation must now be integrated over each dimen-
sion, xi. However, since each dimension has only two
values the multidimensional integral is equivalent to a
multidimensional sum over these two values. This results

in the following equation

1 l l l E
<r(£)> =«—— z z (2 0k H WALH(ki,xi))
2p xl-o x2= x =0 5 — i=1
p
1 1 1 p
= __ 2 ok I 2 . z z ( n WALH(ki.Xi))}
zp g — Xs=0 x2=0 xl=0 i=1

Substituting in the algebraic definition of the one

digit Walsh function results in

1 1 1 ikixi
«(gm—5:0“ z . z <<-1) }
2 5 — xp=0 xl=0
p k x
=chkIn<1+<-1)ii>1
2p 5 — i=1

206

The term in brackets is zero if sun; of the ki's are

nonzero. Hence it functions as a Kronecker delta and we

may simplify the equation accordingly

_1_

I: }=c=c
2‘35

9
<f(£)> - Ck{2 55,9 g oo...o

This shows that the average of an arbitrary function is
the CO coefficients of its Walsh expansion.
The computation of <(f(x))2> is straightforward. First

expressing the square of the function as a Cartesian product.

WALH(k

1 J’XJ')

‘M
O

W
O

E
H

IImnj

WALH(ki,xi)

I
k l J

—-1

which upon substitution of the definition of the Walsh

function results in:

<f(x)>2 =

VIM

This equation may be integrated over each dimension, xi

207

1
2(ki+ki)xi

' i
<<f<gg>>2> -- f z 2: 0k ck. <-1>
ii!
15... .)
+k x
l (kl+k1)xi i=2 1 l i
=—-z 2: CR Ck'{ Z 2‘. [(l+(-1) )((-l) }
2p k k! _ _ X =0 X =0 .
_ _ p 2
E (k +k')x
...}..z z ck ck. { II (1+(-1) 1 1 1)}
2p Ek? _. ... 1:1

The term in brackets acts as a Kronecker delta, requiring

k = 5'. Hence the equation is now easily reduced

0
<(f(x))2> -3; Z 2: C C H 25

29513' 1‘- 1‘3'1 1 ki’ki
l o
= —— C ‘
2p f. f. I. Cr 2 519a
= z (Ck)2
E ...

giving a sum of the squared coefficients.

208

Combining the two results we may directly write the

equation for the variance solely in terms of the expansion

coefficients:
2 = 2 _ = 2 _ 2 = , 2
OTotal <(f(§)) > <f(§)> ECE COD .0 ick-
00...0

where the 2' is a summation over all k except the

k
sequency term.

209

APPENDIX 5
THE CALCULATION OF WALSH PARTIAL VARIANCES

A partial variance is defined as the variance, or
dispersion, in one dimension of a multidimensional function

a = <<r*<xl>>2> - <f*(X1>>2

2
l
where f*(xl) = <f(xl,x2,x3,...,xn)>, the multidimensional
function averaged over all dimensions except the first.
0% is called the partial variance of variable, or parameter,
x1.

If f(xl,...,xn) is expanded in a multidimensional finite

Walsh series with two points along each axis, then ki

and x1 may be represented in binary by one digit.
1 1 Q (
f(x x ...x ) = Z ... Z (3 H WALH k x )
1’ 2 n k1=0 k =0 k1° kn i=1 1’ i
n
n
2 k.x
1 1 i=1 1 i
= Z I C (=1)
k =0 k =0 k1 °'kn

210

The average of the function with respect to x2,...xn

is the sum of all those values divided by the number of

samples
‘K' = < >
f (X1) f(xl°"xn) x2...xn
n
1 1 1 1 1 iilkixi
= “fi3I z . 2 .[ z 2 Ck . k (-l) l
2 x2=0 xn-O kl-O kn=O l n
Switching summations:
n
1 1 1 1 2 kixi
_ l 1:.1
"*N-l z z: [ z X Ck k (-l) J
2 kl=0 kn=0 Xn=0 x2=0 l n

eXpanding the term in brackets:

211

f*(xl) =

n

. Z k x

1 1 k _ 1 4
z z 0(1+(-1> 2><-1)1'3

.-

l l k
0 0 =0 x =
l n n Xn 3

x
l l[

:3

1 1 k1x1 — k1
Z ... 2 0k k (-l) E H (1+(-l) )]
=0 k =0 l" n i=2

 

l k X l k x
Z {-1) l l 2n-l = Z (3 l l

J

Now calculate the second moment, <f*(xl)2>, using the

function f*(xl) and squaring the summing over x1.

1 l k X l k'X
<(f*(xl))2> =% 2 { 2 0k 0 O(-l) l 1 z ck,O O(-1)1 1}
= = . '=
X1 0 kl O l kl O l
'
= % % % Ck 0 O Ck'OO O % (_l)(kl+kl)xl
= '= on o. =
kl 0 kl O l 1 X1 0
l l k +k'
l l l
= " X X C C (l + (-l) )
2 kl=o k': klo O kiO..O
= C2 + C2

212

To calculate the second required term, we need <f*(xl)>X
l

l l k x
(“0‘1”): =% Z Z Ck 0 0(‘l) l 1
1 x =0 k =0 1 °
1 1
-1 k
=55: ck00(1+(-1)1)
k =0 l o.
1
= C0. 0
Hence
2 2
* _
(f (Xl)>xl CO. 0

Subtracting the first moment from the second, we obtain

the desired partial variance

213

APPENDIX 6

DERIVATION OF WALSH COUPLED PARTIAL

VARIANCE FORMULAS

To measure the effect of coupled parameters, say "i"

coupled together at a time, we calculate a coupled partial

variance.

2 2
= * _
01,2,0022 (f (x£+l..x2«) > <f(‘x")>£

where f*(x£+l..x£) is the function f(x) averaged over the

variables x£+l..xn, and <f(x)>: is the average of the multi-

dimensional function over all the variables x1..xn.

Expand f(xl..xn) in an n—dimensional finite Walsh

series with two points along each axis. In this case

Xi and k1 may be represented by a one digit binary number.

)=c

ll ZJID

f(Xl-X WALH (ki.xi)

Z
n E E J l

The average of f(x) is the CO term as previously shown

<f(x)> = C0

21A

)2>.

So we need only calculate the term <f*(xl..x£

First calculating f*(xl..x2)

* =
f (x1..x£) <f(xl..xn)>x

£+l" n

 

 

PVM
I'Pi‘
{\J
:3
I
to

k X k

n
C 1' E (1+(-1) 1)] (~1)i=1
5 2n‘£[i=2+l

1x1

 

I'FM

2

iglkixi
c —1 ' 0
x ( > k2+1,05

WM

f*(x

We must now square this function f*(x1...x£) and average

it over x1...x£.

l
* 'X' = -
f (x1..X£)f (X1...XQ,) Z 2* Ck* Cm“. ( l)

2
‘ 2 1 l 1 film-(mi)Xi
<f*(xl..x£ > = _T Z .. Z {Z Z Ck*C *(-l) }
2 x =0 x =0 k* n* - a
l l --
Rearranging the summations
2 k n '
2 l - ii
<f*(x ..x ) > = 2 C *0 *{-——- TI (1+ (-l) )}
1 Q 5* 2* K .2 21 i=1
i
=2 ZC*C H O
5* 3* 5 - i=1 ni’ki
l l
2 2
=ZC*= Z X C
E* 5 kl=0 k2=0 k1'°k10 0

Substituting the appropriate expressions for the two

moments we obtain the ith coupled partial variance

where {H is a summation of all g* vectors except the one
kx
equal to 0.

216

APPENDIX 7

RELATIONSHIP BETWEEN LINEARLY DEPENDENT EQUATIONS

AND THEIR FOURIER COEFFICIENTS

In chemical kinetics mass balance equations often allow
us to substitute an algebraic equation for a differential
one. These mass balance equations are linear and in enzyme

kinetics they are of the form:

where the X1 are the different types of enzyme—containing
intermediates, and the Vi is the number of enzyme molecules
in specie Xi'

Given N-l "Xi" expansions in fourier series, the fourier
coefficients of the Nth species may be calculated. Using
these fourier coefficients one can calculate the partial
variances of the Nth specie.

This can be seen by inserting the N-l fourier expansions

into the mass balance equation

217

i-l M
a (J)
E 21 VJEM Z aL

X
0 J=

cosLX + b(J)sinLX] + v

i i

N M
+ 2 vEL z 0a£3)cosLX + 0(3)s1an]
J: 1+1 3

Solve for'X1

N—l M
X -03; [ 2 v K{ 2 afiK)cosLX + bé
vi K=1KL=O

K)

sinLX} - E0]

M N-l -v N-l -vK
{ z ( z ——5 aéx))cosLx + ( z (— K)b£K))sinLX} - E
L=0 K=l ”1 x=1V1

O

t 1
aLcosLX + bLsinLX

I
“(‘13

0

Hence the fourier coefficients of X1 are

N-l -v
a' = X J 8.6K) - E
0 Kgl vi 0 0
a' = N21 :35 a(K)
L =1 ‘3. L
. ”‘1 2’1. (K)
bL = 2 b
K=1 " L

218

APPENDIX 8

SENSITIVITY ANALYSIS PROGRAMS

The use of the Sensitivity Analysis Programs at Michigan
State University is a straightforward task. If the mathema-
tical model is composed of ordinary differential equations
or algebraic equations, no modifications to the programs are
necessary. The equations only have to be coded into FORTRAN
66. After this is done, one has to decide: What kind of an
analysis is desired (Walsh or Fourier), the parameters’
nominal values and range of variation, the transformation
function to be used, and the time points of interest. This
data is read by the program SENANAL which does the required
simulations. A second program, TRANS, reads the output from
SENANAL, TAPE3, and computes the expansion coefficients and
partial variances, both single and coupled. Since a Sensi—
tivity Analysis may generate a large amount of data, depending
on the number of output functions, parameters, and time points,
an Optional plotting program, PLTSEN, is provided. This pro-
gram reads the output from TRANS, TAPE9, and plots four sets
of curves for each output function. PLTSEN plots the average
value of the output function, the single partial variances,

the expansion coefficients, and the relative deviation.

219

The following cards execute the programs SENANAL
AND PLTSEN.

PNC CARD

JOB CARD

Pw CARD

ATTACH,MAIN,SENAN LBINARYOPT2.
FTN,I=INPUT,B=SUB.

LOAD,MAIN.

LOAD,SUB.

EXECUTE.

EETUEN,LGO.

10. REWIND,TAPE3.

11. ATTACH,TBIN,TRANSFORMBINARYOPT2.
12. LOAD,TBIN.

13. EXECUTE.

1n. CATALOG,TAPE9,SENSITIVITYANALYSISFILE,RP=30.
15. ATTACH,PLT,PLTSENBINARYOPT2.

16. RETURN,LGO.

17. REWIND,TAPE9.

13. PLT.

20. (789)

\OCDNChUl-LI‘UONH

SUBROUTINE MODEL (TIN, TOUT, YIN, NFUNC, TSTART)
COMMON /PARA/ P(50)

This is a subroutine which on input has TIN as the
initial value at which the output functions have values
(there are NFUNC output functions, YIN(l) is the first output
function). TSTART is Optional and tells MODEL when it is
starting a new parameter vector (IF (TSTART .EQ. TIN) ). The
common block /PARA/ contains the parameters to be varied
in the model.

On output from MODEL, the output functions, sometimes
called object functions, have their values at 'TOUT', the

time on returning from MODEL.

220

Note that this subroutine must change its
FORTRAN code for each different mathematical
model, but not for parameter variations.

(7891

Data cards for program SENANAL- see the comment cards in

SUBROUTINE READIN. I

(789)

Data cards for program PLTSEN, see the comment cards in the

program PLTSEN

(789)

(6789)

The somewhat difficult part is to force SUBROUTINE
MODEL to solve for the output functions given a parameter
set, the initial values of the output functions and the time
at which the solution is desired. The application to
algebraic equations is straightforward. However, solving
differential equations is more difficult. The use of the
EPISODE package (Hindemarsh, 1977) for solving ordinary
differential equations is recommendedzmniis a part of the
SENANAL package. These subprograms are extensively docu-

mented internally with comment cards.

221

Program SENANAL

PROGRAM SENANAL(INPUT=65,0UTPUT=65,TAPE1=INPUT,TAPE2=OUTPUT,
. TAPE3=513)
0*

CG

0* PROGRAM SENANAL IS THE DRIVER PROGRAM OF A SUITE OF CODES

C* WHICH PREFORMS SENSITIVITY ANALYSIS ON A MODEL. SENANAL

C* READS INPUT AND BASED ON THAT INPUT CHOSES WALSH SENSITIVITY
C* ANALYSIS METHOD OR FOURIER SENSITIVITY ANALYSIS METHOD. IT THEN
C* PROCEDES TO SOLVE THE MODEL EQUATIONS OVER THE DESIRED

C* TIME POINTS WITH THE NECESSARY PAPAMETERS. EACH

0* PARAMETER VECTOR WHICH IS TO BE SOLVED IS CALLED A SIMULATION. *
C’ SENANAL SOLVES THE SIMULATIONS BY FIRST CREATING THE PARAMETER *
C’ VECTOR AND THEN SOLVING THE MODEL EQUATIONS OVER ALL THE DESIRED’
C* TIME POINTS. THE MODEL SOLUTIONS ( OBJECT FUNCTIONS ) ARE

0* WRITTEN OUT TO TAPE} AT EACH TIME POINT.

0*

0* AFTER A SIMULATION IS COMPLETED, SENANAL CREATES ANOTHER
C* PARAMETER VECTOR AND SOLVES THE NEXT SIMULATION. THIS IS

C’ REPEATED UNTIL ALL THE NECESSARY SIMULATIONS HAVE BEEN SOLVED.
0*

C* VARIABLES

*tit

l**

c-l»

0* BEGIN(NFUN0) - THE INITIAL CONDITIONS, OR EQUIVALENTLY

0* (REAL) THE VALUES OF THE OBJECT FUNCTIONS AT TSTART.
0*

cl,

0* IACCUR - ORDER OF ACCURACY OF THE FREQUENCY SET
0* (INTEGER)

c-I»

(3*

0* IOMEGA = O IF FOURIER 4TH ORDER SET IS TO BE USED

0* (INTEGER) 1 IF SPECIAL FREQUENCY SET IS TO BE USED

0* -1 IF STANDARD WALSH FREQUENCY SET IS TO BE USED
CAI»

0* . IMETH - METHOD FLAG FOR SENSITIVITY ANALYSIS; - 1 FOR FOURIER
0* (INTEGER) ANALYSIS, = 2 FOR WALSH ANALYSIS.

c-l

0* ITRANS - FLAG FOR TYPE OF TRANSFORMATION FUNCTION

0* (INTEGER) SEE SUBROUTINE PARAM FOR DETAILS.

c!

C’ IW(NPARA) 3 AN ARRAY CONTAINING THE FREQUENCY SET TO BE USED IN
C’ (INTEGER) THE S. A. RUN.

0* IF 'IOMEGA' .EQ. 1, THIS ARRAY MUST BE READ IN FROM
C* CARDS. OTHERWISE THE FREQUENCY SETS ARE CREATED

********#********************‘¥

222

Program SENANAL, CONT'D.

0* INTERNALLY. *
cl» *
0* NFUNC - NUMBER OF OBJECT FUNCTIONS WHICH WILL BE SAVED *
0* (INTEGER) AT EACH TIME POINT. :
C}

0* NLABEL(NFUNC) - THE NAMES (LABELS) OF THE OBJECT FUNCTIONS *
0* NLABEL(1) SHOULD BE THE NAME OF THE FIRST *
0* OBJECT FUNCTION, ETC. *
0* *
0* NPARA - NUMBER OF PARAMETERS TO VARY *
0* (INTEGER) *
0* i
0* NSIMUL - NUMBER OF SIMULATIONS *
0* (INTEGER) *
0* *
0* PHI(NPARA) = MAXIMUM VALUES OF THE PARAMETERs( OR ONE SIGMA MAx)*
0* (REAL) *
cl)

0* PLO(NPARA) . MINIMUM VALUES OF THE PARAMETERs( OR ONE SIGMA MIN)
0* (REAL)

C*

0* TIME(TNPTS) - ARRAY CONTAINS THE TIME POINTS AT WHICH THE

0* (REAL) OUTPUT FUNCTIONS ARE TO BE SAVED AND THE

0* SENSITIVITY ANALYZED.

CAI

0* TSTART - INITIAL TIME POINT, SO THERE ARE NO S. A. VALUES SAVED
0* (REAL) AT THIS POINT

0* TITLE(8) - A ONE CARD TITLE FOR S. A. RUN
0* (INTEGER) (WRITTEN IN 8A1O FORMAT )

0* TNPTs . NUMBER OF TIME POINTS
0* (INTEGER)

0* YIN(NFUNC) . AN ARRAY OF LENGTH NFUNC CONTAINING ON
0* (REAL) INPUT TO SUBROUTINE MODEL THE VALUES OF
0* OBJECT FUNCTIONS AT TIN AND UPON OUTPUT FROM MODEL

0* YIN( ) CONTAINS THE VALUES OF THE OBJECT FUNCTIONS
.0* AT TOUT. '

C’ ERROR CODES

C* STOP "R1" OR STOP 1: IF IOMEGA WAS UNACCEPTABLE,EITHER NOT READ
0* CORRECTLY OR ABS(IOMEGA) .GT. 1

******t**$t*#* **** ** **** *Iitll 33* 13*

0* STOP "R2" OR STOP 2: IMETH WAS UNACCEPTABLE (.LT.1 .OR. .GT. 2)

223

Program SENANAL, CONT'D.

STOP 3: TNPTS WAS TOO LARGE OR TOO SMALL (0,150)

STOP 4: ITRANS WAS OUTSIDE DEFINED RANGE (1,5)

STOP "R4" OR STOP 5: NFUNC HAS A VALUE OUTSIDE THE DEFINED RANGE

( 1.40) ~

STOP "R5" OR STOP 6: NSIMUL HAS A VALUE OUTSIDE THE DEFINED
RANGE ( .GE. 1)

STOP 7: PHI(J) .LE. PLo(J), THIS COULD CAUSE A DIVIDE
BY ZERO IN SUBROUTINE PARAM.

STOP 10: IW(J) .LE. ZERO, FREQUENCIES MUST BE .GE. 1
STOP "R3": NPARA .LT. 1 NUMBER OF PARAMETERS MUST .GE. 1
STOP "F1": NPARA .GT. 50 , USING FOURIER METHOD, NPARA

STOP "ORDER" MEANS THAT THE ORDER OF ACCURACY VARIABLE
WAS LESS THAN 4.

MUST BE .LE. 50 ALSO

STOP "GETER": ERROR IN A FREE FORMAT READ, EITHER AN EOF
OR AN ILLEGAL CHARACTER.

UPON SUCCESFUL COMPLETION OF SENANAL TAPE3 HAS THE
FOLLOWING FORMAT.

1) TITLE -- ( 8A1O FORMAT)

2) METHOD,NPARA,TNPTS,NSIMUL,NFUNC
( A10, 416 FORMAT; ' METHOD =1OHWALSH , 1OHFOURIER

3) FREQUENCY SET ( 15H FORMAT, A LABEL FOR THE FILE )
AND IACCUR IN I3-FORMAT

4) Iw(1),Iw(2),...,IW(NPARA) ( 1616 FORMAT )
5) TIME POINTS ( 15H FORMAT, A LABEL FOR THE FILE )
6) TIME(1),TIME(2),...,TIMB(TNPTS) ( 7E12.6 FORMAT )

7) NLABEL(1),...,NLABEL(NFUN0) (8A1O FORMAT )

******

**** Calcutta: ******** ****** ** ************ ##8##

. , 22M

Program SENANAL, CONT'D.

8) YIN(1),YIN(2),...,YIN(NFUNC) UNFORMATTED WRITE

THERE ARE TNPTS*NSIMUL RECORDS OF TYPE 8, ONE FOR EACH

TIME POINT IN A SIMULATION MULTIPLIED BY THE NUMBER OF
SIMULATIONS.

THESE SIMULATION VECTORS ARE IN AN UNSUITABLE FORM

FOR SENSITIVITY ANALYSIS SINCE TO DO S. A. WE NEED
ALL THE DIFFERENT SIMULATIONS OBJECT FUNCTIONS' VALUES

AT THE SAME TIME POINT.

THE SUITE OF CODES RUN BY PROGRAM TRANS WILL REFORMAT
TAPE3 AND WILL TRANSFORM THE SIMULATION CURVES INTO
SEQUENCY VECTORS( WALSH OR FOURIER ) FOR WHICH PARTIAL
VARIANCES WILL BE COMPUTED.

* Oat #IIII* it* 0:: #13 t t *

CWW

C‘I'

0*

C}

0*
(3*

COMMON /PARA/ P(SO)

REAL PMAx(50),PMIN(5o),PAVE(5O)
REAL TIME(150),PHI(50),PLO(SO)
REAL YIN(40),BEGIN(4O)

INTEGER IOMEGA,IMETH,NFUN0,ITRANS
INTEGER TITLE(8),TRANS(5.2),IW(50)
INTEGER NLABEL(40)

NPARA,NSIMUL,TNPTS,LABEL(2)

DATA PAVE/50*O./, PMIN/50*1.OE+99/, PMAX/50*O./
DATA LABEL/ 10H FOURIER ,1OH WALSH /
DATA TRANs/1OHLOGUNIFORM,1OH UNIFORM ,1OH SINE TEST,

+1OHLOG(P)BELL,1OHBELL-SHAPE,1OHARITHMETIC,1OHMULTPLIER ,
+3*(1OH /

SUBROUTINE TIMES IS A TOTALLY UNNECESSARY BUT SOMEWHAT USEFUL
TIMING ROUTINE

CALL TIMEs(1,O)
CALL READIN(IOMEGA,TIME,TSTART,IMETH,NFUN0,ITRANS,PHI,PLO,

+NPARA,NSIMUL,TNPTS,TITLE,BEGIN,IW,NLABEL,IACCUR)

CALL TIMEs(1,O)
WRITE OUT INPUT

WRITE(2,5) TITLE
FORMAT(1H1,8A10)
WRITE(2,6)

FORMAT<1H )

WRITE(2,10) LABEL(IMETH)

1O

11

12

27

28
31

15

17

18
19

21

22
0*
0*
0*
0*

225

Program SENANAL, CONT'D.

FORMAT(1H ,* THIS IS A SENANAL RUN USING *,A1O,* ANALYSIS*)
WRITE(2,11) IACCUR,(IW(J),J=1,NPARA)
FORMAT(1H ,* WITH THE*,13,*TH ORDER FREQUENCY VECTOR=*,13(2X,15)

+./.30X.
+14(ZX.15))

WRITE(2,12) NSIMUL,TNPTS
FORMAT(1H ,* THERE ARE *,16,* SIMULATIONS IN THIS RUN WITH*,

+,I5,* TIME POINTS*)

WRITE(2,27)

FORMAT(/,/,6x,* FUNCTION *,5x,* INITIAL VALUE *,/)

DO 31 J-1,NFUNC

WRITE(2,2S) NLABEL(J),BEGIN(J)

FORMAT(6X,A10,5X,1PE14.6)

CONTINUE

WRITE(2,6)

WRITE(2,15) TRANS(ITRANS,IMETH)

FORMAT(1H ,* THE PARAMETERS WERE CALCULATED USING *,A1O,

+* TYPE TRANSFORMATION FUNCTIONS*)

WRITE(2,17)
FORMAT(* *,/,* PARAMETER*,2X,* PHI(J) *,7X,* PLO(J)*)
DO 19 J=1,NPARA
WRITE(2,18) J,PHI(J),PL0(J)
FORMAT(3x,Is,2x,2x,1PE13.6,2X,E13.6)
CONTINUE
WRITE(2,21)
FORMAT(* *,/,* TIME POINTS *)
WRITE(2,22)(TIME(J),J=1,TNPTS
FORMAT(* *,1o(1x,E12.6))

CHECK FOR ACCEPTABLE INPUT PARAMETERS

IF(IOMEGA .LT. -1 .OR. IOMEGA .GT. 1 ) STOP 1
IF(IMETH .LT. 1 .OR. IMETH .GT. 2 ) STOP 2
IF( TNPTS .LT. 1 .OR. TNPTS .GT. 150 ) STOP 3
IF(ITRANS .GT. 5 .OR. ITRANS .LT. 1 ) STOP 4
IF(NFUNC .LT. 1 .OR. NFUNC .GT. 40 ) STOP 5
IF( NSIMUL .LT. 1) STOP 6
IF( IACCUR .LT. 4 ) STOP "ORDER"
Do 1 J-1,NPARA
IF( PHI(J) .LE. PLo(J) ) STOP 7
IF( Iw(J) .LE. O ) STOP 1O
CONTINUE
J-1
IF( TIME(1) .LE. TSTART ) WRITE(2,2) J,J
IF( TNPTS .EQ. 1) GO TO 4
DO 3 J=2,TNPTS
IF( TIME(J) .LE. TIME(J-1) ) WRITE(2,2) J,J
FORMAT(1H ,/,* TIME(*,I4,*) .LE. TIME(*,I4,*- 1)*)

20
25
26
30
35

226

Program SENANAL, CONT'D.

CONTINUE
CONTINUE

WRITE THE OUTPUT TAPE LABELS

WRITE(3.23) TITLE

FORMAT(8A1O)

WRITE(3,20) LABEL(IMETH),NPARA,TNPTS,NSIMUL,NFUNC
FORMAT(A1O,416)

WRITE(3,25) IACCUR

FORMAT(* FREQUENCY SET *,13)
WRITE(3.26)(Iw(J),J=1,NPARA)
FORMAT(16I6)

WRITE(3.30)

FORMAT(* TIME POINTS *)
WRITE(3,35)(TIME(J),J=1,TNPTs)
FORMAT(7E12.6)
WRITE(3,23)(NLABEL(J),J=1,NFUNc)
CALL TIMEs(2,O)

LOOP OVER THE DIFFERENTS SIMULATIONS

DO 1000 ISIMUL=1,NSIMUL

IQ ' ISIMUL

CALCULATE THE PARAMETER VECTOR FOR THIS SIMULATION
CALL PARAM(IMETH,IQ,ITRANS,PHI,PLO,NPARA,P,NSIMUL,IW)
CALCULATE THE PARAMETER STATISTICS

DO 100 J=1,NPARA
PMAx(J) - AMAX1(PMAx(J),P(J)g
PMIN(J) - AMIN1(PMIN(J),P(J)
PAVE(J) . (P(J) + FLOAT(ISIMUL - 1)*PAVE(J))/FLOAT(ISIMUL)
CONTINUE
CALL TIMES(3.O)

INITIALIZE THE FUNCTION WITH ITS INITIAL VALUE.

(NECESSARY IF THE FUNCTIONS ARE ODE'S, OTHERWISE ONE CAN SET
BEGIN TO ZERO )

DO98J=1,NFUNC
YIN(J) . BEGIN(J)
CONTINUE
SET INITIAL TIME FOR SIMULATION RUN
TIN - TSTART

227

Program SENANAL, CONT'D.

0* CALCULATE THE OUTPUT FUNCTIONS FOR THE "TNPTS" POINTS
0*
DO 200 KOUNT . 1,TNPTS

TOUT - TIME(KOUNT)

CALL MODEL(TIN, TOUT, YIN, NFUNC, TSTART )
cl.
0* WRITE OUT THE SOLUTION AT TOUT
C".

WRITE(3.10000) (YIN(J),J=1,NFUN0)
1OOOO FORMAT(4020)

.C*

0*
CALL TIMES(4,O)
TIN = TOUT

200 CONTINUE

C1!

0*

1000 CONTINUE

C.

0* SIMULATIONS ARE OVER WITH
0* WRITE OUT THE PARAMETER STATS
C‘.’
WRITE(2,6)
WRITE(2,1500)
1500 FORMAT(1H ,1x,* PARAMETER*,3X,* AVERAGE VALUE *,2x,
+* MAXIMUM VALUE *,2x,* MINIMUM VALUE *)
DO 1620 J=1,NPARA
WRITE(2,1610) J,PAVE(J),PMAx(J),PMIN(J)
1610 FORMAT(1H ,5x,15,5x,2x,3(1PE13.6,4x))
1620 CONTINUE
0* PRINT TIMING DATA
CALL TIMEs(1,1)
END

228

Program SENANAL; CONT'D.
SUBROUTINE READIN

SUBROUTINE READIN(IOMEGA,TIME,TSTART,IMETH,NFUNC,ITRANS,PHI,
+PLO,NPARA,NSIMUL,TNPTS,TITLE,BEGIN,IW,NLABEL,IACCUR)

C)
t
y

C* *
0* SUBROUTINE READIN READS THE INPUT FOR THE PROGRAM SENANAL. ALL *
0* VARIABLES ARE DEFINED IN THE SENANAL COMMENT CARDS. *
C’ *
0* THIS IS INSTALLATION DEPENDENT SECTION OF THE METHOD AS IT USES *
0* FREE FORMAT INPUT. (VARIABLES SEPARATED BY A COMMA ) *
0* *
0* FORMAT OF INPUT CARDS *
C* *
0* SET 1 - TITLE (ONE CARD ) *
0* *
0* SET 2 = IOMEGA,IMETH,NPARA (INTEGERS) *
0* *
0* SET 3 IFF IOMEGA - 1 *
0: SET 3A . IW(1),IW(2),IW(3)....,IW(NPARA),IACCUR (INTEGERS):
0

0* SET 4 . NSIMUL,TNPTS,ITRANS,NFUNC,TSTART (TSTART IS A REAL,

0* THE OTHERS ARE INTEGERS*
C* *
0* SET 5 = TIME(1),TIME(2),...,TIME(TNPTs) ( REALS ) *
0* *
0* SET 6 . PHI(1),PHI(2),...,PHI(NPARA) (REALS) *
C* *
0* SET 7 - PLo(1),PL0(2),...,PL0(NPARA) (REALS) *
0* §
0* SET 8 - BEGIN(1),BEGIN(2),...,BEGIN(NFUN0) (REALS) *
C* *
0* SET 9 . NLABEL(1),...,NLABEL(NFUNC) *
0* TO BE READ IN ONE (1) VALUE TO A CARD (A10 FORMAT) :
0*

C* *
0* VARIABLES *
C* *
0* BEGIN(NFUNC) - THE INITIAL CONDITIONS, 0R EQUIVALENTLY‘ *
0* (REAL) THE VALUES OF THE OBJECT FUNCTIONS AT TSTART. :
ci-

C* *
0* IACCUR . ORDER OF ACCURACY OF THE FREQUENCY SET *
0* (INTEGER) .
C* *
C‘ *
0* IOMEGA . 0 IF FOURIER 4TH ORDER SET IS TO BE USED *
0* (INTEGER) 1 IF SPECIAL FREQUENCY SET IS To BE USED :

C‘ -1 IF STANDARD WALSH FREQUENCY SET IS TO BE USED

229

Program SENANAL, CONT'D.
SUBROUTINE READIN

C’ *
0* IMETH - METHOD FLAG FOR SENSITIVITY ANALYSIS; . 1 FOR FOURIER *
0* (INTEGER) ANALYSIS, . 2 FOR WALSH ANALYSIS. *
C1!

0* ITRANS = FLAG FOR TYPE OF TRANSFORMATION FUNCTION

0* (INTEGER) SEE SUBROUTINE PARAM FOR DETAILS.

C1!»

0* Iw(NPARA) . AN ARRAY CONTAINING THE FREQUENCY SET TO BE USED IN
0* (INTEGER) THE S. A. RUN.

0* IF 'IOMECA' .EQ. 1, THIS ARRAY MUST BE READ IN FROM
0* CARDS. OTHERWISE THE FREQUENCY SETS ARE CREATED

0* INTERNALLY.

Ci

C* NFUNC 8 NUMBER OF OBJECT FUNCTIONS WHICH WILL BE SAVED
0* (INTEGER) AT EACH TIME POINT.

C1!

0* NLABEL(NFUNC) = THE NAMES (LABELs) OF THE OBJECT FUNCTIONS
0* NLABEL(1) SHOULD BE THE NAME OF THE FIRST
0* OBJECT FUNCTION, ETC.

cl-

0* NPARA - NUMBER OF PARAMETERS TO VARY
0* (INTEGER)

0* NSIMUL - NUMBER OF SIMULATIONS
0* (INTEGER)

0* PHI(NPARA) = MAXIMUM VALUES OF THE PARAMETERs( OR ONE SIGMA MAX)
0* REAL

0*

0* PLO(NPARA) = MINIMUM VALUES OF THE PARAMETERs( OR ONE SIGMA MIN)
0* (REAL)

C}

0* TIME(TNPTS) . ARRAY CONTAINS THE TIME POINTS AT WHICH THE

0* (REAL) OUTPUT FUNCTIONS ARE TO BE SAVED AND THE

0* SENSITIVITY ANALYZED.

C1}

0* TSTART - INITIAL TIME POINT, SO THERE ARE NO 3. A. VALUES SAVED
0* (REAL) AT THIS POINT

0* TITLE(B) - A ONE CARD TITLE FOR S. A. RUN
0* (INTEGER) (WRITTEN IN 8A1O FORMAT )

0* TNPTS - NUMBER OF TIME POINTS
0* (INTEGER)

0* '
CW

*ttttttakatttatlktlk**********##********tittttt

REAL TIME(150),BEGIN(40)

0*

C*

C*
C*

0*
C*
C*
10
0*
0*

c-l-

C*

100

230

Program SENANAL, CONT'D.
SUBROUTINE READIN

INTEGER TITLE(8),IOMEGA,IMETH,NFUN0,ITRANS,TNPTS,NPARA,Iw(NPARA)
INTEGER NLABEL(40) «

JCOMMON /GETERR/ IFLAG,NUMVAR,RAB0(40)

EQUIVALENCE (RAB0(1),IRAB0(1))
INTEGER IRAB0(40)

IO IS THE INPUT UNIT ( TAPE1 )
DATA IO/1/

READ IN SET 1

READ(IO,10) (TITLE(J),J=1,8)
FORMAT(8A10)

READ IN SET 2

IACCUR - 4

ICARD = 2

CALL GETNUM(Io)

IF( IFLAG .GE. 0 ) CALL 0ETERR(IFLAG, ICARD, NUMVAR)
IOMEGA = IRAB0(1)

IMETH . IRAB0(2)

NPARA -- IRAB0(3 )

IF(IOMEGA .LT. -1 .OR. IOMEGA .GT. 1 ) STOP "R1"
IF(IMETH .LT. 1 .OR. IMETH .GT. 2 ) STOP "R2"

IF( NPARA .LT. 1 ) STOP "R3"

OBTAIN FREQUENCY SET

IF(IOMECA .EQ. 0) CALL FOURST(IW,NPARA)

IF(IOMEGA .EQ. -1) CALL WALSET(IW,NPARA)

IF(IOMECA .NE. 1) GO TO 100

READ IN SPECIAL FREQUENCY SET (SET 3 )

READ IN THE ACCURACY OF THE SET ALSO

NP1 . NPARA + 1

CALL IREAD(IW,NP1,IO, ICARD)

IACCUR - IW(NP1)

CONTINUE

READ IN SET 4

ICARD . ICARD + 1

CALL GETNUM(Io)

IF( IFLAG .GE. 0 ) CALL GETERR(IFLAO, ICARD, NUMVAR)
NSIMUL . IRAB0(1) '
TNPTS = IRAB0(2)

ITRANS - IRAB0(3)

NFUNC = IRAB0(4)

TSTART . RAB0(5)

Ca»
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0*

Ci

0*
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0*
0*
(3*
0*
cl;

c-lv

150
175

101

231
Program SENANAL, CONT'D.
SUBROUTINE READIN

IF(NFUNC .LT. 1 .OR. NFUNC .GT. 40 ) STOP "R4"
IF( NSIMUL .LT. 1) STOP "R5"

READ IN THE TIME POINTS ( SET 5)
CALL RREAD(TIME,TNPTS,IO, ICARD)

READ IN PHI(J) ( SET 6 )

CALL RREAD(PHI,NPARA,IO, ICARD)

READ IN PLO(J) ( SET 7 )

CALL RREAD(PLO,NPARA,IO, ICARD)

READ IN INITIAL VALUES ( SET 8 )
CALL RREAD(BEGIN,NFUN0,IO, ICARD)

DO 175 J=1,NFUN0

READ(IO,1SO) NLABEL(J)

ICARD . ICARD + 1

FORMAT(A1O)

CONTINUE

WRITE(2,1O1) ICARD

FORMAT(1H ,/,I6,* DATA CARDS READ IN *)

RETURN
END

232

Program SENANAL, CONT'D.
SUBROUTINE FOURST

SUBROUTINE FOURST(IW,NPARA)
0*
GWWWWWWMWW

C* *
0* SUBROUTINE FOURST CALCULATE THE STANDARD 4TH ORDER CORRECT *
0* FOURIER FREQUENCY SET FOR "NPARA" PARAMETERS. *
0* ._ *
0* REFERENCE; CUKIER,SHAILBY,SHULER. JOURNAL OF CHEMICAL PHYSICS, *
0: VOL 63, NO. 3, (1975) PP 1140-1149. :
C

CWMWWWWWW
INTEGER Iw(NPARA),IOMEGA(50),IDN(49)

DATA IOMEGA/O,O,1,5,11,1,17,23,19,25,41,31,23,87,67,
+ 73.85.143.149,99 .119,237,267,283,151.385.
+ 157,215,449,163.337.253.375.441.673,773,875,873,587,849,
+ 623,637,891.943,1171.1225,1335.1725,1663,2019/
DATA IDN/ 4,8,6,1O,2O,22,32,4O,38,26,56,62,46,76,96,
+ 6O,86,126,134,112,92,128,154,196,34,416,106,
+ 208,328,198,382,88,348,186,140,170,284,568,302,438,
+ 41O,248,448,388,596,216,100,488,166/
0*
IF(NPARA .GT. 50 ) STOP "F1"
IW(1) - IOMEGA(NPARA)
DO 100 J=2,NPARA
Iw(J) a IW(J-1) + IDN(NPARA + 1 - J)
100 CONTINUE
RETURN
END

233

Program SENANAL, CONT'D.
SUBROUTINE WALSET

SUBROUTINE WALSET(IW,NPARA)

C* *
C* SUBROUTINE WALSET CALCULATES THE FREQUENCY SET FOR EXACT WALSH *
C* ANALYSIS FOR 'NPARA' PARAMETERS. *
C* *
C* REFERENCE: T.H. PIERCE, PHD THESIS (1980) *
C* *

INTEGER IW(NPARA)

DO 100 J=1,NPARA
Iw(J) . 2**(J-1)
1OO CONTINUE
RETURN
END

23H

Program SENANAL, CONT’D.
SUBROUTINE IREAD

SUBROUTINE IREAD(IRRAY,LAST,IO, ICARD)
0*
memm
0*
0* SUBROUTINE IREAD READS IN A VARIABLE LENGTH (LAST) INTEGER
0* ARRAY USING FREE FORMAT INPUT.
0*

WWW
C

INTEGER IRRAY(LAST)

****

0*

0*
COMMON /GETERR/ IFLAG,NUMVAR RABc(40)
EQUIVALENCE (RABC(1),IRABC(1))
INTEGER IRABc(40)

0*

KOUNT a 1
1O CONTINUE
ICARD =- ICARD + 1
CALL GETNUM(Io)
IF( IFLAG .GE. 0 ) CALL GETERR(IFLAG, ICARD, NUMVAR)
0* IF THE CARD READ WAS BLANK NUMVAR . 0.
IF( NUMVAR .LT. 1 ) GO TO 10
, DO 20 J=1,NUMVAR
IRRAY(KOUNT) . IRAB0(J)
KOUNT - KOUNT + 1
IF(KOUNT .CT. LAST) 00 TO 25
20 CONTINUE
0*
0* RETURN FOR ANOTHER CARD FULL OF VARIABLES
GO TO 10
25 CONTINUE
0* ALL DONE SO STOP
RETURN
END

235

Program SENANAL, CONT'D.
SUBROUTINE RREAD

SUBROUTINE RREAD(ARRAY,LAST,IO, ICARD)
0* f
CHMWWWWWW
0*
0* SUBROUTINE RREAD READS IN A VARIABLE LENGTH (LAST) REAL
0* ARRAY USING THE FREE FORMAT ROUTINE GETNUM.
0*
0W

REAL ARRAY(LAST) '

****

0*

0*
COMMON /GETERR/ IFLA0,NUMVAR,RAB0(40)
EQUIVALENCE (RAB0(1),IRAB0(1))
INTEGER IRAB0(40)

0*

KOUNT = 1
1O CONTINUE
ICARD - ICARD + 1
CALL GETNUM(Io)
IF( IFLAG .GE. 0 ) CALL GETERR(IFLAG, ICARD, NUMVAR)
0* IF THE CARD READ WAS BLANK NUMVAR - 0.
IF( NUMVAR .LT. 1) GO TO 10
DO 20 J=1,NUMVAR
ARRAY(KOUNT) - RABO(J)
KOUNT = KOUNT + 1
IF(KOUNT .GT. LAST) GO TO 25
2O CONTINUE

C* RETURN FOR ANOTHER CARD FULL OF VARIABLES
GO TO 10

25 CONTINUE

0* ALL DONE SO STOP
RETURN
END

236

Program SENANAL, CONT'D.
SUBROUTINE PARAM

SUBROUTINE PARAM(METH,IQ,TRANS,PHI,PLO,NPARA,P,NSIMUL,IW)

CWW

0* u
C* SUBROUTINE PARAM COMPUTES THE IQ"TH PARAMETER *
c* VECTOR. THIS Is DONE USING A PRESELECTED ( TRANS,METH ) *
C* TRANSFORMATION FUNCTION. *
0* *
C* METH . 1 ---- FOURIER METHOD *
0* i
0* TRANS *
C* 1 -> USE FOURIER LOG-UNIFORM TRANSFORMATION *
C* PHI . NOMINAL*EXP(DELTA) *
C* PLO . NOMINAL*EXP( -DELTA ) *
C* WITH LN(P) SPREAD UNIFORMLY OVER @LN(PHI) , LN(PLO)@ *
0* M
C* 2 => USE FOURIER UNIFORM TRANSFORMATION *
C* PHI - NOMINAL + DELTA *
C* PLO - NOMINAL - DELTA *
C* P IS UNIFORMLY SPREAD OVER @ PLO , PHI @ :
ci-

C* 3 => USE THE FOURIER TEST FUNCTION *
C* PHI - NOMINAL*( 1 + DELTA ) *
C* PLO = NOMINAL*( 1 - DELTA) *
C* P(SQ) . NOMINAL*(1. + DELTA*SIN(W*SQ) ) *
0* *
C* 4 => USE THE FOURIER COSH DISTRIBUTION FUNCTION *
C* IN LOG(P)-SPACE *
C* HERE LN(PHI)-LN(PLO) - 4.0/A *
0* WHERE 82.87 OF THE SAMPLES ARE BETWEEN PHI AND PLO :
cl-

C* 5 => USE THE FOURIER COSH DISTRIBUTION FUNCTION *
C* IN P-SPACE *
C* HERE (PHI)-(PLO) = 4.0/A *
C* WHERE 82.87 OF THE SAMPLES ARE BETWEEN PHI AND PLO *
0* i
0* *
C* METH - 2 ---- WALSH METHOD *
0* i
C* TRANS *
0* i
c* 1 => USE ARITHMETIC WALSH TRANSFORMATION *
C* PHI - NOMINAL + DELTA *
C* PLO - NOMINAL - DELTA *
C* P IS EITHER PHI OR PLO *
C* i
C* 2 => USE MULTIPLICATIVE WALSH TRANSFORM *
C* PHI . NOMINAL*1O**( DELTA ) 1

C* PLO a NOMINAL*1O**(-DELTA )

0*

C*

c!
c*
100
0*
CAI»

110

200
C'l'

210
C.

300
C"

310

C“
400

237

Program SENANAL, CONT'D.
SUBROUTINE PARAM

P IS EITHER PHI OR PLO

REAL NOMINAL,DELTA,PHI(NPARA),PLo(NPARA),P(NPARA)

INTEGER TRANS,IQ,METH,NSIMUL,IW(NPARA)
INTEGER WALH '

EXTERNAL WALH

IF ( METH .EQ. 2 ) GO TO 1000

TWODPI = 2.0/ACOS(-1.0)

R . FLOAT(NSIMUL)

SQ . FLOAT(2*IQ -NSIMUL- 1)/(R*TWODPI)

GO TO (100,200,300,400,SOO)TRANS

CONTINUE
FOURIER METHOD WITH LOG-UNIFORM TRANSFORMATION FUNCTION

DO 110 J=1,NPARA

DELTA - O.5*ALOG(PHI(J)/PL0(J))

NOMINAL . SQRT(PHI(J)*PLO(J))

P(J) = NOMINAL*EXP(DELTA*TWODPI*ASIN(SIN(SQ*FLOAT(IW(J)))))
CONTINUE

RETURN

CONTINUE

FOURIER METHOD USING UNIFORM TRANSFORMATION FUNCTION

DO 210 J=1,NPARA

NOMINAL = O.5*(PHI(J)+PLO(J))

DELTA - ( PHI(J) - PLo(J) )*O.5

P(J) - NOMINAL + DELTA*TWODPI*ASIN(SIN(SQ*FLOAT(IW(J))))
CONTINUE

RETURN

CONTINUE

FOURIER METHOD WITH TEST TRANSFORMATION FUNCTION
DO 310 J=1,NPARA

NOMINAL - O.S*(PHI(J)+PLO(J))

DELTA . (PHI(J)-PLo(J))/(PHI(J)+PL0(J))

P(J) . NOMINAL*(1.O + DELTA*SIN(FLOAT(IW(J))*SQ))
CONTINUE

RETURN

CONTINUE

(3*
(2*
C*

410

C*
500

0*
C"
C*

510

0*
0*
1000
0*
0*

1100
C*

1110

0*
1200
0*

238

Program SENANAL, CONT'D.
SUBROUTINE PARAM

COSH-DISTRIBUTION FUNCTION
BELL-SHAPE IN LOG(K)-SPACE

DO 410 J=1,NPARA

AJ = 4.0/(ALOG(PHI(J)) - ALOG(PLO(J)))

THETA = SQ * FLOAT(IW(J))

UJ . (1. O/(2.*AJ))*ALOG((1. + SIN(THETA))/(1. - SIN(THETA)))
NOMINAL = SQRT(PHI(J)/PL0(J))

P(J) - NOMINAL * EXP(UJ)

CONTINUE

RETURN

CONTINUE

COSH-DISTRIBUTION FUNCTION
BELL-SHAPED IN K-SPACE

DO 510 J=1,NPARA

AJ = 4.0/(PHI(J) - PLo(J))

THETA a SQ*FLOAT(Iw(J))

UJ = (1. o/AJ)*ALOG((1. + SIN(THETA))/(1. - SIN(THETA)))
NOMINAL = (PHI(J)+PLO(J))*O. 5

P(J) = NOMINAL + UJ

CONTINUE

RETURN

CONTINUE
ENTRY INTO HERE FOR WALSH ANALYSIS

ISO 8 IQ - 1
GO TO (1100,1200) TRANS

CONTINUE

ARITHMETIC WALSH TRANSFORMATION FUNCTION

Do 1110 J=1,NPARA

NOMINAL - O.5*(PHI(J)+PL0(J))

DELTA - ( PHI(J) - PLo(J) )*O.5

P(J) =.NOMINAL + DELTA*FLOAT(WALH(IW(J),ISQ))
CONTINUE

RETURN

CONTINUE

MULTIPLICATIVE WALSH TRANSFORMATION FUNCTION

Do 1210 J=1,NPARA

DELTA = O.5*ALOG10(PHI(J)/PL0(J))

NOMINAL = SORT(PHI(J)*PL0(J))

P(J) . NOMINAL*10.0**(DELTA*FLOAT(WALH(IW(J),ISQ)))

239

Program SENANAL, CONT'D.
SUBROUTINE PARAM

1210 CONTINUE
RETURN
END

c-l‘l-I-I'

2ND

Program SENANAL, CONT'D.
SUBROUTINE GETNUM

SUBROUTINE GETNUM(LINPUT)

COMMON /GETERR/ ICRK,JREAD,RABC(40)
DIMENSION IRABc(4o)

EQUIVALENCE (RABC(1),IRABC(1))
COMMON /EL/ L(80)

LOGICAL INTR

CWWWWWW

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*** FREE FORM VARIABLE INPUT ROUTINE. ***

ROUTINE ACCEPTS A,F,E AND I FORMAT INPUT.

ALL BLANKS EXCEPT IN HOLLERITH STRINGS ARE IGNORED.
THE ONLY LEGAL DELIMITER IS COMMA (,), ANY

OTHER RESULTS IN ERROR TERMINATION.

KL IS THE MAXIMUM COLUMN WIDTH COUNTER.

ONLY 80 COLUMNS ARE READ, SO ONLY 40 VARIABLES
CAN BE RETURNED. TO ENLARGE THIS, CHANGE THE DATA

AND COMMON STATEMENTS TO REFLECT THE SIZE YOU WISH.

--- INPUT ---
LINPUT -- THE TAPE UNIT BEING READ FROM

--- OUTPUT ---
JREAD IS THE NUMBER OF VARIABLES RETURNED IN
COMMON /GETERR/
RABc(=IRABc) CONTAINS THE READ VARIABLES.
COMMON /EL/ CONTAINS THE LINE AS READ IN 80R1
ERROR CODES: (STANDARD IF(UNIT) VALUEs)

IORK=1 ILLEGAL CHARACTER (MSG PRINTED)
VARIABLES TO POINT OF ERROR RETURNED
ICRK=O EOF ON READ, JREAD=O

ICRK3-1 NORMAL TERMINATION

--- INTERNAL VARIABLES ---

S IS SIGN OF VARIABLE, IFA THE SIGN OF THE EXPONENT
NUM IS THE MANTISSA, IE THE EXPONENT

IP IS THE NUMBER OF DECIMAL PLACES INPUT.

I IS THE CHARACTER COUNTER (1-80)

INTR -- REAL VARIABLE(FALSE)/INTEGER(TRUE) FLAG

HOLLERITH STRINGS OF 10 OR MORE CHARACTERS MAY

BE INPUT WITHOUT COMMAS EVERY 1O CHARACTERS AND WILL
BE INSERTED 1O CHARACTERS PER WORD WITH BLANK FILL
(STANDARD A FORMAT). ANY COMMAS FOUND IN THE HOLLER-
ITH STRING END THE STRING AT THAT POINT. THE FIRST

CHARACTER OF THE STRING MAY NOT BE A COMMAS, ,)
PERIOD(. ) PLUs(+ ) OR MINUS(- ) OR DIGIT(O 9 OR BLANK.

22:21:313231312“11112111““11111111“:

Cm
CM
CM
Cm
Cm

2H1

Program SENANAL, CONT'D.
SUBROUTINE GETNUM

HOLLERITH STRINGS WHICH ARE THE LAST INPUT ON A
LINE, BUT NOT ENDING WITH A COMMA, WILL BE ASSUMED
TO CONTINUE RIGHT OUT TO COLUMN 80(OR KL) AND BLANK
FILLED VARIABLES WILL THUS BE RETURNED.

”it:

CWW

INTEGER PERIOD,COMMA,BLANK,ZERO,PLUS,EE,DD

DATA PERIOD,COMMA,BLANK,ZERO /1R., 1R,, 1R , 1RO/
DATA NINE,PLUS,MINUS,EE,DD /1R9, 1R+, 1R-, 1RE, 1RD/
DATA KL /80/

Cm

0.4%;
10

CW

25

C****
251

26
cifififi
0****

Cm
Cm

READ THE INPUT LINE FROM UNIT LINPUT

READ (LINPUT,1ooo) (L(I),I=1,80)
JREAD=O
IF (EOF(LINPUT) .NE. 0.) GO TO 998
ICRK=-1

I'1

PREPARE FOR A NEW VARIABLE

NUM*IE=IFA=IP=O
INTR=.TRUE.
S=1.0

DECODE THE FIRST CHARACTER IN THE VARIABLE

IF (I .GT. KL) RETURN
IF (L(I) .LE. NINE .AND. L(I) .GE. ZERO) Go TO 35

IF
IF

(L(I)
(L(I)

S'-1.0

GO
IF
IF
IF

DO

TO 30
(L(I)
(L(I)
(L(I)

.EQ. PLUS) GO TO 3O
.NE. MINUS) GO TO 25

.EQ. PERIOD) GO TO 39

.EQ. COMMA) GO TO 60

.EQ. BLANK) GO TO 291
HOLLERITH VARIABLE (A FORMAT)

26 LL=1,1O

JL=LL

IF (I

.GT. KL) GO TO 27

IF (L(I) .EQ. COMMA) GO TO 27

ISH=6

O-6*LL

IE-SHIFT(L(I) ,ISH)

NUM'O
I=I+1
CONTI

R(NUM,IE)

NUE

FULL WORD(1O CHARS) FILLED IF YOU FALL
THRU HERE.

STORE THE HOLLERITH VARIABLE

JREAD=JREAD+1
IRABO(JREAD)=NUM

SKIP THE TRAILING COMMA, OTHERWISE ASSUME
THE HOLLERITH STRING CONTINUES

CM

27

29

291

cm
30
35

C****

39
40

CM

cm
50

51

52

2&2

Program SENANAL, CONT’D.
SUBROUTINE GETNUM

IF (I .GT. KL) RETURN
IF (L(I) .EQ. COMMA) GO To 291
NUM=O
GO TO 251
BLANK FILL WORD
DO 29 LL=JL,1O
ISH=6O-6*LL
IE-SHIFT(BLANK,ISH)
NUM-OR(NUM,IE)
STORE THE PARTIAL HOLLERITH VARIABLE
JREAD-JREAD+1
IRABc(JREAD) - NUM
I-I+1
GO TO 10
INTEGER PORTION OF VARIABLE
I=I+1
IF (L(I) .EQ. PERIOD) GO TO 39
IF (I .GT. KL) GO To 60
IF (L(I) .EQ. BLANK) GO TO 30
IF (L(I) .EQ. EE) GO TO 50
IF (L(I) .EQ. COMMA) Go TO 60
IF (L(I) .LT. ZERO .OR. L(I) .GT. NINE) GO TO 999'
NUM - NUM*1O + (L(I)-27)
GO To 30
EVALUATE DECIMAL PORTION
INTR-.FALSE.
I=I+1
IF (I .GT. KL) Go TO 60
IF (L(I) .EQ. BLANK) GO TO 40
IF (L(I) .EQ. EE) GO To 50
IF (L(I) .EQ. COMMA) GO TO 60
IF (L(I) .LT. ZERO .OR. L(I) .GT. NINE) GO TO 999
INCREMENT THE DECIMAL COUNT
IP=IP+1
NUM - NUM*1O + (L(I)-27)
GO TO 40
EVALUATE EXPONENT
IFA-1
INTR-.FALSE.
I-I+1
IF (L(I) .EQ. PLUS) GO TO 51
IF (L(I) .NE. MINUS) GO TO 52
IFA--1
I=I+1
IF (I .GT. KL) GO TO 60
IF (L(I) .EQ. COMMA) GO TO 60
IF (L(I) .EQ. BLANK) GO TO 51
IF (L(I) .LT. ZERO .OR. L(I) .GT. NINE) GO TO 999

62
64

CW

995
999

CM
CM

998

2M3

Program SENANAL, CONT'D.
SUBROUTINE GETNUM

IE - IE*1O + (L(I)-27)
GO TO 51
STORE THE FINISHED VARIABLE (EXCEPT HOLLERITH)

CONTINUE
CHECK ILLEGAL EXPONENT RANGE
THE CHECK IS NOT PERFECT: DIGITS BEFORE
THE MANTISSA PERIOD ARE NOT CONSIDERED.

IEX=IE*IFA-IP

IF (IEX .GT. 522) GO TO 995

IF (IEX .LT. -294) GO TO 995

I=I+1

JREADsJREAD+1

IF (INTR) GO To 62

RABC(JREAD) . S*(FLOAT(NUM)*1O.**IEX)

GO TO 64

IRABc(JREAD) = S*NUM

IF (I .GT. KL) RETURN

GO TO 10

ERROR CONDITION CODE

I=I-1

CONTINUE

ICRK=1

JM=I-1

IF (JM .LE. 0) JM=1

PRINT 1010, L,(BLANK,LL=1,JM),PLUS

RETURN
EOF ENCOUNTERED, JREAD ALREADY ZEROED, SET
ICRK AND RETURN.

ICRK=O

RETURN

CWWWWW

C****
1000

1010
Can!”

**FORMATS**
FORMAT (80R1)
FORMAT (*OILLEGAL CHARACTER FOUND AT PLUS(+) */1X,80R1/1X,80R1)

END

2AM

Program SENANAL, CONT'D.
FUNCTION WALH

INTEGER FUNCTION WALH(N,IT)

SUBROUTINE WALH(N,IT) COMPUTES THE HADAMARD-ORDERED '
WALSH FUNCTION OF SEQUENCY ' N ' AT TIME POINT ' IT .

WHERE N 18 OF THE RANGE ( 0,1,2,5,...(2**(LENGTH+1) - 1)
AND IT IS OF THE RANGE ( O,1,2.3.4,...,(2**(LENGTH+1) - 1)

VV

#******

CWWWWWWW

1O
C‘I’

27
C‘I’
C‘I’

INTEGER TBIT(6O),NBIT(60),M,I
REAL OLDN,FRAC
DATA LENGTH /15/

DECODE N INTO ITS BINARY REPRESENTATION

OLDN - FLOAT(N)

D0 10 I-1,LENGTH
MsOLDN/2.O

FRAC - OLDN/2.0 - FLOAT(M)
NBIT(I) - FRAC*2.0

OLDN = FLOAT(M)

CONTINUE

DECODE IT INTO ITS BINARY REPRESENTATION

TOLD - IT

DO 27 I=1,LENGTH
M-TOLD/2.0

FRAC a TOLD/2.0 - FLOAT(M)
TBIT(I) . FRAC*2.O

TOLD . FLOAT(M)

CONTINUE

WE NOW KNOW THE BINARY REP FOR T AND N
CALCULATE THE EXPONENT

NSUM . NBIT(1)*TBIT(1)

DO 50 I=2,LENGTH

NSUM . NSUM + NBIT(1)*TBIT(1)

CONTINUE

WRITE(2,2)(((NBIT(L),L=1,LENGTH),(TBIT(K),K=1,LENGTH)).NSUM)
FORMAT(* *,*NBIT=*,1SI1,* TBIT=*,1511,* NSUM -*,I4)

WALH - (-1)**NSUM

RETURN

END

 

2H5

Program SENANAL, CONT'D.
SUBRO UTINE TIMES

SUBROUTINE TIMEs(ISUB,ITYPE)

Cmmmmmi
C* *
C* SUBROUTINE TIMES COMPUTES THE CPU TIME SPENT BETWEEN CALLS. *
C* THIS IS INSTALLATION DEPENDENT. *
C* ITS USE Is FOR DOCUMENTATION PURPOSES ONLY 1
C"
C* ISUB - THE PROCEDURE TO BE TIMED. :
0*
c* ITYPE - FLAG: .LT. 1 FOR TIMING *
C* .GE. 1 FOR FINAL PRINT *
CAI Ir
GWW
C* *
0*

REAL TIMS(15),NEW,LAST

INTEGER NAME(15)
C}

DATA TIMs/15*O./

DATA LAST /O./

DATA NAME/1OHREAD INPUT,1OHSIMULATION,1OHPARAM CALC,1OHMODEL CALC

+,1OHWRITE OUTP/

DATA LENGTH /5/
c!»

IF(ITYPE .GE. 1) GO TO 5

NEW = SECOND(CPU)
C* INITIALIZE THE VALUES IN FIRST ENTRY

IF( LAST .EQ. O. ) LAST . NEW

TIMS(ISUB) = TIMs(ISUB) + NEW - LAST

LAST . NEW

RETURN
5 CONTINUE

WRITE(2,1oo)
1OO FORMAT(* *,/,/,/,5X,* SECONDS*,4X,* PROCEDURE *,/)

TSUM . 0.

DO 10 J=1,LENGTH

TSUM = TSUM + TIMS(J)

WRITE(2,150) TIMs(J),NAME(J)
150 FORMAT(* *,5X,F7.3.5X,A10)
1O CONTINUE

WRITE(2,160) TSUM
160 FORMAT(* *,/,/,5X,F7.5,5X,* TOTAL TIME *)

RETURN
END

2H6

Program SENANAL, CONT'D.
SUBROUTINE GETERR

SUBROUTINE GETERR(IFLAG,ICARD,NUMVAR)
CAI

1

0* SUBROUTINE GETERR IS AN ERROR MESSAGE SUBROUTINE WHICH *
0* TERMINATES SENANAL IF INPUT WAS NOT CORRECTLY *
C’ READ IN. *
0* *
C* IFLAG IS THE FLAG FROM SUBROUTINE GETNUM *
0* 11-
0* ICARD IS THE INPUT CARD NUMBER OF THE ERROR *
0* *
0* NUMVAR IS THE NUMBER OF VARIABLES ON CARD ICARD *

Cflmmmmimmmm
INTEGER IFLAG,ICARD,NUMVAR
WRITE(2,11) IFLAG,ICARD,NUMVAR

11 FORMAT(1H ,5H*****,* ERROR IN GETNUM *,/,15X,* ICRK =*,15,
+/.15X,* ICARD -*,I5,/,15X,* NUMVAR -*,15)
STOP "GETER"
END

2M7

Program TRANS

PROGRAM TRANS(OUTPUT365,TAPE2=OUTPUT,TAPE3=513,TAPE4=513,
+ TAPE53513,TAPE6=513,TAPE7'513,TAPE8=65,TAPE9=6S)

Ct}
CWWWWWWWW
0* PROGRAM TRANS IS THE FOLLOW-UP PROGRAM TO PROGRAM 'SENANAL'.
0* IN THIS PROGRAM 'TAPEB', THE OUTPUT FILE FROM 'SENANAL' IS

0* READ IN AND OPERATED ON.

C*

0* FIRST THE SIMULATION RUNS ARE TRANSPOSED INTO SENSITIVITY-

0* ANALYSIS POINTS. (INSTEAD OF ALL THE TIME POINTS OF ONE FUNCTION
0* A SIMULATION, WE HAVE ALL THE FUNCTIONS AT ONE TIME POINT,

0* A SENSITIVITY-ANALYSIS POINT ) UPON SUCCESSFUL TRANSPOSITION
0* WE ITERATE THRU THE TIME POINTS. EACH S.A. POINT IS THEN

0* TRANSFORMED INTO SEQUENCY SPACE ( FOURIER OR WALSH ).

0* THIS TRANSFORMATION GIVES US THE EXPANSION COEFFICIENTS

0* FROM WHICH WE COMPUTE THE PARTIAL VARIANCES OF THE OBJECT

0* FUNCTIONS.

Ci

0* AFTER THE PARTIAL VARIANCES ARE CALCULATED THEY ARE

0* WRITTEN OUT ONTO TAPE9. THE EXPANSION COEFFICIENTS ARE WRITTEN
0* OUT ONTO TAPE8. THE TRANSPOSED MATRIX IS AVAILABLE ON TAPE7,
0* LOGICAL UNIT 'WRITEUP'.

lit 3* ** ti 13* ****** #* *¥ ***1‘ **** ** ***.*

C-I

C* VARIABLES

C"

C* F( ) - AN ARRAY WHICH HOLDS ONE OBJECT FUNCTION

C* IE AT LEAST OF LENGTH 'NSIMUL'

C*

C*

C* A( ) = A REAL ARRAY WHICH WILL HOLD THE COSINE

C* COEFFICIENTS IN THE FOURIER METHOD. THEREFORE IT

C* MUST BE OF LENGTH (NSIMUL + 1)

c-I

C* B( ) . A REAL ARRAY WHICH WILL HOLD THE SINE COEFFICIENTS

C* IN THE FOURIER METHOD. IT ALSO MUST BE OF LENGTH

c* ( NSIMUL + 1)

0*

C* IWK( ) . THE WORKING STORAGE OF THIS PROGRAM. THIS ARRAY Is
C* USED AS STORAGE FOR DIFFERENT TEMPORARY VARIABLES.

C* *
C* ' MINIMALLY IT MUST BE DIMENSIONED FOR THE FFT ROUTINES*
C* IF NSIMUL IS THE NUMBER OF SIMULATIONS IN THE FOURIER*
C* METHOD THEN THRU SYMMETRY WE HAVE 2*NSIMUL POINTS

C* TO FOURIER TRANSFORM. *

0* THE EQUATION FOR IWK DIMENSIONS IS: *

2&8

Program TRANS CONT'D.

C* MINIMUM LENGTH = 5*( F + N ) + 26

(3*

C* WHERE F IS THE NUMBER OF THE PRIME FACTORS OF NSIMUL
C* EXCLUDING THE TRIVIAL FACTOR 1 .

C* N . 2*NSIMUL

0*

C* IE IF NSIMUL . 21 ( HAS To BE ODD )

0* THEN 21 a 3'7 -> F = 2

0* MINIMUM LENGTH - 5*( 2 + 42 ) + 26 = 158

0*

(3*

C* ERROR CODES:

0*

C* STOP "WALPR" , STOPS EXECUTION IF TWO FREQUENCIES ARE EQUAL
C* SEE SUBROUTINE WALPR.

(3*

c* STOP "MTH" , THE METHOD READ IN ON TAPE3 WAS SOMETHING OTHER
0* THAN WALSH OR FOURIER. SEE PROGRAM TRANS

0*

0* STOP 3 , ERROR IN MATRIX TRANSPOSITION. SEE SUBROUTINE TRANP.
0*

0*

C* OUTPUT FORMAT FOR TAPE? -PARTIAL VARIANCES-

0*

C* 1) LABEL,TIME,AVE,STDDEV,RELDEV,LENGTH,NPARA

C* ( A8,4(2X,E15.7),2I6 -FORMAT )

0*

G: LABEL . NAME OF THE OBJECT FUNCTION

C

C* TIME . TIME VALUE OF OBJECT FUNCTION

0*

C* AVE - AVERAGE VALUE OF OBJECT FUNCTION AT THIS TIME
0*

C* STDDEV - SQUARE ROOT OF TOTAL VARIANCE OF OBJECT FUNCTION
0* AT THIS TIME

(3*

C* RELDEV a STDDEV/AVE: STANDARD DEVIATION DIVIDED BY AVERAGE
C* VALUE

c*

C* LENGTH - NUMBER OF PARTIAL VARIANCES TO WRITE OUT
C* ( NPARA*(NPARA + 1))/2

0*

C* NPARA - NUMBER OF PARAMETERS ANALYZED.

0*

C.

C* 2) ( SWLJ(K),K-1,LENGTH )

C* ( 5(2X,E15.7) -FORMAT )

0*

0* SWLJ(K) = A SINGLE OR COUPLED PARTIAL VARIANCE, WHERE

***... ***ttt ****** *1: slut an: 11* aka: ***!!! 8:11 #:1121113 8* ##1## ***... ***

2M9

Program TRANS CONT'D.

0* CARDS 1 AND 2 ARE REPEATED FOR EACH LABEL-TIME POINT

0* WALSH ANALYSIS

0*

C* 1) LABEL,TIME,NCOEFF ( A8,E15.7,I6 -FORMAT )
0*

C* 2) (F(K),K-1,NCOEFF ) ( 4020 -FORMAT )

0*

0* CARDS 1 AND 2 ARE REPEATED FOR EACH DIFFERENT LABEL-TIME
0* POINT.

C* K = NPARA*(L-1) - (L*(L-1))/2 + J *
C* *
C* *
C* 5) (B(IW(L)+1) L=1,NPARA) *
C* ( 5(2X,E15.7) -FORMAT *
C* *
C* B(Iw(L)+1) . IN FOURIER ANALYSIS, IT IS THE SINE *
C* EXPANSION COEFFICIENT FOR THE Iw(L)TH *
C* FREQUENCY. :1
C* IN WALSH ANALYSIS, IT IS THE EXPANSION *
C* COEFFICIENT FOR THE IW(L)TH FREQUENCY. *
C* *
C* CARDS 1,2 AND 3 ARE REPEATED FOR EACH LABEL-TIME POINT. *
C* *
C* *
C* OUTPUT FORMAT FOR TAPE8 -EXPANSION COEFFICIENTS- :
0*
0* FOURIER ANALYSIS *
0* *
C* 1) LABEL,TIME,NZ ( A8,E15.7,I6 -FORMAT) I
0*
0* 2) ((A(K),B(K)),K=1,N2) ( 4020 -FORMAT ) g
0*
*
*
*
*
*
*
*
*
*
*
*
*

0*WWWWWWWW
REAL TIME(150)
REAL MINSW(10,55),MAXSW(10,55),AVESW(10,55)
REAL A(2O48),B(2O48)
REAL F(2048)
REAL x(2048)
0*
INTEGER TITLE(8),METHOD,NPARA,TNPTS,NSIMUL,IW(50).NFUNC
INTEGER WRITEUP,WRITED,READUP,READOWN
INTEGER IWK(sooo)
INTEGER ITYP(2)
INTEGER ILABEL(1o)
0*
LOGICAL TEST

0*
0*

0*

0*
C'I'

0*
0*
0*
0*
1O
11
20

21

250
Program TRANS CONT'D.

EQUIVALENCE (IWK(1),F(1))
COMMON SWLJ(21O)

DATA IUNIT/3/. IHALF/2048/
DATA IFLAG/O/

DATA READUP/4/, READOWN/S/, WRITEUP/6/, WRITED/7/
DATA MAXSW/550*0.0/, MINSW/SSO*1.O/, AVESW/550*O./

DATA ITYP/1OH FOURIER ,1OH WALSH /
DATA TEST/ .FALSE. / ‘

INITIALIZE THE TIMING ROUTINE
CALL TIMES(1,O)

READ TAPE3

READ(3.1O) (TITLE(J),J=1,8)

FORMAT(8A1O)

WRITE( 2,11) (TITLE(J),J=1,8)

FORMAT(1H ,8A1o) ,

READ(5,20) METHOD,NPARA,TNPTS,NSIMUL,NFUNC
FORMAT(A1O,4I6)

WRITE( 2,21)METHOD,NPARA,NFUNC,NSIMUL,TNPTS

FORMAT(1H ,/,1H ,A1O,* SENSITIVITY ANALYSIS USING :*,/,

+* NUMBER OF PARAMETERS =*,I6,/,* NUMBER OF OBJECT FUNCTIONS =*,
+ I6,/,* NUMBER OF SIMULATIONS =*,I6,/,* NUMBER OF TIME POINTS

+.I6)

3O
4O
41

42

5O

60
0*

READ(3.30) JUNK,IACCUR
FORMAT(A1O,5X,I3)
READ(3.40)(IW(J),J=1,NPARA)
FORMAT(16I6)

WRITE( 2,41)

FORMAT(1H ,* FREQUENCY SET * 1
WRITE( 2,42) (IW(J).J=1.NPARA
FORMAT(15X,16I6)

READ(3.30) JUNK

READ(3.SO) (TIME(J),J=1,TNPTs)
FORMAT(7E12.6)

READ(3.60)(ILABEL(J),J=1,NFUNc)
FORMAT(8(A8,2x))

DETERMINE THE TYPE OF ANALYSIS

IF( METHOD .EQ. ITYP(1) ) TEST a .TRUE.
IF( METHOD .EQ. ITYP(2) ) TEST = .TRUE.
IF(.NOT. TEST ) STOP "MTH"

251

Program TRANS CONT'D.

0*
0* TIME THE INPUT
CALL TIMES(1,0)

0*
0*
0* NOW WE ARE READY TO TRANSPOSE THE MATRIX
0
CALL TRANP( IUNIT, READUP, READOWN, WRITEUP, WRITED,
+ TNPTS, NFUNC, NSIMUL, IWK(1), IWK(1). IWK(IHALFHLIHALFJUI‘BOW)
C

0* TIME THE TRANSPOSE OPERATION
CALL TIMEs(2,O)
0W
0*
C* INITIALIZE N2,LENGTH
0*
C* N2 . LENGTH OF A FOURIER TRANSFORM COEFFICIENT VECTOR
0*
C* LENGTH - LENGTH OF THE PARTIAL VARIANCE MATRIX WHEN FOLDED
C* INTO A LINEAR ARRAY
0*
0mm
N2 = NSIMUL + 1
LENGTH - ((NPARA*(NPARA+1))/2)

0*
0*
DO 1000 ITIME=1,TNPTS
0*
0*
C* NOW TRANSFORM EACH OBJECT FUNCTION AT THE TIME POINT
0* 'TIME(ITIME)'
0*
DO 900 NF = 1, NFUNC
0*
0*
READ(WRITEUP)(F(K),K=1,NUMRow)
0*
0* NOW HAVING SET UP THE ARRAY F TRANSFORM IT
0
IF(METHOD .EQ. ITYP(2)) CALL WHT( NSIMUL, F, IFLAG,IWK(IHALF+1) )
IF( METHOD.EQ.ITYP(1)) CALL FFAST(F,NSIMUL,X,IWK,A,B)
0*

C* CALCULATE THE TIME SPENT IN TRANSFORMATION
CALL TIMES(3.O)

C

0* NOW CALCULATE THE PARTIAL VARIANCES

0*

IF(METHOD.EQ.ITYP(2))CALL WALPAR(F,NSIMUL,IW,NPARA,SWLJ,TOTVAR)
' IF(METHOD.EQ.ITYP(1))CALL FORPAR(A,B,NSIMUL,IW,NPARA,SWLJ,

252
Program TRANS CONT'D.

0*

C* CALC THE TIME SPENT IN PARTIAL VARIANCES CALCULATIONS
CALL TIMEs(4,o)

0*

0*

C* CALC THE PARTIAL VARIANCE STATISTICS

0*
DO 300 L=1,NPARA
D0 250 J=L,NPARA
INDEX - NPARA*(L-1) - (L*(L-1))/2 + J ,
MINSW(NF,INDEX) - AMIN1( MINSW(NF,INDEx), SWLJ(INDEx) )
MAXSW(NF,INDEX) - AMAX1( MAXSW(NF,INDEX), SWLJ(INDEx) )
AVESW(NF,INDEX)=((ITIME-1.)*AVEsw(NF,INDEx)+SWLJ(INDEx))/FLOAT(ITI
+ME)

250 CONTINUE

3OO CONTINUE

C* WRITE OUT THE EXPANSION COEFFICIENTS
0*
IF( METHOD .EQ. ITYP(2) ) GO TO 375
C* FOURIER METHOD
CALL OUTCF( A, B, N2, TIME(ITIME), ILABEL(NF) )
CALL OUTP( SWLJ, A(1), TOTVAR,TIME(ITIME),LENGTH,ILABEL(NF),B,IW,
+NPARA)
GO TO 400

375 CONTINUE
C* WALSH METHOD
CALL OUTCW(F, NSIMUL, TIME(ITIME), ILABEL(NF) )
CALL OUTP( SWLJ, F(1), TOTVAR, TIME(ITIME), LENGTH,ILABEL(NF),F,IW

+,NPARA)
4OO CONTINUE
0*
0*

0* COMPUTE TIME SPENT IN WRITING OUTPUT
CALL TIMES(5,0)

0*

0*

900 CONTINUE

1000 CONTINUE

0*
C* WRITE OUT DIAGNOSTICS
0*
DO 1100 NF=1,NFUNC
SUM - 0.0
WRITE( 2,1400) ILABEL(NF)
DO 1050 L=1,NPARA

253
Program TRANS CONT'D.

DO 1050 J=L,NPARA
INDEX = NPARA*(L-1) - (L*(L-1))/2 + J
WRITE( 2,1500) L,J,AVEsw(NE,INDEx),MINsw(NE,INDEX),MAXSW(NP,INDEx)
SUM . SUM + AVESW(NP,INDEx)
1050 CONTINUE .
WRITE( 2,1600) SUM
1100 CONTINUE
1400 E0RMAT(1H1,1OX,A1O,* CONCENTRATION STATISTICS *,/)
1500 FORMAT(1H ,* (*,Iz,*,*,I2,*) *,* AWESW =*,1PE14.6,
+3X,* MIN -*,E14.6,3X,* MAX . *,E14.6)
1600 FORMAT(/,/,/,1OX,* SUM OE AVERAGES -*,G14.6)
CALL TIMEs(1,1)
END

25“

Program TRANS CONT'D.
SUBROUTINE OUTP

SUBROUTINE OUTP(SWLJ,AVE,TOTVAR,TIME,LENGTH,LABEL,B,IW,NPARA)

(3*
QWWWWWWWW
C* SUBROUTINE OUTP WRITES OUT THE PARTIAL VARIANCES ON LOGICAL *
0* UNIT 'IUNIT’. *
CWWWWW
0*

REAL SWLJ(LENGTH)

REAL 3(1)

INTEGER Iw(1)

DATA IUNIT/9/
0*

STDDEV . SORT(TOTVAR)
RELDEv - 0.0
IF( AVE .EQ. 0.0 ) GO To 5
RELDEv . STDDEV/AVE
5 CONTINUE
WRITE(IUNIT , 10) LABEL , TIME, AVE , STDDEV, RELDEV, LENGTH , NPARA
1O PORMAT(A8,4(2X,E15.7).216)
WRITE(IUNIT,20)(SWLJ(L),L=1,LENGTH)
20 FORMAT(5(2X,E15.7))
WRITE(IUNIT,20)(R(Iw(L)+1),L=1,NPARA)
RETURN
END

255

Program TRANS CONT'D.
SUBROUTINE OUTCW

SUBROUTINE OUTcw(F,NCOEFF,TIME,LABEL)
CWWWWWWWMW*
0* SUBROUTINE OUTCW WRITES OUT THE WALSH EXPANSION COEFFICIENTS *
0* TO LOGICAL UNIT 'IUNIT' *
0W

REAL F(NCOEFF)

DATA IUNIT/B/

WRITE(IUNIT,10) LABEL,TIME,NCOEFF
1O FORMAT(A8,E15.7,16)

WRITE(IUNIT,20)(F(K),K=1,NCOEFF)

20 FORMAT(4020)

RETURN

END

256

Program TRANS CONT'D.
SUBROUTINE OUTCF

SUBROUTINE OUTCF(A,B,N2,TIME,LABEL)

GWWMWWWWW
C* ‘ *
c: SUBROUTINE OUTCF WRITES OUT THE FOURIER COEFFICIENTS *
C *
CMWWWMWW
REAL A(N2),B(N2)
DATA IUNIT/B/
c*

WRITE(IUNIT,10) LABEL,TIME,Nz
1O FORMAT(A8,E15.7,I6)
,WRITE(IUNIT,20)((A(X),B(K)),X=1,N2)
20 FORMAT(4020)
RETURN
END

257

Program TRANS CONT'D.
SUBROUTINE WALPAR

SUBROUTINE WALPAR(A, N, IW, NPARA, SWLJ ,TOTVAR)

I

SUBROUTINE WALPAR CALCULATES THE TOTAL VARIANCE AND *
0* PARTIAL VARIANCES GIVEN THE WALSH EXPANSIONS COEFFICIENTS *
0* AND THE FREQUENCY SET. ONLY THE SINGLE PARTIAL VARIANCES AND *
0: COUPLED PARTIAL VARIANCES, S(L,J) ARE COMPUTED. *
C *
0*. ALL PARTIAL VARIANCES ARE STORED IN A LINEAR ARRAY - *
0* (SWLJ( ) ), WHICH IS AN UNFOLDED UPPER TRIANGULAR MATRIX. *
0* THE DIAGONAL ELEMENTS, SWLJ( I,I ), ARE THE I'TH ISINGLE *
0* PARTIAL VARIANCES, AND THE (L,J)'TH ELEMENT IS THE COUPLED *
0* PARTIAL VARIANCE OF THE L'TH ND J'TH PARAMETERS. THIS IS *
0* LINEARLY FOLDED BY *
0* *
0* INDEX - NPARA*(L - 1) - (L*(L - 1))/2 + J *
0* *
0* REFERENCE: T.H. PIERCE PHD. THESIS, M.S.U. 1980 *
0* *
0* INPUT *
0* *
0* A - AN ARRAY OF THE WALSH EXPANSION COEFFICIENTS *
0* *
0* N - THE NUMBER OF EXPANSION COEFFICIENTS IN 'A' *
0* *
0* IV - AN INTEGER ARRAY OF THE FREQUENCY SET USED. *
0* *
0* NPARA . THE NUMBER OF PARAMETERS TO BE ANALYZED *
0* *
0* OUTPUT *
0* *
0* SWLJ . AN ARRAY OF THE SINGLE AND COUPLED PARTIAL VARIANCES *
0* *
0* TOTVAR - THE TOTAL VARIANCE OF THE EXPANDED FUNCTION *
0* PARSEVAL'S FORMULA - AO**2 *
0* *
0* *
0* RESTRICTIONS *
0* *
0* 1) Iw(J) MUST NEVER BE EQUAL TO Iw(X) FOR ANY J,K *
0* *
0: 2) SWLJ MUST BE DIMENSIONED AT LEAST (NPARA*(NPARA+1))/2 *
C *

REAL A(N)
REAL SWLJ(1)

258

Program TRANS CONT'D.
SUBROUTINE WALPAR

INTEGER IW(NPARA)

CWWWWWWW*

(3*
C‘.

CALCULATE THE TOTAL VARIANCE

REMEMBER TO ADD ONE(1) TO THE FREQUENCIES TO ACCOUNT FOR
THE FREQUENCY AO STORED AS A(1)

CWWWWWWW*

0*

100
C‘I'
0*

200

0*

C‘.

300
400

TOTVAR - 0.

SKIP A(1) AS THIS IS THE AVERAGE VALUE
D0 100 J=2,N .

TOTVAR - TOTVAR + A(J)*A(J)

CONTINUE

CALCULATE THE SINGLE PARTIAL VARIANCES

DO 200 Ja1,NPARA

INDEX - NPARA*(J-1) - (J*(J-1))/2 + J
SWLJ(INDEx) - (A(Iw(J)+1)*A(Iw(J)+1))/TOTVAR
CONTINUE

CALCULATE THE COUPLED PARTIAL VARIANCES

NPARM1 . NPARA - 1

DO 400 L=1, NPARM1

JSTART . L + 1

D0 500 J = JSTART, NPARA

IF THE FREQUENCIES ARE EQUAL; IVAL=O (MISTAKE)
IF(Iw(L) .EQ. Iw(J) ) STOP "WALPR"

IVAL a XOR(IW(L),Iw(J))

INDEX . NPARA*(L-1) - (L*(L-1))/2 + J
SWLJ(INDEx) . (A(IVAL + 1)*A(IVAL + 1))/TOTVAR
CONTINUE

CONTINUE

RETURN

END

259

Program TRANS CONT'D.
SUBROUTINE WHT

SUBROUTINE WHT(NUM,X,II,Y)
CWMWWWWW

*****
C**** II . 0 HADAMARD-ORDERED WHT **
0**** II - 1 INVERSE HADAMARD-ORDERED WHT **
0**** II - 2 WALSH-ORDERED WHT **
0**** II - 3 INVERSE WALSH-ORDERED WHT **
0**** **
0**** THIS ROUTINE CALCULATES THE PAST WALSH-HADAMARD **
.0**** TRANSFORMS (WHT) FOR ANY GIVEN NUMBER WHICH **
C**** IS A POWER OF TWO. **
0**** **
C**** NUM . NUMBER OF POINTS **
0**** X(NUM) - ARRAY OF DATA TO BE TRANSFORMED **
C**** ON OUTPUT X(NUM) IS THE TRANSFORMED **
C**** EXPANSION COEFFICIENTS **
0* *
0* - REFERENCE: AHMED AND RAO, "ORTHOGONAL TRANSFORMS FOR *
0* DIGITAL SIGNAL PROCESSING ", SPRINGER- *
0* VERLAG, (1975). *
0* i

0

DIMENSION IPOWER(20),X(NUM),Y(NUM)
IF(II.LE.1) GO TO 14

0**** BIT REVERSE THE INPUT
DO 11 I=1,NUM
IB a I - 1
VIL = 1

9 IBD = IB/2

IPOWER(IL) = 1
IF(IB.EQ.(IBD*2)) IPOWER(IL) = O
IF(IBD.EQ. 0) GO TO 10
IB = IBD
IL = IL + 1
GO TO 9
1o CONTINUE
IP . 1
IFAC . NUM
DO 12 I1 - 1,IL
IFAC . IFA0/2
12 IP . IP + IFAO*IPOWER(I1)
11 Y(IP) - x(I)
D0 13 I - 1,NUM
13 X(I) = Y(I)
14 CONTINUE
0**** CALCULATE THE NUMBER OF ITERATIONS
55 ITER . 0

c****
c****

c****
c****
c****

c****
c****
c****

48

49
C****
0****

15

260

Program TRANS CONT'D.
SUBROUTINE WHT

IREM = NUM

IREM = IREM/2
IF(IREM.EQ.O) GO TO 2
ITER=ITER + 1

GO TO 1

CONTINUE

BEGIN A LOOP FOR (LOG TO BASE TWO OF NUM) ITERATIONS
D0 50 M81,ITER

CALCULATE THE NUMBER OF PARTIONS
IF(M.EQ.1) NUMP . 1
IF(M.NE.1) NUMP = NUMP*2
MNUM . NUM/NUMP
MNUM2 = MNUM/2

BEGIN A LOOP FOR THE NUMBER OF PARTITIONS.

ALPH . 1.
D0 49 MP . 1,NUMP
IB . (MP-1)*MNUM

BEGIN A LOOP THROUGH THIS PARTITION.

DO 48 MP2 . 1,MNUM2

MNUM21= MNUM2 + MP2 + IB

IBA = IB +MP2

Y(IBA) - x(IBA) + ALPH*X(MNUM21)
Y(MNUM21) = x(IBA) - ALPH*X(MNUM21)
CONTINUE

IF(II.GE.2) ALPH = -ALPH

CONTINUE

Do 7 I=1,NUM

x(1) = Y(I)

CONTINUE -

IF(II.EQ.1 .OR. II.EQ.3) RETURN
R-1./FLOAT(NUM)

D0 15 I-1,NUM

X(I) . x(I)*R

RETURN

END

261

Program TRANS CONT'D.
TRANP

SUBROUTINE TRANP( IUNIT, READUP, READOWN, WRITEUP WRITED,

+ TNPTS, NFUNC, NSIMUL, 0, UP, DOWN, LENGTH,NUMROW3
0*
C**********************************************************************

0* SUBROUTINE TRANP TAKES THE MATRIX STORED ON 'IUNIT' AND *
0* TRANSPOSES IT. ( A(I,J) => A(J,I) ) RETURNING THE *
0* TRANSPOSED MATRIX ON LOGICAL UNIT 'WRITEUP'. *
0* , *
0* NOTES *
0* *
0* 1) THE ARRAY '0' MAY BE EQUIVALENCED TO EITHER THE ARRAY *
0* 'UP' OR THE ARRAY 'DOWN'. *
0* *

*

*

0* 2) LENGTH MUST BE ONE POWER OF TWO GREATER THAN NROW, UNLESS
0* NROW IS A POWER OF TWO.

CWWWWWWW
INTEGER READUP,READOWN,WRITEUP,WRITED
INTEGER TNPTS,NSIMUL,NFUNC,IUNIT
Cmmmmmmm
0* THE VECTOR 'C' SHOULD BE OF DIMENSION ONE POWER OF TWO
0* GREATER THAN NROW( UNLESS NROW IS A POWER OF TWO )
REAL 0(LENGTH)
* REAL UP(LENGTH),DOWN(LENGTH)
0
DATA KOUNT/O/, ZERO/O./, LCOUNT/O/, NUMADD/O/
DATA NUMADD2/O/
NCOL . TNPTS*NFUNC
NROW , NSIMUL
0* READ IN THE MATRIX
1 CONTINUE
D0 1000 J=1, TNPTS
ISTR a (J-1)*NFUNC + 1
ISTOP = ISTR + NFUNC - 1
READ(IUNIT,10000)(C(K),X=ISTR,ISTOP)
10000 FORMAT(4020)
1000 CONTINUE
0* IF KOUNT=0 ,THEN WE NEED TO WRITE THE UP-TAPE(TAPE1 INITIAL Y)
0* IF KOUNT . 1 , THEN WE WRITE THE DOWN-TAPE( TAPE2 INITIALLY
0*
IF(XOUNT .NE. 0 ) GO TO 5
0* WRITE THE ODD ROWS
DO 500 K=1,NCOL
500 WRITE(READUP) C(K)
KOUNT - 1
c*
0: LCOUNT COUNTS THE NUMBER OF ROWS WRITTEN, BOTH UP AND DOWN
C
LCOUNT = LCOUNT + 1

262

Program TRANS CONT'D.
TRANP

0* IF DONE, IE LCOUNT = NUMBER OF ROWS, THEN GO TO NEXT TASK
0* IF NOT DONE, THEN CONTINUE READ-WRITE

(3*
IF( LCOUNT .EQ. NROW ) GO TO 9
Go TO 1

5 CONTINUE

0* WRITE THE EVEN ROWS
D0 510 K=1,NCOL
510 WRITE(READOWN) 0(K)
0* SET KOUNT=O so NEXT WRITE IS 'DOWN'
KOUNT - O
LCOUNT . LCOUNT + 1
0* CHECK FOR END OF DATA
IF(LCOUNT .EQ. NRow) GO TO
GO TO
9 CONTINUE
0* TAPE 1,2 ARE WRITTEN WITH THE MATRIX NOW WE NEED TO MAKE SURE
0* THAT THE ROW-DIMENSION IS A POWER OF TWO,AND IF NOT THEN WE
0* MUST ADD SUFFICIENT ZEROS TO MAKE THE ROW-DIMENSION A POWER OF 2

0* CHECK FOR HAVING WRIITEN AN EVEN NUMBER OF ROWS
IF(KOUNT .EQ. o ) GO TO 120

0* ODD NUMBER OF ROWS WERE WRITTEN

0* ADD ONE ROW 0F ZEROS
D0 115 K-1,NCOL
WRITE(READOWN ) ZERO

115 CONTINUE
LCOUNT - LCOUNT + 1

0* EVEN NUMBER OF ROWS WRITTEN

120 CONTINUE

0* FIGURE OUT EXPONENT OF NEAREST POWER OF TWO LARGER
D0 116 M=1,50
MDIVID = M ,
RTEST - FLOAT(LCOUNT)/(2.**M)
IF ( RTEST .LE. 1 ) GO TO 118

116 CONTINUE
STOP 2

118 CONTINUE

0* CHECK FOR EXACT POWER OF TWO
IF (RTEST .EQ. 1. ) GO TO 200

0* NOW CALCULATE THE NUMBER OF ROWS WE MUST ADD
NUMADD - 2**MDIVID - LCOUNT

0* NUMADD SHOULD ALSO BE DIVISIBLE BY TWO
IF(NUMADD .NE. 2*(NUMADD/2) ) STOP 3

0* WRITE ZEROS INTO DUMMY ROWS
NUMADD2 . NUMADD/z

130
200
0*
0*
0*

263

Program TRANS CONT'D.
TRANP

D0 130 L=1,NUMADD2
DO 130 K=1,NCOL
WRITE(READUP ) ZERO
WRITE(READOWN ) ZERO
CONTINUE

CONTINUE

LET NUMADD BE THE TOTAL NUMBER OF ADDED ROWS
REMEMBER WE MAY HAVE ADDED A ROW EARLIER

IF( NROW .NE. LCOUNT )NUMADD - NUMADD + 1

0* THE FINAL CHECK

0*

0*
C$

101

112

1O
0'.
C‘.’
C"
0*

20

(3*
0*

3O

NUMROW . NUMADD + NROW

IF(NUMROW .NE. (2**MDIVID)) STOP 4
EVEN AND ODD TAPE WRITTEN

LOOP = 1

REWIND READUP

REWIND READOWN

REWIND WRITEUP

REWIND WRITED

DIAGNOSTICS

WRITE( 2,101)

F0RMAT(/,/,/ * SUBROUTINE TRANP STATISTICS *,/)
WRITE( 2,112 NROW,NCOL,NUMADD,NUMADD2

FORMAT(* *,* NR0W=*,I5,* NCOL=*,I4,* NUMADD-*,I6,

+* NUMADD2-*,I5)

NINSERT - NCOL
NCHECK = NUMROW/2
CONTINUE
INITIALIZE FOR THE READ-WRITE
KOUNT - NUMBER OF INSERTS DONE
LCOUNT = TOTAL NUMBER OF READ-WRITES DONE THIS
ITERATION
KOUNT - O
LCOUNT = O
CONTINUE
READ(READUP )(UP(L),L=1,LOOP)
READ(READOWN)(DOWN(L),L=1,LOOP)
WRITE(WRITEUP)(UP(L),L-1,IOOP),(DOWN(M),M=1,LOOP)
KOUNT . KOUNT + 1
IF( KOUNT .NE. NINSERT ) GO TO 20
LCOUNT - LCOUNT + 1
LCOUNT SHOULD EQUALNCHECK HERE
ONLY IF NCHECK - 1 AND IT IS THE LAST MIX
IF( LCOUNT .EQ. NCHECK ) GO TO 65
KOUNT . 0
CONTINUE
READ(READUP )(UP(L),L=1,LOOP)

5O

(3*

65

264

Program TRANS CONT'D.
TRANP

READ(READOWN )(DOWN(L),L=1,LOOP)

WRITE(WRITED)(UP(L),L=1,LOOP),(DOWN(M),M=1,LOOP)
KOUNT . KOUNT + 1

_IF( KOUNT .NE. NINSERT ) GO TO 30

LCOUNT = LCOUNT + 1

IF( LCOUNT .EQ. NCHECK ) GO TO 50
KOUNT 8 0

GO TO 20

CONTINUE

LOOP 8 LOOP*2

NCHECK a NCHECK/2

ERROR CHECKING -

IF( NCHECK .LE. 0 ) STOP 2
ISAVUP = READUP
ISAVD =READOWN
READUP . WRITEUP
READOWN . WRITED
WRITEUPuISAVUP
WRITED= ISAVD
REWIND READUP
REWIND READOWN
REWIND WRITEUP
REWIND WRITED

GO TO 10
CONTINUE

REWIND WRITEUP
RETURN

END

265

Program TRANS CONT'D.
SUBROUTINE TIMES

SUBROUTINE TIMES(ISUB,ITYPE)

CWWWWWWWW

0*
0*
0*
0*
0*
0*
0*
0*
0*
0*

SUBROUTINE TIMES COMPUTES THE CPU TIME SPENT BETWEEN CALLS.

THIS IS INSTALLATION DEPENDENT.
ITS USE IS FOR DOCUMENTATION PURPOSES ONLY

ISUB . THE PROCEDURE TO BE TIMED.

ITYPE 8 FLAG: .LT. 1 FOR TIMING
.GE. 1 FOR FINAL PRINT

*#***** ***

CWWWWWWW

C.

0*

0*

0*

5

100

150
10

160

REAL TIMS(15).NEW,LAST
INTEGER NAME(15)

DATA TIM3/15*O./
DATA LAST /O./

DATA NAME/1OHREAD INPUT,1OHTRANSPOSE ,1OHTRANSFORM ,1OHPARTIALVAR

+,1OHWRITE OUTP/

DATA LENGTH /5/

IF(ITYPE .GE. 1) GO TO 5
NEW a SECOND(CPU)
INITIALIZE THE VALUES IN FIRST ENTRY
IF( LAST .EQ. o. ) LAST . NEW
TIMS(ISUB) - TIMs(ISUB) + NEW - LAST
LAST . NEW
RETURN
CONTINUE
WRITE( 2,100)
FORMAT(* *,/,/,/,5X,* SECONDS*,4X,* PROCEDURE *,/)
TSUM = 0.
DO 10 J=1,LENGTH
TSUM - TSUM + TIMs(J)
WRITE( 2,150) TIMs(J),NAME(J)
FORMAT(* *,5X,F7.3,5X,A10)
CONTINUE
WRITE( 2,160) TSUM
FORMAT(* *,/,/,6X,F7.3.4x,* TOTAL TIME *)
RETURN
END

266

Program TRANS CONT'D.
SUBROUTINE FFAST

SUBROUTINE FFAST(F,NPTS,X,IWK,A,B)

i
i

0* SUBROUTINE FFAST COMPUTES THE FOURIER TRANSFORM OF A VECTOR *
0* IN THIS CASE 'F' IS LENGTHENED FROM ( -PI/2 , PI/2 ) TO *
0* ( O , 2PI ) AND THEN FOURIER-TRANSFORMED INTO COSINE AND SINE *
0* COEFFICIENTS, A AND B RESPECTIVELY. *
0* *
0* NOTE *
0* *
0* 1) SINCE 'F' Is NEVER USED IN FFCSIN IT MAY BE EQUIVALENCED *
0* To 'IWK'. *
0* *
0* 2) THE ROUTINES ALSO ALLOW THE EQUIVALENCING OF 'A' AND *
0* 'X'. *
0* *
0* 3) A0, THE AVERAGE VALUE, IS STORED AS 'A(1)'. *
0

REAL F(NPTs)

REAL x(1),A(1),B(1)
0*

INTEGER IWK(1)
0* NPTS MUST BE AN ODD INTEGER AND NOTE WE ARE GOING FROM

0* -PI/2 TO PI/2 AND TRANSFORMING TO (O,2*PI)
c******
0*

NPTSP1=NPTS + 1

NPTSZ=NPTS*2

N2=NPTS+1

RNPTS2=(1.O/FLOAT(NPT52))
IQ=(NPTS-1)/2

IQP1=IQ+1
0*
C
L=0
C
0* TRANSFORM F(-PI/2 , PI/2) TO X(O , 2*PI)
0

DO 1000 J=IQP1,NPTS

L=L+1

1000 x(L)-F(J)
DO 2000 J=1,IQP1
L=L+1
JJ=NPTSP1-J
x(L)=F(JJ)

2000 CONTINUE
DO 3000 J=1,IQ
L=L+1

267

Program TRANS CONT'D.
SUBROUTINE FFAST

JJ=IQP1-J
x(L)=F(JJ)
3000 CONTINUE
D0 4000 J=1,IQ

L=L+1
x(L)=F(J)
4000 CONTINUE
C
C

0* CALL IMSL ROUTINES T0 CALCULATE FOURIER COEFFICIENTS
0
CALL FFCSIN(X,NFT52,A,E,IVH)
0* . ,
0* SCALE THE COEFFICIENTS TO THEIR CORRECT VALUES.
DO 350 J=1,N2
A(J)=RNFT52*A(J)
B(J)=RNFT52*B(J)
350 CONTINUE
A(1)-A(1)/2.0

A(N2)=A(N2)/2.0
B(1)=O.O
B(N2)=0.0

C

c!-
RETURN

END

 

268

Program TRANS CONT'D.
SUBROUTINE FORPAR

SUBROUTINE FORPAR(A, B, NPTS, IW, NPARA, SWLJ, T0TVAR,IACCUR)
CWWW
0* CALCULATE THE PARTIAL VARIANCES
0* THIS ROUTINE IS SET UP FOR 4TH ORDER ACCURATE FREQUENCY
0* SETS
0* FIRST CALCULATE THE VARIANCE ( PARSEVAL"S FORMULA - A0**2

C* THEN CALCULATE THE SUM OF HARMONICS NOTING THAT ALL BUT THE
C’ NPARA'TH ARE SUMMED TO THE FIRST HARMONIC AND THE NPARA'TH ONLY T

t It!‘ 8*

t

0: THE FUNDAMENTAL HARMONIC ' z
0 .

0* ALL PARTIAL VARIANCES ARE STORED IN A LINEAR ARRAY *
0* (SWLJ( ) ), WHICH IS AN UNFOLDED UPPER TRIANGULAR MATRIX. *
0* THE DIAGONAL ELEMENTS, SWLJ( I,I ), ARE THE I'TH ISINGLE *
0* PARTIAL VARIANCES, AND THE (L,J)'TH ELEMENT IS THE COUPLED *
0* PARTIAL VARIANCE OF THE L'TH ND J'TH PARAMETERS. THIS IS *
0* LINEARLY FOLDED BY *
C* *
0* INDEX - NPARA*(L - 1) - (L*(L - 1))/2 + J *
C* i
0* REFERENCE: T.H. PIERCE PHD. THESIS, M.S.U. 1980 *
0* n
C’ INPUT a
C* a
0* A - AN ARRAY OF THE COSINE EXPANSION COEFFICIENTS *
C’ *
0* B . AN ARRAY OF THE SINE EXPANSION COEFFICIENTS *
C* *
0* NPTS - NUMBER OF SIMULATIONS *
C* i
0* IW . AN INTEGER ARRAY OF THE FREQUENCY SET USED. *
C* *
0* NPARA . THE NUMBER OF PARAMETERS TO BE ANALYzED *
C* i
0* OUTPUT *
C* *
0* SWLJ = AN ARRAY OF THE SINGLE AND COUPLED PARTIAL VARIANCES *
0* a
0* TOTVAR - THE TOTAL VARIANCE OF THE EXPANDED FUNCTION *
0: PARSEVAL'S FORMULA - A0**2 *
C it
0* *
0* RESTRICTIONS *
C’ *
0* 1) Iw(J) MUST NEVER BE EQUAL T0 Iw(K) FOR ANY J,X *
C‘ *
0* 2) SWLJ MUST BE DIMENSIONED AT LEAST (NPARA*(NPARA+1))/2 *

CWWWWWWWW
REAL A(1),B(1),SWLJ(1)

0*

ci-

0*
C"

600

269

Program TRANS CONT'D.
SUBROUTINE FORPAR

INTEGER Iw(NPARA)

IRANGE = IACCUR - 1

IMID = IACCUR/2

N2-NPTS+1

NPARM1 = NPARA - 1

TOTVAR = 0.0

SKIP A(1) AND B(1) AS A0, THE AVERAGE VALUE, IS
STORED AS A(1); B(1) - 0.

D0 600 JJ=2,N2 -

TOTVAR = A(JJ)*A(JJ) + B(JJ)*B(JJ) +TOTVAR
CONTINUE

C* LP IS THE HARMONICS

625

650

' 675

C*
C*

Ci’

D0 650 L=1,NPARM1
SUM a 0.0

DC 625 LP . 1,2

LPWP1=LP*Iw(L) + 1

SUM . A(LPWP1)*A(LPWP1) + B(LPHP1)*B(LPWP1) + SUM
CONTINUE

INDEX = NPARA*(L-1) - (L*(L-1))/2 + L

SWLJ(INDEX) = SUM/TOTVAR

CONTINUE

SUM=O. 0

D0 675 L=1,1

LPWP1 = L*Iw(NPARA) + 1

SUM = SUM + A(LPWP1)*A(LPWP1) + B(LPWP1)*B(LPWP1)
CONTINUE

INDEX = (NPARA*(NPARA+1))/2

SWLJ(INDEX) = SUM/TOTVAR

D0 900 L21, NPARM1

JSTART = L + 1

DO 800 J=JSTART, NPARA

SUM . 0.

DO 750 HP=1,IRANGE

IP - IMID - KP

D0 700 IK - 1, IRANGE

K = IMID - IK

IF(IP .EQ. 0) GO TO 750

IF( K .EQ. 0) GO TO 700

ADD ONE (1) TO THE FREQUENCY COUNT TO ACCOUNT FOR A0 B0
9

IFREQ a IP*Iw(L) + K*IW(J) + 1

IF(IFREQ .LE. 1 ) 00 TO 700

IF(IFREQ .GE. N2) 00 TO 700

700
750

900

270

Program TRANS CONT'D.
SUBROUTINE FORPAR

SUM = A(IFREQ)*A(IFREQ) + B(IFRE0)*B(IFREQ) + SUM
CONTINUE

CONTINUE

INDEX = NPARA*(L-1) - (L*(L-1))/2 + J

SWLJ(INDEX) = SUM/TOTVAR

CONTINUE

CONTINUE

RETURN

END

271

Program PLOTSEN

PROGRAM PLTSEN(INPUT,OUTPUT,TAPE1=INPUT,TAPE2=OUTPUT,
+ TAPE9)
CW*W*WWWWW

0* THIS PROGRAM PLOTS RESULTS OF TAPE9 SENSITIVITY ANALYSIS FILE. *
0* THE PROGRAM READS CARDS FOR INFORMATION ON WHAT TO PLOT. *
0* IT THEN SEARCHES THE FILE (TAPE9) FOR THE DESIRED VALUES, *
0* AND PLOTS IT ON A LINE PRINTER. *
0* IF MORE THAN ONE PLOT IS DESIRED, IT REWINDS THE FILE *
0* AND REPEATS. *
0* a
0* INPUT *
0* cum1 *
0* *
0* NPLOT, NCONC (2I5 FORMAT) *
0* i
0* NPLOT a THE TOTAL NUMBER OF PLOTS *
0* DESIRED. *
0* *
0* NCONC= THE NUMBER OF DIFFERENT OUTPUT *
0* FUNCTIONS IN TAPE9. *
0* *
0* NOTE CARDS 2-6 ARE TO BE REPEATED FOR ALL THE DESIRED OBJECT *
0* FUNCTIONS. *
0* *
0* CARD 2 *
0* ITEST ( A10 FORMAT) *
0* i
0* ITEST . THE LABEL OF THE OBJECT (OR *
0* OUTPUT) FUNCTION TO BE PLOTTED :
0*

0* CARD 3 *
0* *
0* NFUN0,NPOINT *
cl» - I»
0* NFUNC - THE NUMBER OF FUNCTIONS T0 PLOT *
0* FOR THE LPT'TH PLOT. *
0* *
0* NPOINT - THE NUMBER OF POINTS TO BE PLOTTED *
0* IN THE LPT'TH PLOT (X-AXIS) :
C"

0* CARD 4 *
0* *
0* ITITLE(8) (8A1O FORMAT ) *
0* i
0* ITITLE = THE PLOT TITLE FOR THE LPT'TH *

272

Program PLOTSEN CONT'D.

0* CARD 5
0* SYM(K) (1OA1 FORMAT)

0* , SYM(K) . THE SYMBOLS TO BE USED IN THE PLOT.
0* ( IE THE SYMBOL FOR THE KTH FUNCTION).

0* CARD 6 - CARD(5+NFUNC)

0* NAME(K) (A10 FORMAT)

* t * t t *1! Cat *1! t t

0* NAME<K> ' THE NAME (LABEL) OF THE KTH FUNCTION*
0* TO BE PLOTTED. *
0* *
Ci§iii§fiiQfiflGflflQ§i§§i*fifi§§i*fl§G§iG§Q§é*§i*§§fifliiiififlfii§§§§i§§§§§§§§§§§i§

REAL TIME(1OO)
REAL PLT(1OO,10)
REAL DELx(100)

cl»
0HARAOTER*1O JNAME(10),NAME(10),ITITLE(B)
CHARACTER*1O ITEST
CHI,
COMMON /PLTPTs/ sm(10)
0*
DATA IO/2/, IS/1/, DELX/1OO*0./
DATA MAX/100/
Ca»
0* READ IN CARD INPUT
c-I-
cl»
0* READ IN:
0* NFUNC = NUMBER OF FUNCTION TO PLOT
0* NPOINT a NUMBER OF POINTS PER PLOT
0* NPLOT . NUMBER OF PLOTS
C1}

READ(1,2O) NPLOT,NCONC
20 F0RMAT(3IS)
WRITE(2,30) NPLOT
30 FORMAT(‘1',’ THERE ARE ‘,I5,' PLOTS')

DO 1000 LPT=1,NPLOT
0* READ IN CORRECT OBJECT FUNCTIONS

READ(1,35) ITEST
3S FORMAT(A10)

273

Program PLOTSEN CONT'D.

0* READ IN THE NUMBER OF FUNCTIONS TO BE PLOTTED AND

A‘A—J

0* THE NUMBER OF POINTS TO PLOT PER FUNCTION

READ(1,*) NFUNC,NPOINT
45 FORMAT(2I5)
0*
0* READ IN PLOT TITLE
READ(1,10)(ITITLE(K),K=1,8)
1O F0RMAT(BA10)
0* READ IN PLOT SYMBOLS
READ(1,55)(SYM(K),K=1,NFUN0)
55 FORMAT(1OA1)
(3*
0* READ IN THE NAME OF THE PLOTTED FUNCTIONS
DO 80 K=1,NFUNC
READ(1.75) NAME(K)
75 FORMAT(A10)
80 CONTINUE
NKOUNT - NCONC*NPOINT
0*
LPLOTsLPT
CALL READ9(NKOUNT, NFUNC,TIME,ITEST,NCONC,LPLOT,PLT,ITPTs)
IFLAG . 0
IF( ITPTS .NE. NPOINT ) IFLAG - 1
IF ( IFLAG .EQ. 1) WRITE(2,310) NPOINT,ITPTS
310 FORMAT(‘1',’ NUMBER OF POINTS EXPECTED =',I6./.SX,
+ ' NUMBER OF POINTS READ =',I6)
IF( IFLAG .EQ. 1 ) NPOINT - ITPTS

0*
CALL PLOT(IO,PLT,DELX,IS,ITITLE,NAME,TIME(1),TIME(NPOINT),NPOIII,
+ NFUNO,MAX)

c*

0* WRITE OUT THE PLOTTED POINTS.

c*

WRITE(2,300)(NAME(K),K=1,NFUNc)
300 FORMAT(‘1',’ POINT',10(1X,A1O,1x))

D0 350 Ka1,NPOINT

WRITE(2,325)( K,(PLT(K,L),L=1,NFUN0))
325 FORMAT(' ',I4,1X,1O(1X,1PE10.3,1x))
350 CONTINUE

REWIND 9
1000 CONTINUE

END

27M

Program PLOTSEN CONT'D.
SUBROUTINE READ9

SUBROUTINE READ9 (NKOUNT , NFUNC , TIM, ITEST , NCONC , LPLOT , PLT , ITIME)

CWWWWWWMW

0*
(3*
cl-
0*
0*
0*
0*
(3*
0*
0*
0*
c*
0*
C*
(3*
C1!

SUBROUTINE READ READS IN THE OUTPUT TAPE7 FROM PROGRAM
TRANS. THIS IS READ IN SO THAT IT MAY BE PLOTTED.

INPUT UNIT ' 9

OUTPUT UNIT 3 2

VARIABLES :

NKOUNT = NUMBER OF TOTAL SENSITIVITY POINTS
NCONC = NUMBER OF CONCENTRATIONS

TIM(70) = TIME POINT OF S. A. POINT

****************

CW

C"

0*

C"

200

350
C

CHARACTER*1O LABEL,ITEST
REAL PLT(100,10)

REAL TIM(100),SWLK(50),B(50)
DATA IIN/9/

ITIME 8 O
ISCALE = 1

DO 1000 KOUNT = 1,NKOUNT

READ(IIN,10) LABEL,TIME,AVE,STDDEV,RELDEV,LENGTH,NPARA
FORMAT(A1O,4(2X,E15.7).2I6)

READ(IIN,2O)(SWLK(K),K=1,LENGTH)
F0RMAT(5(2X,E15.7))

READ(IIN,20)(B(L),L=1,NPARA)

FINDS CORRECT CONCENTRATION LABEL

DO 200 I=1,NCONC
IVAL=I
IF( LABEL .EQ. ITEST) GO TO 350
CONTINUE
GO TO 1000
CONTINUE
NORMILIZE LPLOT TO ( 1,2,3,4)

*

275

Program PLOTSEN CONT'D.
SUBRO UTINE READ9

c*

IF(LPLOT .LE. 4) GO TO 400

. LPLOT =- LPLOT - 4

00 T0 350
400 CONTINUE
0* FOUND THE CORRECT OBJECT FUNCTION
0* SAVE THE DESIRED VALUES.

ITIME . ITIME + 1

TIM(ITIME) - TIME

GO TO ( 500. 550, 600, 650 ) LPLOT
500 CONTINUE

C SAVE THE AVERAGE VALUE
0
PLT(ITIME,1) . AVE
00 TO 1000
C
550 CONTINUE
0 SAVE THE RELATIVE DEVIATION CURVE
PLT(ITIME,1) = RELDEV
00 TO 1000
C
600 CONTINUE
C SAVE THE SINGLE PARTIAL VARIANCES

D0 610 NP=1,NPARA
INDEX = NPARA*(NP-1) - (NP*(NP-1))/2 + NP
PLT(ITIME,NP) s SWLK(INDEX)
610 CONTINUE

00 TO 1000
C
650 CONTINUE
0
0 SAVE THE EXPANSION COEFFICIENTS
D0 660 NP - 1, NPARA
PLT(ITIME,NP) = B(NP)
660 CONTINUE
0
1000 CONTINUE
0*

RETURN
END

C*

276

Program PLOTSEN CONT'D.
SUBROUTINE PLOT

SUBROUTINE PLOT(IO,ARAY,XARAY,ISCALE,JNAME,NAME,BLOW,BHI,NPT,NF,

+ MAX)

i

0*
(2*
0*
c*
C

0*
0*
C*
0*
(3*
(1*
C*
(3*
0*
0*
0*
0*
0*
0*
0*
C

0*
0*
C*
0*
C*
c*
0*
0*
C*
0*
0*
C*
0*
0*

FUNCTION TO BE READ IN USING A1 FORMAT OR DEFINED WITH 1H FORMAT

IO 8 THE OUTPUT UNIT

ISCALE 8 TYPE OF PLOT DESIRED, 1=II LINEAR SCALE ,2= SEMILOG,

3' LOG-LOG. FOR LOG-LOG READ IN EQUAL INTERVALS ON A LOG
SCALE.

NO MIXING OF 1,2,0R 3 ALLOWED IN THE SAME PLOT.

BLOW= THE LOWER BOUND OF THE PLOT FOR THE X-AXIS
BHI ' THE UPPER BOUND OF THE PLOT FOR THE X-AXIS

NPT = THE NUMBER OF POINTS PER FUNCTION TO BE PLOTTED

************

MAX 3 THE INNER DIMENSION OF THE ARAY DEFINED IN THE MAIN PROGRAM*

NF - THE NUMBER OF FUNCTIONS TO BE PLOTTED

33* lit

ARAY(MAX,NF) - ARRAY OF POINTS TO BE PLOTTED

*

XARAY(MAX) ' THE ERRORS ASSOCIATED WITH THE POINTS FOR THE FIRST *
FUNCTION. ONLY THE FIRST FUNCTION WILL BE PLOTTED WITH ERROR*

BARS *

*

JNAME(8) ' THE TITLE OF THE PLOT. THIS WILL BE PRINTED AT THE TOP*
OF THE PLOT (8A1O FORMAT ) :
NAME(10) = THE NAME OF EACH FUNCTION TO BE PLOTTED(USE A10 FORMAT*
OR 10H ) *

H

A LABELED COMMON BLOCK IS ALSO REQUIRED *
THIS BLOCK CONTAINS THE SYMBOLS TO BE USED IN THE *
PLOT FOR EACH FUNCTION *
USE *
COMMON /PLTPTS/ POINT(10) *

WHERE POINT(I) IS THE SYMBOL FOR THE ITH :

CWflmWWWWWW

Ci

C‘I’

DIMENSION ARAY(MAX,NF)
REAL' XMAx(10),XMIN(1o)
0HARACTER*7 30(3)
0HARACTER*1O NAME(10),JNAME(10)

XARAY(MAX)

DIMENSION VAL(106)
DIMENSION XDIv(3),XMIN1(11)

0*

C“

C“

160
50

277

Program PLOTSEN CONT'D.
SUBROUTINE PLOT

COMMON /PLTPTs/ POINT(10)

DATA BLANK/1H /,

* DASH/1H-/,STAR/1H*/

DATA XDIV/1.,2.,5./
DATA SC/‘LINEAR','LOC','LOC LOG'/

IF( NPT .LE. MAX ) GO To 5
WRITE(IO,1)(JNAME(K),K=1,8)

FORMAT(' ARRAY SIZE TOO LARGE IN PLOT OF',/,3x,8A1O)
RETURN

CONTINUE

IF( BLOW .LT. BHI) GO TO so

WRITE(IO,160) BLow,BRI

FORMAT(' X-AXIS MINIMUM AND MAXIMUM ARE NOT',E1S.7,'.GE.',E15.7)
CONTINUE

C’ TO PAGE OR NOT TO PAGE

0\

OKONCD

12
13

O

CO

630

IF(NPT.LE.40) GO TO 8
WRITE(IO,6)
PORMAT('1')
GO TO 9
WRITE(IO.7)
FORMAT(/////)
CONTINUE

WRITE HEADER FOR PLOTS
WRITE(IO,10)((JNAME(K),K=1,8),SC(ISCALE))
FORMAT(' PLOT OF ',8A10,5X,A10,' SCALE',/)
wRITE(IO,13)
WRITE(IO,12)(POINT(I),NAME(I),I=1,NF)
FORMAT(1OX,A2,5X,A10)
FORMAT(SOH ----------------------------- ,/)
WRITE(IO,13)

FIND MAXIMUM AND MINIMUM

M-o
INITIALIZE XMAX AND XMIN
DO 630 LD-1,NF
XMAx(LD)-ARAY(1,LD)
XMIN(LD)=ARAY(1,LD)
CONTINUE
DO 20 LDS=1,NP
DO 20 L-1,NPT
IF(ARAY(L,LDS).GT.XMAX(LDS)) XMAx(LDS)=ARAY(L,LDs)
IF(ARAY(L,LDS).LT.XMIN(LDS)) XMIN(LDs)=ARAY(L,LDs)
CONTINUE
CHECK FOR ZERO ARRAYS

640
21

156

650
25

22

23

26
29
27

28

3O
31
32
C‘.
0*

C‘.’
80

278

Program PLOTSEN CONT'D.
SUBROUTINE PLOT

DO 640 NSE=1,NE
IF(XMAX(NSF).EQ.O..AND.XMAX(NSF).EQ.XMIN(NSF)) WRITE(IO,21)NSF
CONTINUE

FORMAT(' ALL POINTS IN THE',I3,' GRAPH OF THIS PLOT ARE ZERO')
IP(ISCALE.EQ.3) GO To 2
DIv=(BHI-BLow)/(NPT—1)
IF(ISCALE.NE.1) GO TO 3
FAC=1

XMIN2=2.**4O
XMAX1--XMIN2

Do 650 JL-1,NE
IF(XMAX(JL).GT.XMAX1)XMAX18XMAXEJLg
IF(XMIN(JL).LT.XMIN2)XMIN2-XMIN JL
CONTINUE

XDIE=XMAx1-XMIN2

DO 22 L-1,3

IF((XDIF/XDIV(L)).LE.100.) GO To 23
CONTINUE

FAC=FAC*10.

XDIF=XDIF/10.

GO To 25

XSCALE=FAC*XDIV(L)

IF(XSCALE.GT.1.) GO TO 28
IF(XSCALE.EQ.1..AND.XDIF.GT.SO.) CO To 28
DO 29 LL=1,7

DO 26 L=1,3

IF(XDIF*XDIV(L).GT.100.) GO TO 27

CONTINUE

PACsPAC*1O.

XDIE=XDIE*1O.

IF(L.EQ.1) L=4

IF(L.EQ.4.AND.FAC.NE.1.) FAC=FAC/10.
XSCALE=1./(EAC*XDIv(L-1))

CONTINUE

XMIN2=XMIN2/XSCALE
XMIN2=INT(XMIN2/10.+(SIGN(1.,XMIN2)-1.)/2.)*10.*XSCALE
DO 30 L=1,11
XMIN1(L)=XMIN2+RLOAT(L-1)*1O.*XSCALE
WRITE(IO,31) XMIN1
FORMAT(SX,G12.5,2X.9(G9.2,1X),G12.5)
WRITE(IO,32)

FORMAT(13X,21('I....'),'I.I')
XVAL=BLOW-DIV

JO=1

HERE WE LOOP OVER THE POINT PLOTTINC ONE LINE AT A TIME
IARAY IS THE ARRAY INDEX OF VAL( ) WHERE A SYMBOL SHOULD BE
DO 40 L=1,NPT

800

500

510

730
42

710

720
760
700

44
40

501
502
503
504

505
810

279

Program PLOTSEN CONT'D.
SUBROUTINE PLOT

DO 800 LS=1,106

VAL(Ls)=BLANK

IE(ISCALE.EQ.3) GO TO 500
XVAL=XVAL+DIV

GO TO 510

VALOC=VALOC+DIV
XVAL=10.**(VALOG)

CONTINUE

DO 700 Jz=1,NP
IARAY-(ARAY(L,Jz) - XMIN2)/XSCALE + 0.5
IE(IARAY.CT.106) GO TO 700
IF(IARAY.LT.O) IARAY . o
IF(JZ.GT.1)GO TO 730
IERR=ABS(XARAY(L)/XSCALE)+O.5
JERR-106-IERR

KERR-IARAY-1

IE(IARAY.NE.o)CO To 42
IF(IERR.EQ.O)GO TO 700

CO To 710

IE(IARAY.Eo.O)CO TO 700
WAL(IARAY)=POINT(Jz)
IF(IERR.EQ.0.0R.JZ.GT.1)GO TO 700
LERR-IARAY-IERR
IE(IERR.CE.IARAY)LERR=1
JOHN=IARAY+IERR
IF(JOHN.GT.106)JOHN=106
KERR1=IARAY+1

Do 720 JTZ=LERR,KERR
VAL(JTz)-DASH

CONTINUE

DO 760 JTY=KERR1,JOHN
VAL(JTY)=DASH

CONTINUE

CONTINUE
WRITE(IO,44)XVAL,VAL,ARAY(L,1)
FORMAT(1X,E11.4,1X,'I',106A1,‘I',1X,E11.4)
CONTINUE

Go To (501.502.503.504,505).J0
WRITE(IO,32)

GO TO 810

WRITE(IO,106)

GO To 810

WRITE(IO,1O9)

CO To 810

WRITE(IO,113)

GO TO 810

WRITE(IO,116)

CONTINUE

280

Program PLOTSEN CONT'D.
'SUBROUTINE PLOT

IF( ISCALE .EQ. 1) GO TO 1000
C* RETURN THE VALUES TO NORMAL SPACE
DO 965 K=1,NPT
XARAY(K) - 10.0**(XARAY(K))
965 CONTINUE
DO 975 L-1,NF
DO 975 K=1,NPT
ARAY(K,L) . 1O.O**(ARAY(K,L))

975 CONTINUE
1000 CONTINUE
RETURN
C
C LOG SCALE
C
3 DO 900 MAA'1,NF

IF(XMIN(MAA).LE.O.) GO To 110
900 CONTINUE
XVAL-BLOW-DIV
150 CONTINUE
Do 100 L=1,NPT
IF( XARAY(L) .LE. 0 ) XARAY(L) - 1.0
XARAY(L) - ALOG1o(XARAY(L))
DO 100 LL-1,NF
1OO ARAY(L,LL)=ALOG10(ARAY(L,LL))
XMIN2=2.**4O
XMAX1--XMIN2
DO 910 JL-1,NF
IF(XMAX(JL).GT.XMAX1)XMAX1=XMAX(JL)
IE(XMIN(JL).LT.XMIN2)XMIN2=XMIN(JL)
91 O CONTINUE
XMAx1=ALOG1O(XMAx1)
XMIN2=ALOG10(XMIN2)
XDIF=XMAX1-XMIN2
IXDIE=INT<XDIP)+1
IXMIN=INT(XMIN2+(SIGN(1.,XMIN2)-1.)/2.)
IF( IXDIF .GT. 5 ) Go TO 10000
GO To (1o1,102,1O3,1O3,1O4),IXDIP
1OOOO CONTINUE
IF (IXDIF.LE.10) GO To 1500
0* TO LARGE A RANGE OF Y VALUES
WRITE(IO,45) IXDIF
45 FORMAT(' TO LARGE A RANGE ON THE Y AXIS, MAGNITUDE=',IB,'.GT.10')
1500 CONTINUE
DO 105 L=1,11
105 XMIN1(L)-10.**(IXMIN+L-1)
WRITE(IO,31) XMIN1
WRITE(IO,32)
XSCALE=O.1

281

Program PLOTSEN CONT'D.
SUBROUTINE PLOT

IXMIN-IXMIN*1O
J0=1
GO TO 80

101 XMIN1(1)=10.**IXMIN
XMIN1(11)=10.**(IXMIN+1)
WRITE(IO,250) XMIN1(1),XMIN1(11)

250 P0RMAT(9X,E8.1,25x,'2',19x,'3',11x,'4',9x,'5',6x,'6',
*sx,'7',5x.'8'.4x.'9';ZX.E8-1)

WRITE(IO,106)

106 FORMAT(13X,'I',29('.'),'I',19('.'),'I' 11('.'),'I',
*9('o'),.I.,6('o'),.I',5(.o'),'I.,5('o.3,
*‘I....I...I',5('.'),'I')

XSCALE-0.01
IXMIN-IXMIN*1OO
J0=2
GO T0 80

102 Do 107 L=1,3

107 XMIN1(L)=10.**(IXMIN+L-1)
WRITE(IO,108)(XMIN1(L),L-1,3)

108 P0RMAT(9X,E8.1,30x,E8.1,42x,E8.1)
WRITE(IO,109)

109 F0RMAT(13X,2('I..............I.........I.....I....I',

*'...I..I..I.I.'),'I.....I')
XSCALE-1./50.
IXMIN-IXMIN*SO
J0-3
GO T0 80

103 D0 111 L=1,4

111 XMIN1(L)-10.**(IXMIN+L-1)

WRITE(IO,112)(XMIN1(L),L-1
112 P0RMAT(9X,E8.1,3(18x,E8.1)$
J0=4
WRITE(IO,113)
113 P0RMAT(13X,4('I',24('.')),'I.....I')
XSCALE-1./25.
IXMIN=IXMIN*25.
GO TO 80
104 Do 114 L-1.5
114 IMIN1(L)-10.**(IXMIN+L-1)
WRITE(IO,115)(XMIN1(L),L=1 5)
115 P0RMAT(9X,E8.1,4(13x,E8.1)$
J0=5
WRITE(IO,116)
116 P0RMAT(13X,5('I',19('.')),'I.....I')
XSCALE=1./20.
IXMIN=IXMIN*20
GO TO 80

4)

O0

950

110

155

120
159

282

Program PLOTSEN CONT'D.
SUBROUTINE PLOT

LOG-LOG SCALE

D0 950 ND=1,NP
IVAL = ND

IF(XMIN(ND).LE.O.) GO TO 110

CONTINUE

IP(BL0W.LE.0.) GO TO 120

VALOG=ALOG10$BLOW)

VBLOCsALOG10 BHI)

DIV-(VBLOG-VALOG)/NPT

VAL0G=VALOG-DIV

GO TO 150

WRITE(IO,155) NAME(IVAL)

FORMAT(' PLOT OF ',A10,' CONTAINS NEGATIVE VALUEs',/,
*' AND WILL BE DONE WITH A LINEAR SCALE')

ISCALE=1

GO TO 156

WRITE(IO,159) BLOW

FORMAT(' THE LOWER BOUND =‘,E15.7,' Is NEGATIVE' )
RETURN

END

283

APPENDIX 9

Uth Order Accurate WALSH Sequenoy Set

l2:

BINARY EXPANSION

D7
‘0‘

l l 1

2 10 2

u 100 3

8 1000 U

16 10000 5
31 11111 6
32 100000 7
6M 1000000 8
124 1111100 0
128 10000000 IO
256 100000000 ll
“96 111110000 12
512 1000000000 13
102“ lOOOOOOOOOO In
1849 11100111001 15
198“ 11110000000 15
2OU8 100000000000 7
23Ul 100100100101 13
2730 101010101010 19
3699 111001110011 20

MOON 111110100100 1

REFERENCES

REFERENCES

ACM Signum, (1979):.l3 (2), 22.

Acton, Forman 8., (1966), Analysis of Straight Lines, Dover,
New York. '

Ahmed, N. and Rao, K. R. (1975), Transforms for Digital
Signal Processing, Springer, New York.

Ainslie, G. R., Shill, J. P. and Neet, K. E., (1972),
J. Biol. Chem., 247, 7080.

 

Andrews, H. C. and Caspari, K. L. (1970), IEEE Trans. Comp.
10, 16.

Bgtgs, D. J. and Frieden, C. (1973), J. Biol. Chem. 248,
7 7 . "'

Beauchamp, K. G. (1975), Walsh Functions and Their Applica-
tions, Academic Press, New YorE.

Beck, J. V. and Arnold, K. J. (1977), Parameter Estimation
in Engineering and Science, J. Wiley and Sons, New York.

Benacerraf, P. and Putnam, H. eds. (196M), Philosophy of .
Mathematics, Selected Readings, Prentice-Hall, New Jersey.

Carslaw, H. S. (1950), Introduction to the Theory of
Fourier's Series and Integrals 3rd ed.,*Dover, New York.

 

Chrestenson, H. E. (1952), Pac. J. Math. i, 17.

Cooley, J. w. and Tukey, J. w. (1965), Math. Comp. 12,
297.

Cukier, R. I., Fortuin, C. M., Schuler, K. E., Persohek,
A. G. and Schaibly, J. H. (1973). J. Chem. Phyfl. 52, 38h3.

 

Cukier, R. I., Schailby, J. H., Schuler, K. E. (1975),
J. Chem. Phys. 0;, llUO.

Cukier, R. I., Levine, H. B. and Schuler, K. E. (1978),
J. Comp. Phys. 20, l.

Dahlquist, G. and Bjorck, A. (197”), Numerical Methods,
Prentice-Hall, New York.

28M

285

De Boor, Carl (1978), A Practical Guide to Splines,
Springer, New York.

Dickenson, R. P. and Gelinas, R. J. (1976), J. Comp.
Phys. 2;, 123. _—

Dye, J. L. and Nicely, V. A. (1971), J. Chem. Ed. A8,
AA3-AA8. _—

 

Fall, A., McRae, G., and Seinfeld, J. (1979), Int. J. Chem.
Kinetics 11, 1137.

 

 

Fersht, A. (1977), Enzyme Structure and Mechanism, Freeman,
New York.

 

Fine, N. J. (19u9), Trans. Amer. Math. 800., £2, 372.

 

Fine, N. J. (1955), Pacific J. Math. 5, 51.

 

Fine, N. J. (1955), Pacific J. Math. 2, 61.

Garfinkel D. and Marbach, C. B. (1977), Ann. Rev. Biophys.
Bioeng. 6, 525.

Gear, C. W. (1971), Numerical Initial Value Problem in
Ordinary DifferentiaI EguatIOn, Prentice-Hall, New Jersey.

 

 

Hamming, R. W. (1973). Numerical Methods for Scientists
and Engineers, McGraw—HilI, New York.

 

Hill, R. (1925), Proc. R. Soc. B100, H19.

Hindmarsh, A. C. (1977), L. L. L. Report, UCRL-51186,
Rev. 1.

Ho Guan (1976), Ph.D. Thesis, Michigan State University.

International Mathematical and Statistical Libraries
(1977), Houston, Texas.

Jerri, A. J. (1977), Proceedings of the IEEE 62, 1565.

June, D. S., Kennedy, ., Pierce, T. H., Elias, S. V.,
Halaka, F., Behbahani, NeJad, I., El-Bayoumi, A., Suelter,
C. H., and Dye, J. L. (1979), J. Amer. Chem. Soc., lgl,
2218—2219.

 

June, D. S., Dye, J. L. and Suelter, C. H. (1980), Bio-
chemistry. “’

 

Kak, S. C. (1970), Electronic Letters, 6

 

286

Kanyar, B. (1978), Acta Biochem. et Biophys. Acad. Hung.
13, 153.

Knuth, Donald (1973). The Art of Computer Programming,
Sorting and Searching,“V01. 3, AddisonewesIey, San Fran—
EISco.

 

 

 

Koshland, D. E., Nemethy, G. and Filmer, D. (1966),
Biochemistry 5, 365.

 

Kunz, H. 0. (1979), IEEE Trans. on Comp. C-28,

 

Lanczos, C. (1955). Applied Analysis, Prentice-Hall, New
York.

 

Lackey, R. B. and Meltzer, D. (1971), IEEE Trans. on Comp.
C-20, 211.

 

Levine, H. (l97h), "Sensitivity Analysis of a Linear Pro-
gramming Model of Petroleum Economics", Systems, Science,
and Software Report SSS-R-722526.

Michaelis, L. and Menten, M. L. (1913), Biochem. 2. A9,
333. ‘—

Monod, J., Wyman, J. and Changeux, J. P. (1965), J. Molec.
Biol. 12, 88.

 

 

 

Nordsieck, A. (1962), Math. of Computation 10, 22.

 

Polge, R. J. and Bhagavan, B. K. (1976), IEEE Trans. on
Comp. , 53A.

#

RedliCh, O. (1972), J. Chem. Ed., 49, 222.

 

 

Schailby, J. H., and Schuler, K. E. (1973), J. Chem. Phys.
52, 3879.

Segel, I. (1975). Enzyme Kinetics, J. Wiley and Sons, New
York.

 

Shampine, L. F. and Gear, C. w. (1979), SIAM Review 21,
1-17. “

 

Singleton, R. C. (1967), Comm. of the ACM 10, 6U7—65A.

 

Singleton, R. C. (1968)., Comm. of the ACM 11, 773-779.

 

Tomovic, R. and Vukobratovic, M. (1972), General Sensi-
tivity Theory, American Elsevier, New York.

 

 

287

Walsh, J. L. (1923), Amer. J. Math. us, 5-2u.

Weyl, Hermann, (1938), Amer. J. Math. 60, 887.

Whitehead, E. (1970), Prgg. in BiOphys. and Mol. Biol.
g1, 321.

Zygmund, A. (1959), Trignometric Series 2ed Vol. 1,
Cambridge University Press, New York.