Although the concept of causality is intuitive, an universally accepted objective measure to quantify causal relationships does not exist. In complex systems where the internal mechanism is not well understood, it is helpful to estimate how different parts of the system are related. In the context of time-series data, Granger Causality (GC) has long been used as a way to quantify such relationships, having been successfully been applied in fields as diverse as econometrics and neurology. Multiple Granger-like and extensions to GC have also been proposed. A recent measure developed to address limitations of GC, New Causality (NC), offers several advantages over GC, such as normalization and better proportionality with respect to internal mechanisms. However, NC is limited in scope by its seminal definition being based on parametric linear models. In this work, a critical analysis of NC is presented, NC is extended to a wide range of nonlinear models and finally, enhancements to a method of estimating nonlinear models for use with NC are reported.A critical analysis is conducted to study the relationship between NC values and model estimation errors. It is shown that NC is much more sensitive to overfitting in comparison to GC. Although the variance of NC estimates is reduced by applying regularization techniques, NC estimates are also prone to bias. In this work, diverse case-studies are presented showing the behavior of NC estimation in the presence of regularization. A mathematical study of the sources of bias in the estimates is given.For systems that cannot be modeled well by linear models, the seminal definition of NC performs poorly. This works gives examples in which nonlinear observation models cause NC values obtained with the seminal definition to behave contrary to intuitive expectations. A nonlinear extension of NC to all linear-in-parameters models is then developed and shown to address these limitations. The extension reduces to the seminal definition of NC for linear models and offers a flexible weighting mechanism to distribute contributions among nonlinear terms. The nonlinear extension is applied to a range of synthetic data and real EEG data with promising results.The sensitivity of NC to parameter estimation errors demands that special care be taken when using NC with nonlinear models. As a complement to nonlinear NC, enhancements to a algorithm for nonlinear parametric model estimation are presented. The algorithm combines a genetic search element for regressor selection with a set-theoretic optimal bounded ellipsoid algorithm for parameter estimation. The enhancements to the genetic search make use of sparsity and information theoretic measures to reduce the computational cost of the algorithm. Significant reductions are shown and direction for further improvements of the algorithm are given. The main contributions of this work are providing a method for estimating causal relationships between signals using nonlinear estimated models, and a framework for estimating the relationships using an enhanced algorithm for model structure search and parameter estimation.
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