Electrical and Electronic Engineering - Theses
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ItemGranger Causality Under Quantized and Partially Observed Measurements( 2022)In this thesis, we investigate Granger causality assessment and preservation under quantized signals and partially observed measurements, including filtered, or noisy measurements. To do so, we first present a necessary and sufficient rank criterion for the equality of two conditional Gaussian distributions. We introduce a partial finite-order Markov assumption, and under this assumption, we find a characterization of Granger causality in terms of the rank of a matrix involving the covariances. We call this the causality matrix. We show that the smallest singular value of the causality matrix gives a lower bound on the distance between the two conditional Gaussian distributions appearing in the definition of Granger causality and yields a new measure of causality. Unlike most literature, this approach does not require system models to be explicitly identified and their parameters to be examined. To the best of our knowledge, this causality matrix and the measure of causality, which is different from previously introduced measures such as Geweke's and Kullback–Leibler divergence-based measures, have not been introduced in the literature. Furthermore, we introduce conditions to preserve and also assess causality under general quantization schemes. In the causality preservation case, we present conditions under which a variation of Wiener-Granger causality between quantized signals implies Granger causality between the underlying Gaussian signals. We also show that assessing causality between Gaussian signals through quantized data can be achieved by more relaxed conditions and assumptions. We introduce explicit conditions for non-uniform, uniform, and high-resolution quantizers. Apart from the assumed partial Markov order and joint Gaussianity, this approach does not require the parameters of a system model to be identified. No assumptions are made on the identifiability of the jointly Gaussian random processes through the quantized observations. To the best of our knowledge, this is the first work on causality investigation under quantization, or indeed under any nonlinear effect. Moreover, we consider two different setups of the effects of filtering on Granger causality. First, we do not make any assumptions on Markovanity of the Gaussian processes, and we assume all the past information is available. Under such situations, we present necessary and sufficient conditions to preserve Granger causality under causal and non-causal linear time-invariant filtering. These results are generalizations and improvements of previously presented results in the literature. We then consider the next setup of the problem where finite information about the processes is available, and a partial finite-order Markov property holds. We derive conditions under which causality can be reliably preserved and assessed from the second-order moments of filtered measurements. The filters may be noncausal or nonminimum-phase, as long as they are stable. To the best of our knowledge, the preservation of causality for such a general class of filters and its connection to causality assessment have not been considered before. We also investigate the effects of additive noise on causality and present conditions to preserve and assess causality. The additive noise signals are not required to be Gaussian or independent. For the first time, we introduce results for preservation and assessment of causality under general conditions.