Ciccone, Valentina (2019) Entropic methods in learning stochastic systems with latent variables and homogeneous Gaussian random fields. [Ph.D. thesis]
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Abstract (italian or english)
This dissertation is divided into two main parts, the common thread being the prominent role of entropy-based methods in the identification and estimation of stochastic models and systems.
The first part of the dissertation deals with the problem of robustness in the identification of stochastic models with latent variables, namely variables that, although influencing the behaviour of some other manifest variables, are not directly observable. These models boast a long tradition and find natural application in many disciplines within engineering and applied science including psychology, econometrics, system engineering, machine learning and statistics, to name but a few. In this part of the dissertation, relying on certain invariance properties of the relative entropy and inspired by the previous contributions on robust estimation, we propose, for the case of zero-mean Gaussian random variables and processes, a novel approach for constructing a confidence region for the underlying model from a given finite sample estimate. This region depends only on the number of data and, by construction, contains the true model with a user-chosen probability. This paradigm is applied to the identification of two classes of latent variable models, namely factor models and graphical models with latent variables, for which we search the most parsimonious model in the confidence region by solving a convex optimization problem.
The second part of this dissertation focuses on homogeneous Gaussian random fields, namely stationary Gaussian processes defined over a multidimensional lattice, which find application, for instance, in multidimensional signal processing, spatial statistics and image analysis. In this part of the dissertation, relying on the properties of multilevel circulant and multi-level Toeplitz matrices, we derive an explicit formula for the computation of the relative entropy rate between two homogeneous random fields in terms of their spectral densities. Moreover, we establish a correspondence between the relative entropy rate for homogeneous Gaussian random fields and the relative entropy rate for their spectral domain representation. Both the cases of general and periodic homogeneous random fields are considered.
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