Reduction of Markov Chains using a Value-of-Information-Based Approach

In this paper, we propose an approach to obtain reduced-order models of Markov chains. Our approach is composed of two information-theoretic processes. The first is a means of comparing pairs of stationary chains on different state spaces, which is done via the negative Kullback-Leibler divergence defined on a model joint space. Model reduction is achieved by solving a value-of-information criterion with respect to this divergence. Optimizing the criterion leads to a probabilistic partitioning of the states in the high-order Markov chain. A single free parameter that emerges through the optimization process dictates both the partition uncertainty and the number of state groups. We provide a data-driven means of choosing the `optimal' value of this free parameter, which sidesteps needing to a priori know the number of state groups in an arbitrary chain.Comment: Submitted to Entrop

Approximation of Markov Processes by Lower Dimensional Processes via Total Variation Metrics

The aim of this paper is to approximate a finite-state Markov process by another process with fewer states, called herein the approximating process. The approximation problem is formulated using two different methods. The first method, utilizes the total variation distance to discriminate the transition probabilities of a high dimensional Markov process and a reduced order Markov process. The approximation is obtained by optimizing a linear functional defined in terms of transition probabilities of the reduced order Markov process over a total variation distance constraint. The transition probabilities of the approximated Markov process are given by a water-filling solution. The second method, utilizes total variation distance to discriminate the invariant probability of a Markov process and that of the approximating process. The approximation is obtained via two alternative formulations: (a) maximizing a functional of the occupancy distribution of the Markov process, and (b) maximizing the entropy of the approximating process invariant probability. For both formulations, once the reduced invariant probability is obtained, which does not correspond to a Markov process, a further approximation by a Markov process is proposed which minimizes the Kullback-Leibler divergence. These approximations are given by water-filling solutions. Finally, the theoretical results of both methods are applied to specific examples to illustrate the methodology, and the water-filling behavior of the approximations.Comment: 38 pages, 17 figures, submitted to IEEE-TA

Horseshoe regularization for wavelet-based lensing inversion

Gravitational lensing, a phenomenon in astronomy, occurs when the gravitational field of a massive object, such as a galaxy or a black hole, bends the path of light from a distant object behind it. This bending results in a distortion or magnification of the distant object's image, often seen as arcs or rings surrounding the foreground object. The Starlet wavelet transform offers a robust approach to representing galaxy images sparsely. This technique breaks down an image into wavelet coefficients at various scales and orientations, effectively capturing both large-scale structures and fine details. The Starlet wavelet transform offers a robust approach to representing galaxy images sparsely. This technique breaks down an image into wavelet coefficients at various scales and orientations, effectively capturing both large-scale structures and fine details. The horseshoe prior has emerged as a highly effective Bayesian technique for promoting sparsity and regularization in statistical modeling. It aggressively shrinks negligible values while preserving important features, making it particularly useful in situations where the reconstruction of an original image from limited noisy observations is inherently challenging. The main objective of this thesis is to apply sparse regularization techniques, particularly the horseshoe prior, to reconstruct the background source galaxy from gravitationally lensed images. By demonstrating the effectiveness of the horseshoe prior in this context, this thesis tackles the challenging inverse problem of reconstructing lensed galaxy images. Our proposed methodology involves applying the horseshoe prior to the wavelet coefficients of lensed galaxy images. By exploiting the sparsity of the wavelet representation and the noise-suppressing behavior of the horseshoe prior, we achieve well-regularized reconstructions that reduce noise and artifacts while preserving structural details. Experiments conducted on simulated lensed galaxy images demonstrate lower mean squared error and higher structural similarity with the horseshoe prior compared to alternative methods, validating its efficacy as an efficient sparse modeling technique.Les lentilles gravitationnelles se produisent lorsque le champ gravitationnel d'un objet massif dévie la trajectoire de la lumière provenant d'un objet lointain, entraînant une distorsion ou une amplification de l'image de l'objet lointain. La transformation Starlet fournit une méthode robuste pour obtenir une représentation éparse des images de galaxies, capturant efficacement leurs caractéristiques essentielles avec un minimum de données. Cette représentation réduit les besoins de stockage et de calcul, et facilite des tâches telles que le débruitage, la compression et l'extraction de caractéristiques. La distribution a priori de fer à cheval est une technique bayésienne efficace pour promouvoir la sparsité et la régularisation dans la modélisation statistique. Elle réduit de manière agressive les valeurs négligeables tout en préservant les caractéristiques importantes, ce qui la rend particulièrement utile dans les situations où la reconstruction d'une image originale à partir d'observations bruitées est difficile. Étant donné la nature mal posée de la reconstruction des images de galaxies à partir de données bruitées, l'utilisation de la distribution a priori devient cruciale pour résoudre les ambiguïtés. Les techniques utilisant une distribution a priori favorisant la sparsité ont été efficaces pour relever des défis similaires dans divers domaines. L'objectif principal de cette thèse est d'appliquer des techniques de régularisation favorisant la sparsité, en particulier la distribution a priori de fer à cheval, pour reconstruire les galaxies d'arrière-plan à partir d'images de lentilles gravitationnelles. Notre méthodologie proposée consiste à appliquer la distribution a priori de fer à cheval aux coefficients d'ondelettes des images de galaxies lentillées. En exploitant la sparsité de la représentation en ondelettes et le comportement de suppression du bruit de la distribution a priori de fer à cheval, nous obtenons des reconstructions bien régularisées qui réduisent le bruit et les artefacts tout en préservant les détails structurels. Des expériences menées sur des images simulées de galaxies lentillées montrent une erreur quadratique moyenne inférieure et une similarité structurelle plus élevée avec la distribution a priori de fer à cheval par rapport à d'autres méthodes, validant son efficacité

Dynamic temperature estimation of power electronics systems

This thesis proposes a method for accurate temperature estimation of thermally-aware power electronics systems. The duality between electrical systems and thermal systems was considered for thermal modeling. High dimensional thermal models present a challenge for online estimation. RC (resistor-capacitor) circuits that create a tradeoff between accuracy and complexity were used to simulate the dynamic thermal behavior of power electronics. The complexity of the thermal network was further reduced by applying a structure-preserving model order reduction technique. The reduced order thermal model was an RC circuit with fewer capacitors. Preserving the physical correspondence between the reduced order model and the physical system allows the user to use the reduced order thermal model in the sensor placement optimization process. The accuracy of the thermal estimates can be easily increased by increasing the number of sensors in the system. However, a large number of sensors increases the cost and complexity of the system. It might also interfere with the circuit design and create packaging problems. An optimal number and optimal placement of temperature sensors was found. The optimal sensor placement problem was solved by maximizing the trace of observability Gramian. The optimal number of temperature sensors was based on the state estimation error obtained from a Kalman filter. The dynamic thermal behavior of the power electronics systems was represented by a linear state space model by applying the conservation of energy principle. Therefore, assuming Gaussian noise, it is well-known that a Kalman filter is an optimal estimator for such systems. A continuous-discrete Kalman filter was used to estimate the dynamic thermal behavior of power electronics systems using an optimal number of temperature sensors placed at optimal locations. The proposed method was applied on 2-D and 3-D power electronics systems. Theoretical results were validated experimentally using IR thermal imaging and thermocouples. It was shown that the proposed method can accurately reconstruct the dynamic temperature profile of power electronics systems using a small number of temperature sensors

An information-theoretic framework to aggregate a Markov chain

Abstract — This paper is concerned with an information-theoretic framework to aggregate a large-scale Markov chain to obtain a reduced order Markov model. The Kullback-Leibler (K-L) divergence rate is employed as a metric to measure the distance between two stationary Markov chains. Model reduction is obtained by considering an optimization problem with respect to this metric. The solution is just the optimal aggregated Markov model. We show that the solution of the bi-partition problem is given by an eigenvalue problem. To construct a reduced order model with m super-states, a recursive algorithm is proposed and illustrated with examples. I