Density estimation is a fundamental technique employed in various fields to model and to understand the underlying distribution of data. The primary objective of density estimation is to estimate the probability density function of a random variable. This process is particularly valuable when dealing with univariate or multivariate data and is essential for tasks such as clustering, anomaly detection, and generative modeling. In this paper we propose the mono-variate approximation of the density using spline quasi interpolation and we applied it in the context of clustering modeling. The clustering technique used is based on the construction of suitable multivariate distributions which rely on the estimation of the monovariate empirical densities (marginals). Such an approximation is achieved by using the proposed spline quasi-interpolation, while the joint distributions to model the sought clustering partition is constructed with the use of copulas functions. In particular, since copulas can capture the dependence between the features of the data independently from the marginal distributions, a finite mixture copula model is proposed. The presented algorithm is validated on artificial and real datasets.

Images are ambiguous at each of many levels of a contextual hi(cid:173) erarchy. Nevertheless, the high-level interpretation of most scenes is unambiguous, as evidenced by the superior performance of hu(cid:173) mans. This observation argues for global vision models, such as de(cid:173) formable templates. Unfortunately, such models are computation(cid:173) ally intractable for unconstrained problems. We propose a composi(cid:173) tional model in which primitives are recursively composed, subject to syntactic restrictions, to form tree-structured objects and object groupings.

In previous work [6, 9, 10], we advanced a new technique for direct visual matching of images for the purposes of face recognition and image retrieval, using a probabilistic measure of similarity based primarily on a Bayesian (MAP) analysis of image differ(cid:173) ences, leading to a "dual" basis similar to eigenfaces [13]. The performance advantage of this probabilistic matching technique over standard Euclidean nearest-neighbor eigenface matching was recently demonstrated using results from DARPA's 1996 "FERET" face recognition competition, in which this probabilistic matching algorithm was found to be the top performer. We have further developed a simple method of replacing the costly com put ion of nonlinear (online) Bayesian similarity measures by the relatively inexpensive computation of linear (offline) subspace projections and simple (online) Euclidean norms, thus resulting in a significant computational speed-up for implementation with very large image databases as typically encountered in real-world applications.

We describe Maximum-Likelihood Continuity Mapping (MALCOM), an alternative to hidden Markov models (HMMs) for processing sequence data such as speech. While HMMs have a discrete "hidden" space con(cid:173) strained by a fixed finite-automaton architecture, MALCOM has a con(cid:173) tinuous hidden space-a continuity map-that is constrained only by a smoothness requirement on paths through the space. MALCOM fits into the same probabilistic framework for speech recognition as HMMs, but it represents a more realistic model of the speech production process. To evaluate the extent to which MALCOM captures speech production information, we generated continuous speech continuity maps for three speakers and used the paths through them to predict measured speech articulator data. The median correlation between the MALCOM paths obtained from only the speech acoustics and articulator measurements was 0.77 on an independent test set not used to train MALCOM or the predictor.

The thesis focuses on developing a data-driven algorithm, based on machine learning, to solve the stochastic alternating current (AC) chance-constrained (CC) Optimal Power Flow (OPF) problem. Although the AC CC-OPF problem has been successful in academic circles, it is highly nonlinear and computationally demanding, which limits its practical impact. The proposed approach aims to address this limitation and demonstrate its empirical efficiency through applications to multiple IEEE test cases. To solve the non-convex and computationally challenging CC AC-OPF problem, the proposed approach relies on a machine learning Gaussian process regression (GPR) model. The full Gaussian process (GP) approach is capable of learning a simple yet non-convex data-driven approximation to the AC power flow equations that can incorporate uncertain inputs. The proposed approach uses various approximations for GP-uncertainty propagation. The full GP CC-OPF approach exhibits highly competitive and promising results, outperforming the state-of-the-art sample-based chance constraint approaches. To further improve the robustness and complexity/accuracy trade-off of the full GP CC-OPF, a fast data-driven setup is proposed. This setup relies on the sparse and hybrid Gaussian processes (GP) framework to model the power flow equations with input uncertainty.

This survey paper presents a comprehensive analysis of crime prediction methodologies, exploring the various techniques and technologies utilized in this area. The paper covers the statistical methods, machine learning algorithms, and deep learning techniques employed to analyze crime data, while also examining their effectiveness and limitations. We propose a methodological taxonomy that classifies crime prediction algorithms into specific techniques. This taxonomy is structured into four tiers, including methodology category, methodology sub-category, methodology techniques, and methodology sub-techniques. Empirical and experimental evaluations are provided to rank the different techniques. The empirical evaluation assesses the crime prediction techniques based on four criteria, while the experimental evaluation ranks the algorithms that employ the same sub-technique, the different sub-techniques that employ the same technique, the different techniques that employ the same methodology sub-category, the different methodology sub-categories within the same category, and the different methodology categories. The combination of methodological taxonomy, empirical evaluations, and experimental comparisons allows for a nuanced and comprehensive understanding of crime prediction algorithms, aiding researchers in making informed decisions. Finally, the paper provides a glimpse into the future of crime prediction techniques, highlighting potential advancements and opportunities for further research in this field

Bayesian optimization is a widely used technique for optimizing black-box functions, with Expected Improvement (EI) being the most commonly utilized acquisition function in this domain. While EI is often viewed as distinct from other information-theoretic acquisition functions, such as entropy search (ES) and max-value entropy search (MES), our work reveals that EI can be considered a special case of MES when approached through variational inference (VI). In this context, we have developed the Variational Entropy Search (VES) methodology and the VES-Gamma algorithm, which adapts EI by incorporating principles from information-theoretic concepts. The efficacy of VES-Gamma is demonstrated across a variety of test functions and read datasets, highlighting its theoretical and practical utilities in Bayesian optimization scenarios.

We address the problem of non-convergence of online reinforcement learning algorithms (e.g., Q learning and SARSA(A)) by adopt(cid:173) ing an incremental-batch approach that separates the exploration process from the function fitting process. Our BFBP (Batch Fit to Best Paths) algorithm alternates between an exploration phase (during which trajectories are generated to try to find fragments of the optimal policy) and a function fitting phase (during which a function approximator is fit to the best known paths from start states to terminal states). An advantage of this approach is that batch value-function fitting is a global process, which allows it to address the tradeoffs in function approximation that cannot be handled by local, online algorithms. This approach was pioneered by Boyan and Moore with their GROWSUPPORT and ROUT al(cid:173) gorithms. We show how to improve upon their work by applying a better exploration process and by enriching the function fitting procedure to incorporate Bellman error and advantage error mea(cid:173) sures into the objective function.

The standard approach to Computational Learning Theory is usually formulated within the so-called frequentist approach to Statistics. Within this paradigm one is interested in constructing an estimator, based on a flnite sample, which possesses a small loss (generalization error). While many algorithms have been constructed and analyzed within this context, it is not clear how these approaches relate to standard optimality criteria within the frequentist framework. Two classic optimality criteria within the latter approach are the minimax and admissibility criteria, which charac- terize optimality of estimators in a rigorous and precise fashion [9]. Except in some special cases [12], it is not known whether any of the approaches used within the Learning community lead to optimality in either of the above senses of the word. On the other hand, it is known that under certain regularity conditions, Bayesian estimators lead to either minimax or admissible estimators, and thus to well-deflned optimality in the classical (frequentist) sense. In fact, it can be shown that Bayes estimators are essentially the only estimators which can achieve optimality in the above senses [9]. This optimality feature provides strong motivation for the study of Bayesian approaches in a frequentist setting.

Machine learning has reached a point where many probabilistic meth- ods can be understood as variations, extensions and combinations of a much smaller set of abstract themes, e.g., as different instances of the EM algorithm. This enables the systematic derivation of algorithms cus- tomized for different models. Here, we describe the AUTO BAYES sys- tem which takes a high-level statistical model specification, uses power- ful symbolic techniques based on schema-based program synthesis and computer algebra to derive an efficient specialized algorithm for learning that model, and generates executable code implementing that algorithm. This capability is far beyond that of code collections such as Matlab tool- boxes or even tools for model-independent optimization such as BUGS for Gibbs sampling: complex new algorithms can be generated with- out new programming, algorithms can be highly specialized and tightly crafted for the exact structure of the model and data, and efficient and commented code can be generated for different languages or systems. We describe a symbolic program synthesis system which works as a "statistical algorithm compiler:" it compiles a statistical model specification into a custom algorithm design and from that further down into a working program implementing the algorithm design.