Alejandro de la Concha Duarte

Assuming we have i.i.d observations from two unknown probability density functions (pdfs), p and q, the density/likelihood-ratio estimation (LRE) is an elegant approach to compare the two pdfs only by relying on the available data. In this paper, we introduce the first — to the best of our knowledge — graph-based extension of this problem, which reads as follows: Suppose each node v of a fixed graph has access to observations coming from two unknown node-specific pdfs, p_v and q_v, and the goal is to estimate for each node the likelihood-ratio between both pdfs by also taking into account the information provided by the graph structure. The node-level estimation tasks are supposed to exhibit similarities conveyed by the graph, which suggests that the nodes could collaborate to solve them more efficiently. We develop this idea in a concrete non-parametric method that we call Graph-based Relative Unconstrained Least-squares Importance Fitting (GRULSIF). We derive convergence rates for our collaborative approach that highlights the role played by variables such as the number of available observations per node, the size of the graph, and how accurately the graph structure encodes the similarity between tasks. These theoretical results explicit the situations where collaborative estimation effectively leads to an improvement in performance compared to solving each problem independently.

Multiple two-sample test problem in a graph-structured setting is a common scenario in fields such as Spatial Statistics and Neuroscience. Each node v in fixed graph deals with a two-sample testing problem between two node-specific probability density functions, p_v and q_v. The goal is to identify nodes where the null hypothesis p_v = q_v should be rejected, under the assumption that connected nodes would yield similar test outcomes. We propose the non-parametric collaborative two-sample testing (CTST) framework that efficiently leverages the graph structure and minimizes the assumptions over p_v and q_v. CTST integrates elements from f-div estimation, Kernel Methods, and Multitask Learning.

The density/likelihood-ratio estimation problem between two pdfs p and q, has been investigated mainly for the offline case. This paper contributes by introducing a new framework for online non-parametric density/likelihood-ratio estimation for the setting where pairs of i.i.d observations (x_t ∼p, x’_t ∼q) are observed over time. Our online estimator named OLRE capitalizes on recent advances in Kernel Methods and functional minimization. The non-parametric and online nature of OLRE makes it a significant step in the development of sequential change-point detectors based on the likelihood-ratio.

This paper addresses the problem of segmenting a stream of graph signals: we aim to detect changes in the mean of a multivariate signal defined over the nodes of a known graph. We propose an offline change-point detection method that relies on the concept of graph stationarity. This approach allows the problem to be translated from the original vertex domain to the spectral domain induced by the Graph Fourier Transform, making the change-point detection problem much easier to solve. Our change-point detection method adopts a model selection approach that assumes the mean of the graph signals at each segment admits a sparse spectral representation and determines automatically the number of change-points.

Complex network models have proven to be useful tools for studying systemic risk, particularly in characterizing and describing the level of interconnectedness in a financial system. Nevertheless, most existing work has focused primarily on a single phenomenon: interbank (exposure) networks. In this paper, we study the Mexican banking system using a comprehensive set of market interactions, including transactions in securities markets, payment system flows, interbank loans, foreign exchange, and derivatives exposures. To the best of our knowledge, this is the first attempt to describe so comprehensively the complexity and interconnectedness of a banking system.

This book chapter describes the Structural Equation Modeling approach using Partial Least Squares (SEM-PLS). Different software alternatives for implementing this type of model are presented. Finally, a practical case is solved step by step to better illustrate both the theory and the use of the software.