arXiv subjects

Carlos Matrán

Publications and source records attributed to Carlos Matrán.

9 recordsLinked to original sources

arXiv2024

Improving Model Choice in Classification: An Approach Based on Clustering of Covariance Matrices

This work introduces a refinement of the Parsimonious Model for fitting a Gaussian Mixture. The improvement is based on the consideration of clusters of the involved covariance matrices according to a criterion, such as sharing Principal Directions. This and other similarity criteria that arise from the spectral decomposition of a matrix are the bases of the Parsimonious Model. We show that such groupings of covariance matrices can be achieved through simple modifications of the CEM (Classification Expectation Maximization) algorithm. Our approach leads to propose Gaussian Mixture Models for model-based clustering and discriminant analysis, in which covariance matrices are clustered according to a parsimonious criterion, creating intermediate steps between the fourteen widely known parsimonious models. The added versatility not only allows us to obtain models with fewer parameters for fitting the data, but also provides greater interpretability. We show its usefulness for model-based clustering and discriminant analysis, providing algorithms to find approximate solutions verifying suitable size, shape and orientation constraints, and applying them to both simulation and real data examples.

stat.ME↗

arXiv2020

optimalFlow: Optimal-transport approach to flow cytometry gating and population matching

Data obtained from Flow Cytometry present pronounced variability due to biological and technical reasons. Biological variability is a well-known phenomenon produced by measurements on different individuals, with different characteristics such as illness, age, sex, etc. The use of different settings for measurement, the variation of the conditions during experiments and the different types of flow cytometers are some of the technical causes of variability. This mixture of sources of variability makes the use of supervised machine learning for identification of cell populations difficult. The present work is conceived as a combination of strategies to facilitate the task of supervised gating. We propose $optimalFlowTemplates$, based on a similarity distance and $\text{Wasserstein barycenters}$, which clusters cytometries and produces prototype cytometries for the different groups. We show that supervised learning, restricted to the new groups, performs better than the same techniques applied to the whole collection. We also present $optimalFlowClassification$, which uses a database of gated cytometries and optimalFlowTemplates to assign cell types to a new cytometry. We show that this procedure can outperform state of the art techniques in the proposed datasets. Our code is freely available as $optimalFlow$ a Bioconductor R package at https://bioconductor.org/packages/optimalFlow. optimalFlowTemplates+optimalFlowClassification addresses the problem of using supervised learning while accounting for biological and technical variability. Our methodology provides a robust automated gating workflow that handles the intrinsic variability of flow cytometry data well. Our main innovation is the methodology itself and the optimal-transport techniques that we apply to flow cytometry analysis.

stat.ML↗

arXiv2020

Center-Outward Distribution Functions, Quantiles, Ranks, and Signs in $\mathbb{R}^d$

Univariate concepts as quantile and distribution functions involving ranks and signs, do not canonically extend to $\mathbb{R}^d, d\geq 2$. Palliating that has generated an abundant literature. Chapter 1 shows that, unlike the many definitions that have been proposed so far, the measure transportation-based ones introduced in Chernozhukov et al. (2017) enjoy all the properties that make univariate quantiles and ranks successful tools for semiparametric statistical inference. We therefore propose a new center-outward definition of multivariate distribution and quantile functions, along with their empirical counterparts, for which we obtain a Glivenko-Cantelli result. Our approach is geometric and, contrary to the Monge-Kantorovich one in Chernozhukov et al. (2017), does not require any moment assumptions. The resulting ranks and signs are strictly distribution-free, and maximal invariant under the action of a data-driven class of (order-preserving) transformations generating the family of absolutely continuous distributions; that property is the theoretical foundation of the semiparametric efficiency preservation property of ranks. The corresponding quantiles are equivariant under the same transformations. The empirical proposed distribution functions are defined at observed values only. A continuous extension to the entire $\mathbb{R}^d$, yielding continuous empirical quantile contours while preserving the monotonicity and Glivenko-Cantelli features is desirable. Such extension requires solving a nontrivial problem of smooth interpolation under cyclical monotonicity constraints. A complete solution of that problem is given in Chapter 2; we show that the resulting distribution and quantile functions are Lipschitz, and provide a sharp lower bound for the Lipschitz constants. A numerical study of empirical center-outward quantile contours and their consistency is conducted.

stat.ME↗

arXiv2019

Box-constrained monotone $L_\infty$-approximations to Lipschitz regularizations, with applications to robust testing

Tests of fit to exact models in statistical analysis often lead to rejections even when the model is a useful approximate description of the random generator of the data. Among possible relaxations of a fixed model, the one defined by contamination neighbourhoods, namely, $\mathcal{V}_α(P_0)=\{(1-α)P_0+αQ: Q \in \mathcal{P}\}$, where $\mathcal{P}$ is the set of all probabilities in the sample space, has received much attention, from its central role in Robust Statistics. For probabilities on the real line, consistent tests of fit to $\mathcal{V}_α(P_0)$ can be based on $d_K(P_0,R_α(P))$, the minimal Kolmogorov distance between $P_0$ and the set of trimmings of $P$, $R_α(P)=\big\{\tilde P\in\mathcal{P}:\tilde P\ll P,\,{\textstyle \frac{d\tilde P}{dP}\leq\frac{1}{1-α}}\, P\text{-a.s.}\big\}$. We show that this functional admits equivalent formulations in terms of, either best approximation in uniform norm by $L$-Lipschitz functions satisfying a box constraint, or as the best monotone approximation in uniform norm to the $L$-Lipschitz regularization, which is seen to be expressable in terms of the average of the Pasch-Hausdorff envelopes. This representation for the solution of the variational problem allows to obtain results showing stability of the functional $d_K(P_0,R_α(P))$, as well as directional differentiability, providing the basis for a Central Limit Theorem for that functional.

math.OC↗

arXiv2019

On approximate validation of models: A Kolmogorov-Smirnov based approach

Classical tests of fit typically reject a model for large enough real data samples. In contrast, often in statistical practice a model offers a good description of the data even though it is not the "true" random generator. We consider a more flexible approach based on contamination neighbourhoods around a model. Using trimming methods and the Kolmogorov metric we introduce a functional statistic measuring departures from a contaminated model and the associated estimator corresponding to its sample version. We show how this estimator allows testing of fit for the (slightly) contaminated model vs sensible deviations from it, with uniformly exponentially small type I and type II error probabilities. We also address the asymptotic behavior of the estimator showing that, under suitable regularity conditions, it asymptotically behaves as the supremum of a Gaussian process. As an application we explore methods of comparison between descriptive models based on the paradigm of model falseness. We also include some connections of our approach with the False-Discovery-Rate setting, showing competitive behavior when estimating the contamination level, although applicable in a wider framework.

math.ST↗

arXiv2013

The empirical cost of optimal incomplete transportation

We consider the problem of optimal incomplete transportation between the empirical measure on an i.i.d. uniform sample on the d-dimensional unit cube $[0,1]^d$ and the true measure. This is a family of problems lying in between classical optimal transportation and nearest neighbor problems. We show that the empirical cost of optimal incomplete transportation vanishes at rate $O_P(n^{-1/d})$, where n denotes the sample size. In dimension $d\geq3$ the rate is the same as in classical optimal transportation, but in low dimension it is (much) higher than the classical rate.

math.PR↗

arXiv2012

Similarity of samples and trimming

We say that two probabilities are similar at level $α$ if they are contaminated versions (up to an $α$ fraction) of the same common probability. We show how this model is related to minimal distances between sets of trimmed probabilities. Empirical versions turn out to present an overfitting effect in the sense that trimming beyond the similarity level results in trimmed samples that are closer than expected to each other. We show how this can be combined with a bootstrap approach to assess similarity from two data samples.

math.ST↗

arXiv2008

Trimming and likelihood: Robust location and dispersion estimation in the elliptical model

Robust estimators of location and dispersion are often used in the elliptical model to obtain an uncontaminated and highly representative subsample by trimming the data outside an ellipsoid based in the associated Mahalanobis distance. Here we analyze some one (or $k$)-step Maximum Likelihood Estimators computed on a subsample obtained with such a procedure. We introduce different models which arise naturally from the ways in which the discarded data can be treated, leading to truncated or censored likelihoods, as well as to a likelihood based on an only outliers gross errors model. Results on existence, uniqueness, robustness and asymptotic properties of the proposed estimators are included. A remarkable fact is that the proposed estimators generally keep the breakdown point of the initial (robust) estimators, but they could improve the rate of convergence of the initial estimator because our estimators always converge at rate $n^{1/2}$, independently of the rate of convergence of the initial estimator.

math.ST↗

arXiv2008

A general trimming approach to robust Cluster Analysis

We introduce a new method for performing clustering with the aim of fitting clusters with different scatters and weights. It is designed by allowing to handle a proportion $α$ of contaminating data to guarantee the robustness of the method. As a characteristic feature, restrictions on the ratio between the maximum and the minimum eigenvalues of the groups scatter matrices are introduced. This makes the problem to be well defined and guarantees the consistency of the sample solutions to the population ones. The method covers a wide range of clustering approaches depending on the strength of the chosen restrictions. Our proposal includes an algorithm for approximately solving the sample problem.

math.ST↗