arXiv subjects

Álvaro Gajardo

Publications and source records attributed to Álvaro Gajardo.

3 recordsLinked to original sources

arXiv2026

Predictive Distributions and the Transition from Sparse to Dense Functional Data

A representation of Gaussian distributed sparsely sampled longitudinal data in terms of predictive distributions for their functional principal component scores (FPCs) maps available data for each subject to a multivariate Gaussian predictive distribution. Of special interest is the case where the number of observations per subject increases in the transition from sparse (longitudinal) to dense (functional) sampling of underlying stochastic processes. We study the convergence of the predicted scores given noisy longitudinal observations towards the true but unobservable FPCs, and under Gaussianity demonstrate the shrinkage of the entire predictive distribution towards a point mass located at the true FPCs and also extensions to the shrinkage of functional $K$-truncated predictive distributions when the truncation point $K=K(n)$ diverges with sample size $n$. To address the problem of non-consistency of point predictions, we construct predictive distributions aimed at predicting outcomes for the case of sparsely sampled longitudinal predictors in functional linear models and derive asymptotic rates of convergence for the $2$-Wasserstein metric between true and estimated predictive distributions. Predictive distributions are illustrated for longitudinal data from the Baltimore Longitudinal Study of Aging.

stat.ME↗

arXiv2021

Learning delay dynamics for multivariate stochastic processes, with application to the prediction of the growth rate of COVID-19 cases in the United States

Delay differential equations form the underpinning of many complex dynamical systems. The forward problem of solving random differential equations with delay has received increasing attention in recent years. Motivated by the challenge to predict the COVID-19 caseload trajectories for individual states in the U.S., we target here the inverse problem. Given a sample of observed random trajectories obeying an unknown random differential equation model with delay, we use a functional data analysis framework to learn the model parameters that govern the underlying dynamics from the data. We show existence and uniqueness of the analytical solutions of the population delay random differential equation model when one has discrete time delays in the functional concurrent regression model and also for a second scenario where one has a delay continuum or distributed delay. The latter involves a functional linear regression model with history index. The derivative of the process of interest is modeled using the process itself as predictor and also other functional predictors with predictor-specific delayed impacts. This dynamics learning approach is shown to be well suited to model the growth rate of COVID-19 for the states that are part of the U.S., by pooling information from the individual states, using the case process and concurrently observed economic and mobility data as predictors.

math.AP↗

arXiv2021

Cox Point Process Regression

Point processes in time have a wide range of applications that include the claims arrival process in insurance or the analysis of queues in operations research. Due to advances in technology, such samples of point processes are increasingly encountered. A key object of interest is the local intensity function. It has a straightforward interpretation that allows to understand and explore point process data. We consider functional approaches for point processes, where one has a sample of repeated realizations of the point process. This situation is inherently connected with Cox processes, where the intensity functions of the replications are modeled as random functions. Here we study a situation where one records covariates for each replication of the process, such as the daily temperature for bike rentals. For modeling point processes as responses with vector covariates as predictors we propose a novel regression approach for the intensity function that is intrinsically nonparametric. While the intensity function of a point process that is only observed once on a fixed domain cannot be identified, we show how covariates and repeated observations of the process can be utilized to make consistent estimation possible, and we also derive asymptotic rates of convergence without invoking parametric assumptions.

stat.ME↗