arXiv subjects

Ricardo Carrizo Vergara

Publications and source records attributed to Ricardo Carrizo Vergara.

7 recordsLinked to original sources

arXiv2025

Karhunen-Loève expansion of Random Measures

We present an orthogonal expansion for real, function-regulated, second-order random measures over $\mathbb{R}^{d}$ with measure covariance. Such a expansion, which can be seen as a Karhunen-Loève decomposition, consists in a series of deterministic real measures weighted by uncorrelated real random variables with the variances forming a convergent series. The convergence of the series is in a mean-square sense stochastically and against measurable bounded test functions (with compact support if the random measure is not finite) in the measure sense, which implies set-wise convergence. This is proven taking advantage of the extra requirement of having a covariance measure over $\mathbb{R}^{d}\times\mathbb{R}^{d}$ describing the covariance structure of the random measure, for which we also provide a series expansion. These results cover for instance the cases of Gaussian White Noise, Poisson and Cox point processes, and can be used to obtain expansions for trawl processes.

math.PR↗

arXiv2025

The evolving categories multinomial distribution: introduction with applications to movement ecology and vote transfer

We introduce the evolving categories multinomial (ECM) distribution for multivariate count data taken over time. This distribution models the counts of individuals following iid stochastic dynamics among categories, with the number and identity of the categories also evolving over time. We specify the one-time and two-times marginal distributions of the counts and the first and second order moments. When the total number of individuals is unknown, placing a Poisson prior on it yields a new distribution (ECM-Poisson), whose main properties we also describe. Since likelihoods are intractable or impractical, we propose two estimating functions for parameter estimation: a Gaussian pseudo-likelihood and a pairwise composite likelihood. We show two application scenarios: the inference of movement parameters of animals moving continuously in space-time with irregular survey regions, and the inference of vote transfer in two-rounds elections. We give three illustrations: a simulation study with Ornstein-Uhlenbeck moving individuals, paying special attention to the autocorrelation parameter; the inference of movement and behavior parameters of lesser prairie-chickens; and the estimation of vote transfer in the 2021 Chilean presidential election.

stat.AP↗

arXiv2024

Movement-based models for abundance data

We develop two statistical models for space-time abundance data based on a stochastic underlying continuous individual movement. In contrast to current models for abundance in statistical ecology, our models exploit the explicit connection between movement and counts, including the induced space-time auto-correlation. Our first model, called Snapshot, describes the counts of free moving individuals with a false-negative detection error. Our second model, called Capture, describes the capture and retention of moving individuals, and it follows an axiomatic approach based on three simple principles from which it is deduced that the density of the capture time is the solution of a Volterra integral equation of the second kind. Mild conditions are imposed to the underlying stochastic movement model, which is free to choose. We develop simulation methods for both models. The joint distribution of the space-time counts provides an example of a new multivariate distribution, here named the evolving categories multinomial distribution, for which we establish key properties. Since the general likelihood is intractable, we propose a pseudo-likelihood fitting method assuming multivariate Gaussianity respecting mean and covariance structures, justified by the central limit theorem. We conduct simulation studies to validate the method, and we fit our models to experimental data of a spreading population. We estimate movement parameters and compare our models to a basic ecological diffusion model. Movement parameters can be estimated using abundance data, but one must be aware of the necessary conditions to avoid underestimation of spread parameters.

stat.AP↗

arXiv2023

Function-measure kernels, self-integrability and uniquely-defined stochastic integrals

In this work we study the self-integral of a function-measure kernel and its importance on stochastic integration. A continuous-function measure kernel $K$ over $D \subset \mathbb{R}^{d}$ is a function of two variables which acts as a continuous function in the first variable and as a real Radon measure in the second. Some analytical properties of such kernels are studied, particularly in the case of cross-positive-definite type kernels. The self-integral of $K$ over a bounded set $D$ is the "integral of $K$ with respect to itself". It is defined using Riemann sums and denoted $\int_{D}K(x,dx)$. Some examples where such notion is well-defined are presented. This concept turns out to be crucial for unique-definiteness of stochastic integrals, that is, when the integral is independent of the way of approaching it. If $K$ is the cross-covariance kernel between a mean-square continuous stochastic process $Z$ and a random measure with measure covariance structure $M$, $\int_{D}K(x,dx)$ is the expectation of the stochastic integral $\int_{D} ZdM$ when both are uniquely-defined. It is also proven that when $Z$ and $M$ are jointly Gaussian, self-integrability properties on $K$ are necessary and sufficient to guarantee the unique-definiteness of $\int_{D}ZdM$. Results on integrations over subsets, as well as potential $σ$-additive structures are obtained. Three applications of these results are proposed, involving tensor products of Gaussian random measures, the study of a uniquely-defined stochastic integral with respect to fractional Brownian motion with Hurst index $H > \frac{1}{2}$, and the non-uniquely-defined stochastic integrals with respect to orthogonal random measures. The studied stochastic integrals are defined without use of martingale-type conditions, providing a potential filtration-free approach to stochastic calculus grounded on covariance structures.

math.PR↗

arXiv2022

Necessary and sufficient conditions for a family of continuous functions to form a Karhunen-Loève basis

Given an orthonormal system of $L^{2}(D)$ consistent of continuous functions $(f_{n})_{n}$, with $D \subset \mathbb{R}^{d}$ compact, and given a sequence of strictly positive coefficients $(λ_{n})_{n}$ forming a convergent series, we prove that they consist in the eigenfunctions and eigenvectors of a covariance operator associated to a continuous positive-definite Kernel if and only if the sequence of partial sums $ \sum_{j \leq n} λ_{j} f_{j}^{2} $ is equicontinuous over $D$.

math.FA↗

arXiv2019

First-order linear evolution equations with càdlàg-in-time solutions

In this work we study first-order linear parabolic evolution PDEs over $\mathbb{R}^{d}\times\mathbb{R}$ and $\mathbb{R}^{d}\times\mathbb{R}^{+}$ comprising a spatial operator defined through a symbol function and a source term such that its spatial Fourier transform is a slow-growing measure over $\mathbb{R}^{d}\times\mathbb{R}$. When the source term is required to has its support on $\mathbb{R}^{d}\times\mathbb{R}^{+}$, it is shown that there exists a unique solution such that its spatial Fourier transform is a slow-growing measure with support in $\mathbb{R}^{d}\times\mathbb{R}^{+}$, which in addition has a càdlàg-in-time behaviour. This allows to well-pose and analyse an initial value problem associated to this class of equations and to consider cases where the spatial operator can be a pseudo-differential operator. We also look at for solutions to the cases where the source term is such that its spatial and spatio-temporal Fourier transforms are slow-growing measures over $\mathbb{R}^{d}\times\mathbb{R}$. In such a case, it is shown that when the real part of the symbol function of the spatial operator is inferiorly bounded by a strictly positive constant, there exists a unique solution whose both spatial and spatio-temporal Fourier transforms are slow-growing measures over $\mathbb{R}^{d}\times\mathbb{R}$, and which also has a càdlàg-in-time behaviour. In addition, it is proven that the solution to an associated Cauchy problem converges spatio-temporally asymptotically to this unique solution as the time flows long enough.

math.AP↗

arXiv2018

A general framework for SPDE-based stationary random fields

This paper presents theoretical advances in the application of the Stochastic Partial Differential Equation (SPDE) approach in geostatistics. We show a general approach to construct stationary models related to a wide class of linear SPDEs, with applications to spatio-temporal models having non-trivial properties. Within the framework of Generalized Random Fields, a criterion for existence and uniqueness of stationary solutions for this class of SPDEs is proposed and proven. Their covariance are then obtained through their spectral measure. We present a result relating the covariance in the case of a White Noise source term with that of a generic case through convolution. Then, we obtain a variety of SPDE-based stationary random fields. In particular, well-known results regarding the Matérn Model and Markovian models are recovered. A new relationship between the Stein model and a particular SPDE is obtained. New spatio-temporal models obtained from evolution SPDEs of arbitrary temporal derivative order are then obtained, for which properties of separability and symmetry can be controlled. We also obtain results concerning stationary solutions for physically inspired models, such as solutions to the heat equation, the advection-diffusion equation, some Langevin's equations and the wave equation.

math.ST↗