Publication: ESTADÍSTICA ESPACIAL CON MATRICES DE COVARIANZA CASI SINGULARES
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Date
2019-12
Authors
PÉREZ OJEDA, JAVIER IGNACIO
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Abstract
El problema de tratar procesos con matrices de covarianza singulares o casi singula-
res ha sido largamente estudiado desde hace décadas (Penrose, 1955), donde se han
propuesto distintas estimaciones para las matrices de covarianza, especialmente para
la inversa.
En este trabajo se consideran distintos enfoques para tratar matrices de covarian-
za singulares o mal condicionadas, desarrollados principalmente por Ayyıldız et al.
(2012), Marzetta et al. (2011) y Tucci y Wang (2011), llevando cada enfoque a pro-
cesos espaciales, que poseen una estructura de covarianza paramétrica que es sensible
a distancias pequeñas entre ubicaciones, lo cual conlleva a una matriz singular o casi
singular. Luego se procede a realizar estimaciones de la matriz de covarianza, para
hacer inferencia en la estimación de parámetros y kriging. Dada una matriz de cova-
rianza paramétrica casi singular, se hace la estimación de valores propios mediante el
método invcov, con el cual se prodece a hacer kriging en nuevas ubicaciones. Final-
mente, se estiman los parámetros del modelo paramétrico usado. Además, se utilizan
los resultados en aplicaciones con datos reales, para un conjunto de datos de micro
algas.
Estimating spatial processes with singular or near singular covariance matrices has been studied since several decades ago (Penrose, 1955), and many estimators for the inverse covariance matrix has been proposed. In this work we consider several approaches in treating singular and nearly singular covariance matrices, and matrices with a large condition number (ill-conditioned), proposed mainly by Ayyıldız et al. (2012), Marzetta et al. (2011) and Tucci y Wang (2011), taking the idea and adapting it to spatial processes with parametric cova- riances, which are highly dependent of the locations proximity, then the resulting covariance is singular or nearly singular. Then, we proceed to estimate the covariance matrix parameter, and therefore be able to perform parameter estimation and spa- tial prediction through kriging. Given a nearly singular parametric covariance matrix, eigenvalues estimation is carried out via invcov method, and then kriging is perfor- med for new locations. Finally, we estimate the covariance matrix parameters with the invcov method in several grids, and the results of the estimations are used in an application to marine microalgae.
Estimating spatial processes with singular or near singular covariance matrices has been studied since several decades ago (Penrose, 1955), and many estimators for the inverse covariance matrix has been proposed. In this work we consider several approaches in treating singular and nearly singular covariance matrices, and matrices with a large condition number (ill-conditioned), proposed mainly by Ayyıldız et al. (2012), Marzetta et al. (2011) and Tucci y Wang (2011), taking the idea and adapting it to spatial processes with parametric cova- riances, which are highly dependent of the locations proximity, then the resulting covariance is singular or nearly singular. Then, we proceed to estimate the covariance matrix parameter, and therefore be able to perform parameter estimation and spa- tial prediction through kriging. Given a nearly singular parametric covariance matrix, eigenvalues estimation is carried out via invcov method, and then kriging is perfor- med for new locations. Finally, we estimate the covariance matrix parameters with the invcov method in several grids, and the results of the estimations are used in an application to marine microalgae.
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Keywords
ESTADÍSTICA ESPACIAL , MATRICES CASI SINGULARES