Testing for Normality in Geostatistics. A New Approach Based on the Mahalanobis Distance

Testing for Normality in Geostatistics. A New Approach Based on the Mahalanobis Distance

Simple kriging is a best linear predictor (BLP) and ordinary kriging is a best linear unbiased predictor (BLUP). When the underlying process is normal, simple kriging is not only a BLP but a best predictor (BT) as well, that is, under squared loss, this predictor coincides with the conditional ex...

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Main Authors: Giraldo, Ramón, Porcu, Emilio
Format: Online
Language:spa
Published: Universidad Pedagógica y Tecnológica de Colombia 2022
Subjects:
Distribución chi-cuadrado, distribución normal multivariada, distancia de Mahalanobis, prueba de normalidad, campo aleatorio, simulación de Monte Carlo
Chi-square distribution, Multivariate normal distribution, Mahalanobis distance, Normality test, Random field, Monte Carlo simulation.
Online Access:https://revistas.uptc.edu.co/index.php/ciencia_en_desarrollo/article/view/13650
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author Giraldo, Ramón
Porcu, Emilio
author_facet Giraldo, Ramón
Porcu, Emilio
author_sort Giraldo, Ramón
collection OJS
description Simple kriging is a best linear predictor (BLP) and ordinary kriging is a best linear unbiased predictor (BLUP). When the underlying process is normal, simple kriging is not only a BLP but a best predictor (BT) as well, that is, under squared loss, this predictor coincides with the conditional expectation of the predictor given the information. In this scenario, ordinary kriging provides an approximation to the BP. For this reason, in applied geostatistics, it is important to test for normality. Given a realization of a spatial random process, the simple kriging predictor will be optimal if the random vector follows a multivariate normal distribution. Some classical tests, such as Shapiro-Wilk (SW), Shapiro-Francia (SF), or Anderson-Darling (AD) are frequently used to evaluate the normality assumption. Such approaches assume independence and hence are not effective for at least two reasons. On the one hand, observations in a geostatistical analysis are typically spatially correlated. On the other hand, kriging optimality as mentioned above is based on multivariate rather than univariate normality. In this work, we provide a simulation study to describe the negative effect of using normality univariate tests with geostatistical data. We also show how the Mahalanobis distance can be adapted to the geostatistical context to test for normality.
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spelling oai:oai.revistas.uptc.edu.co:article-136502023-06-26T20:43:15Z Testing for Normality in Geostatistics. A New Approach Based on the Mahalanobis Distance Pruebas de Normalidad en Geoestadística. Un nuevo enfoque basado en la distancia de Mahalanobis Giraldo, Ramón Porcu, Emilio Distribución chi-cuadrado, distribución normal multivariada, distancia de Mahalanobis, prueba de normalidad, campo aleatorio, simulación de Monte Carlo Chi-square distribution, Multivariate normal distribution, Mahalanobis distance, Normality test, Random field, Monte Carlo simulation. Simple kriging is a best linear predictor (BLP) and ordinary kriging is a best linear unbiased predictor (BLUP). When the underlying process is normal, simple kriging is not only a BLP but a best predictor (BT) as well, that is, under squared loss, this predictor coincides with the conditional expectation of the predictor given the information. In this scenario, ordinary kriging provides an approximation to the BP. For this reason, in applied geostatistics, it is important to test for normality. Given a realization of a spatial random process, the simple kriging predictor will be optimal if the random vector follows a multivariate normal distribution. Some classical tests, such as Shapiro-Wilk (SW), Shapiro-Francia (SF), or Anderson-Darling (AD) are frequently used to evaluate the normality assumption. Such approaches assume independence and hence are not effective for at least two reasons. On the one hand, observations in a geostatistical analysis are typically spatially correlated. On the other hand, kriging optimality as mentioned above is based on multivariate rather than univariate normality. In this work, we provide a simulation study to describe the negative effect of using normality univariate tests with geostatistical data. We also show how the Mahalanobis distance can be adapted to the geostatistical context to test for normality. En geoestadística, bajo estacionariedad, kriging simple (KS) es el mejor predictor lineal (MPL) y kriging ordinario (KO) es el mejor predictor lineal insesgado (MPLI). Cuando el proceso estocástico es Normal, KS no es solo un MPL sino un mejor predictor (MP), es decir que bajo la función de pe ́rdida cuadrática, éste coincide con la esperanza condicional del predictor dada la información. En este escenario, el predictor KO sirve como aproximación del MP. Por esta razón, en geoestadística aplicada, es importante probar el supuesto de normalidad. Dada una realización de un proceso espacial, KS será un predictor óptimo si el vector aleatorio subyacente sigue una distribución normal multivariada. Algunas pruebas de normalidad clásicas como Shapiro-Wilk (SW), Shapiro-Francia (SF), o Anderson-Darling (AD) son usadas para evaluar este supuesto. Estas asumen independencia y por ello no son apropiadas en geoestadística (y en general en estadística espacial). Por un lado, las observaciones en geoestadística son espacialmente correlacionadas. Por otro lado la optimalidad del kriging es fundamentada en normalidad multivariada (no en normalidad univariada). En este trabajo se presenta un estudio de simulación para mostrar por qué es inapropiado el uso de pruebas univaridas de normalidad con datos geoestadísticos. También, como solución al problema anterior, se propone una adaptación de la prueba de Mahalanobis al contexto geoestadístico para hacer de manera correcta el test de normalidad en este ambito. Universidad Pedagógica y Tecnológica de Colombia 2022-07-12 info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion application/pdf https://revistas.uptc.edu.co/index.php/ciencia_en_desarrollo/article/view/13650 10.19053/01217488.v13.n2.2022.13650 Ciencia En Desarrollo; Vol. 13 No. 2 (2022): Vol. 13 Núm. 2 (2022): Vol. 13 Núm. 2 (2022): Vol 13, Núm.2 (2022): Julio-Diciembre; 99-112 Ciencia en Desarrollo; Vol. 13 Núm. 2 (2022): Vol. 13 Núm. 2 (2022): Vol. 13 Núm. 2 (2022): Vol 13, Núm.2 (2022): Julio-Diciembre; 99-112 2462-7658 0121-7488 spa https://revistas.uptc.edu.co/index.php/ciencia_en_desarrollo/article/view/13650/12649
spellingShingle Distribución chi-cuadrado, distribución normal multivariada, distancia de Mahalanobis, prueba de normalidad, campo aleatorio, simulación de Monte Carlo
Chi-square distribution, Multivariate normal distribution, Mahalanobis distance, Normality test, Random field, Monte Carlo simulation.
Giraldo, Ramón
Porcu, Emilio
Testing for Normality in Geostatistics. A New Approach Based on the Mahalanobis Distance
title Testing for Normality in Geostatistics. A New Approach Based on the Mahalanobis Distance
title_alt Pruebas de Normalidad en Geoestadística. Un nuevo enfoque basado en la distancia de Mahalanobis
title_full Testing for Normality in Geostatistics. A New Approach Based on the Mahalanobis Distance
title_fullStr Testing for Normality in Geostatistics. A New Approach Based on the Mahalanobis Distance
title_full_unstemmed Testing for Normality in Geostatistics. A New Approach Based on the Mahalanobis Distance
title_short Testing for Normality in Geostatistics. A New Approach Based on the Mahalanobis Distance
title_sort testing for normality in geostatistics a new approach based on the mahalanobis distance
topic Distribución chi-cuadrado, distribución normal multivariada, distancia de Mahalanobis, prueba de normalidad, campo aleatorio, simulación de Monte Carlo
Chi-square distribution, Multivariate normal distribution, Mahalanobis distance, Normality test, Random field, Monte Carlo simulation.
topic_facet Distribución chi-cuadrado, distribución normal multivariada, distancia de Mahalanobis, prueba de normalidad, campo aleatorio, simulación de Monte Carlo
Chi-square distribution, Multivariate normal distribution, Mahalanobis distance, Normality test, Random field, Monte Carlo simulation.
url https://revistas.uptc.edu.co/index.php/ciencia_en_desarrollo/article/view/13650
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