Detection of Influential Observations in Spatial Regression Model Based on Outliers and Bad Leverage Classification

Influential observations (IOs), which are outliers in the x direction, y direction or both, remain a problem in the classical regression model fitting. Spatial regression models have a peculiar kind of outliers because they are local in nature. Spatial regression models are also not free from the ef...

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Autores principales:	Ali Mohammed Baba, Habshah Midi, Mohd Bakri Adam, Nur Haizum Abd Rahman
Formato:	article
Lenguaje:	EN
Publicado:	MDPI AG 2021
Materias:	spatial regression model influential observation outlier leverage prediction residual masking and swamping Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/3bcd8390d89d4cc888ff60c819805569
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id	oai:doaj.org-article:3bcd8390d89d4cc888ff60c819805569
record_format	dspace
institution	DOAJ
collection	DOAJ
language	EN
topic	spatial regression model influential observation outlier leverage prediction residual masking and swamping Mathematics QA1-939
spellingShingle	spatial regression model influential observation outlier leverage prediction residual masking and swamping Mathematics QA1-939 Ali Mohammed Baba Habshah Midi Mohd Bakri Adam Nur Haizum Abd Rahman Detection of Influential Observations in Spatial Regression Model Based on Outliers and Bad Leverage Classification
description	Influential observations (IOs), which are outliers in the x direction, y direction or both, remain a problem in the classical regression model fitting. Spatial regression models have a peculiar kind of outliers because they are local in nature. Spatial regression models are also not free from the effect of influential observations. Researchers have adapted some classical regression techniques to spatial models and obtained satisfactory results. However, masking or/and swamping remains a stumbling block for such methods. In this article, we obtain a measure of spatial Studentized prediction residuals that incorporate spatial information on the dependent variable and the residuals. We propose a robust spatial diagnostic plot to classify observations into regular observations, vertical outliers, good and bad leverage points using a classification based on spatial Studentized prediction residuals and spatial diagnostic potentials, which we refer to as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>S</mi><mi>R</mi><mi>s</mi><mo>−</mo><msub><mi>P</mi><mrow><mi>o</mi><mi>s</mi><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>S</mi><mi>R</mi><mi>s</mi><mo>−</mo><msub><mi>P</mi><mrow><mi>o</mi><mi>s</mi><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula>. Observations that fall into the vertical outliers and bad leverage points categories are referred to as IOs. Representations of some classical regression measures of diagnostic in general spatial models are presented. The commonly used diagnostic measure in spatial diagnostics, the Cook’s distance, is compared to some robust methods, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>H</mi><mi>i</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> (using robust and non-robust measures), and our proposed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>S</mi><mi>R</mi><mi>s</mi><mo>−</mo><msub><mi>P</mi><mrow><mi>o</mi><mi>s</mi><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>S</mi><mi>R</mi><mi>s</mi><mo>−</mo><msub><mi>P</mi><mrow><mi>o</mi><mi>s</mi><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula> plots. Results of our simulation study and applications to real data showed that the Cook’s distance, non-robust <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="normal">H</mi><mrow><mi>si</mi><mn>1</mn></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> and robust <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>H</mi><mrow><mi>s</mi><mi>i</mi><mn>2</mn></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> were not very successful in detecting IOs. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="normal">H</mi><mrow><mi>si</mi><mn>1</mn></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> suffered from the masking effect, and the robust <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>H</mi><mrow><mi>s</mi><mi>i</mi><mn>2</mn></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> suffered from swamping in general spatial models. Interestingly, the results showed that the proposed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>S</mi><mi>R</mi><mi>s</mi><mo>−</mo><msub><mi>P</mi><mrow><mi>o</mi><mi>s</mi><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula> plot, followed by the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>S</mi><mi>R</mi><mi>s</mi><mo>−</mo><msub><mi>P</mi><mrow><mi>o</mi><mi>s</mi><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula> plot, was very successful in classifying observations into the correct groups, hence correctly detecting the real IOs.
format	article
author	Ali Mohammed Baba Habshah Midi Mohd Bakri Adam Nur Haizum Abd Rahman
author_facet	Ali Mohammed Baba Habshah Midi Mohd Bakri Adam Nur Haizum Abd Rahman
author_sort	Ali Mohammed Baba
title	Detection of Influential Observations in Spatial Regression Model Based on Outliers and Bad Leverage Classification
title_short	Detection of Influential Observations in Spatial Regression Model Based on Outliers and Bad Leverage Classification
title_full	Detection of Influential Observations in Spatial Regression Model Based on Outliers and Bad Leverage Classification
title_fullStr	Detection of Influential Observations in Spatial Regression Model Based on Outliers and Bad Leverage Classification
title_full_unstemmed	Detection of Influential Observations in Spatial Regression Model Based on Outliers and Bad Leverage Classification
title_sort	detection of influential observations in spatial regression model based on outliers and bad leverage classification
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/3bcd8390d89d4cc888ff60c819805569
work_keys_str_mv	AT alimohammedbaba detectionofinfluentialobservationsinspatialregressionmodelbasedonoutliersandbadleverageclassification AT habshahmidi detectionofinfluentialobservationsinspatialregressionmodelbasedonoutliersandbadleverageclassification AT mohdbakriadam detectionofinfluentialobservationsinspatialregressionmodelbasedonoutliersandbadleverageclassification AT nurhaizumabdrahman detectionofinfluentialobservationsinspatialregressionmodelbasedonoutliersandbadleverageclassification
_version_	1718410271419006976
spelling	oai:doaj.org-article:3bcd8390d89d4cc888ff60c8198055692021-11-25T19:06:10ZDetection of Influential Observations in Spatial Regression Model Based on Outliers and Bad Leverage Classification10.3390/sym131120302073-8994https://doaj.org/article/3bcd8390d89d4cc888ff60c8198055692021-10-01T00:00:00Zhttps://www.mdpi.com/2073-8994/13/11/2030https://doaj.org/toc/2073-8994Influential observations (IOs), which are outliers in the x direction, y direction or both, remain a problem in the classical regression model fitting. Spatial regression models have a peculiar kind of outliers because they are local in nature. Spatial regression models are also not free from the effect of influential observations. Researchers have adapted some classical regression techniques to spatial models and obtained satisfactory results. However, masking or/and swamping remains a stumbling block for such methods. In this article, we obtain a measure of spatial Studentized prediction residuals that incorporate spatial information on the dependent variable and the residuals. We propose a robust spatial diagnostic plot to classify observations into regular observations, vertical outliers, good and bad leverage points using a classification based on spatial Studentized prediction residuals and spatial diagnostic potentials, which we refer to as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>S</mi><mi>R</mi><mi>s</mi><mo>−</mo><msub><mi>P</mi><mrow><mi>o</mi><mi>s</mi><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>S</mi><mi>R</mi><mi>s</mi><mo>−</mo><msub><mi>P</mi><mrow><mi>o</mi><mi>s</mi><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula>. Observations that fall into the vertical outliers and bad leverage points categories are referred to as IOs. Representations of some classical regression measures of diagnostic in general spatial models are presented. The commonly used diagnostic measure in spatial diagnostics, the Cook’s distance, is compared to some robust methods, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>H</mi><mi>i</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> (using robust and non-robust measures), and our proposed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>S</mi><mi>R</mi><mi>s</mi><mo>−</mo><msub><mi>P</mi><mrow><mi>o</mi><mi>s</mi><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>S</mi><mi>R</mi><mi>s</mi><mo>−</mo><msub><mi>P</mi><mrow><mi>o</mi><mi>s</mi><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula> plots. Results of our simulation study and applications to real data showed that the Cook’s distance, non-robust <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="normal">H</mi><mrow><mi>si</mi><mn>1</mn></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> and robust <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>H</mi><mrow><mi>s</mi><mi>i</mi><mn>2</mn></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> were not very successful in detecting IOs. The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="normal">H</mi><mrow><mi>si</mi><mn>1</mn></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> suffered from the masking effect, and the robust <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>H</mi><mrow><mi>s</mi><mi>i</mi><mn>2</mn></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> suffered from swamping in general spatial models. Interestingly, the results showed that the proposed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mi>S</mi><mi>R</mi><mi>s</mi><mo>−</mo><msub><mi>P</mi><mrow><mi>o</mi><mi>s</mi><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula> plot, followed by the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mi>S</mi><mi>R</mi><mi>s</mi><mo>−</mo><msub><mi>P</mi><mrow><mi>o</mi><mi>s</mi><mi>i</mi></mrow></msub></mrow></semantics></math></inline-formula> plot, was very successful in classifying observations into the correct groups, hence correctly detecting the real IOs.Ali Mohammed BabaHabshah MidiMohd Bakri AdamNur Haizum Abd RahmanMDPI AGarticlespatial regression modelinfluential observationoutlierleverageprediction residualmasking and swampingMathematicsQA1-939ENSymmetry, Vol 13, Iss 2030, p 2030 (2021)

Detection of Influential Observations in Spatial Regression Model Based on Outliers and Bad Leverage Classification

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