Diagnostic Tools and a Remedial Method for Collinearity in Geographically Weighted Regression

2007 ◽  
Vol 39 (10) ◽  
pp. 2464-2481 ◽  
Author(s):  
David C Wheeler
Keyword(s):  
Ridge Regression ◽  
Geographically Weighted Regression ◽  
Model Building ◽  
Weighted Regression ◽  
Diagnostic Tools ◽  
Local Regression ◽  
Explanatory Variables ◽  
Building Process ◽  
Global Regression ◽  
Spatially Varying

Geographically weighted regression (GWR) is drawing attention as a statistical method to estimate regression models with spatially varying relationships between explanatory variables and a response variable. Local collinearity in weighted explanatory variables leads to GWR coefficient estimates that are correlated locally and across space, have inflated variances, and are at times counterintuitive and contradictory in sign to the global regression estimates. The presence of local collinearity in the absence of global collinearity necessitates the use of diagnostic tools in the local regression model building process to highlight areas in which the results are not reliable for statistical inference. The method of ridge regression can also be integrated into the GWR framework to constrain and stabilize regression coefficients and lower prediction error. This paper presents numerous diagnostic tools and ridge regression in GWR and demonstrates the utility of these techniques with an example using the Columbus crime dataset.

scholarly journals Exploring spatial nonstationary environmental effects on Yellow Perch distribution in Lake Erie

PeerJ ◽  
2019 ◽  
Vol 7 ◽  
pp. e7350
Author(s):  
Changdong Liu ◽  
Junchao Liu ◽  
Yan Jiao ◽  
Yanli Tang ◽  
Kevin B. Reid
Keyword(s):  
Environmental Effects ◽  
Geographically Weighted Regression ◽  
Regression Models ◽  
Yellow Perch ◽  
Lake Erie ◽  
Regression Coefficients ◽  
Weighted Regression ◽  
Local Regression ◽  
Spatially Varying ◽  
Spatial Locations

Background Global regression models under an implicit assumption of spatial stationarity were commonly applied to estimate the environmental effects on aquatic species distribution. However, the relationships between species distribution and environmental variables may change among spatial locations, especially at large spatial scales with complicated habitat. Local regression models are appropriate supplementary tools to explore species-environment relationships at finer scales. Method We applied geographically weighted regression (GWR) models on Yellow Perch in Lake Erie to estimate spatially-varying environmental effects on the presence probabilities of this species. Outputs from GWR were compared with those from generalized additive models (GAMs) in exploring the Yellow Perch distribution. Local regression coefficients from the GWR were mapped to visualize spatially-varying species-environment relationships. K-means cluster analyses based on the t-values of GWR local regression coefficients were used to characterize the distinct zones of ecological relationships. Results Geographically weighted regression resulted in a significant improvement over the GAM in goodness-of-fit and accuracy of model prediction. Results from the GWR revealed the magnitude and direction of environmental effects on Yellow Perch distribution changed among spatial locations. Consistent species-environment relationships were found in the west and east basins for adults. The different kinds of species-environment relationships found in the central management unit (MU) implied the variation of relationships at a scale finer than the MU. Conclusions This study draws attention to the importance of accounting for spatial nonstationarity in exploring species-environment relationships. The GWR results can provide support for identification of unique stocks and potential refinement of the current jurisdictional MU structure toward more ecologically relevant MUs for the sustainable management of Yellow Perch in Lake Erie.

scholarly journals Distance metric choice can both reduce and induce collinearity in geographically weighted regression

2018 ◽  
Vol 47 (3) ◽  
pp. 489-507 ◽  
Author(s):  
Alexis Comber ◽  
Khanh Chi ◽  
Man Q Huy ◽  
Quan Nguyen ◽  
Binbin Lu ◽  
...  
Keyword(s):  
Geographically Weighted Regression ◽  
House Price ◽  
Elastic Net ◽  
Weighted Regression ◽  
Manhattan Distance ◽  
Distance Metrics ◽  
Distance Metric ◽  
Local Regression ◽  
Spatially Varying ◽  
The Impact

This paper explores the impact of different distance metrics on collinearity in local regression models such as geographically weighted regression. Using a case study of house price data collected in Hà Nội, Vietnam, and by fully varying both power and rotation parameters to create different Minkowski distances, the analysis shows that local collinearity can be both negatively and positively affected by distance metric choice. The Minkowski distance that maximised collinearity in a geographically weighted regression was approximate to a Manhattan distance with (power =  0.70) with a rotation of 30°, and that which minimised collinearity was parameterised with power  = 0.05 and a rotation of 70°. The results indicate that distance metric choice can provide a useful extra tuning component to address local collinearity issues in spatially varying coefficient modelling and that understanding the interaction of distance metric and collinearity can provide insight into the nature and structure of the data relationships. The discussion considers first, the exploration and selection of different distance metrics to minimise collinearity as an alternative to localised ridge regression, lasso and elastic net approaches. Second, it discusses the how distance metric choice could extend the methods that additionally optimise local model fit (lasso and elastic net) by selecting a distance metric that further helped minimise local collinearity. Third, it identifies the need to investigate the relationship between kernel bandwidth, distance metrics and collinearity as an area of further work.

A General Framework for Estimation and Inference of Geographically Weighted Regression Models: 1. Location-Specific Kernel Bandwidths and a Test for Locational Heterogeneity

2002 ◽  
Vol 34 (4) ◽  
pp. 733-754 ◽  
Author(s):  
Antonio Páez ◽  
Takashi Uchida ◽  
Kazuaki Miyamoto
Keyword(s):  
Geographically Weighted Regression ◽  
Degrees Of Freedom ◽  
Likelihood Function ◽  
Error Variance ◽  
Model Specification ◽  
Weighted Regression ◽  
Regularity Conditions ◽  
Local Regression ◽  
Specification Tests ◽  
Geographically Weighted Regression Model

Geographically weighted regression (GWR) has been proposed as a technique to explore spatial parametric nonstationarity. The method has been developed mainly along the lines of local regression and smoothing techniques, a strategy that has led to a number of difficult questions about the regularity conditions of the likelihood function, the effective number of degrees of freedom, and in general the relevance of extending the method to derive inference and model specification tests. In this paper we argue that placing GWR within a different statistical context, as a spatial model of error variance heterogeneity, or what might be termed locational heterogeneity, solves these difficulties. A maximum-likelihood-based framework for estimation and inference of a general geographically weighted regression model is presented that leads to a method to estimate location-specific kernel bandwidths. Moreover, a test for locational heterogeneity is derived and its use exemplified with a case study.

A Simulation-Based Study of Geographically Weighted Regression as a Method for Investigating Spatially Varying Relationships

2011 ◽  
Vol 43 (12) ◽  
pp. 2992-3010 ◽  
Author(s):  
Antonio Páez ◽  
Steven Farber ◽  
David Wheeler
Keyword(s):  
Geographically Weighted Regression ◽  
Numerical Experiments ◽  
Cross Validation ◽  
Weighted Regression ◽  
Small Samples ◽  
Large Variability ◽  
Underlying Process ◽  
Simulation Based ◽  
Multivariate Relationships ◽  
Spatially Varying

Large variability and correlations among the coefficients obtained from the method of geographically weighted regression (GWR) have been identified in previous research. This is an issue that poses a serious challenge for the utility of the method as a tool to investigate multivariate relationships. The objectives of this paper are to assess: (1) the ability of GWR to discriminate between a spatially constant processes and one with spatially varying relationships; and (2) to accurately retrieve spatially varying relationships. Extensive numerical experiments are used to investigate situations where the underlying process is stationary and nonstationary, and to assess the degree to which spurious intercoefficient correlations are introduced. Two different implementations of GWR and cross-validation approaches are assessed. Results suggest that judicious application of GWR can be used to discern whether the underlying process is nonstationary. Furthermore, evidence of spurious correlations indicates that caution must be exercised when drawing conclusions regarding spatial relationships retrieved using this approach, particularly when working with small samples.

scholarly journals A Comparison of Nonergodic Ground-Motion Models based on Geographically Weighted Regression and the Integrated Nested Laplace Approximation

2021 ◽  
Author(s):  
Nicolas Kuehn
Keyword(s):  
Ground Motion ◽  
Geographically Weighted Regression ◽  
Seismic Hazard Assessment ◽  
Predictive Performance ◽  
Laplace Approximation ◽  
Weighted Regression ◽  
Integrated Nested Laplace Approximation ◽  
Motion Models ◽  
Spatially Varying ◽  
Ground Motion Models

Different nonergodic Ground-Motion Models based on spatially varying coefficient models are compared for ground-motion data in Italy. The models are based different methodologies: Multi-source geographically weighted regression (Caramenti et al., 2020), and Bayesian hierarchical models estimated with the integrated nested Laplace approximation (Rue et al., 2009). The different models are compared in terms of their predictive performance, their spatial coefficients, and their predictions. Models that include spatial terms perform slightly better than a simple base model that includes only event and station terms, in terms of out-of sample error based on cross-validation. The Bayesian spatial models have slightly lower generalization error, which can be attributed to the fact that they can include random effects for events and stations. The different methodologies give rise to different dependencies of the spatially varying terms on event and station locations, leading to between-model uncertainty in their predictions, which should be accommodated in a nonergodic seismic hazard assessment.

scholarly journals mgwr: A Python Implementation of Multiscale Geographically Weighted Regression for Investigating Process Spatial Heterogeneity and Scale

2019 ◽  
Vol 8 (6) ◽  
pp. 269 ◽  
Author(s):  
Taylor Oshan ◽  
Ziqi Li ◽  
Wei Kang ◽  
Levi Wolf ◽  
A. Fotheringham
Keyword(s):  
Spatial Heterogeneity ◽  
Geographically Weighted Regression ◽  
Linear Models ◽  
Statistical Technique ◽  
Weighted Regression ◽  
Parameter Estimates ◽  
Local Models ◽  
Spatial Processes ◽  
Geographic Scale ◽  
Global Regression

Geographically weighted regression (GWR) is a spatial statistical technique that recognizes that traditional ‘global’ regression models may be limited when spatial processes vary with spatial context. GWR captures process spatial heterogeneity by allowing effects to vary over space. To do this, GWR calibrates an ensemble of local linear models at any number of locations using ‘borrowed’ nearby data. This provides a surface of location-specific parameter estimates for each relationship in the model that is allowed to vary spatially, as well as a single bandwidth parameter that provides intuition about the geographic scale of the processes. A recent extension to this framework allows each relationship to vary according to a distinct spatial scale parameter, and is therefore known as multiscale (M)GWR. This paper introduces mgwr, a Python-based implementation of MGWR that explicitly focuses on the multiscale analysis of spatial heterogeneity. It provides novel functionality for inference and exploratory analysis of local spatial processes, new diagnostics unique to multi-scale local models, and drastic improvements to efficiency in estimation routines. We provide two case studies using mgwr, in addition to reviewing core concepts of local models. We present this in a literate programming style, providing an overview of the primary software functionality and demonstrations of suggested usage alongside the discussion of primary concepts and demonstration of the improvements made in mgwr.

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