(A)Spatial Regression

(A)Spatial Regression

Copyright: © 2018 |Pages: 30
DOI: 10.4018/978-1-5225-3270-5.ch004


The global spatial lag model for stationary processes (Y=?WY+Xß+e, where W represents the neighborhood matrix while ? is the spatial autoregressive coefficient) and the spatial regimes (individual regression for each sub-region), developed within SpaceStat®, are two possibilities. An extra option is the local continuous variation framework computed by the Geographically Weighted Regression (GWR). The idea here is to adjust a regression model (the ßs of the regression model are not the same everywhere) by weighting the neighborhood observations. In this way, the estimation computation will reflect automatic adjustments, according to the distance and values of the available samples. Section “Geographically Weighted Regression” introduces this matter while the next two sections review the conventional bivariate and multivariate OLS regression, respectively. As an extension of the OLS family, non-linear relationships, regression with dummy variables, path analysis and logistic regression are briefly explained under the following section three although myGeoffice© does not cover these practices. Spearman Rank Correlation for ordinal data is presented in the last section.
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Geographically Weighted Regression

GWR is a specific spatial model, which allows representing non-stationary local phenomena by generating a separate regression equation for every feature analyzed as a means to address spatial variation (Fotheringham, Brunsdon & Charlton, 2002). Hence, GWR allows the modelling of processes that vary over space (non-stationary regions). Since it usually works with aggregated data, this inferential model considers that spatial data may change abruptly (or not) at region boundaries only.

For Lloyd (2007), one key decision emerges from this local inferential approach for spatial non-stationary situations (when the same stimulus provokes a different response in different parts of the study region): The choice of a weighting function (kernel shape VS kernel bandwidth) regarding neighborhood samples (which plays a central role towards each prediction).

Figure 1.

The adaptive (left) and fixed weighted scheme (right) for the weighted matrix. According to Yu & Wei (2011), the fixed scheme might produce large estimate variances where data is sparse. On the other hand, it can mask subtle local variations where data is dense. Indeed, this is the key of GWR in order to measure spatial variation in any study region.

In a conventional form, according to Mennis (2006), the GWR equation can be expressed as , where represents the estimate of the dependent study variable for i observation, m equals the number of independent variables, is the intercept value, stands for the parameter estimate for the independent k variable, xik denotes the value of the k variable for i observation, (ui, vi) regards the 2D coordinate location of i sample while is the residual error. Instead of calibrating a single and global OLS conventional regression equation for the overall area, GWR generates a separate regression equation for each observation. Each equality is then calibrated with different weights of the neighboring observations contained in the dataset, based on the assumption that observations nearby one another have a greater influence on one another’s parameter estimates than observations farther apart (Mennis, 2006). In essence, only the nearest samples of each estimated site will be used for the local estimation regression.

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