# Spatial Autocorrelation

DOI: 10.4018/978-1-5225-3270-5.ch007

## Abstract

This chapter respects the six procedures of spatial autocorrelation covered by myGeoffice©, including variogram setup and fitness, Moran I correlograms, an innovative version of the conventional Moran scatterplot and the recent Moran variance scatterplot.
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## Introduction

If the finding of spatial structures is fundamental for geography then spatial interpolation, spatial autoregressive models and spatial autocorrelation exists as three major techniques for a better understanding of spatial relationships among spatial entities.

Kriging means optimal interpolation of unknown values from observed data at known locations. After the variogram has been defined, this algebraic relationship between values at different distances is used to estimate Kriging weights based on four factors: (A) Closeness to the location being estimated; (B) Redundancy between data values; (C) Anisotropic continuity; (D) Magnitude of continuity. Kriging is BLUP (best linear unbiased predictor), whether or not data is normally distributed. It is linear since estimations are weighted linear combinations of the available data. It is unbiased because the error of the mean tries to be close to zero, i.e., there are no over- or under-estimates. It is best since its goal is to minimize error variance.

Spatial autoregressive models expand the standard linear model regression with additional terms that account for local patterns (Banerjee, Carlin & Gelfand, 2015). Essentially, a spatial lag model expresses the notion that the variable value at (x,y) location is related to the values of the same variable measured at nearby locations (the interaction effect and measured by the spatial weights matrix, i.e., W). The basic spatial lagged autoregression equation equals yi=, where the spatial autoregression parameter (ρ) should typically be estimated from the available data (Smith, 2015).

Spatial autocorrelation measures the way things are distributed in space (clustering) and it can be seen as a casual process by measuring the degree of influence (correlation) exerted over its neighbors. This means location leads to spatial dependence (correlation or variation that each neighbor holds in relation to a particular point) and spatial heterogeneity (clustering, concentration or proportion of neighborhood average in relation to a specific point) established by Tobler’s First Law of Geography. Unfortunately, this situation creates a conflict between the goals of classical statistics and its assumptions (samples are precise, representative, taken randomly from the population and constitute a small portion of a very large one) because classical estimators tend to be inconsistent as the degree of spatial dependence increases (either positive or negative). It appears that Mother Nature does not care about the assumptions of statisticians (Negreiros, Costa, Painho & Santos, 2007).

Supported by Griffith and Layne (1999), spatial autoregression plus spatial autocorrelation provides yet another linkage with Kriging: The spatial interpolation issue. According to both authors, several relationships among these three concepts emerge:

• Spatial autocorrelation is the required condition of Kriging and spatial regression, i.e., spatial autocorrelation is a physical reality and is a necessity for interpolating nearby values.

• Spatial autocorrelation seeks spatial identification while spatial regression and Kriging seek spatial prediction.

• The variance-covariance matrix is included in spatial regression and Kriging.

• Once a variogram is computed, Kriging can be used for spatial interpolation.

• Incorporating spatial autocorrelation within spatial regression results in a 5-10% improvement in the statistical description for a given georeferenced dataset.

• Kriging is primarily applied with continuous regions while spatial autoregression involves aggregations of phenomena into discrete regions such as ward units.

• Spatial autoregressive methods assume that spatial interpolation follows an underlying trend plus random residuals. However, Kriging presumes two other views: (A) If Universal Kriging is assumed then spatial interpolation is interrelated to a background trend; (B) If Ordinary Kriging is chosen, spatial interpolation is consistent with the samples global average plus random residuals.

• Autoregressive residuals can be used for a reasonable reality approximation test to verify existing patterns among residuals and a key information source of possible assumption violations, variable transformations, outliers and trends surface.

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