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Spatial Filtering

Spatial filtering is the application of formulae to obtain enhanced or improved images in remote sensing and more robust findings in data analytic work. Mathematical operators are utilized to separate geographically structured noise from both trend and random noise in georeferenced data, enhancing results by allowing clearer visualization and sounder statistical inference. Nearby/adjacent values are manipulated to adjust the value at a given point, smoothing, reducing variability, and retaining the local features of georeferenced data.

This manipulation is similar to focusing a camera in order to avoid a blurred picture. Any real-world data for a given geographic resolution will display noticeable discrepancies between individual ground truth measures and visualization sharpness, producing a map that is similar to a fuzzy picture. Possible causes of this blurriness include geographic scale; averaging of aggregated data within a locational unit; a tendency for similar data values to cluster on a map, but at a different resolution; and the arbitrariness of locational units forming a map. Spatial filtering mathematically manipulates data in order to correct for distortions introduced by these sources. A wide range of image filters, but a more limited range of data analytic filters that are the topic here, are available. Four types of spatial filters exist for georeferenced data analysis.

Autoregressive Linear Operators

Filtering time-series data with impulse-response function specifications predates spatial filtering and motivated the development of spatial simultaneous autoregressive linear operators, whose error term is correlated with some response variable y. These spatial filters take the matrix form (I − ρC), where ρ is a spatial autocorrelation parameter, n is the number of areal units, I is an n × n identity matrix, and C is an n× n geographic connectivity/weights matrix (e.g., cij = 1 if areal units i and j are nearby/adjacent and cij = 0; otherwise cii = 0). The parameter ρ is estimated for y and then is used in the multiplications (I − ρC)Y, for the n× 1 vector of response values, and (I − ρC)X, for the n× (p+ 1) vector of p covariates and intercept term.

This spatial filter is written in terms of a variance component, is almost always coupled with the normal probability model, and if properly specified renders independent and identically distributed random-error terms. Smoothing occurs in that each data set value is rewritten as the difference between the observed value and a linear combination of neighboring values. A spatial lag (i.e., the values of y for neighboring locations) autoregressive linear operator will be nearly equivalent, being applied to the response variable y but not to the x covariates. Furthermore, a spatial conditional autoregressive linear operator (the conditional expectation of y at a particular location given its values at all other locations) is the mean plus a weighted sum of the mean-centered values at immediate neighboring locations and can be constructed from matrix (I − ρC).

Getis's GiSpecification

The other three data-analytic spatial filters shift attention from a variance to a mean response component that can be employed with conventional and generalized linear regression techniques. The Getis specification is a multistep procedure based upon Ripley's second-order statistic or the range of a geostatistical semivariogram model coupled with the Getis-Ord Gi statistic, and it converts each spatially autocorrelated variable into a pair of variates, one capturing spatial dependencies and one capturing nonspatial systematic and random effects. Regressing the response variable on the set of spatial and aspatial variates allows geographically structured noise to be separated from trend and random noise in georeferenced data.

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