Fundamental assumption of spatial analysis: "everything is related to everything else, but near things are more related than distant things" (first law of geography Tobler, 1970).

In other terms, observations present regularities (correlation, clustering, local/global trends) and spatial statistics methods explicitly use spatial regularities to "improve" the information produced by statistical analysis.

Types of spatial regularity (dependence)

• large scale (trend): regularity in the variability of the phenomenon considered over the entire study region
• small scale: Autocorrelation: the variability is such that values relative to nearby sites tend to be similar (positive dependence); Clustering: presence of groups of similar values, valleys or 'bridges' in surfaces and regions of rapid transition

Objectives of spatial analysis

• Highlight the presence of clustering and identify clusters in space
• Predict the phenomenon of interest in unsampled areas: build maps
• Measure the spatial regularity/variability of a phenomenon: how much variability is due to the spatial regularity of the phenomenon and how much is due to 'disturbance factors'?
• Explain the variability of the phenomenon through measurable factors by appropriately adjusting the inference to account for the spatial dependence of the phenomenon: spatial regression.
• Sample observations taking into account their spatial dependence in order to appropriately calibrate the sample information

Spatial Data

\(Y\) indicates the variable under study

\(s\) represents the location where Y is detected (coordinate vector e.g. geographic or cartographic)

\(Y(s)\) or \(Y_s\) indicates the value that variable Y assumes at s. The notation suggests that attribute Y varies in space

"Point-referenced" data (or geostatistical data): \(Y(s)\) varies continuously in \(D\), continuous and fixed subset of \(R^2\) of positive volume. We will only consider situations where $d = 2 $ and \(s\) is a coordinate vector e.g. geographic or cartographic

• Areal data (or regional data) are data where the fixed subset \(D\) is partitioned into a finite number of sites; these can be arranged on regular grids (for example in agricultural experiments or image reconstruction) or irregular (for example administrative geographic areas). Where \(s\) represents in this case the single area/site typically identified through a code)

• A point pattern is a finite set of random locations, \(s_1,..,s_n\) relative to an event that manifests in space; where \(s\) represents a vector of point coordinates.