Spatial Statistics

Introduction

Often we find ourselves with a dataset which has a location / spatial information that should be part of the modelling. This can be useful to model the dependence structure of datasets, and this dependence structure can be positively / negatively correlated.

• Air Pollution Measurements: Air pollution levels of nearby locations are positively correlated.
• Plant growths: Plants in the nearby locations compete with each other for the same resources, growth of one kind of plant may be negatively correlated with the growth of another.

How spatial analysis is different?

• When we observe say \(Y_n = (Y(s_1), \dots, Y(s_n))\) where \(s_1, \dots, s_n\) are the spatial locations, think of this as a single realization of a \(n\)-variable random vector, instead of the usual \(n\) replications of a univariate random vector. It is not clear at this point, whether any inference is possible based on a single sample observation[1].

• When time series methodologies are concerned, the sample indices have a natural ordering, hence \(Y_t\) depends only on the past values \(\{ Y_s : s < t \}\). However, for the spatial case, say even when agricultural plots are arranged in a line to produce a natural ordering, the value \(Y_{t}\) may depend on both \(Y_{t-1}\) and \(Y_{t+1}\).

Types of Spatial Data

In the spatial data, there are two things in concern.

• There is a spatial horizon \(S\). From this spatial horizon, some locations \(s_1, \dots, s_n\) are where the samples are observed.
• For each of these locations, some variables \(Y(s)\) is observed.

Therefore, we assume the existence of a stochastic process \(\{ Y(s): s \in S \}\).

There are 3 major types of spatial data based on the characteristics of this stochastic process.

1. Point Patterns: The location themselves are realizations of some stochastic process, e.g., events of earthquakes / volcano.
   • The main question here is to try to find patterns / clusters on which locations these events happen or are they happen randomly throughout the spatial horizon.
   • Usually, if we just observe the locations alone, these are called unmarked point process. Example is location of volcanos.
   • If we also observe some variables associated with these event locations, these are called marked point process. Example is along with the location of earthquake, we also measure its intensity.
2. Geostatistical Data: The underlying stochastic process for the \(Y\)-variable is defined on the continuous domain \(S\). However, due to fixed sampling design, we observe \(Y(s)\) only at few designated locations, e.g. \(Y(s_1), \dots, Y(s_n)\).
   • The aim is to use the observed values to predict the continuous surface \(Y(s)\), and use the neighbouring correlation to improve the prediction.
   • Example is to predict the underground oil reserves based on the quantity of oil at few mining locations.
3. Lattice Data: The underlying sampling frame is a fixed designed areal units. In each of these areal units, some \(Y\)-observations are made. These areal units could be subplots, counties, census tracts, etc.
   • Example is agricultural plots are sub-divided into subplots, where different types of crop-yields are observed.
   • If it is census-tract data, then often the aim is to do some kind of regression of one spatial-process on another to explain its variation.
   • Another aim is to predict the observation (called “kriging”) at the unobserved lattice areal units based on the observed values.

Inference for Spatial Linear Models

References

[1]: Schabenberger, Oliver. “Statistical methods for spatial data analysis.” Chapman Hall/CRC (2005).

[2]: Whittle, Peter. “On stationary processes in the plane.” Biometrika (1954): 434-449.