Lecture 40 —Monday, April 23, 2007

What was covered?

• Introduction to spatial analysis
• Types of spatial data

Spatial Data

• Formally spatial data entail recording an attribute’s value, z, along with the attribute’s location, s. We represent this as Z(s), where typically . D is called the spatial domain and is typically a subset of the Cartesian plane.
• Spatial patterns are of interest in their own right, but they can also arise to complicate standard statistical analysis. Here's an illustration.
  • In a typical statistical experiment, say a simple linear regression, we manipulate the levels of a variable x for different experimental units while observing the value of a second variable y (that we call the response).
  • But in an observational study, carried out because an experiment might be difficult to perform, we often use the natural variability in x as a surrogate for experimental manipulation. This natural variability is captured by obtaining observational units from different spatial locations.
  • Thus we have a logical dilemma in an observational study with a spatial dimension. Having failed to randomly assign treatment to observational units we have no legitimate basis for inferring causation. If x and y are seen to vary together it could be because an unmeasured spatially varying variable is affecting both of them.
• Spatial location can serve as a catch-all for all such unmeasured variables. Tobler’s law of geography (Tobler 1970) states “Everything is related to everything else, but near things are more related than distant things.” So we expect the spatial dimension to be important in most observational studies.

Two Viewpoints

• There are two typical ways of approaching the spatial dimension in data analysis.
  1. The spatial dimension is a nuisance. If we take n spatially referenced samples then due to possible spatial correlation among sites our observations are unlikely to be independent. Hence our sample size is less than we hoped for and statistical power is reduced.
  2. The spatial dimension is the focus of the study. Here our problem concerns sample size. If we take n spatially referenced samples then in terms of the spatial dimension we are trying to study we have a sample of size 1. Our sample estimates the function Z(s), a spatial surface of which we have obtained a single realization with our sample. What’s random in this case are the individual values obtained at given locations, not the locations themselves. This randomness arises from measurement error, temporal changes in the spatial surface, etc.

Types of Spatial Data

• Regardless of the viewpoint one takes three distinct categories of spatial data are recognized: geostatistical data, lattice data, and point process data (spatial point patterns).
• Geostatistical data: The spatial domain D here is fixed and continuous. With geostatistical datas the number of locations at which observations can be made is uncountable. Between any two sample locations s_i and s_j are an infinite (in theory) number of other potential sample points. How we choose to construct our sample of D has no bearing on whether the data are geostatistical or not. We can lay down a grid and sample at the intersection of gridlines, or we can randomly select points. The attribute itself Z(s) can be categorical or continuous.
• Lattice data: Here the set D is fixed and countable (usually finite). For instance, summary statistics on a county level, species richness measured on contiguous quadrats are all examples of lattice data. With lattice data there is no space between these discrete units at which additional sample points could be obtained.
• Point process data: Unlike geostatistical and lattice data here the domain D is random. A typical example is the spatial location of a plant species, i.e., the places where the plant species is observed to grow. With point process data the focus is on the location. Thus rather than analyzing the Z process we analyze the D process.

The categories of spatial data are somewhat fluid

• Geostatistical data under certain circumstances can be treated as point process data. Define

 Now suppose we record the locations where U(s) = 1. These locations are clearly random and thus we now have point process data.

• If we aggregate geostatistical data by area and summarize the results over the area we generate lattice data.
• Going from lattice to geostatistical data is harder. Typically we have to pretend that the value of Z(s) on a lattice occurrs at the center of the lattice units and then treat D as if it were continuous so that intermediate values are possible.

How are spatial data used?

• Point process data
  1. With point process data the first question is typically, are the points distributed randomly? This amounts to looking at the distances between neighbors. For geostatistical data such a question would be nonsensical because the locations of points are determined by design.
  2. If there is more than one point process, the next obvious question is are the two processes related to each other in some way?
• Geostatistical data
  1. Typically the goal with geostatistical data is to characterize the spatial process that produced the data either to predict values at locations not sampled or for purposes of general modeling.
  2. In modeling the spatial process is often used to account for any lingering statistical dependence in the response after the basic regression model has been formulated.
• Lattice data
  1. The same objectives hold for lattice data as for geostatistical data except now questions of spatial connectivity and neighborhood structure arise too.

Cited Reference

Tobler, W. 1970. A computer movie simulating urban growth in the Detroit region. Economic Geography 46: 234–240.

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