2 The Spatial Linear Model
Goals:
- Explain how the spatial linear model differs from the linear model with independent random errors.
- Explain how modeling for point-referenced data with distance-based model covariances differs from modeling for areal data with neighborhood-based model covariances.
2.1 Introducing the Spatial Linear Model
Statistical linear models are often parameterized as
\[ \mathbf{y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\epsilon}, \tag{2.1}\]
where for a sample size \(n\), \(\mathbf{y}\) is an \(n \times 1\) column vector of response variables, \(\mathbf{X}\) is an \(n \times p\) design (model) matrix of explanatory variables, \(\boldsymbol{\beta}\) is a \(p \times 1\) column vector of fixed effects controlling the impact of \(\mathbf{X}\) on \(\mathbf{y}\), and \(\boldsymbol{\epsilon}\) is an \(n \times 1\) column vector of random errors. We typically assume that \(\text{E}(\boldsymbol{\epsilon}) = \mathbf{0}\) and \(\text{Cov}(\boldsymbol{\epsilon}) = \sigma^2_\epsilon \mathbf{I}\), where \(\text{E}(\cdot)\) denotes expectation, \(\text{Cov}(\cdot)\) denotes covariance, \(\sigma^2_\epsilon\) denotes a variance parameter, and \(\mathbf{I}\) denotes the identity matrix.
To accommodate spatial dependence in \(\mathbf{y}\), an \(n \times 1\) spatial random effect, \(\boldsymbol{\tau}\), is added to Equation 2.1, yielding the model
\[ \mathbf{y} = \mathbf{X} \boldsymbol{\beta} + \boldsymbol{\tau} + \boldsymbol{\epsilon}, \tag{2.2}\]
where \(\boldsymbol{\tau}\) is independent of \(\boldsymbol{\epsilon}\), \(\text{E}(\boldsymbol{\tau}) = \mathbf{0}\), \(\text{Cov}(\boldsymbol{\tau}) = \sigma^2_\tau \mathbf{R}\), and \(\mathbf{R}\) is a matrix that determines the spatial dependence structure in \(\mathbf{y}\) and depends on a range parameter, \(\phi\), which controls the behavior of the spatial covariance as a function of distance. We discuss \(\mathbf{R}\) in more detail shortly. The parameter \(\sigma^2_\tau\) is called the spatially dependent random error variance or partial sill. The parameter \(\sigma^2_\epsilon\) is called the spatially independent random error variance or nugget. These two variance parameters are henceforth more intuitively written as \(\sigma^2_{de}\) and \(\sigma^2_{ie}\), respectively. The covariance of \(\mathbf{y}\) is denoted \(\boldsymbol{\Sigma}\) and given by \(\sigma^2_{de} \mathbf{R} + \sigma^2_{ie} \mathbf{I}\). The parameters that compose this covariance are contained in the vector \(\boldsymbol{\theta}\), which is called the covariance parameter vector.
Equation 2.2 is called the spatial linear model. The spatial linear model applies to both point-referenced and areal (i.e., lattice) data. Spatial data are point-referenced when the elements in \(\mathbf{y}\) are observed at point-locations indexed by x-coordinates and y-coordinates on a spatially continuous surface with an infinite number of locations. For example, consider sampling soil at any point-location in a field. Spatial data are areal when the elements in \(\mathbf{y}\) are observed as part of a finite network of polygons whose connections are indexed by a neighborhood structure. For example, the polygons may represent states in a country who are neighbors if they share at least one boundary.
2.2 Modeling Covariance in the Spatial Linear Model
A primary way in which the model in Equation 2.2 differs for point-referenced and areal data is the way in which \(\mathbf{R}\) in \(\text{Cov}(\boldsymbol{\tau}) = \sigma^2_{de} \mathbf{R}\) is modeled. For point-referenced data, the \(\mathbf{R}\) matrix is generally constructed using the Euclidean distance between spatial locations. For example, the exponential spatial covariance function generates an \(\mathbf{R}\) matrix given by
\[ \mathbf{R} = \exp(-\mathbf{H} / \phi), \tag{2.3}\]
where \(\mathbf{H}\) is a matrix of Euclidean distances among observations and \(\phi\) is the range parameter. Some spatial covariance functions have an extra parameter – one example is the Matérn covariance. Spatial models for point-referenced data are fit in spmodel using the splm() function.
On the other hand, \(\mathbf{R}\) for areal data is often constructed from how the areal polygons are oriented in space. Commonly, a neighborhood structure is used to construct \(\mathbf{R}\), where two observations are considered to be “neighbors” if they share a common boundary. In the simultaneous auto-regressive (SAR) model,
\[ \mathbf{R} = [(\mathbf{I} - \phi \mathbf{W}) (\mathbf{I} - \phi \mathbf{W}^\top)]^{-1} \tag{2.4}\]
where \(\mathbf{I}\) is the identity matrix and \(\mathbf{W}\) is a weight matrix that describes the neighborhood structure among observations. A popular neighborhood structure is queen contiguity, in which two polygons are neighbors if they share a boundary. It is important to clarify that observations are not considered neighbors with themselves. Spatial models for areal data are fit in spmodel using the spautor() function.