Description

Two autoregressive(AR) models are currently implemented: the adjacency model (a conditional AR, i.e., CAR), and the AR1 model for time series.

An AR1 random effect is specified as AR1(1|<grouping factor>). It describes correlations between realizations of the random effect for (typically) successive time-steps by a correlation φ, denoted ARphi in function calls. Nested AR1 effects can be specified by a nested grouping factor, as in AR1(1|<time index> %in% <nesting factor>).

A CAR random effect is specified as adjacency(1|<grouping factor>). The correlations among levels of the random effect form a matrix (I-ρ adjMatrix)^{-1}, in terms of an adjMatrix matrix which must be provided, and of the scalar ρ, denoted rho in function calls. The rows and columns of adjMatrix must have names matching those of levels of the random effect or else be ordered as increasing values of the levels of the geographic location index specifying the spatial random effect. For example, if the model formula is
y ~ adjacency(1|geo.loc) and <data>$geo.loc is 2,4,3,1,... the first row/column of the matrix refers to geo.loc=1, i.e. to the fourth row of the data.

Details

Efficient algorithms for CAR models have been widely discussed in particular in the econometric literature (e.g., LeSage and Pace 2009), but these models are not necessarily recommended for irregular lattices (see Wall, 2004 and Martellosio, 2012 for some insights on the implications of autoregressive models).

In CAR models, the covariance matrix of random effects u can be described as λ(I-ρ W)^{-1} where W is the (symmetric) adjacency matrix. HLCor uses the spectral decomposition of the adjacency matrix, written as boldW=VDV' where D is a diagonal matrix of eigenvalues d_i. The covariance of V'u is λ(I-ρ D)^{-1}, which is a diagonal matrix with elements λ_i=λ/(1-ρ d_i). Hence 1/λ_i is in the linear predictor form α+β d_i This can be used to fit λ and ρ efficiently. A call to corrHLfit with the additional argument init.HLfit=list(rho=0) should be equivalent in speed and result to the HLCor call.

This is fast for small datasets (as in the example below) but more generic maximization algorithms may be preferable for large ones. It is suggested to use fitme generally unless one has a large number of small data sets to analyze. A call to fitme or corrHLfit without that initial value does not use the spectral decomposition. It performs numerical maximization of the likelihood (or restricted likelihood) as function of the correlation parameter ρ. The choice of fitting function may slightly impact the results. The ML fits by corrHLfit and HLCor should be practically equivalent. The REML fits should slightly differ from each other, due to the fact that the REML approximation for GLMMs does not maximize a single likelihood function.

If HLCor is used, the results are reported as the coefficients α ((Intercept)) and β (adjd) of the predictor for 1/λ_i, in addition to the resulting values of ρ and of the common λ factor.

Different fits may also differ in using or not algorithms that exploit the sparsity of the precision matrix of the autoregressive random effect. By default, spaMM tends to select sparse-precision algorithms for large datasets and large (i.e. many-level) random effects (details are complex). However, for AR1 models, the dimension of the implied precision matrix is determined by the extreme values of grouping factor (typically interpreted as a time index), as all intermediate values must be considered. Then, the correlation-based algorithms may be more efficient if only a few levels are present in the data, as only a small correlation matrix is required in that case.

References

LeSage, J., Pace, R.K. (2009) Introduction to Spatial Econometrics. Chapman & Hall/CRC.

Martellosio, F. (2012) The correlation structure of spatial autoregressions, Econometric Theory 28, 1373-1391.

Wall M.M. (2004) A close look at the spatial structure implied by the CAR and SAR models: Journal of Statistical Planning and Inference 121: 311-324.

Examples