Estimation of trends in spatial data

Often data collected over space exhibits a trend, that is, the mean varies with spatial location. After formally defining spatial trend, a review and comparison of the current statistical methods for spatial trend analysis is presented. A new data analytic and exploratory method for estimating trends in spatial count data is developed. A nonhomogeneous Poisson likelihood is used with the generalized additive models of Hastie and Tibshirani (1986). These models, which are an extension of generalized linear models, allow the nonparametric estimation of the mean of the Poisson distribution as a function of spatial location. The only restriction on the mean function is that the log of the mean is a sum of unspecified but smooth functions, one function for each spatial coordinate. Unlike many of the current methods for trend analysis this method allows the incorporation of smooth functions of other external variables that may help define the spatial trend. Estimation of the smooth functions is through empirical maximization of the expected loglikelihood. The estimation algorithm requires scatterplot smoothers and new results on the consistency of running line smoothers with dependent data are shown. The bootstrap method is used to provide variability estimates for the smooth trend functions. Significance of the trend in a spatial direction can be assessed with permutation tests. A method based on the minimum average squared prediction error is used for the conversion of mapped spatial point data into count data so that the trend estimation technique can be applied. The methods are illustrated with tree and wombat warren location data and with data on king crab densities in the Bering Sea.