5Surface Estimation
5.1 Introduction
The problem of mean surface estimation is common in many large scale investigations. There is in particular a vast literature on geostatistics (Kriging). Cressie (1993), Cressie and Huang (1999), Diggle and Ribeiro 2007), Gelfand et al. (2010), Isaaks and Srivastava (1989), Ripley (1981), and Opsomer et al. (1999) are some of the references where background information can be found on this topic. In the literature, of typical interest has been situations where the observations (after removing any spatial trend) are either spatially uncorrelated or have stationary covariances. In this chapter, we start with a nonparametric regression model, where the stationarity assumption for the errors need not hold. In particular, there may be substantial heterogeneity in the data with spatial autocorrelations. To introduce the topic, consider some spatial observations on a real-valued random variable of interest. Our primary aim is kernel estimation of the expected value of this random variable. We also consider estimation of non-exceedance probabilities and estimation of the spatial Gini index.
To give some examples of probability estimation for spatial data, consider for instance a forest monitoring data set from Switzerland (Source: Swiss National Forest Inventory) from the regions Jura and the Swiss Plateau. The observations are of the type (xi, yi, zi), i = 1, 2, …, n, where xi and yi denote respectively the West-East and the South-North coordinates ...