Academic journal article Geographical Analysis

The Effect on Attribute Prediction of Location Uncertainty in Spatial Data

Academic journal article Geographical Analysis

The Effect on Attribute Prediction of Location Uncertainty in Spatial Data

Article excerpt

A datum is considered spatial if it contains location information. Typically, there is also attribute information, whose distribution depends on its location. Thus, error in location information can lead to error in attribute information, which is reflected ultimately in the inference drawn from the data. We propose a statistical model for incorporating location error into spatial data analysis. We investigate the effect of location error on the spatial lag, the covariance function, and optimal spatial linear prediction (that is, kriging). We show that the form of kriging after adjusting for location error is the same as that of kriging without adjusting for location error. However, location error changes entries in the matrix of explanatory variables, the matrix of covariances between the sample sites, and the vector of covariances between the sample sites and the prediction location. We investigate, through simulation, the effect that varying trend, measurement error, location error, range of spatial depend ence, sample size, and prediction location have on kriging after and without adjusting for location error. When the location error is large, kriging after adjusting for location error performs markedly better than kriging without adjusting for location error, in terms of both the prediction bias and the mean squared prediction erorr.

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Data are considered spatial if they contain location information. Typically, there is also attribute information available. The distribution of the attribute varies from location to location. Attribute information consists of the measured response (or responses), which can be either discrete (for example, counts of animal populations) or continuous (for example, soil pH). With the advent of optimal spatial linear prediction (that is, kriging), the analysis of spatially dependent data has progressed rapidly in the past forty years. Cressie (1990) lists three components necessary for the development of ordinary kriging: (i) use of covariances (or variograms) to weight observations, (ii) use of the best linear unbiased estimator (BLUE) for an unknown constant mean [mu], and (iii) use of spatial locations in the determination of covariances (or variograms).

Since locations are used to determine covariances, and covariances are used to weight observations, uncertainty in locations leads to a further component of variability in kriging predictions. Almost without fail, texts and articles about kriging assume that locations are known without error. Here, we explicitly account for location uncertainty and quantify its effect on statistical inference. We show how to incorporate location error by adjusting the component parts of the kriging equations and then proceeding as if the spatial process were sampled without location error.

Suppose that we have one (spatially incomplete) observation on the random process {Z(s): s [member of] D [subset] [R.sup.d]}. Without further assumptions about Z(*), data {(Z([s.sub.1]),...,Z([s.sub.n])} represent a sample from a single realization and statistical inference is not possible. In subsequent sections of this article, we shall assume that Z(*) has a linear mean and a stationary covariance function:

(i) var(Z(s))<[infinity]; for all s [member of] D,

(ii) [mu](s) = E(Z(s)) = f(s)'[beta]; for all s [member of] D,

(iii) cov(Z([s.sub.1]),Z([s.sub.2])) = C([s.sub.2]-[s.sub.1]); for all [s.sub.1],[s.sub.2] [member of] D. (1)

Condition (i) ensures that the first two moments of the process are defined. Condition (ii) assumes that the mean function is linear in the (q + 1) ) x 1 vector of unknown parameters [beta], where the explanatory variables f(s) are known functions of s a D. Condition (iii) states that the covariance between locations [s.sub.1] and [s.sub.2] depends only on the spatial lag h = [s.sub.2] - [s.sub.1]. The lag plays a central role in geostatistics; however, little attention has been paid to the effect of location uncertainty (that is, error in s and thus in h) on spatial methods such as kriging (see, for example, Clark, 1979, for a brief discussion). …

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