Image differencing is used for many applications involving change detection. Although it is usually followed by a thresholding operation to isolate regions of change there are few methods available in the literature specific to (and appropriate for) change detection. We describe four different methods for selecting thresholds that work on very different principles. Either the noise or the signal is modelled, and the model covers either the spatial or intensity distribution characteristics. The methods are: 1/ a Normal model is used for the noise intensity distribution, 2/ signal intensities are tested by making local intensity distribution comparisons in the two image frames (i.e. the difference map is not used), 3/ the spatial properties of the noise are modelled by a Poisson distribution, and 4/ the spatial properties of the signal are modelled as a stable number of regions (or stable Euler number).

Spatial econometric methods deal with the incorporation of spatial interaction and spatial structure into regression analysis. The field has seen a recent and rapid growth spurred both by theoretical concerns as well as by the need to be able to apply econometric models to emerging large geocoded data bases. The review presented in this chapter outlines the basic terminology and discusses in some detail the specification of spatial effects, estimation of spatial regression models, and specification tests for spatial effects.

This paper reviews a number of conceptual issues pertaining to the implementation of an explicit “spatial ” perspective in applied econometrics. It provides an overview of the motivation for including spatial effects in regression models, both from a theory-driven as well as from a data-driven perspective. Considerable attention is paid to the inferential framework necessary to carry out estimation and testing and the different assumptions, constraints and implications embedded in the various specifications available in the literature. The review combines insights from the traditional spatial econometrics literature as well as from geostatistics, biostatistics and medical image analysis.

Landscape ecologists often deal with aggregated data and multiscaled spatial phenomena. Recognizing the sensitivity of the results of spatial analyses to the definition of units for which data are collected is critical to characterizinglandscapes with minimal bias and avoidance of spurious relationships.We introduce and exam- ine the effect of data aggregation on analysis of landscape structure as exemplified through what has become known, in the statistical and geographical literature, as the Modifiable Areal Unit Problem (MAUP). The MAUP applies to two separate, but interrelated, problems with spatial data analysis. The first is the "scale problem", where the same set of areal data is aggregated into several sets of larger areal units, with each com- bination leading to different data values and inferences. The second aspect of the MAUP is the "zoning prob- lem", where a given set of areal units is recombined into zones that are of the same size but located different- ly, again resulting in variation in data values and, consequently, different conclusions. We conduct a series of spatial autocorrelation analyses based on NDVI (Normalized Difference Vegetation Index) to demonstrate how the MAUP may affect the results of landscape analysis. We conclude with a discussion of the broader- scale implicationsfor the MAUP in landscape ecology and suggest approaches for dealing with this issue.

Ripley’s K(t) function is a tool to analyze completely mapped spatial point process data. Spatial point process data consists of the locations of events. These are usually recorded in 2 dimensions, but they may be locations along a line or in space. Here I will only describe K(t) for 2 dimensional spatial data. Completely mapped data includes the locations of all events in a predefined study area. Ripley’s K(t) function can be used to summarize a point pattern, test hypotheses about the pattern, estimate parameters and fit models. Bivariate or multivariate generalizations can be used to describe relationships between two or more point patterns. Applications include spatial patterns of trees [29, 20, 10], herbaceous plants [28], bird nests [11], and disease cases [7]. Details of various theoretical aspects of K(t) are in books by Ripley [26], Diggle [6], Cressie [4], Stoyan and Stoyan [31]. Examples of computation and interpretation can be found in those books and also Upton and Fingleton [32]. Theoretical K(t) function The K function is K(t) =λ −1 E [ # extra events within distance t of a randomly chosen event] (1) [23, 24], where λ is the density (number per unit area) of events. K(t) describes characteristics of the point processes at many distance scales. Alternative summaries (e.g.

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We propose a new approach to the multivariate analysis of data sets with known sampling site spatial positions. A between-sites neighbouring relationship must be derived from site positions and this relationship is introduced into the multivariate analyses through neighbouring weights (number of neighbours at each site) and through the matrix of the neighbouring graph. Eigenvector analysis methods (e.g., principal component analysis, correspondence analysis) can then be used to detect total, local and global structures. The introduction of the D-centring (centring with respect to the neighbouring weights) allows us to write a total variance decomposition into local and global components, and to propose a unified view of several methods. After a brief review of the matrix approach to this problem, we present the results obtained on both simulated and real data sets, showing how spatial structure can be detected and analysed. Freely available computer programs to perform computations and graphical displays are proposed.

Spatial regression deals with the specification, estimation and diagnostic checking of regression models that incorporate spatial effects. Two broad classes of spatial effects may be distinguished, referred to as spatial depen-dence and spatial heterogeneity (Anselin 1988b). In this chapter, attention

“Nearest neighbor methods ” include at least six different groups of statistical methods. All have in common the idea that some aspect of the similarity between a point and its nearest neighbor can be used to make useful inferences. In some cases, the similarity is the distance between the point and its nearest neighbor; in others, the appropriate similarity is based on other identifying characteristics of the points. I will discuss in detail nearest neighbor methods for spatial point processes and field experiments because these are commonly used in biology and environmetrics. I will very briefly discuss nearest-neighbor designs for field experiments, in which each pair of treatments occurs as neighbors equally frequently. I will not discuss nearest neighbor estimates of probability density functions [23], nearest neighbor methods for discrimination or classification [59, pp. 191-201], or nearest neighbor linkage (i.e. simple linkage) in hierarchical clustering [32, , pp. 57-60]. Although these last three methods have been applied to environmetric data, they are much more general. Nearest neighbor methods for spatial point processes Spatial point process data describe the locations of “interesting ” events, and (possibly) some information about each event. Some examples include locations of tree trunks (e.g. [52]), locations of bird nests (e.g. [11]), locations of pottery shards, and locations of cancer cases (e.g. [20]). I will focus on the most common case where the location is recorded in two dimensions (x, y). Similar techniques can be used for three dimensional data (e.g.

R objects to be used in the examples To make the subsequent R-examples a bit less messy, a few new R-objects are defined here that will be used in those examples without repeating the definitions. trees1 and trees2 were originally data frames containing information on all trees in two 50m × 50m square sample plots in Lapland (plotted on pp. 54–57). Components (columns) named x and y give the locations in metres from the plot centre, so x and y ∈ [−25,25]. The other components are not used in this part of the chapter, except in extracting the subset pines1 of all pines in trees1. 77 For simpler use of splancs functions, the coordinates are extracted into objects in splancs’s spatial point data format:> pts1 <- as.points(trees1)> pts2 <- as.points(trees2)> pts3 <- as.points(pines1) Another new object is made of the plot boundaries> poly <- as.points(c(-25,25,25,-25),c(-25,-25,25,25)) Corresponding ppp-objects ppts1, ppts2 and ppts3, for spatstat were produced by> ppts1 <- as.ppp(pts1,owin(poly=list(x=poly[,1],y=poly[,2]))) etc. There may be easier ways, but this is a general recipe for conversion from splancs-type objects to ppp-objects. 78 In addition, from now on obvious library commands will not be repeated in the examples. However, the libraries in use are usually mentioned in the text. Graphical settings such as postscript and dev.off commands and things like par(mfrow=c(1,2)) will also be omitted from the rest of the examples unless the feature has not been used in earlier examples. 79 A couple of kernels kde2d and ksmooth.ppp use the Gaussian kernel k(h) = (2π) −1 exp(−|h | 2 /2) (although kde2d also allows for different smoothing parameters in x- and y-directions, which is reasonable in density estimation if x and y are in different scales) kernel2d uses the quartic kernel