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This 3 page Class Notes was uploaded by Dorris Borer on Monday September 28, 2015. The Class Notes belongs to ESE502 at University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 4 views. For similar materials see /class/215451/ese502-university-of-pennsylvania in Electrical Engineering at University of Pennsylvania.
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Date Created: 09/28/15
CONTINUOUS SPATIAL DATA ANALYSIS 1 Overview of Spatial Stochastic Processes The key difference between continuous spatial data and point patterns is that there is now assumed to be a meaningful value Y s at every location s in the region of interest For example Y s might be the temperature at s or the level of air pollution at s We shall consider a number of illustrative examples in the next section But before doing so it is convenient to outline the basic analytical framework to be used throughout this part of the NOTEBOOK If the region of interest is again denoted by R and if the value Y s at each location s e R is treated as a random variable then the collection of random variables 111 YsseR is designated as a spatial stochastic process on R also called a random field on R It should be clear from the outset that such uncountably infinite collections of random variables cannot be analyzed in any meaningful way without making a number of strong assumptions We shall make these assumptions explicit as we proceed Observe next that there is a clear parallel between spatial stochastic processes and temporal stochastic processes 112 YtteT where the set T is some continuous possibly unbounded interval of time In many respects the only substantive difference between 11 and 12 is the dimension of the underlying domain Hence it is not surprising that most of the assumptions and analytical methods to be employed here have their roots in time series analysis One key difference that should be mentioned here is that time is naturally ordered from past to presen to future whereas physical space generally has no preferred directions This will have a number of important consequences that will be discussed as we proceed 11 Standard Notation The key to studying the infinite collections of random variables such as 11 is of course to take finite samples of Y s values and attempt to draw inferences on the basis of this information To do so we shall employ the following standard notation For any given set of sample K R locations s1 ilnCR as in Figure 11 let the random vector Fig11 Sample Locations ESE 502 II ll Tony E Smith NOTEBOOK FOR SPATIAL DA TA ANALYSIS Part I Spatial Point Pattern Analysis YSl Y1 113 Y m Y n represent the possible list of values that may be observed at these locations Note that following standard matrix conventions we always take vectors to be column vectors unless otherwise stated The second representation in 13 will usually be used when the speci c locations of these samples are not relevant Note also that it is often more convenient to write vectors in transpose form as Y Y1Yn39 thus yielding a more compact inline representation Each possible realization yl 114 y y1y39 y of the random vector Y then denotes a possible set of speci c observations such as the temperatures at each location i l n Most of our analysis will focus on the means and variances of these random variables as well as the covariances between them Again following standard notation we shall usually denote the mean of each random variable Y SI by 115 EYslus1ul iln so that the corresponding mean vector for Y is given by 116 EY EKaEYl39 M39 1 Similarly the variance of random variable Y 3 can be denoted in a number of alternative ways as 117 VaIYEY M2O39ZS0392 01 The last representation facilitates comparison with the covariance of two random variables Ys1 and Ysj as defined by 118 covms ms 1 EY Y we 1 a The full matrix of variances and covariances for the components of Y is then designated as the covariance matrix for Y and is written alternatively as ESE 502 112 Tony E Smith NOTEBOOK FOR SPATIAL DA TA ANALYSIS Part I Spatial Point Pattern Analysis 119 covY E 5111 2 covYY1covYnYn 039 n covY1Y1covYlYn a n1 rm where by de nition covKY varK As we shall see below spatial stochastic processes can be often be usefully studied in terms of these first and second moments means and covariances This is especially true for the important case of multivariate normally distributed random vectors that will be discussed in some detail below For the present it suffices to say that much of our effort to model spatial stochastic processes will focus on the structure of these means and covariances for finite samples To do so it is convenient to start with the following overall conceptual framework 12 Basic Modeling Framework Essentially all spatial statistical models that we shall consider start by decomposing the statistical variation of random variables Y s into a deterministic trend term us and a stochastic residual term 8s as follows 121 Ysuscs seR Here us is almost always take to be the mean of Y s so that by definition 122 so Inn ms 2 E8s Emu ms 3 Ecs seR Expressions 121 and 122 together constitute the basic modeling framework to be used throughout the analyses to follow It should be emphasized that this framework is simply a convenient representation of Y s and involves no substantive assumptions But it is nonetheless very suggestive In particular since u defines a deterministic function on R it is useful to think of u as a spatial trend function representing the typical values of the given spatial stochastic process over all R ie the global structure of the Y process Similarly since c is by definition a spatial stochastic process on R with mean identically zero it is useful to think of c as a spatial residual process representing local variations about u ie the local structure of the Y process Within this framework our basic modeling strategy will be to identify a spatial trend function u that fits the Y process so well that the resulting residual process c is not statistically distinguishable from random noise To make this strategy precise we must of course develop appropriate models of random noise in a manner paralleling the CSR hypothesis above But first we consider a range of motivating examples ESE 502 113 Tony E Smith
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