Penn ESE 502 - Spatially Dependent Random Effects

Unformatted text preview:

NOTEBOOK FOR SPATIAL DATA ANALYSIS Part II Continuous Spatial Data Analysis 3 Spatially Dependent Random Effects Observe that all regressions in the illustrations above starting with expression 2 1 3 in the Sudan rainfall example have relied on an implicit model of unobserved random effects i e regression residuals as a collection i i 1 n of independently and identically distributed normal random variables where for our purposes individual sample points i are taken to represent different spatial locations si But recall from the introductory discussion in Section 1 2 above that for more realistic spatial statistical models we must allow for possible spatial dependencies among these residuals Hence the main objective of the present section is to extend this model to one that is sufficiently broad to cover the types of spatial dependencies we shall need To do so we begin in Section 3 1 by examining random effects at a single location and show that normality can be motivated by the classical Central Limit Theorem In Section 3 2 these results will be extended to random effects at multiple locations by applying the Multivariate Central Limit Theorem to motivate multivariate normality of such joint random effects This multi normal model will form the statistical underpinning for all subsequent analyses Finally in Section 3 3 we introduce the notion of spatial stationarity to model covariances among these spatial random effects i i 1 n 3 1 Random Effects at a Single Location First recall that the unobserved random effects i at each location or sample point si are assumed to fluctuate around zero with E i 0 Now imagine that this overall random effect i is composed of many independent factors 3 1 1 i ei1 ei 2 eim m e k 1 ik where in typical realizations some of these factors eik will be positive and others negative Suppose moreover that each individual factor contributes only a very small part of total Then no matter how these individual random factors are distributed their cumulative effect i must eventually have a bell shaped distribution centered around zero This can be illustrated by a simple example in which each random component eik assumes the values 1 m and 1 m with equal probability so that E eik 0 for all k 1 m Then each is distributed as shown for the m 1 case in Figure 3 1 a below Now even though this distribution is clearly flat if we consider the m 2 case 3 1 2 i ei1 e12 then it is seen in Figure 3 1 b that the distribution is already starting to be bell shaped around zero In particular the value 0 is much more likely than either of the extremes 1 and 1 The reason of course is that this value can be achieved in two ways namely ei1 12 ei 2 12 and ei1 12 ei 2 12 whereas the extreme values can each occur in ESE 502 II 3 1 Tony E Smith NOTEBOOK FOR SPATIAL DATA ANALYSIS Part II Continuous Spatial Data Analysis only one way This simple observation reveals a fundamental fact about sums of independent random variables intermediate values of sums can occur in more ways than extreme values and hence tend to be more likely It is this property of independent sums that gives rise to their bell shaped distributions as can be seen in parts c and d of Figure 3 1 0 7 0 7 0 6 0 6 0 5 0 5 0 4 0 4 0 3 0 3 0 2 0 2 0 1 0 1 0 1 0 8 1 0 6 0 4 0 2 0 0 0 2 0 4 0 6 1 0 8 0 1 1 0 1 1 0 b m 2 a m 1 1 0 25 0 3 0 25 0 2 0 2 0 15 0 15 0 1 0 1 0 05 0 05 0 2 1 5 1 1 0 5 0 0 0 5 c m 10 1 1 5 1 2 0 2 1 5 1 1 0 5 0 0 5 0 d m 20 1 1 5 1 2 Figure 3 1 Cumulative Binary Errors But while this basic shape property is easily understood the truly amazing fact is that the limiting form if this bell shape always corresponds to essentially the same distribution namely the normal distribution To state this precisely it is important to notice first that ESE 502 II 3 2 Tony E Smith NOTEBOOK FOR SPATIAL DATA ANALYSIS Part II Continuous Spatial Data Analysis while the distributions in Figure 3 1 start to become bell shaped they are also starting to concentrate around zero Indeed the limiting form of this particular distribution must necessarily be a unit point mass at zero 1 and is certainly not normally distributed Here is turns out that the individual values of these factors eik 1 m or eik 1 m become too small as m increases so that eventually even their sum i will almost certainly vanish At the other extreme suppose that these values are independent of m say eik 1 or eik 1 Then while these individual values will eventually become small relative to their sum i the variance of i itself will increase without bound 2 In a similar manner observe that if the common means of these individual factors were not identically zero then the limiting mean of i would also be unbounded 3 So it should be clear that precise analysis of limiting random sums is rather delicate 3 1 1 Standardized Random Variables The time honored solution to these difficulties is to rescale these random sums in a manner which ensures that both their mean and variance remain constant as m increases To do so we begin by observing that for any random variable X with mean E X and variance 2 var X the transformed random variable 3 1 3 Z X 1 X necessarily has zero mean since by the linearity of expectations 3 1 4 E Z 1 E X 0 Moreover Z also has unit variance since by 3 4 3 1 5 1 X 2 E X 2 1 2 1 var Z E Z 2 E E X 2 2 Simply observe that if xik is a binary random variable with Pr xik 1 5 Pr xik 1 then by definition eik xik m so that i xi1 xim m is seen to be the average of m samples from this binary distribution But by the Law of Large Numbers such sample averages must eventually concentrate at the population mean E xik 0 2 In particular since var eik E eik 5 1 5 1 1 for all k it would then follow from the 2 2 independence of individual factors that var i 2 m k 1 var eik m var e1k m and hence that var i as m 3 Since E i m k 1 E eik m E ei1 implies E i m E ei1 it follows that if E ei 1 0 then E i as m ESE 502 II 3 …


View Full Document

Penn ESE 502 - Spatially Dependent Random Effects

Download Spatially Dependent Random Effects
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Spatially Dependent Random Effects and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Spatially Dependent Random Effects 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?