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Unformatted text preview: University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou Data with trend • As we discussed earlier, if the intrinsic stationarity assumption holds, which implies E ( Z ( s + h ) Z ( s )) = 0 and V ar ( Z ( s + h ) Z ( s )) = 2 γ ( h ) we can write Var( Z ( s + h ) Z ( s )) = E ( Z ( s + h ) Z ( s )) 2 and therefore we can use the method of moments estimator for the variogram (also called the classical estimator): 2ˆ γ ( h ) = 1 N ( h ) X N ( h ) ( Z ( s i ) Z ( s j )) 2 , where the sum is over N ( h ) such that s i s j = h . • But what if there is a tend in our data? For example, the values may increase from north to south, or northeast to southwest, etc. We will have to take this into account when computing the variogram. Why? It can be shown that the formula for computing the sample variogram is also equal to: 2ˆ γ ( h ) = 1 N ( h ) X N ( h ) ( Z ( s i ) Z ( s j )) 2 ˆ μ 2 diff By assuming a constant mean ( μ diff = 0) it is like adding a positive quantity to the variogram. Adding a square term will result to a parabola, and therefore a parabolic variogram is an indication of a presence of a trend in our data. 1 • Example 1: The Wolfcamp aquifer data: See Cressie (1993, pp. 212 214). The U.S. Department of Energy proposed (in the 1980s) a nuclear waste site to be in Deaf Smith County in Texas (bordering New Mexico). The contamination of the aquifer was a concern, and therefore the piezometrichead data were obtained at 85 locations by drilling a narrow pipe through the aquifer. The measures are in feet abovelocations by drilling a narrow pipe through the aquifer....
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 Spring '11
 NicolasChristou
 Statistics, Unemployment, Robust statistics, Cressie

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