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Unformatted text preview: University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou Universal kriging The Ordinary Kriging (OK) that was discussed earlier is based on the constant mean model given by Z ( s ) = μ + δ ( s ) where δ ( · ) has mean zero and variogram 2 γ ( · ). Many times this is a too simple model to use. The mean can be a function of the coordinates X,Y , in some linear, quadratic, or higher form. For example the value of Z at location s can be expressed now as Z ( s i ) = β + β 1 X i + β 2 Y i + δ ( s i ) , linear Or Z ( s i ) = β + β 1 X i + β 2 Y i + β 3 X 2 i + β 4 X i Y i + β 5 Y 2 i + δ ( s i ) , quadratic, etc. If this is the case then we say that there is a trend of the polynomial type. We need to take this into account when we find the kriging weights. The predicted value Z ( s ) at location s will be again a linear combination of the observed Z ( s i ) ,i = 1 , · · · ,n values: ˆ Z ( s ) = w 1 Z ( s 1 ) + w 2 Z ( s 2 ) + · · · + w n Z ( s n ) = n X i =1 w i Z i where w 1 + w 2 + · · · + w n = 1 Suppose now that a trend of the linear form is present. Then the value ˆ Z ( s ) can be expressed as ˆ Z ( s ) = n X i =1 w i Z i = n X i =1 w i β + n X i =1 w i β 1 X i + n X i =1 w i β 2 Y i + n X i =1 w i δ ( s i ) or ˆ Z ( s ) = n X i =1 w i Z i = β + β 1 n X i =1 w i X i + β 2 n X i =1 w i Y i + n X i =1 w i δ ( s i ) (1) But also the value of Z ( s ) can be expressed (based on the linear trend) as Z ( s ) = β + β 1 X + β 2 Y + δ ( s ) (2) Compare (1) and (2). In order to ensure that we have an unbiased estimator we will need the following conditions: 1 n X i =1 w i X i = X n X i =1 w i Y i = Y and n X i =1 w i = 1 As with ordinary kriging, to find the weights when a trend is present we need to minimize the mean square error (MSE) min Z ( s )- n X i =1 w i Z ( s i ) ! 2 subject to the above constraints. This minimization will be unconstrained if we incorporate the 3 constraints in the objective function. The result is a system of n + 3 equations for the linear trend. If the trend is quadratic we will need n + 6 equations, and n + 10 equations for a three- dimentional quadratic trend, etc. On the next page the system of equations for the linear trend example is presented in matrix form. Note: We should try to understand why the trend exists based on the nature of our data, use a simple form of the trend if possible, and avoid extrapolation beyond the available data. Once we decided about which trend to use, we subtract this trend from the observed data to obtain the residuals. We then use the residuals to compute the sample variogram, fit a model variogram to it, predict the values at the unsampled locations (“kriged” the residuals), and finally add the kriged residuals back to the trend....
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- Spring '11
- Statistics, WI, Variogram, Kriging, raster map