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c173c273_lec9_w11[1]

# c173c273_lec9_w11[1] - University of California Los Angeles...

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University of California, Los Angeles Department of Statistics Statistics C173/C273 Instructor: Nicolas Christou Spatial statistics - prediction Spatial prediction: a. One of our goals in geostatistics is to predict values at unsampled loacations. This will create a “raster map” of the area under consideration. b. Idea: The value at an unsampled location is related to the values at sampled locations. If not, then the predicted value can be simply the average of the sampled locations. c. There is some relationship between the point being estimated and the samples around it based on its location and the distance between the points. d. Diﬀerent methods for spatial prediction. All methods assume that nearby samples are more important (they have more weight) in predicting the unknown value. Some simple methods of spatial prediction: 1. Method of polygons. 2. Triangulation. 3. Weighted average based on triangulation. 4. Inverse distance method. A. Method of polygons: Very simple. The point being estimated is equal to the nearest observed sampled data point. 1

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B. Triangulation Form a triangle around the point being estimated with three observed data points. For these three data points we know their x,y coordinates and the value z of the variable of interest as follows: x y z x 1 y 1 z 1 x 2 y 2 z 2 x 3 y 3 z 3 We use these points to ﬁnd the equation of a plane by solving the system of equations: ax 1 + by 1 + c = z 1 ax 2 + by 2 + c = z 2 ax 3 + by 3 + c = z 3 Suppose now we want to ﬁnd the value of z 0 at location given by the coordinates ( x 0 ,y 0 ) of a point within the triangle. Example: Estimate the value at (65, 137) if the surrounding three points are: z 1 = 696 at (63, 140), z 2 = 227 at (64, 129), and z 3 = 606 at (71, 140). 2
C. Weighted average based on triangulation: Suppose we want to estimate the value at point O based on the values z A ,z B ,z C from the ﬁgure below: A weighted average is used as follows: ˆ z O = z A × Area( OBC ) Area( ABC ) + z B × Area( OCA ) Area( ABC ) + z C × Area( OAB ) Area( ABC ) This method gives more weight to the data point closer to the point being estimated. 3

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D. Inverse distance method: Suppose we want to estimate data point from the observed points as shown in the ﬁgure below: Each observed data point has an impact on the point being predicted with the clos- est points having larger impact. We can consider as a predictor the following linear combination (weighted average): ˆ z ( s 0 ) = w 1 z 1 + w 2 z 2 + ··· + w n z n = n X i =1 w i z i The weights w i ,i = 1 , ··· ,n are inversely related to the distance of point i to the point being estimated. For example,
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c173c273_lec9_w11[1] - University of California Los Angeles...

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