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11.5.11 Smoothing with More than One Pre... > 11.5.11 Smoothing with More than One... - Pg. 573

Chapter 11 More Regression Methods 573 11.5.11 Smoothing with More than One Predictor The running interval smoother can be generalized to more than one predictor by replacing MADN with the minimum volume ellipsoid estimate of scatter, M, introduced in Chapter 6, and by measuring the distance between x i and x j with D i j = (x i - x j ) M -1 (x i - x j ). When trying to predict y, given x i , simply compute the trimmed mean of all y j values such that x j is close to x i . More formally, compute the trimmed mean of all the y j values for which the subscript j satisfies D i j f . The choice f = 1 or 0.8 often gives good results. When there are only two predictors, adjustments can be made as in the previous subsection. That is, start with f = 1, generate a graph of the three-dimensional smooth, and try other choices for f to see how the graph is affected. (For p = 2 and when estimating quantiles, also see He, Ng, & Portnoy, 1998.) To provide some indication of how well the method performs, first suppose y = x 1 + x 2 + . The left panel of Figure 11.7 shows a smooth based on f = 1 and n = 20 observations, where x 1 , x 2 , and all have a standard normal distribution. As can be seen, the shape of the