ESTIMATING THE REGRESSION COEFFICIENTS

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of its points and a plane is completely determined by specifying three of its points, a hyperplane is completely determined by specifying p + 1 of its points. (This geometric fact corresponds to the algebraic fact that the p + 1 coefficients b o, bi' ... ' bp are determined by p + 1 equations.) If a hyperplane contains fewer than p + 1 data points, it can be "pivoted" about these points until it meets another data point. By pivoting in an appropriate direction, the sum of absolute deviations of the data points from the hyperplane can be decreased (or at least not increased). (This is easiest to picture in the case p = 1, which is discussed in Section 4.2) Therefore, in order to minimize the sum of absolute deviations, we need only look at hyperplanes that pass through at least p + 1 of the data points.

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Justification of the Algorithm. In light of the preceding subsection, a reasonable algorithm would be to look at a sequence of hyperplanes, each hyperplane being determined by a set of p + 1 data points. Given the current hyperplane and its set of determining data points, we could select one data point to remove from the set and select another data point to replace it. The selection should be done so that the hyperplane determined by the new set of data points has a smaller sum of absolute deviations. Actually, the algorithm we presented can be described in this way. Finding a suitable direction amounts to selecting a data point to remove, and finding the best value of t in (4.10) amounts to selecting a replacement. To see this, consider the case p = 3 and suppose the current set of data points is XI' x 2 , x 3 , x 4 Let

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so that the current coefficient vector is b = A -Ie. If data point X3 is removed and replaced by X k, the new coefficient vector is b* = A* - I e* where

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This amounts to using direction vector d 3 and finding t* = (Yk - b'xk)/d;x k , because b + t*d 3 = A* - le*, which can be verified by using the fact that d 3 is the third column of A-I. This justifies the use of the columns of A-I as direction vectors.

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LEAST-ABSOLUTE-DEVIATIONS REGRESSION

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Of course it is sensible to choose the direction vector that gives the most negative derivative for (4.10) since this promises the most decrease in the sum of absolute deviations. The procedure for finding t* is justified in Section 4.2.

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The Fire Data. To apply the LAD multiple regression algorithm to the fire data in Table 4.3, we start by choosing four of the areas, say, areas 1, 2, 3, and 4. Using the data for these four areas, form

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The initial vector of estimates is

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and the initial matrix of direction vectors is -284.9 164.5 0.5233 14.59 -308.3 176.6 0.6744 15.51 157.8 -79.71 -0.4656 -8.185

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A-I =

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Now we want to find a vector of estimates that is better than b. To do this we calculate the right-hand derivative of (4.10) for the eight directions represented by the four columns of A - I and their negatives. Denote the first column of A - I by d l Calculations for obtaining the derivative in the direction d I are shown in Table 4.4. Recall that areas 7 and 24 have been omitted from the model. Note that Z, = y; - b'x; and w, = d;x;. For example,

= 2.152 - [47.93 -23.26 -0.1161 -2.443]

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