ESTIMATING THE REGRESSION LINE in Java

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ESTIMATING THE REGRESSION LINE
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that minimizes the sum
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n + 1] - bx i ) - - 2 - (Yi - bx i )
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(6.3)
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In particular, this shows that minimization of (6.2) can only determine an estimate of {3 and not of a. The estimate of a is calculated afterward by a different procedure. Regard (6.3) as a function of b. We want to know why this function is minimized by choosing b to be the weighted median of the pairwise slopes bij with weights proportional to IXi - x/ For the forearm length data, the function is [rank(165.8 - 28.1b) - 17](165.8 - 28.1b)
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+ [rank(169.8 - 29.1b) - 17](169.8 - 29.1b) + ... + [rank(167.2 - 29.7b) - 17](167.2 - 29.7b)
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(6.4 )
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The pairwise slopes bij and their weights are shown in Table 6.2. The graph of function (6.4) is shown in Figure 6.2. It consists of a series of line segments.
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+--___-+-_ _ _-+____+-___-+-_ _ _---1
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2.50 2.55 Figure 6.2 2.60 2.65 2.70 2.75 Graph of function (6.4).
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More generally, the graph of function (6.3) consists of a series of line segments joined together at the points b = b;j. In discussing this graph it is important to note that we are dealing with two different sorts of slopes: (1) the slope of function (6.3), which will tell us where the function has its minimum, and (2) the pairwise slopes b;j' which are the points b at which slope (1) changes value. The slope of (6.3) is the coefficient of b, which is - Uri - t(n + 1)]x;, where r; = rank(y; - bxJ If b is close to b;j' then y; - bx; and Yj - bXj are close to each other (because bij = (Y; - Y)/(x; - x) and hence their ranks differ only by 1. As b increases from slightly less than b;j to slightly greater than b;j' their ranks are interchanged, and hence r; increases by 1 and rj decreases by 1 (or vice versa, depending on whether x; < Xj or x; > x). The resulting change in the slope of (6.3) is Ix; - x j I. As b varies from - 00 to + 00, the slope of (6.3) goes from - t T to t T, where T = L Ix; - x j I, increasing by Ix; - x j I at each point b;j. At some point the slope must change from a negative value to a positive value, that is, the function stops decreasing and starts increasing. This of course is the minimizing value of b. Put the pairwise slopes in increasing order and, for each b;j' let T;j be the cumulative sum of all Ix f - x g I for which bfg < b;j. The slope of (6.3) for b slightly less than b;j is - t T + T;j and the slope for b slightly greater than bij is - t T + Tij + Ix; - xjl. The minimizing value of b is b km , where - t T + Tkm < 0 and - t T + Tkm + IXk - xm I > O. Dividing these conditions by T, we see that they are equivalent to (6.1), and hence bkm is the weighted median of the pairwise slopes bij with weights W;j = Ix; - xjl/T. For the forearm length data, the slope of (6.4) in the interval from b 2,16 = 2.667 to b 17, 33 = 2.683 is - t T + T 17 ,33 = - 401.1 + 400.7 = - 0.4 < 0 (401.1 = 802.2/2,802.2 is the sum of all the Ix; - xjl, and 400.7 is the sum of the Ix; - xjl for which bll < 2.683). And the slope in the interval + T 17,33 + IXl7 - x331 = -0.4 from b 17 ,33 = 2.683 to b4 ,24 = 2.684 is + 4.1 = 3.7 > O. So the function stops decreasing and starts increasing at ~ = 2.683.
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