barcode font reporting services Curve Fitting, Regression, and Correlation in Software

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CHAPTER 8 Curve Fitting, Regression, and Correlation
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Fig. 8-10
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The least-squares parabola 8.15. Derive the normal equations (19), page 269, for the least-squares parabola.
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y a bx cx2
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Let the sample points be (x1, y1), (x2, y2), . . . , (xn, yn). Then the values of y on the least-squares parabola corresponding to x1, x2, . . . , xn are a bx1 cx2, 1 a bx2 cx2, 2 c, a bxn cx2 n
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Therefore, the deviations from y1, y2, . . . , yn are given by d1 a bx1 cx2 1 y1, d2 a bx2 cx2 2 y2, c, dn a bxn cx2 n yn
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and the sum of the squares of the deviations is given by ad 2 This is a function of a, b, and c, i.e., F(a, b, c) To minimize this function, we must have 'F 'a Now 'F 'a 'F 'b 'F 'c ' a 'a (a ' a 'b (a ' a 'c (a bx bx bx 0, cx2 cx2 cx2 'F 'b y)2 y)2 y)2 0, 'F 'c a 2(a a 2x(a a 2x2(a 0 bx bx bx cx2 cx2 cx2 y) y) y) a (a bx cx2 y)2 a (a bx cx2 y)2
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Simplifying each of these summations and setting them equal to zero yields the equations (19), page 269.
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CHAPTER 8 Curve Fitting, Regression, and Correlation
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8.16. Fit a least-squares parabola having the form y a bx cx2 to the data in Table 8-8.
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Table 8-8 x y 1.2 4.5 1.8 5.9 3.1 7.0 4.9 7.8 5.7 7.2 7.1 6.8 8.6 4.5 9.8 2.7
Then normal equations are ay (1) a xy
2 ax y
an a ax
b ax
c a x2 c a x3 c a x4
b a x2 b a x3
a a x2
The work involved in computing the sums can be arranged as in Table 8-9. Table 8-9 x 1.2 1.8 3.1 4.9 5.7 7.1 8.6 9.8 gx 42.2 y 4.5 5.9 7.0 7.8 7.2 6.8 4.5 2.7 gy 46.4 x2 1.44 3.24 9.61 24.01 32.49 50.41 73.96 96.04 gx2 291.20 x3 1.73 5.83 29.79 117.65 185.19 357.91 636.06 941.19 gx3 2275.35 8, 291.20c 2275.35c 18971.92c 46.4 230.42 1449.00 x4 2.08 10.49 92.35 576.48 1055.58 2541.16 5470.12 9223.66 gx4 18,971.92 xy 5.40 10.62 21.70 38.22 41.04 48.28 38.70 26.46 gxy 230.42 x2y 6.48 19.12 67.27 187.28 233.93 342.79 332.82 259.31 gx2y 1449.00
Then the normal equations (1) become, since n 8a (2) 42.2a 291.20a Solving, a 2.588, b 2.065, c y 42.2b 291.20b 2275.35b
0.2110; hence the required least-squares parabola has the equation 2.588 2.065x 0.2110x2
8.17. Use the least-squares parabola of Problem 8.16 to estimate the values of y from the given values of x.
For x 1.2, yest 2.588 2.065(1.2) 0.2110(1.2)2 4.762. Similarly, other estimated values are obtained. The results are shown in Table 8-10 together with the actual values of y. Table 8-10 yest y 4.762 4.5 5.621 5.9 6.962 7.0 7.640 7.8 7.503 7.2 6.613 6.8 4.741 4.5 2.561 2.7
Multiple regression 8.18. A variable z is to be estimated from variables x and y by means of a regression equation having the form z a bx cy. Show that the least-squares regression equation is obtained by determining a, b, and c so that they satisfy (21), page 269.
CHAPTER 8 Curve Fitting, Regression, and Correlation
Let the sample points be (x1, y1, z1), . . . , (xn, yn, zn). Then the values of z on the least-squares regression plane corresponding to (x1, y1), . . . , (xn, yn) are, respectively, a bx1 cy1, c, a bxn cyn
Therefore, the deviations from z1, . . . , zn are given by d1 a bx1 cy1 z1, c, dn a bxn cyn zn
and the sum of the squares of the deviations is given by ad 2 a (a bx cy z)2
Considering this as a function of a, b, c and setting the partial derivatives with respect to a, b, and c equal to zero, the required normal equations (21) on page 269, are obtained.
8.19. Table 8-11 shows the weights z to the nearest pound, heights x to the nearest inch, and ages y to the nearest year, of 12 boys, (a) Find the least-squares regression equation of z on x and y. (b) Determine the estimated values of z from the given values of x and y. (c) Estimate the weight of a boy who is 9 years old and 54 inches tall. Table 8-11 Weight (z) Height (x) Age (y) 64 57 8 71 59 10 53 49 6 67 62 11 55 51 8 58 50 7 77 55 10 57 48 9 56 52 10 51 42 6 76 61 12 68 57 9
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