Introduction to Probability and Statistics for Engineers and Scientists
384 Chapter 9: Regression TABLE 9.3 xPPPˆ −Pˆ 5 .061 .063 −.002 10 .113 .109 .040 20 .192 .193 −.001 30 .259 .269 −.010 40 .339 ...
9.8Weighted Least Squares 385 On taking partial derivatives with respect toAandBand setting them equal to 0, we obtain the follo ...
386 Chapter 9: Regression Now let us determine the estimator produced by minimizing the weighted sum of squares. That is, let us ...
9.8Weighted Least Squares 387 (b)The weighted sum of squares can also be seen as the relevant quantity to be minimized by multip ...
388 Chapter 9: Regression Now in many applications it is probably reasonable to suppose that the Yi are independent random varia ...
9.8Weighted Least Squares 389 50 45 40 35 30 25 20 15 Travel time 02 46 810 Distance (miles) FIGURE 9.12 Example 9.8b. The ratio ...
390 Chapter 9: Regression (b)Proof that Var √ Y≈.25 whenYis Poisson with meanλ. Consider the Taylor series expansion ofg(y) =√ya ...
9.9Polynomial Regression 391 9.9Polynomial Regression InsituationswherethefunctionalrelationshipbetweentheresponseYandtheindepen ...
392 Chapter 9: Regression Even more so than in linear regression, it is extremely risky to use a polynomial fit to predict the v ...
9.9Polynomial Regression 393 might hold. Since ∑ i xi=55, ∑ i xi^2 =385, ∑ i xi^3 =3,025, ∑ i xi^4 =25, 333 ∑ i Yi=1,291.1, ∑ i ...
394 Chapter 9: Regression REMARK In matrix notation Equation 9.9.1 can be written as 1,291.1 9,549.3 77,758.9 = 10 55 ...
*9.10Multiple Linear Regression 395 To determine the least squares estimators, we repeatedly take partial derivatives of the pre ...
396 Chapter 9: Regression β= β 0 β 1 .. . βk , e= e 1 e 2 .. . en thenYis ann×1,Xann×p,βap×1, and ...
*9.10Multiple Linear Regression 397 It is now easy to see that the matrix equation X′XB=X′Y is equivalent to the set of normal E ...
398 Chapter 9: Regression Multiple Linear Regression Enter the number of rows Enter the number of columns 8 3 Begin Data Entry Q ...
*9.10Multiple Linear Regression 399 Multiple Linear Regression Compute Inverse Back 1 Step 1 1 1 1 1 1 679 1420 1349 296 6975 32 ...
400 Chapter 9: Regression Multiple Linear Regression Compute coeffs. Back 1 Step Enter 8 response values: 8.4 Add This Value To ...
*9.10Multiple Linear Regression 401 Since Var(Yr)=σ^2 , we see that Cov(Bi− 1 ,Bj− 1 )=σ^2 ∑n r= 1 CirCjr (9.10.4) =σ^2 (CC′)ij ...
402 Chapter 9: Regression The quantityσ^2 can be estimated by using the sum of squares of the residuals. That is, if we let SSR= ...
*9.10Multiple Linear Regression 403 =(Y′−B′X′)(Y−XB) =Y′Y−Y′XB−B′X′Y+B′X′XB =Y′Y−Y′XB where the last equality follows from the n ...
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