By Richard Arnold Johnson, Dean W. Wichern
This industry chief bargains a readable creation to the statistical research of multivariate observations. offers readers the information essential to make right interpretations and choose acceptable innovations for reading multivariate info. starts off with a formula of the inhabitants versions, delineates the corresponding pattern effects, and liberally illustrates every thing with examples. Offers an abundance of examples and workouts in keeping with actual data. acceptable for experimental scientists in various disciplines.
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I. ·.... -:. 10 Modified scatter plots for the paper-quality data with outlier (25) (a) selected and (b) deleted. 22 Chapter 1 Aspects of Multivariate Analysis , .... ; ···~· ..... I r----:-1 .... , .. ,: :~· ... ·.... ' .. r ' 135 ~ ···: ~. Machine ... ·. 9 .. · • ' \. ·,... ' .. (x,) .. ~. ·.. ·. :· ~ ',4 .. (a) .. 3 ·. · ·. 1 ·. 845 Density (x,) ... ·. ..... 1 I Modified scatter plots with (a) group of points selected and (b) points, including specimen 25, deleted and the scatter plots rescaled.
What is important is the appearance of the crossproduct term 2a 12x 1x 2 necessitated by the nonzero correlation r12 . Equation (1-19) can be compared with (1-13). The expression in (1-13) can be regarded as a special case of (1-19) with a 11 = 1/s11 , a22 = 1/s22 , and a 12 = 0. Yl) for situations in which the variables are correlated has the general form d(P,Q) = Vau(x 1 - + 2an(xl- y1 )(xz- Yz) + a 2 z(Xz- Yz) 2 yi) 2 (1-20) and can always be computed once a 1 b a 12 , and a 22 are known. In addition, the coordinates of all points P = (x1 , x 2 ) that are a constant squared distance c 2 from Q satisfy a 11 (x 1 yJ) 2 + 2an(x 1 - - y1 ) (x 2 Yl) + a 22 (x 2 - Y2) 2 = c 2 - (1-21) By definition, this is the equation of an ellipse centered at Q.
Yp)· Suppose Q is a fixed point [it may be the origin 0 = (0, 0, ... , 0)] and the coordinate variables vary independently of one another. • , sPP be sample variances constructed from n measurements on x 1, x 2 , ••. , xP, respectively. Then the statistical distance from P to Q is d(P,Q) = /(xl- Yt)2 + (xz- Yz)2 + ... + (xp- Yp)Z Y su s22 sPP (1-16) 34 Cbaptef 1 ;\spects of Multivariate Analysis All points P that are a constant squared distance from Q lie on a hyperellipsoid centered at Q whose major and minor axes are parallel to the coordinate axes.
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