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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where a, b, c are determined from the normal equations ay a xy
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c a x2 c a x3 c a x4 (19)
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These are obtained formally by summing both sides of (18) after multiplying successively by 1, x and x2, respectively.
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The above ideas can also be generalized to more variables. For example, if we feel that there is a linear relationship between a dependent variable z and two independent variables x and y, then we would seek an equation connecting the variables that has the form z a bx cy (20)
This is called a regression equation of z on x and y. If x is the dependent variable, a similar equation would be called a regression equation of x on y and z. Because (20) represents a plane in a three-dimensional rectangular coordinate system, it is often called a regression plane. To find the least-squares regression plane, we determine a, b, c in (20) so that az a xz a yz na a ax a ay b ax c ay c a xy c a y2 (21)
b a x2 b a xy
These equations, called the normal equations corresponding to (20), are obtained as a result of applying a definition similar to that on page 266. Note that they can be obtained formally from (20) on multiplying by 1, x, y, respectively, and summing. Generalizations to more variables, involving linear or nonlinear equations leading to regression surfaces in three- or higher-dimensional spaces, are easily made.
Standard Error of Estimate
If we let yest denote the estimated value of y for a given value of x, as obtained from the regression curve of y on x, then a measure of the scatter about the regression curve is supplied by the quantity sy.x a( y B n yest)2 (22)
which is called the standard error of estimate of y on x. Since g(y yest)2 gd2, as used in the Definition on page 266, we see that out of all possible regression curves the least-squares curve has the smallest standard error of estimate. In the case of a regression line yest a bx, with a and b given by (4), we have s2 y.x or s2 y.x a( y a y2 y)2 # a ay n b a (x n b a xy x)( y # y) # (23) (24)
We can also express s2 for the least-squares line in terms of the variance and correlation coefficient as y.x s2 y.x s2(1 y r2) 1, i.e., 1 r 1. (25)
from which it incidentally follows as a corollary that r2
CHAPTER 8 Curve Fitting, Regression, and Correlation
The standard error of estimate has properties analogous to those of standard deviation. For example, if we construct pairs of lines parallel to the regression line of y on x at respective vertical distances sy.x, and 2sy.x, and 3sy.x from it, we should find if n is large enough that there would be included between these pairs of lines about 68%, 95%, and 99.7% of the sample points, respectively. See Problem 8.23. s Just as there is an unbiased estimate of population variance given by ^2 ns2 >(n 1), so there is an unbis y.x ns2 >(n 2). For this reaased estimate of the square of the standard error of estimate. This is given by ^2 y.x son some statisticians prefer to give (22) with n 2 instead of n in the denominator. The above remarks are easily modified for the regression line of x on y (in which case the standard error of estimate is denoted by sx.y) or for nonlinear or multiple regression.
The Linear Correlation Coefficient
Up to now we have defined the correlation coefficient formally by (15) but have not examined its significance. In attempting to do this, let us note that from (25) and the definitions of sy.x and sy, we have r2 Now we can show that (see Problem 8.24) a( y y)2 # a( y yest)2 a ( yest y)2 # (27) 1 a( y a( y yest)2 y)2 # (26)
The quantity on the left of (27) is called the total variation. The first sum on the right of (27) is then called the unexplained variation, while the second sum is called the explained variation. This terminology arises because the deviations y yest behave in a random or unpredictable manner while the deviations yest y are explained # by the least-squares regression line and so tend to follow a definite pattern. It follows from (26) and (27) that r2 a ( yest a( y y)2 # y)2 # explained variation total variation (28)
Therefore, r2 can be interpreted as the fraction of the total variation that is explained by the least-squares regression line. In other words, r measures how well the least-squares regression line fits the sample data. If the total 1, we say that there is perfect linear corvariation is all explained by the regression line, i.e., if r2 1 or r relation (and in such case also perfect linear regression). On the other hand, if the total variation is all unexplained, then the explained variation is zero and so r 0. In practice the quantity r2, sometimes called the coefficient of determination, lies between 0 and 1. The correlation coefficient can be computed from either of the results r sxy sx sy a (x $a (x x)( y # y) # y)2 # y)2 # y)2 # (30) (29)
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