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One of the main purposes of curve fitting is to estimate one of the variables (the dependent variable) from the other (the independent variable). The process of estimation is often referred to as regression. If y is to be estimated from x by means of some equation, we call the equation a regression equation of y on x and the corresponding curve a regression curve of y on x.
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CHAPTER 8 Curve Fitting, Regression, and Correlation
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The Method of Least Squares
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Generally, more than one curve of a given type will appear to fit a set of data. To avoid individual judgment in constructing lines, parabolas, or other approximating curves, it is necessary to agree on a definition of a bestfitting line, best-fitting parabola, etc. To motivate a possible definition, consider Fig. 8-4 in which the data points are (x1, y1), . . . , (xn, yn). For a given value of x, say, x1, there will be a difference between the value y1 and the corresponding value as determined from the curve C. We denote this difference by d1, which is sometimes referred to as a deviation, error, or residual and may be positive, negative, or zero. Similarly, corresponding to the values x2, . . . , xn, we obtain the deviations d2, . . . , dn.
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A measure of the goodness of fit of the curve C to the set of data is provided by the quantity d 2 d 2 c d 2. If this is small, the fit is good, if it is large, the fit is bad. We therefore make the follow1 2 n ing definition. Definition Of all curves in a given family of curves approximating a set of n data points, a curve having the property that d2 1 d2 2 c d2 n a minimum
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is called a best-fitting curve in the family. A curve having this property is said to fit the data in the least-squares sense and is called a least-squares regression curve, or simply a least-squares curve. A line having this property is called a least-squares line; a parabola with this property is called a least-squares parabola, etc. It is customary to employ the above definition when x is the independent variable and y is the dependent variable. If x is the dependent variable, the definition is modified by considering horizontal instead of vertical deviations, which amounts to interchanging the x and y axes. These two definitions lead in general to two different least-squares curves. Unless otherwise specified, we shall consider y as the dependent and x as the independent variable.
CHAPTER 8 Curve Fitting, Regression, and Correlation
It is possible to define another least-squares curve by considering perpendicular distances from the data points to the curve instead of either vertical or horizontal distances. However, this is not used very often.
The Least-Squares Line
By using the above definition, we can show (see Problem 8.3) that the least-squares line approximating the set of points (x1, y1), . . . , (xn, yn) has the equation y a bx (3)
where the constants a and b are determined by solving simultaneously the equations ay a xy an a ax b ax b a x2 (4)
which are called the normal equations for the least-squares line. Note that we have for brevity used gy, gxy instead of g n 1 yj, g n 1xj yj. The normal equations (4) are easily remembered by observing that the first equation j j can be obtained formally by summing on both sides of (3), while the second equation is obtained formally by first multiplying both sides of (3) by x and then summing. Of course, this is not a derivation of the normal equations but only a means for remembering them. The values of a and b obtained from (4) are given by Q a yR Q a x2 R a n a x2 The result for b in (5) can also be written b Here, as usual, a bar indicates mean, e.g., x # by n yields a (x a (x x)( y # x)2 # y) # (6) Q a xR Q a xyR Q a xR
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