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Suppose that a scientist has reason to believe that two quantities x and y are related linearly, that is, y = mx + b, at least approximately, for some values of m and b. The scientist performs an experiment and collects data in the form of points (x1, x2), (x2, y2) ,…, (xn, yn) and then plots these points. The points don’t lie exactly on a straight line, so the scientist wants to find constants m and b so that the line y = mx + b “fits” the points as well as possible (see the figure).
Let di = yi − (mxi, + b) be the vertical deviation of the point (xi, yi) from the line. The method of least squares determines m and b so as to minimize
and
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Chapter 14 Solutions
Multivariable Calculus