Solved: 7071. Least squares approximation In its many

Chapter 12, Problem 71

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7071. Least squares approximation In its many guises, the least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example, height and shoe size) in the form of pairs 1x1, y12, 1x2, y22, c, 1xn, yn2. The data may be plotted as a scatterplot in the xy-plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that best fits the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum. Generalize the procedure in Exercise 70 by assuming that n data points 1x1, y12, 1x2, y22, c, 1xn, yn2 are given. Write the function E1m, b2 (summation notation allows for a more compact calculation). Show that the coefficients of the best-fit line are m = 1gxk21gyk2 - ngxkyk 1gxk22 - ngxk 2 and b = 1 n 1gyk - mgxk2, where all sums run from k = 1 to k = n.