Chapter 15.7, Problem 75E

Least squares approximation In its many guises, 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 (x₁, y₁), (x₂, y₂), …., (x_n, y_n). 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 be a minimum.

•

75. Generalize the procedure in Exercise 74 by assuming n data points (x₁, y₁), (x₂, y₂), …, (x_n, y_n) are given. Write the function E(m, b) (summation notation allows for a more compact calculation). Show that the coefficients of the best-fit line are

$m=\frac{\left({\displaystyle \sum x_k}\right)\left({\displaystyle \sum y_k}\right)-n{\displaystyle \sum x_k{y}_k}}{{\left({\displaystyle \sum x_k}\right)}^2-n{\displaystyle \sum x^2{}_k}}$

$b=\frac{1}{n}\left({\displaystyle \sum yk-m{\displaystyle \sum xk}}\right)$

where all sums run from k = 1 to k = n