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Suppose that a scientist has reason to believe that two quantities x and y are related and b. linearly, that is, y = mx + b, at least approximately, for some values of The scientist performs an experiment and collects data in the form of points (x₁, y₁), (x2, y2), (Xn, yn), and then plots these points. The points do not 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, as shown in the figure. Let di y₁ − (mx¡ + b) be the vertical deviation of the point (x₁, y₁) from the line. The method of least squares determines m and b so as to minimize f(m, b) = [₁ d, the sum of the squares of these deviations. Show that, according to this method, the line of - best fit is obtained when =WI Xi m+nb = n Σx²m+ Exib i=1 = n Σyi i=1 Exiyi- i=1 = Thus, the line is found by solving these two equations in the unknowns m and b.

Suppose that a scientist has reason to believe that two quantities x and y are related and b. linearly, that is, y = mx + b, at least approximately, for some values of The scientist performs an experiment and collects data in the form of points (x₁, y₁), (x2, y2), (Xn, yn), and then plots these points. The points do not 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, as shown in the figure. Let di y₁ − (mx¡ + b) be the vertical deviation of the point (x₁, y₁) from the line. The method of least squares determines m and b so as to minimize f(m, b) = [₁ d, the sum of the squares of these deviations. Show that, according to this method, the line of - best fit is obtained when =WI Xi m+nb = n Σx²m+ Exib i=1 = n Σyi i=1 Exiyi- i=1 = Thus, the line is found by solving these two equations in the unknowns m and b.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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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, y₁), (x2, y2), (Xn, yn), and then plots these points. The points do not 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, as shown in the figure. Let di yi - (mx; + 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 f(m, b) = of the squares of these deviations. Show that, according to this method, the line of 1 d, the sum best fit is obtained when n (2₁) Xi m+nb (237) " Σx)m+(Σxi b n = i=1 n £x i=1 - = Thus, the line is found by solving these two equations in the unknowns m and b. Σyi = n Exiyi i=1
Transcribed Image Text: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, y₁), (x2, y2), (Xn, yn), and then plots these points. The points do not 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, as shown in the figure. Let di yi - (mx; + 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 f(m, b) = of the squares of these deviations. Show that, according to this method, the line of 1 d, the sum best fit is obtained when n (2₁) Xi m+nb (237) " Σx)m+(Σxi b n = i=1 n £x i=1 - = Thus, the line is found by solving these two equations in the unknowns m and b. Σyi = n Exiyi i=1
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