Deep Dives 1. Least Squares (#differentiation, #theoreticaltools) Adapted from Stewart's "Cal- culus" 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, 9₁), (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. 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 1 d², the sum of the squares of these vertical deviations. Show that, according to this method, the line of best fit is obtained when m and b satisfy the following system of equations: 43-4+234 i=1 n n n m • Σx² + bΣxi = ΣXiyi i=1 i=1 i=1

Deep Dives 1. Least Squares (#differentiation, #theoreticaltools) Adapted from Stewart's "Cal- culus" 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, 9₁), (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. 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 1 d², the sum of the squares of these vertical deviations. Show that, according to this method, the line of best fit is obtained when m and b satisfy the following system of equations: 43-4+234 i=1 n n n m • Σx² + bΣxi = ΣXiyi i=1 i=1 i=1

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.3: Least Squares Approximation

Problem 30EQ

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