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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text: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 - (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 Σ1 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
m
• Σx² + bΣxi = ΣXiyi
i=1
i=1
i=1
n
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