By using the method of least squares, find the best line through the points: (-1,1), (1,0). (2,2). Step 1. The general equation of a line is co + c₁ = y. Plugging the data points into this formula gives a matrix equation Ac= Step 2. The matrix equation Ac=has no solution, so instead we use the normal equation ATACATÿ ATA= AT- Step 3. Solving the normal equation gives the vector answer which corresponds to the formula for the line Analysis. Compute the vector of the predicted y values: ŷ = Ac Compute the error vector, which is the difference between the actual y values of the points, and the predicted or tweaked values (new values we get from the list of best fit): --ŷ Compute the total error (often considered to be the error we "make" when using least squares): SSE = e² + e + e} or SSE = ||||² SSE-
By using the method of least squares, find the best line through the points: (-1,1), (1,0). (2,2). Step 1. The general equation of a line is co + c₁ = y. Plugging the data points into this formula gives a matrix equation Ac= Step 2. The matrix equation Ac=has no solution, so instead we use the normal equation ATACATÿ ATA= AT- Step 3. Solving the normal equation gives the vector answer which corresponds to the formula for the line Analysis. Compute the vector of the predicted y values: ŷ = Ac Compute the error vector, which is the difference between the actual y values of the points, and the predicted or tweaked values (new values we get from the list of best fit): --ŷ Compute the total error (often considered to be the error we "make" when using least squares): SSE = e² + e + e} or SSE = ||||² SSE-
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 16EQ
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Transcribed Image Text:By using the method of least squares, find the best line through the points:
(-1, 1), (1,0), (2,2).
Step 1. The general equation of a line is co+c1x = y. Plugging the data points into this formula gives a matrix equation Ac = ý.
Step 2. The matrix equation Ac = y has no solution, so instead we use the normal equation ATA ĉ = ATÿ
ATA=
ATy=
Step 3. Solving the normal equation gives the vector answer
ĉ =
which corresponds to the formula for the line
y =
Analysis. Compute the vector of the predicted y values: ŷ = Aĉ.
ŷ =
Compute the error vector, which is the difference between the actual y values of the points, and the predicted or tweaked values (new values we get from the list of best fit): e = ÿ — ŷ.
ē =
Compute the total error (often considered to be the error we "make" when using least squares): SSE = e² + e² + e² or SSE = ||e||2
SSE =
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