2. Assume that the ri, i = 1,...,n, are not all equal. Find the parameters a* and b* for the line of best fit in terms of X, Y, X², Y², XY. Show that the point (a*, b*) is the only local minimizer of f. (Hint: X² - (X)² = ₁₁(x₁ - X)².) n 3. Show that if a* and b* are the parameters of the line of best fit, then Y = a*X+b* (and hence once we have computed a*, we can compute b* using the formula b* = Y - a*X).
2. Assume that the ri, i = 1,...,n, are not all equal. Find the parameters a* and b* for the line of best fit in terms of X, Y, X², Y², XY. Show that the point (a*, b*) is the only local minimizer of f. (Hint: X² - (X)² = ₁₁(x₁ - X)².) n 3. Show that if a* and b* are the parameters of the line of best fit, then Y = a*X+b* (and hence once we have computed a*, we can compute b* using the formula b* = Y - a*X).
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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Please do 2 or 3
![Let (x1, y1),..., (xn, Yn), n ≥ 2, be points on the R2 plane (each xi, Yi ER). We wish to find the straight
line of "best fit" through these points ("best" in the sense that the average squared error is minimized);
that is, we wish to find a, b E R to minimize
1. Let
f(a, b)
=
n
1
n
X
Y
X²
Y2
i=1
(ax₂ + b - y₁) ².
||
||
=
||
XY =
BIH
1-2
n
1-2
*WI *WI=WI WI-WI
Xi
Yi
Xiyi.
=
Show that f(a, b) can be written in the form z¹Qz 2cTz + d, where z = [a, b]T, Q
CER² and dER, and find expressions for Q, c, and d in terms of X, Y, X², Y², XY.
QTE R2X2.
2. Assume that the ri, i = 1,...,n, are not all equal. Find the parameters a* and b* for the line of
best fit in terms of X, Y, X², Y², XY. Show that the point (a*, b*) is the only local minimizer of
f. (Hint: X2_ (X)² = ²/²₁ (x₁ - x)².)
12=
3. Show that if a* and b* are the parameters of the line of best fit, then Y = a* X + b* (and hence once
we have computed a*, we can compute b* using the formula b* =
Y - a*X).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a546a6d-05a4-444d-accb-a68b7cfcdafa%2Fd6eb1e14-5fc9-442e-badc-42d249083d4a%2Fjp1nfa8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let (x1, y1),..., (xn, Yn), n ≥ 2, be points on the R2 plane (each xi, Yi ER). We wish to find the straight
line of "best fit" through these points ("best" in the sense that the average squared error is minimized);
that is, we wish to find a, b E R to minimize
1. Let
f(a, b)
=
n
1
n
X
Y
X²
Y2
i=1
(ax₂ + b - y₁) ².
||
||
=
||
XY =
BIH
1-2
n
1-2
*WI *WI=WI WI-WI
Xi
Yi
Xiyi.
=
Show that f(a, b) can be written in the form z¹Qz 2cTz + d, where z = [a, b]T, Q
CER² and dER, and find expressions for Q, c, and d in terms of X, Y, X², Y², XY.
QTE R2X2.
2. Assume that the ri, i = 1,...,n, are not all equal. Find the parameters a* and b* for the line of
best fit in terms of X, Y, X², Y², XY. Show that the point (a*, b*) is the only local minimizer of
f. (Hint: X2_ (X)² = ²/²₁ (x₁ - x)².)
12=
3. Show that if a* and b* are the parameters of the line of best fit, then Y = a* X + b* (and hence once
we have computed a*, we can compute b* using the formula b* =
Y - a*X).
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