Let (x₁, y₁), ..., (Xn, Yn), n ≥ 2, be points on the R² plane (each xį, Yi € R). 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 € R to minimize 1. Let f(a, b) n X Y X² Y2 XY n i=1 (axį+b-yi) ². || || || 1-2 1-2 n WIWI WIWIWI i=1 Xi Xiyi. - Show that f(a, b) can be written in the form z¹Qz — 2c¹z+d, where z = [a, b]T, Q = Q¹ € R²×², cɛ R² and dɛ R, and find expressions for Q, c, and d in terms of X, Y, X², Y², XY. 2. Assume that the xi, 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*)¹ is the only local minimizer of f. (Hint: X² – (X)² = ½ Σï1 (xi − x)².) 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).
Let (x₁, y₁), ..., (Xn, Yn), n ≥ 2, be points on the R² plane (each xį, Yi € R). 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 € R to minimize 1. Let f(a, b) n X Y X² Y2 XY n i=1 (axį+b-yi) ². || || || 1-2 1-2 n WIWI WIWIWI i=1 Xi Xiyi. - Show that f(a, b) can be written in the form z¹Qz — 2c¹z+d, where z = [a, b]T, Q = Q¹ € R²×², cɛ R² and dɛ R, and find expressions for Q, c, and d in terms of X, Y, X², Y², XY. 2. Assume that the xi, 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*)¹ is the only local minimizer of f. (Hint: X² – (X)² = ½ Σï1 (xi − x)².) 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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Answer all parts with work please
![Let \((x_1, y_1), \ldots, (x_n, y_n), n \geq 2\), be points on the \(\mathbb{R}^2\) plane (each \(x_i, y_i \in \mathbb{R}\)). 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 \(a, b \in \mathbb{R}\) to minimize
\[
f(a, b) = \frac{1}{n} \sum_{i=1}^n (ax_i + b - y_i)^2.
\]
1. Let
\[
\begin{aligned}
\overline{X} &= \frac{1}{n} \sum_{i=1}^n x_i \\
\overline{Y} &= \frac{1}{n} \sum_{i=1}^n y_i \\
\overline{X^2} &= \frac{1}{n} \sum_{i=1}^n x_i^2 \\
\overline{Y^2} &= \frac{1}{n} \sum_{i=1}^n y_i^2 \\
\overline{XY} &= \frac{1}{n} \sum_{i=1}^n x_i y_i.
\end{aligned}
\]
Show that \(f(a, b)\) can be written in the form \(z^T Q z - 2c^T z + d\), where \(z = [a, b]^T\), \(Q = Q^T \in \mathbb{R}^{2 \times 2}\), \(c \in \mathbb{R}^2\) and \(d \in \mathbb{R}\), and find expressions for \(Q\), \(c\), and \(d\) in terms of \(\overline{X}\), \(\overline{Y}\), \(\overline{X^2}\), \(\overline{Y^2}\), \(\overline{XY}\).
2. Assume that the \(x_i, i = 1, \ldots, n\), are not all equal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F44cf4428-d50c-4dc1-a095-73926c558c67%2F305c491b-4b60-48bc-afba-264457180143%2Fzcnj4yu_processed.png&w=3840&q=75)
Transcribed Image Text:Let \((x_1, y_1), \ldots, (x_n, y_n), n \geq 2\), be points on the \(\mathbb{R}^2\) plane (each \(x_i, y_i \in \mathbb{R}\)). 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 \(a, b \in \mathbb{R}\) to minimize
\[
f(a, b) = \frac{1}{n} \sum_{i=1}^n (ax_i + b - y_i)^2.
\]
1. Let
\[
\begin{aligned}
\overline{X} &= \frac{1}{n} \sum_{i=1}^n x_i \\
\overline{Y} &= \frac{1}{n} \sum_{i=1}^n y_i \\
\overline{X^2} &= \frac{1}{n} \sum_{i=1}^n x_i^2 \\
\overline{Y^2} &= \frac{1}{n} \sum_{i=1}^n y_i^2 \\
\overline{XY} &= \frac{1}{n} \sum_{i=1}^n x_i y_i.
\end{aligned}
\]
Show that \(f(a, b)\) can be written in the form \(z^T Q z - 2c^T z + d\), where \(z = [a, b]^T\), \(Q = Q^T \in \mathbb{R}^{2 \times 2}\), \(c \in \mathbb{R}^2\) and \(d \in \mathbb{R}\), and find expressions for \(Q\), \(c\), and \(d\) in terms of \(\overline{X}\), \(\overline{Y}\), \(\overline{X^2}\), \(\overline{Y^2}\), \(\overline{XY}\).
2. Assume that the \(x_i, i = 1, \ldots, n\), are not all equal.
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