Please answer this 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 \[ \overline{X} = \frac{1}{n} \sum_{i=1}^{n} x_i \] \[ \overline{Y} = \frac{1}{n} \sum_{i=1}^{n} y_i \] \[ \overline{X^2} = \frac{1}{n} \sum_{i=1}^{n} x_i^2 \] \[ \overline{Y^2} = \frac{1}{n} \sum_{i=1}^{n} y_i^2 \] \[ \overline{XY} = \frac{1}{n} \sum_{i=1}^{n} x_i y_i \]Show that $f(a, b)$ can be written in the form $z^T Q z - 2c^T z + d$, where $z \equiv [a, b]^T$, $Q = Q^T \in \mathbb{R}^{2 \times 2}$, $c \in \mathbb{R}^2$ 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\

Answered: 1. Let f(a, b) = n X Y X² i=1 XY (ax₂ +…

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 answer this

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
\[
\overline{X} = \frac{1}{n} \sum_{i=1}^{n} x_i
\]
\[
\overline{Y} = \frac{1}{n} \sum_{i=1}^{n} y_i
\]
\[
\overline{X^2} = \frac{1}{n} \sum_{i=1}^{n} x_i^2
\]
\[
\overline{Y^2} = \frac{1}{n} \sum_{i=1}^{n} y_i^2
\]
\[
\overline{XY} = \frac{1}{n} \sum_{i=1}^{n} x_i y_i
\]

Show that $f(a, b)$ can be written in the form $z^T Q z - 2c^T z + d$, where $z \equiv [a, b]^T$, $Q = Q^T \in \mathbb{R}^{2 \times 2}$, $c \in \mathbb{R}^2$ 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\

Expert Solution

Step 1: Step 1

Note: “Since you have posted multiple questions, we will provide the solution only to the first question as per our Q&A guidelines. Please repost the remaining questions separately.”

Given $f (a, b) = 1 / n \sum_{i = 1}^{n} (a x_{i} + b - y_{i})^{2}$

$= 1 / n \sum_{i = 1}^{n} (a^{2} x_{i}^{2} + b^{2} + y_{i}^{2} + 2 a b x_{i} - 2 a x_{i} y_{i} - 2 b y_{i})$

$= 1 / n \sum_{i = 1}^{n} a^{2} x_{i}^{2} + 1 / n \sum_{i = 1}^{n} b^{2} + 1 / n \sum_{i = 1}^{n} y_{i}^{2} + 1 / n \sum_{i = 1}^{n} 2 a b x_{i} - 1 / n \sum_{i = 1}^{n} 2 a x_{i} y_{i} - 1 / n \sum_{i = 1}^{n} 2 b y_{i}$ $= a^{2} (1 / n) \sum_{i = 1}^{n} x_{i}^{2} + b^{2} (1 / n) \sum_{i = 1}^{n} + 1 / n \sum_{i = 1}^{n} y_{i}^{2} + 2 a b (1 / n) \sum_{i = 1}^{n} x_{i} - 2 a (1 / n) \sum_{i = 1}^{n} x_{i} y_{i} - 2 b (1 / n) \sum_{i = 1}^{n} y_{i}$ $= a^{2} \bar{X^{2}} + b^{2} + \bar{Y^{2}} + 2 a b \bar{X} - 2 a \bar{X Y} - 2 b \bar{Y}$