We are given m data points of the form (a¿, b¿) for each i € [m], where a¿ € R² and bį ER, and wish to build a model that predicts the value of the variable b from knowledge of the vector a. In such a situation, one often uses a linear model of the form b = a√x, where x is a parameter vector to be determined. Given a particular parameter vector x, the residual, or prediction error, at the ith data point is defined as axl. | bi - Given a choice between alternative models, one should choose a model that “explains” the available data as best as possible, i.e., a model that results in small residuals. One possibility is to minimize the largest residual. This is the problem of minimizing - max |b; — a√x|, iЄ [m] with respect to x, subject to no additional constraints. Write a linear program to solve for the optimal x € R².
We are given m data points of the form (a¿, b¿) for each i € [m], where a¿ € R² and bį ER, and wish to build a model that predicts the value of the variable b from knowledge of the vector a. In such a situation, one often uses a linear model of the form b = a√x, where x is a parameter vector to be determined. Given a particular parameter vector x, the residual, or prediction error, at the ith data point is defined as axl. | bi - Given a choice between alternative models, one should choose a model that “explains” the available data as best as possible, i.e., a model that results in small residuals. One possibility is to minimize the largest residual. This is the problem of minimizing - max |b; — a√x|, iЄ [m] with respect to x, subject to no additional constraints. Write a linear program to solve for the optimal x € R².
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 31E
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![### Residual Minimization in Linear Models
We are given \( m \) data points of the form \((\mathbf{a}_i, b_i)\) for each \( i \in [m] \), where \(\mathbf{a}_i \in \mathbb{R}^2\) and \(b_i \in \mathbb{R}\), and we wish to build a model that predicts the value of the variable \( b \) from knowledge of the vector \(\mathbf{a}\). In such a situation, one often uses a linear model of the form \(b = \mathbf{a}^\top \mathbf{x}\), where \(\mathbf{x}\) is a parameter vector to be determined.
Given a particular parameter vector \(\mathbf{x}\), the residual, or prediction error, at the \(i\)th data point is defined as:
\[| b_i - \mathbf{a}_i^\top \mathbf{x} |.\]
Given a choice between alternative models, one should choose a model that "explains" the available data as best as possible, i.e., a model that results in small residuals. One possibility is to minimize the largest residual. This is the problem of minimizing:
\[
\max_{i \in [m]} | b_i - \mathbf{a}_i^\top \mathbf{x} |,
\]
with respect to \(\mathbf{x}\), subject to no additional constraints.
### Linear Programming Solution
Write a linear program to solve for the optimal \(\mathbf{x} \in \mathbb{R}^2\).
The task is to find the optimal vector \(\mathbf{x}\) that minimizes the maximum prediction error across all data points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb9e27b5d-0ab8-428e-954d-97d64fc14c61%2F61cd1839-dca1-47d7-8634-f3c269856787%2F8txpg94_processed.png&w=3840&q=75)
Transcribed Image Text:### Residual Minimization in Linear Models
We are given \( m \) data points of the form \((\mathbf{a}_i, b_i)\) for each \( i \in [m] \), where \(\mathbf{a}_i \in \mathbb{R}^2\) and \(b_i \in \mathbb{R}\), and we wish to build a model that predicts the value of the variable \( b \) from knowledge of the vector \(\mathbf{a}\). In such a situation, one often uses a linear model of the form \(b = \mathbf{a}^\top \mathbf{x}\), where \(\mathbf{x}\) is a parameter vector to be determined.
Given a particular parameter vector \(\mathbf{x}\), the residual, or prediction error, at the \(i\)th data point is defined as:
\[| b_i - \mathbf{a}_i^\top \mathbf{x} |.\]
Given a choice between alternative models, one should choose a model that "explains" the available data as best as possible, i.e., a model that results in small residuals. One possibility is to minimize the largest residual. This is the problem of minimizing:
\[
\max_{i \in [m]} | b_i - \mathbf{a}_i^\top \mathbf{x} |,
\]
with respect to \(\mathbf{x}\), subject to no additional constraints.
### Linear Programming Solution
Write a linear program to solve for the optimal \(\mathbf{x} \in \mathbb{R}^2\).
The task is to find the optimal vector \(\mathbf{x}\) that minimizes the maximum prediction error across all data points.
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