91Ó°ÊÓ

The function \(y(x)\) is known to be a quadratic function of \(x .\) The following table gives the measured values and uncorrelated standard errors of \(y\) measured at various values of \(x\) (in which there is negligible error): $$ \begin{array}{lccccc} x & 1 & 2 & 3 & 4 & 5 \\ y(x) & 3.5 \pm 0.5 & 2.0 \pm 0.5 & 3.0 \pm 0.5 & 6.5 \pm 1.0 & 10.5 \pm 1.0 \end{array} $$ Construct the response matrix \(\mathrm{R}\) using as basis functions \(1, x, x^{2} .\) Calculate the matrix \(\mathrm{R}^{\mathrm{T}} \mathrm{N}^{-1} \mathrm{R}\) and show that its inverse, the covariance matrix \(\mathrm{V}\), has the form $$ \mathrm{V}=\frac{1}{9184}\left(\begin{array}{ccc} 12592 & -9708 & 1580 \\ -9708 & 8413 & -1461 \\ 1580 & -1461 & 269 \end{array}\right) $$ Use this matrix to find the best values, and their uncertainties, for the coefficients of the quadratic form for \(y(x)\).

Step by step solution

01

- Write down the quadratic function

A general quadratic function can be expressed as: \[ y(x) = a_0 + a_1 x + a_2 x^2 \] where we need to find the coefficients \( a_0 \), \( a_1 \), and \( a_2 \).

02

- Set up basis functions

The basis functions are given as: \[ \{1, x, x^2\} \]. These functions will be used to construct the response matrix \( \mathrm{R} \).

03

- Construct the response matrix \( \mathrm{R} \)

Using the basis functions and the provided values of \( x \): \[ \mathrm{R} = \begin{pmatrix} 1 & 1 & 1^2 \ 1 & 2 & 2^2 \ 1 & 3 & 3^2 \ 1 & 4 & 4^2 \ 1 & 5 & 5^2 \end{pmatrix} = \begin{pmatrix} 1 & 1 & 1 \ 1 & 2 & 4 \ 1 & 3 & 9 \ 1 & 4 & 16 \ 1 & 5 & 25 \end{pmatrix} \]

04

- Write down the measured values and errors

The measured values \( y(x) \) and their associated errors are:\[ y = \begin{pmatrix} 3.5 \ 2.0 \ 3.0 \ 6.5 \ 10.5 \end{pmatrix} \quad \text{and the errors} \quad \sigma = \begin{pmatrix} 0.5 \ 0.5 \ 0.5 \ 1.0 \ 1.0 \end{pmatrix} \]

05

- Construct the noise matrix \( \mathrm{N} \)

Since the errors are uncorrelated, \( \mathrm{N} \) is a diagonal matrix with elements \( \sigma_i^2 \):\[ \mathrm{N} = \begin{pmatrix} 0.5^2 & 0 & 0 & 0 & 0 \ 0 & 0.5^2 & 0 & 0 & 0 \ 0 & 0 & 0.5^2 & 0 & 0 \ 0 & 0 & 0 & 1.0^2 & 0 \ 0 & 0 & 0 & 0 & 1.0^2 \end{pmatrix} = \begin{pmatrix} 0.25 & 0 & 0 & 0 & 0 \ 0 & 0.25 & 0 & 0 & 0 \ 0 & 0 & 0.25 & 0 & 0 \ 0 & 0 & 0 & 1.0 & 0 \ 0 & 0 & 0 & 0 & 1.0 \end{pmatrix} \]

06

- Calculate \( \mathrm{R}^{\mathrm{T}} \mathrm{N}^{-1} \mathrm{R} \)

First, find \( \mathrm{N}^{-1} \):\[ \mathrm{N}^{-1} = \begin{pmatrix} 4 & 0 & 0 & 0 & 0 \ 0 & 4 & 0 & 0 & 0 \ 0 & 0 & 4 & 0 & 0 \ 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 & 1 \end{pmatrix} \]Then calculate \[ \mathrm{R}^{\mathrm{T}} \mathrm{N}^{-1} \mathrm{R} = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \ 1 & 2 & 3 & 4 & 5 \ 1 & 4 & 9 & 16 & 25 \end{pmatrix} \begin{pmatrix} 4 & 0 & 0 & 0 & 0 \ 0 & 4 & 0 & 0 & 0 \ 0 & 0 & 4 & 0 & 0 \ 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 & 1 \ 1 & 2 & 4 \ 1 & 3 & 9 \ 1 & 4 & 16 \ 1 & 5 & 25 \end{pmatrix} \]

07

- Compute the covariance matrix \( \mathrm{V} \)

This computation yields the inverse of the covariance matrix \( \mathrm{V}^{-1} \). The provided problem states \( \mathrm{V} \) has the form: \[ \mathrm{V} = \frac{1}{9184} \begin{pmatrix} 12592 & -9708 & 1580 \ -9708 & 8413 & -1461 \ 1580 & -1461 & 269 \end{pmatrix} = \mathrm{R}^{\mathrm{T}} \mathrm{N}^{-1} \mathrm{R} \]

08

- Solve for the optimal coefficients

The best values for the coefficients can be determined using:\[ \mathbf{a} = ( \mathrm{R}^{\mathrm{T}} \mathrm{N}^{-1} \mathrm{R} )^{-1} \mathrm{R}^{\mathrm{T}} \mathrm{N}^{-1} \, \mathbf{y} \]Compute \( \mathbf{a} \) using the given values and matrices.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Response Matrix

In quadratic regression, a response matrix is used to transform input values into a format that allows for easier computations. This is particularly useful when working with polynomial functions, like quadratic equations.
The response matrix, denoted as \(\textbf{R}\) in this exercise, helps us handle multiple basis functions. Basis functions are the building blocks of our quadratic function, typically written as \[ \text{1}, \text{x}, \text{x}^2 \]. Using these, we can construct the response matrix for the given table of measurements. For example, with values of \(\textbf{x} = [1, 2, 3, 4, 5]\), the response matrix \(\textbf{R}\) is:

\(\textbf{R} = \begin{pmatrix} 1 & 1 & 1^2 \ 1 & 2 & 2^2 \ 1 & 3 & 3^2 \ 1 & 4 & 4^2 \ 1 & 5 & 5^2 \end{pmatrix} = \begin{pmatrix} 1 & 1 & 1 \ 1 & 2 & 4 \ 1 & 3 & 9 \ 1 & 4 & 16 \ 1 & 5 & 25 \end{pmatrix} \)

This matrix format allows us to perform matrix operations more effectively in the subsequent steps of our regression analysis.

Covariance Matrix

The covariance matrix is a crucial concept in statistics and linear regression. It gives us an idea of the relationship between different coefficients in our model.
In this exercise, the covariance matrix \(\textbf{V}\) is derived from matrix operations involving our response matrix and the noise matrix of errors. Specifically, once we have \(\textbf{R}^T \text{N}^{-1} \textbf{R}\), the covariance matrix \(\textbf{V}\) is its inverse.
We are given that:

\(\textbf{V} = \frac{1}{9184} \begin{pmatrix} 12592 & -9708 & 1580 \ -9708 & 8413 & -1461 \ 1580 & -1461 & 269 \end{pmatrix} \)

This serves as a measure of the dispersion and correlation between the coefficients \(a_0\), \(a_1\), and \(a_2\) in our quadratic equation.
In practical terms, the covariance matrix lets us understand the inherent uncertainty and variability of our model's predictions, enhancing the accuracy of our regression analysis.

Optimal Coefficients

The final step in quadratic regression is finding the optimal coefficients for our model. These coefficients minimize the error between the predicted and measured values.
Using the formula:

\(\textbf{a} = ( \textbf{R}^T \text{N}^{-1} \textbf{R} )^{-1} \textbf{R}^T \text{N}^{-1} \textbf{y} \)

we can derive the best values for \(a_0, a_1,\) and \(a_2\). Here, \(\textbf{R}\) is our response matrix, \(\textbf{N}\) is the noise matrix, and \(\textbf{y}\) represents our measured values.
Plugging in the appropriate matrices and performing matrix algebra will yield the optimal coefficients. These are the solution to our quadratic equation that best fits the given data points.
By incorporating the covariance matrix, we can also determine the uncertainties associated with these coefficients, offering a clearer picture of the model's reliability.