Question

A certain experiment produce the data \(\left( {1,7.9} \right),\left( {2,5.4} \right)\) and \(\left( {3, - .9} \right)\). Describe the model that produces a least-squares fit of these points by a function of the form

\(y = A\cos x + B\sin x\)

Short answer

The required matrix and vectors are shown as:

Design Matrix: \(X = \left( {\begin{aligned}{\cos 1}&{\sin 1}\\{\cos 2}&{\sin 2}\\{\cos 3}&{\sin 3}\end{aligned}} \right)\)

Observation vector: \({\bf{y}} = \left( {\begin{aligned}{7.9}\\{5.4}\\{ - 0.9}\end{aligned}} \right)\)

Parameter vector: \(\beta = \left( {\begin{aligned}A\\B\end{aligned}} \right)\)

Step-by-step solution

The General Linear Model

The equation of the general linear model is defined as:

\({\bf{y}} = X\beta + \in \)

Here, \({\bf{y}} = \left( {\begin{aligned}{{y_1}}\\{{y_2}}\\ \vdots \\{{y_n}}\end{aligned}} \right)\) is an observational vector, \(X = \left( {\begin{aligned}1&{{x_1}}& \cdots &{x_1^n}\\1&{{x_2}}& \cdots &{x_2^n}\\ \vdots & \vdots & \ddots & \vdots \\1&{{x_n}}& \cdots &{x_n^n}\end{aligned}} \right)\) is the design matrix, \(\beta = \left( {\begin{aligned}{{\beta _1}}\\{{\beta _2}}\\ \vdots \\{{\beta _n}}\end{aligned}} \right)\) is parameter vector, and \( \in = \left( {\begin{aligned}{{ \in _1}}\\{{ \in _2}}\\ \vdots \\{{ \in _n}}\end{aligned}} \right)\) is a residual vector.

Find design matrix, observation vector, parameter vector for given data

The given equation is\(y = A\cos x + B\sin x\), and the given data sets are \(\left( {1,7.9} \right)\), \(\left( {2,5.4} \right)\) and \(\left( {3, - 0.9} \right)\).

Write the Design matrix, observational vector, and the parameter vector for the given equation and data set by using the information given in step 1.

Design matrix:

\(X = \left( {\begin{aligned}{\cos 1}&{\sin 1}\\{\cos 2}&{\sin 2}\\{\cos 3}&{\sin 3}\end{aligned}} \right)\)

Observational vector:

\({\bf{y}} = \left( {\begin{aligned}{7.9}\\{5.4}\\{ - 0.9}\end{aligned}} \right)\)

And, the parameter vectorfor the given equation is:

\(\beta = \left( {\begin{aligned}A\\B\end{aligned}} \right)\)

So, the above values are the best fit for the given data set and equation.