Question

Consider fitting the function f(x)= α sin(2x) + β cos(5x) to the data (xᵢ, yᵢ), i = 1, 2, …, n.

Using the Least-squares criterion, derive two linear equations in terms of α and β.

Answer 1

To derive the equations for α and β using the least-squares method, one must set up the sum of squared errors based on the given function, take partial derivatives with respect to α and β, set them to zero, and solve the resulting linear equations.

Step-by-step explanation:

The task involves fitting a function of the form f(x) = α sin(2x) + β cos(5x) using the least-squares criterion to the given data set (x_i, y_i), where i ranges from 1 to n. To derive the linear equations in terms of α and β that minimize the sum of squared errors (SSE), one would set up the following minimization problem:

SSE = Σ(y_i - α sin(2x_i) - β cos(5x_i))², for i = 1, 2, …, n.

To find the minimum SSE, we take partial derivatives with respect to α and β and set them equal to zero, resulting in two linear equations:

Solving these equations simultaneously will give the values of α and β that minimize the SSE for the least-squares fitting.