linear least squares problem::
fit f(t) = sin(t) on the interval [0, π] with polynomials of degree n with n = 1, 2, 3, · · · , 10. Here, we use equally spaced nodes.
Solve the normal equation with a backslash in Matlab. Save the norm of the residual and the condition numbers of AT A??
Could anybody please tell me how can find x, y, and A in that case ??
As I understood, you made the linearization for the sine function, if not you should do it first and then you can put the linearization coefficients as the elements of your design matrix A. (check the MATLAB syms and diff functions for the partial derivative in MATLAB).
regarding the Y vector, it must contain your observations. To get x(parameters vector), you should apply the least squares solution:
x = inv(ATA) * (ATY)
please remember that as you solve a non-linear problem, you might apply an iterative solution that stops whenever the difference between two consecutive iterations is less than a specific threshold.
Why insist on linear least squares? Simpler to estimate amplitude and phase coefficients by minimizing SSE deviations with nonlinear optimization. Easy to set up in a spreadsheet and use solver.
Hi,
I'm totaly agree with Prof. Stuart S. Schwartz, SSE is the sum of the squared differences between each observation and its group's mean.
For more information about this method, please see the links.
https://hlab.stanford.edu/brian/error_sum_of_squares.html
https://www.wikihow.com/Calculate-the-Sum-of-Squares-for-Error-(SSE)
https://365datascience.com/tutorials/statistics-tutorials/sum-squares/
Best regards
I understand that you want a solution for an exercise task in your study. The x are the node values which in the task are specified as equispaced in the given interval (their number is not given, so I think you should try several numbers), the y are simply sin(x). You should know what A is - it's the matrix in the normal equations!
The question seems incorrect. We needn't find x and y.
We have {(xi, yi = sin(xi))}, i=1,...,N and seek (a_n, ..., a_0) for a polynomial function
f(x) = a_n * x^n + ... + a_0.
Least Square minimization gives
A=
( Sum(xi^n) Sum(xi^(n-1)) ... Sum(xi^0);
Sum(xi^(n+1)) Sum(xi^(n-2)) ... Sum(xi^1);
.............................................
Sum(xi^(2*n)) Sum(xi^(2*n-1)) ... Sum(xi^n);
);
b = (Sum(yi); Sum(xi*yi);); ... Sum(xi^n*yi) );
At last,
(a_n a_(n-1) ... a_0) = A^(-1)*b
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