Linear Regression and Curve Fitting

I have a model and some data I'd like to fit to it: X_t = B1*cos(2*pi*omega*t) + B2*sin(2*pi*omega*t) + eta_t

What function would I use to conduct linear regression here, to find B1 and B2?

Answer 1

[Wayne King]

Fs = 1000;
t = 0:1/Fs:1-1/Fs;
y = 1.5*cos(2*pi*100*t)+0.5*sin(2*pi*100*t)+randn(size(t));
y = y(:);
X = ones(length(y),3);
X(:,2) = cos(2*pi*100*t)';
X(:,3) = sin(2*pi*100*t)';
beta = X\y;

beta(1) is the estimate of the constant term, beta(2) the estimate of B1 and beta(3) the estimate of B2.

If you set the random number generator to its default for reproducible results:

rng default
Fs = 1000;
t = 0:1/Fs:1-1/Fs;
y = 1.5*cos(2*pi*100*t)+0.5*sin(2*pi*100*t)+randn(size(t));
y = y(:);
X = ones(length(y),3);
X(:,2) = cos(2*pi*100*t)';
X(:,3) = sin(2*pi*100*t)';
beta = X\y;

The results are:

beta =

   -0.0326
    1.5284
    0.4643

pretty good.

[Sam]

Thank-you Wayne. In my case, where I have the time series data in a .dat file, am I right in thinking that I would load that in as 'y' in place of the y = "1.5*cos(2*pi*100*t)+0.5*sin(2*pi*100*t)+randn(size(t));"?

[Wayne King]

Yes, that's correct. Make sure you flip it to a column vector if it isn't already. Obviously you have to change the frequencies in the design matrix to suit your problem.

If you are trying to estimate other frequencies you need to add two columns to the design matrix per frequency --- one for cosine and one for sine

Answer 2

[Jos (10584)]

You might be interested in the function REGRESS

X = [cos(2*pi*omega*t(:)) sin(2*pi*omega*t(:))]
B = regress(Y,X)

If you want to specify an offset B(3), add a column of ones to X

X(:,3) = 1