How can I fit a cosine function?

Question

I wrote a python function to get the parameters of the following cosine function:

param = Parameters()    
param.add( 'amp', value = amp_guess, min = 0.1 * amp_guess, max = amp_guess )

param.add( 'off', value = off_guess, min = -10, max = 10 )

param.add( 'shift', value = shift_guess[0], min = 0, max = 2 * np.pi, )

fit_values = minimize( self.residual, param, args = ( azi_unique, los_unique ) )

def residual( self, param, azi, data ):
        """
        Parameters
        ----------

        Returns
        -------
        """
        amp = param['amp'].value
        off = param['off'].value
        shift = param['shift'].value
        model = off + amp * np.cos( azi - shift )
        return model - data

In Matlab how can get the amplitude, offset and shift of the cosine function?

Answer 1

My experience tells me that it's always good to depend as little as possible on toolboxes. For your particular case, the model is simple and doing it manually is pretty straightforward.

Assuming that you have the following model:

y = B + A*cos(w*x + phi)

and that your data is equally-spaced, then:

%// Create some bogus data

A   = 8;
B   = -4;
w   = 0.2;
phi = 1.8;

x = 0 : 0.1 : 8.4*pi;
y = B + A*cos(w*x + phi) + 0.5*randn(size(x));

%// Find kick-ass initial estimates
L = length(y);
N = 2^nextpow2(L);

B0 = (max(y(:))+min(y(:)))/2;

Y = fft(y-B0, N)/L;
f = 5/(x(2)-x(1)) * linspace(0,1,N/2+1);

[A0,I] = max( 2*abs(Y(1:N/2+1)) );
w0   = f(I);
phi0 = 2*imag(Y(I));

%// Refine the fit
sol = fminsearch(@(t) sum( (y(:)-t(1)-t(2)*cos(t(3)*x(:)+t(4))).^2 ), [B0 A0 w0 phi0])

Results:
 

sol = %// B was -4      A was 8       w was 0.2     phi was 1.8                
         -4.0097e+000   7.9913e+000   1.9998e-001   1.7961e+000