How to find phase difference between two monthly time series ?

Dear All,

I have two monthly time series and I want to find phase difference between these two time series. Any lead will be helpful. I tried xcorr but dont know how to interpret the results.

An example of the time series are provided below :

T1=[ 2002.2464 -0.0336111001

2002.3285 -0.0340004094

2002.5804 -0.0461633684

2002.6653 -0.0424074503

2002.7474 -0.0373439799

2002.8323 -0.0371806156

2002.9144 -0.036254101

2003 -0.0346929255

2003.0849 -0.0314283391

2003.1615 -0.032081157

2003.2464 -0.030455973

2003.3285 -0.0350536673

2003.4956 -0.0428239858

2003.5804 -0.043654849

2003.6653 -0.0414177932

2003.7474 -0.0402629993

2003.8323 -0.0397198955

2003.9144 -0.0372285111

2004 -0.0386886688]

and

T2=[ 2002.2464 0.0283014575

2002.3285 0.0297969597

2002.5804 0.0356608957

2002.6653 0.0332108386

2002.7474 0.0297147113

2002.8323 0.0288490848

2002.9144 0.027610488

2003 0.026754985

2003.0849 0.0252721022

2003.1615 0.0246729691

2003.2464 0.0247586197

2003.3285 0.0278359036

2003.4956 0.0335587753

2003.5804 0.0341979019

2003.6653 0.0330755219

2003.7474 0.031480688

2003.8323 0.0305930355

2003.9144 0.0287892619

2004 0.0289496422]

Answer 1

[Star Strider]

This would indicate to me that they are completely out of phase:

[r,lags] = xcorr(T1(:,2), T2(:,2));

figure
plot(lags, r)
grid

.

[Smith J]

How to interpret the result, how to use the lag information to convert it to degree

[Smith J]

I obtained the plot like this after using the above formula

[Star Strider]

I doubt that it is possible to convert the lags to degrees, since there is no explicit relationship between them.

This is the best I can do in that respect (the sampling times are not contant, something I did not originally test for, however to use the Fourier transform, they must be, thus the resample calls):

Fs = 10;
[T12r,T11r] = resample(T1(:,2), T1(:,1), Fs);                   % Resample To Constant Sampling Interval, 10 Cy/Yr
[T22r,T21r] = resample(T2(:,2), T2(:,1), Fs);                   % Resample To Constant Sampling Interval, 10 Cy/Yr

L = numel(T12r);
Fn = Fs/2;
FTT12r = fft(T12r)/L;
FTT22r = fft(T22r)/L;
Fv1 = linspace(0, 1, fix(L/2)+1)*Fn;
Iv = 1:numel(Fv1);
Fv2 = linspace(-Fn, Fn, L);

Fconv = FTT12r .*  FTT22r;
[Amax,idx] = max(abs(fftshift(Fconv)));

figure
subplot(2,1,1)
plot(Fv2, abs(fftshift(Fconv)), 'LineWidth',1.5)
grid
ylabel('Amplitude')
text(Fv2(idx), Amax, sprintf('\\uparrow\nAmp = %.2E\nFreq = %.2f', Amax, Fv2(idx)), 'HorizontalAlignment','left', 'VerticalAlignment','top')
subplot(2,1,2)
plot(Fv2, rad2deg(angle(fftshift(Fconv))), 'LineWidth',1)
grid
xlabel('Frequency (Cy/Yr)')
ylabel('Phase (°)')
text(Fv2(idx), rad2deg(angle(fftshift(Fconv(idx)))), sprintf('\\uparrow\nPhase = %.2f°\nFreq = %.2f', rad2deg(angle(fftshift(Fconv(idx)))), Fv2(idx)), 'HorizontalAlignment','left', 'VerticalAlignment','top')

That demonstrates that they are actually 180° out of phase.