Find likeliest periodicity for time series with numpy's Fourier Transform?

Question

Assume I have a time series t with one hundred measurements, each entry representing the measured value for each day. I assume there is some periodicity in the signal -- it might repeat daily, weekly or monthly.

Translating the time series into the Fourier domain might help to find such a periodicity?

How could I use numpy's fft module to find the likeliest period for my time series?

Answer 1

I will aim to answer my own question. You may correct me where appropriate.

Asume our time series t is t = [2,1,0,1,2,3,2,1,0,1,2,3,2,1,0,1,2,3] with 18 measurements. A rather simple example: It seems likely that the length of the period is six time units.

Taking this time series into the Frequency Domain yields us:

    w = numpy.fft.fft(t)
    print numpy.absolute(w)
    array([27.000000, 0.000000, 0.000000, 12.000000, 0.000000, 0.000000,
   0.000000, 0.000000, 0.000000, 3.000000, 0.000000, 0.000000,
   0.000000, 0.000000, 0.000000, 12.000000, 0.000000, 0.000000])

We ignore frequency 0 and observe that the value is largest for index 3 -- this indicates that within our time series t the signal repeats 3 times. Hence the length of the signal -- the period -- would be 18/3 = 6. And indeed:

numpy.fft.fftfreq(18)
array([ 0.      ,  0.055556,  0.111111,  0.166667,  0.222222,  0.277778,
    0.333333,  0.388889,  0.444444, -0.5     , -0.444444, -0.388889,
   -0.333333, -0.277778, -0.222222, -0.166667, -0.111111, -0.055556])

The frequency at index 3 is exactly 1/6, i.e. the frequency for one time unit is 1/6, meaning the signal takes six time units for a full period.

Please let me know if my understanding is correct.

Answer 2

Note than an FFT finds a sinusoidal decomposition, which is different from a periodicity estimator (as any fundamental period can be completely missing from a periodic signal's frequency spectrum. See missing fundamental )

So you will need to post-process your FFT result with something like a cepstrum (using complex cepstral analysis, etc.), or perhaps use a Harmonic Product Spectrum estimator.