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?
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.