transform time series different length and sampling rate with frequency domain

Question

i want to transform my time series (each same scale) to frequency domain. There are 2 things to conside:

1. some time series are longer then the other
2. i have different measuring intervals which means e.g. one time series measures sometimes 2 times an hour, another one measures 3 times an our and so on.

i have 20 time series and the sampling rate vary for each time series. i want to do hierarchical clustering with these 20 time series (ID). Here is an example how my data looks like:

ID  DAY  MONTH HOUR  VALUE
1    1     6   00:30   5
1    1     6   00:58   5.1
1    1     6   1:23    4.2
........................
2    1     6   00:02   2
2    1     6   00:28   1.8
............................
20   1     6   23:55    1.4

can i use frequency transformation and which transformation is useful (fourier transformation). I am new to this topic and i saw someone already asking a similar question, but the situation is not the same (for example i have different sampling rate)

Answer 1

Consider using the Lomb-Scargle Periodogram which is useful for spectral analysis of unevenly spaced data in the time domain. Jacob VanderPlas provides an intuitive overview of the algorithm in his publication in the Astrophysical Journal, May 11 2018 "Understanding the Lomb-Scargle Periodogram".

The algorithm is implemented in MATLAB as plomb and in Python's scipy.signal as lombscargle

Another Jacob, Jacob Svensson, has provided a python implementation to have an equivalent interface to MATLAB's plomb, also copied here:

import numpy as np
from scipy.signal import lombscargle

def lomb(t, y):
    """
    Compute the Lomb-Scargle periodogram
    Jacob Svensson, https://jcbsv.net/about/ 

    Inputs
        t : sample times
        y : measurement values

    Outputs
        f   : frequency vector
        pxx : power spectral density (PSD) estimate

    Inputs are assumed to be vectors of type numpy.ndarray.
    """
    n = t.size
    ofac = 4  # Oversampling factor
    hifac = 1
    T = t[-1] - t[0]
    Ts = T / (n - 1)
    nout = np.round(0.5 * ofac * hifac * n) 
    f = np.arange(1, nout+1) / (n * Ts * ofac)
    f_ang = f * 2 * np.pi
    pxx = lombscargle(t, y, f_ang, precenter=True)
    pxx = pxx * 2 / n
    return f, pxx