根据时间序列数据创建正弦波（Python）

Question

I am trying to implement a Discrete Fourier Transform with time series data from a CSV. I have been able to generate a sine wave (and cosine wave) in Python with SciPy and have gotten back the magnitude and phase information I want. However, I am struggling with real data. My CSV looks like this, simulating events that occur Mondays at 9am.

id1 ?2018-01-05T23:00:00.000Z
id1 ?2018-01-08T09:20:00.000Z
id1 ?2018-01-15T09:43:00.000Z
id1 ?2018-01-22T09:02:00.000Z
id1 ?2018-01-29T09:50:00.000Z
id1 ?2018-02-05T09:47:00.000Z
id1 ?2018-02-12T09:18:00.000Z
id1 ?2018-02-19T09:02:04.000Z
id1 ?2018-02-26T09:54:00.000Z
id1 ?2018-03-05T09:12:00.000Z

After getting it all cleaned up, it looks like this after binning by day (I hope to bin at the hour and minute levels eventually):

ID..Date............Event
id1 2018-01-08 1
id1 2018-01-09 0
id1 2018-01-10 0
id1 2018-01-11 0
id1 2018-01-12 0
id1 2018-01-13 0
id1 2018-01-14 0
id1 2018-01-15 1
id1 2018-01-16 0

...etc. What can I do to transform this into a sine wave? Currently, I am creating a sine wave and running an fft like this:

A = 5 # amplitude
fc = 10 # frequency
fs = 32 * fc # sampling frequency with oversampling factor 32
t = np.arange(0, 2, 1/fs) # time array
phi = 30 # phase shift

x = A * np.cos(2 * np.pi * fc * t + phi)
fourier = fft(x)

I am able to get the phase information from this in the frequency I want which leads me to believe if I can just get a signal from my CSV file and replace that with x, then I'd be able to extract phase information from that. My understanding is that with the binary data (the 'Event' column), magnitude information will not be helpful - is that correct?

How do I turn this CSV file into a sine or cosine wave?

Answer 1

The current dataset that you show indeed looks nothing like a sine-wave but since all mathematically nice functions can be written as a superposition of sines and cosines, this need not be a problem.

More details in the docs: https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.fft.fft.html#numpy.fft.fft

First let us do an example:

import numpy as np
import matplotlib.pyplot as plt
T = 100
x = np.arange(0,T)
y=  np.sin(4*np.pi*x/T)+np.cos(8*np.pi*x/T)

We have thus have a superposition of a sine and cosine with frequencies twice and four-times per step x. We now perform the Fourier Transform:

sp   = np.fft.fft(y)               # the discrete fourier transform
freq = np.fft.fftfreq(y.shape[-1]) # the accompanying frequencies

Now we can reconstruct the original function 'y' through the fourier transform as a superposition of sines and cosines and check whether we succeeded by plotting.

cos=np.sum([(sp[-i]+sp[i]).real/(2*T)*np.cos(2.*np.pi*freq[i]*x)\
             for i in range(len(freq))],axis=0)
sin=np.sum([(sp[-i]-sp[i]).imag/200.*np.sin(2.*np.pi*freq[i]*x)\
              for i in range(len(freq))],axis=0)

plt.plot(x, y,x,cos+sin)
plt.show()

You should see that the two curves match perfectly. Now your actual problem.

T=9
x=np.arange(0,T,0.01) # the interspacing of the datpoints for the (co)sines is 0.01
y = np.array([1,0,0,0,0,0,0,1,0]) # the input data you suggested
sp = np.fft.fft(y)
freq = np.fft.fftfreq(y.shape[-1])
cos=np.sum([(sp[-i]+sp[i]).real/(2*T)*np.cos(2.*np.pi*freq[i]*x)\
             for i in range(len(freq))],axis=0)
sin=np.sum([(sp[-i]-sp[i]).imag/200.*np.sin(2.*np.pi*freq[i]*x)\
              for i in range(len(freq))],axis=0)

plt.plot(np.arange(0,9), y,x,cos+sin)
plt.show() 

The amplitudes of the frequencies are: 'freq[i]' are given by '(sp[-i]-sp[i]).real/(2*T)' and '(sp[-i]+sp[i]).real/(2*T)' for sines and cosines respectively.