Estimating small time shift between two time series

Question

I have two time series, and i suspect that there is a time shift between them, and i want to estimate this time shift.

This question has been asked before in: Find phase

Answer 1

One of the links you provided has the right idea (in fact I am doing pretty much the same thing here)

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import correlate

a,b, N = 0, 10, 1000        #Boundaries, datapoints
shift = -3                  #Shift, note 3/10 of L = b-a

x = np.linspace(a,b,N)
x1 = 1*x + shift
time = np.arange(1-N,N)     #Theoritical definition, time is centered at 0

y1 = sum([np.sin(2*np.pi*i*x/b) for i in range(1,5)])
y2 = sum([np.sin(2*np.pi*i*x1/b) for i in range(1,5)])

#Really only helps with large irregular data, try it
# y1 -= y1.mean()
# y2 -= y2.mean()
# y1 /= y1.std()
# y2 /= y2.std()

cross_correlation = correlate(y1,y2)
shift_calculated = time[cross_correlation.argmax()] *1.0* b/N
y3 = sum([np.sin(2*np.pi*i*(x1-shift_calculated)/b) for i in range(1,5)])
print "Preset shift: ", shift, "\nCalculated shift: ", shift_calculated



plt.plot(x,y1)
plt.plot(x,y2)
plt.plot(x,y3)
plt.legend(("Regular", "Shifted", "Recovered"))
plt.savefig("SO_timeshift.png")
plt.show()

This has the following output:

Preset shift:  -3
Calculated shift:  -2.99

It might be necessary to check

1. Scipy Correlate
2. Time Delay Analaysis

Note that the the argmax() of the correlation shows the position of the alignment, it has to be scaled by the length of b-a = 10-0 = 10 and N to get the actual value.

Checking the source of correlate Source it is not entirely obvious what the imported function from sigtools behaves. For large datasets circular correlation (via Fast Fourier Transforms) is much faster than the straight-forward method. I suspect this is what is implemented in sigtools but I cannot tell for sure. A search for the file in my python2.7 folder only returned the compiled C pyd file.