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

Indeed, interesting problem, but no satisfying answer yet. Let's try to change that...

You say that you prefer not to use interpolation, but, as I understand from your comment, what you really mean is that you would like to avoid upsampling to a higher resolution. A basic solution makes use of a least squares fit with a linear interpolation function, but without upsampling to a higher resolution:

import numpy as np
from scipy.interpolate import interp1d
from scipy.optimize import leastsq

def yvals(x):
    return np.sin(x)+np.sin(2*x)+np.sin(3*x)

dx = .1
X = np.arange(0,2*np.pi,dx)
Y = yvals(X)

unknown_shift = np.random.random() * dx
Y_shifted = yvals(X + unknown_shift)

def err_func(p):
    return interp1d(X,Y)(X[1:-1]+p[0]) - Y_shifted[1:-1]

p0 = [0,] # Inital guess of no shift
found_shift = leastsq(err_func,p0)[0][0]

print "Unknown shift: ", unknown_shift
print "Found   shift: ", found_shift

A sample run gives a quite accurate solution:

Unknown shift:  0.0695701123582
Found   shift:  0.0696105501967

If one includes noise in the shifted Y:

Y_shifted += .1*np.random.normal(size=X.shape)

One gets somewhat less precise results:

Unknown shift:  0.0695701123582
Found   shift:  0.0746643381744

The accuracy under presence of noise improves when more data is available, e.g. with:

X = np.arange(0,200*np.pi,dx)

A typical result is:

Unknown shift:  0.0695701123582
Found   shift:  0.0698527939193