Estimate Autocorrelation using Python

Question

I would like to perform Autocorrelation on the signal shown below. The time between two consecutive points is 2.5ms (or a repetition rate of 400Hz).

This is the equation for estimating autoacrrelation that I would like to use (Taken from http://en.wikipedia.org/wiki/Autocorrelation, section Estimation):

What is the simplest method of finding the estimated autocorrelation of my data in python? Is there something similar to numpy.correlate that I can use?

Or should I just calculate the mean and variance?

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Edit:

With help from unutbu, I have written:

from numpy import *
import numpy as N
import pylab as P

fn = 'data.txt'
x = loadtxt(fn,unpack=True,usecols=[1])
time = loadtxt(fn,unpack=True,usecols=[0]) 

def estimated_autocorrelation(x):
    n = len(x)
    variance = x.var()
    x = x-x.mean()
    r = N.correlate(x, x, mode = 'full')[-n:]
    #assert N.allclose(r, N.array([(x[:n-k]*x[-(n-k):]).sum() for k in range(n)]))
    result = r/(variance*(N.arange(n, 0, -1)))
    return result

P.plot(time,estimated_autocorrelation(x))
P.xlabel('time (s)')
P.ylabel('autocorrelation')
P.show()

Answer 1

I don't think there is a NumPy function for this particular calculation. Here is how I would write it:

def estimated_autocorrelation(x):
    """
    http://stackoverflow.com/q/14297012/190597
    http://en.wikipedia.org/wiki/Autocorrelation#Estimation
    """
    n = len(x)
    variance = x.var()
    x = x-x.mean()
    r = np.correlate(x, x, mode = 'full')[-n:]
    assert np.allclose(r, np.array([(x[:n-k]*x[-(n-k):]).sum() for k in range(n)]))
    result = r/(variance*(np.arange(n, 0, -1)))
    return result

The assert statement is there to both check the calculation and to document its intent.

When you are confident this function is behaving as expected, you can comment-out the assert statement, or run your script with python -O. (The -O flag tells Python to ignore assert statements.)