tf.contrib.distributions.auto_correlation

View source on GitHub

Auto correlation along one axis.

tf.contrib.distributions.auto_correlation(
    x, axis=-1, max_lags=None, center=True, normalize=True, name='auto_correlation'
)

Given a 1-D wide sense stationary (WSS) sequence X, the auto correlation RXX may be defined as (with E expectation and Conj complex conjugate)

RXX[m] := E{ W[m] Conj(W[0]) } = E{ W[0] Conj(W[-m]) },
W[n]   := (X[n] - MU) / S,
MU     := E{ X[0] },
S**2   := E{ (X[0] - MU) Conj(X[0] - MU) }.

This function takes the viewpoint that x is (along one axis) a finite sub-sequence of a realization of (WSS) X, and then uses x to produce an estimate of RXX[m] as follows:

After extending x from length L to inf by zero padding, the auto correlation estimate rxx[m] is computed for m = 0, 1, ..., max_lags as

rxx[m] := (L - m)**-1 sum_n w[n + m] Conj(w[n]),
w[n]   := (x[n] - mu) / s,
mu     := L**-1 sum_n x[n],
s**2   := L**-1 sum_n (x[n] - mu) Conj(x[n] - mu)

The error in this estimate is proportional to 1 / sqrt(len(x) - m), so users often set max_lags small enough so that the entire output is meaningful.

Note that since mu is an imperfect estimate of E{ X[0] }, and we divide by len(x) - m rather than len(x) - m - 1, our estimate of auto correlation contains a slight bias, which goes to zero as len(x) - m --> infinity.