Autocovariance

autocov computes the autocovariance between two column vectors X and Y with same length N using the Fast Fourier Transform algorithm from 0 to N-2.

The resulting autocovariance column vector acv is given by the formula:

acv(p,1) = 1/(N-p) * \sum_{i=1}^{N}(X_{i} - X_bar) * (Y_{i+p} - Y_bar)

where X_bar and Y_bar are the mean estimates:

X_bar = 1/N * \sum_{i=1}^{N} X_{i}; Y_bar = 1/N * \sum_{i=1}^{N} Y_{i}

It satisfies the following identities:
1. variance consistency: if acv = autocov(X,X), then acv(1,1) = var(X)
2. covariance consistence: if acv = autocov(X,Y), then acv(1,1) = cov(X,Y)