acf.acf()

Compute the empirical autocorrelation function up to max_lag.

Usage

Source

acf.acf(
    y,
    max_lag,
)

The lag-:math:k autocorrelation of a series :math:y_1, \ldots, y_T with sample mean :math:\bar{y} is estimated with the biased normalization

.. math::

\hat{\rho}_k =
\frac{\sum_{t=k+1}^{T} (y_t - \bar{y})(y_{t-k} - \bar{y})}
     {\sum_{t=1}^{T} (y_t - \bar{y})^2},

computed for all lags at once via the FFT of the centered series (the Wiener-Khinchin identity), so the cost is :math:O(T \log T) instead of :math:O(T \, k_{\max}). The computation runs along the last axis and broadcasts over any leading batch axes.

Parameters

y: Float[Array, " *batch time"] | Float[np.ndarray, " *batch time"]

Time series with time on the last axis, shape (*batch, time).

max_lag: int

Largest lag to evaluate; must satisfy 1 <= max_lag < time.

Returns

Float[Array, "*batch lags"]

Autocorrelations for lags 0, 1, ..., max_lag (max_lag + 1 values; lag 0 is identically 1.0).

Raises

ValueError

If max_lag is not in [1, time).

Notes

A constant series has zero variance, so the result is meaningless: all NaN in eager execution (a 0 / 0), while under jax.jit() the rounding of the mean can instead yield arbitrary finite values.

References

Adapted from numpyro.diagnostics.autocorrelation(), itself adapted from the Stan implementation.