acf.pacf()

Compute the empirical partial autocorrelation function up to max_lag.

Usage

acf.pacf(
    y,
    max_lag,
)

The lag-:math:k partial autocorrelation is the correlation between :math:y_t and :math:y_{t-k} after removing the linear effect of the intermediate observations :math:y_{t-1}, \ldots, y_{t-k+1}; it equals the last coefficient :math:\phi_{kk} of the best linear predictor of order :math:k. The coefficients are obtained from the empirical autocorrelations (see acf()) with the Durbin-Levinson recursion

.. math::

\phi_{kk} =
\frac{\hat{\rho}_k - \sum_{j=1}^{k-1} \phi_{k-1,j}\, \hat{\rho}_{k-j}}
     {1 - \sum_{j=1}^{k-1} \phi_{k-1,j}\, \hat{\rho}_j}.

The recursion 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"]: Partial 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 (see acf()); a near-deterministic series can likewise overflow the recursion (the denominator above approaches zero) and produce infinities or NaNs.

References

James Durbin (1960). “The Fitting of Time-Series Models”. Revue de l’Institut International de Statistique.