## acf.pacf()


Compute the empirical partial autocorrelation function up to `max_lag`.


Usage

``` python
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()](acf.acf.md#numpyro_forecast.acf.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, `<span class="st">`" *batch time"``]`</span>  
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()](acf.acf.md#numpyro_forecast.acf.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*.