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The partial autocorrelation function, with its confidence band.

public static AutocorrelationResult PartialAutocorrelation(ReadOnlySpan<double> series, int lagCount, AutocorrelationOptions options = null)

Parameters — series are the observations, in time order. lagCount is how many lags past zero to report, at most half the series length — required rather than defaulted, for the reason Autocorrelation gives; the reference defaults it to min(10·log10(n), n/2 - 1) here, and to min(10·log10(n), n - 1) for acf. options is read for AutocorrelationOptions.ConfidenceLevel only — Adjusted and BartlettConfidenceInterval have no meaning for a partial autocorrelation and are ignored.

Returns — AutocorrelationResult: the reflection coefficients at each lag from zero upward, and a flat band around them.

Exceptions — ArgumentException when series holds fewer than two points, is constant, or carries a non-finite value; or when lagCount is above half the series length. ArgumentOutOfRangeException when lagCount is below one.

Example — the same sawtooth series Autocorrelation reads, and its first four partial lags.

using Lodestar.Stats.TimeSeries;

double[] series = [1.0, 3.0, 2.0, 5.0, 4.0, 7.0, 6.0, 9.0, 8.0, 11.0, 10.0, 13.0];

AutocorrelationResult result = SerialCorrelation.PartialAutocorrelation(series, lagCount: 4);

double p1 = Math.Round(result.Values[1], 4);              // => 0.646
double p3 = Math.Round(result.Values[3], 4);              // => -0.6651
double lower3 = Math.Round(result.ConfidenceLower[3], 4);  // => -1.2309
double upper3 = Math.Round(result.ConfidenceUpper[3], 4);  // => -0.0993

Remarks — ywadjusted (also spelled yw, ywa, yw_adjusted) is the reference's own default and the only method shipped; the reference offers seven other estimators — ywm, ols, ols-inefficient, ols-adjusted, ld, ldbiased, burg — across four families. Yule-Walker with the adjusted autocovariance, solved by the Levinson-Durbin recursion, is what a caller gets by not choosing, and it is the reference's default too. Widening this to an options enum is a later change, not a reason to hold this one back.

The band does not use Bartlett's formula. Its variance is 1/n at every lag past zero — Quenouille's result, that a partial autocorrelation past the true order is asymptotically N(0, 1/n) — unlike Autocorrelation's band, which widens with the lag by default. Lag zero's interval is fixed at the point [1, 1] regardless of ConfidenceLevel, which is what the reference also reports.

using Lodestar.Stats.TimeSeries;

double[] series = [1.0, 3.0, 2.0, 5.0, 4.0, 7.0, 6.0, 9.0, 8.0, 11.0, 10.0, 13.0];

AutocorrelationResult result = SerialCorrelation.PartialAutocorrelation(series, lagCount: 4);

bool lagZeroIsAPoint = result.ConfidenceLower[0] == result.ConfidenceUpper[0];   // => True
double bandWidth = Math.Round(result.ConfidenceUpper[1] - result.ConfidenceLower[1], 4);  // => 1.1316

Applies to — net10.0, netstandard2.0.

See also — SerialCorrelation.Autocorrelation, SerialCorrelation.LjungBox, AutocorrelationOptions, the Python equivalence table.