QuanTAlib/lib/statistics/pacf/Pacf.md

PACF: Partial Autocorrelation Function

Strip away the intermediaries, and you'll see the true direct relationship.

Property	Value
Category	Statistic
Inputs	Source (close)
Parameters	period, lag (default 1)
Outputs	Single series (Pacf)
Output range	Varies (see docs)
Warmup	period bars
PineScript	pacf.pine

• The Partial Autocorrelation Function (PACF) measures the correlation between a time series and its lagged values, after removing the effects of all...
• Similar: ACF, LinReg | Trading note: Partial autocorrelation; isolates direct lag relationships. Used for ARIMA model order selection.
• Validated against mathematical properties and Durbin-Levinson recursion expectations.

The Partial Autocorrelation Function (PACF) measures the correlation between a time series and its lagged values, after removing the effects of all intermediate lags. While ACF shows total correlation at each lag, PACF isolates the direct correlation, making it essential for AR model identification.

Historical Context

The partial autocorrelation concept emerged from regression theory, where researchers needed to isolate the direct effect of a variable while controlling for confounding factors. The Durbin-Levinson algorithm (1960) provided an efficient recursive method to compute PACF, reducing the computational burden from solving a new system of equations for each lag.

In time series analysis, PACF became a cornerstone of the Box-Jenkins methodology (1970) for ARIMA model identification. While ACF helps identify MA order, PACF is the primary tool for identifying AR order.

Architecture & Physics

The PACF indicator uses the Durbin-Levinson recursion to efficiently compute partial autocorrelations. This avoids the need to solve separate regression equations for each lag, instead building up the solution recursively from ACF values.

Core Components

1. RingBuffer: Maintains the sliding window of period values
2. ACF Computation: Calculates all autocorrelations up to the target lag
3. Durbin-Levinson Recursion: Computes PACF from ACF values
4. Coefficient Arrays: Temporary storage for recursion (stack-allocated for small lags)

Mathematical Foundation

Partial Autocorrelation Definition

The partial autocorrelation at lag k, denoted \phi_{kk}, is the correlation between X_t and X_{t-k} after removing the linear dependence on X_{t-1}, X_{t-2}, \ldots, X_{t-k+1}.

Equivalently, \phi_{kk} is the last coefficient in the AR(k) regression:

X_t = \phi_{k1} X_{t-1} + \phi_{k2} X_{t-2} + \cdots + \phi_{kk} X_{t-k} + \epsilon_t

Durbin-Levinson Algorithm

The algorithm recursively computes PACF from ACF values:

Initialization:

\phi_{11} = r_1

Recursion for k = 2, 3, ..., K:

\phi_{kk} = \frac{r_k - \sum_{j=1}^{k-1} \phi_{k-1,j} \cdot r_{k-j}}{1 - \sum_{j=1}^{k-1} \phi_{k-1,j} \cdot r_j}

Coefficient Update:

\phi_{kj} = \phi_{k-1,j} - \phi_{kk} \cdot \phi_{k-1,k-j} \quad \text{for } j = 1, \ldots, k-1

Key Properties

• \phi_{11} = r_1 (PACF at lag 1 equals ACF at lag 1)
• -1 \leq \phi_{kk} \leq 1 for all k
• For AR(p) processes, PACF cuts off after lag p (\phi_{kk} = 0 for k > p)
• For MA(q) processes, PACF decays exponentially or sinusoidally
• For ARMA(p,q) processes, PACF exhibits complex behavior after lag p-q

AR Process Identification

For an AR(p) process:

X_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + \cdots + \phi_p X_{t-p} + \epsilon_t

The PACF exhibits:

• \phi_{kk} \neq 0 for k \leq p (significant values)
• \phi_{kk} = 0 for k > p (cuts off sharply)

This cutoff property makes PACF the primary diagnostic for AR order selection.

AR(1) Example

For AR(1) process X_t = \phi X_{t-1} + \epsilon_t:

• \phi_{11} = \phi (the AR coefficient)
• \phi_{kk} = 0 for k > 1

AR(2) Example

For AR(2) process X_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + \epsilon_t:

• \phi_{11} and \phi_{22} are non-zero
• \phi_{kk} = 0 for k > 2

Performance Profile

Metric	Score	Notes
Throughput	~100 ns/bar	ACF loop + Durbin-Levinson recursion
Allocations	0	Zero-allocation for lag ≤ 64 (stackalloc)
Complexity	O(period + lag²)	ACF computation + Durbin-Levinson
Accuracy	8	Good accuracy; numerical stability from recursion

Operation Count (per update)

Operation	Count	Notes
ADD/SUB	~3N + K²	Mean, variance, autocovariance, recursion
MUL	~2N + 2K²	Cross products, coefficient updates
DIV	K + 2	ACF ratios, recursion denominators

Where N = period, K = lag.

Validation

Library	Status	Notes
TA-Lib	N/A	Not available in TA-Lib
Skender	N/A	Not available in Skender
Tulip	N/A	Not available in Tulip
Mathematical	✅	Validated against theoretical properties

PACF is validated through mathematical properties:

• \phi_{11} = r_1 (PACF at lag 1 equals ACF at lag 1)
• Bounded output [-1, 1]
• Constant series returns 0
• AR(1) process produces PACF ≈ φ at lag 1, ≈ 0 for higher lags

Common Pitfalls

1. Period vs Lag Constraint: Period must be greater than lag + 1. Insufficient data produces undefined or unstable results.

2. PACF ≠ ACF: A common confusion is treating PACF and ACF identically. While \phi_{11} = r_1, higher-order PACF values differ significantly from ACF.

3. Warmup Period: PACF requires a full window (period values) plus sufficient data for stable ACF estimates. Values during warmup are unreliable.

4. Numerical Stability: For very high lags, the Durbin-Levinson recursion can accumulate numerical errors. The denominator approaching zero indicates potential instability.

5. AR vs MA Confusion: Sharp PACF cutoff indicates AR; sharp ACF cutoff indicates MA. Using the wrong criterion leads to model misspecification.

6. Significance Testing: PACF values should be tested against confidence bounds. For white noise, 95% confidence bounds are approximately \pm 1.96/\sqrt{n}.

7. Non-Stationarity: Like ACF, PACF assumes stationarity. Trending data should be differenced first.

Usage

using QuanTAlib;

// Create a 20-period PACF indicator with lag 1
var pacf = new Pacf(period: 20, lag: 1);

// Update with new values
var result = pacf.Update(new TValue(DateTime.UtcNow, 100.0));

// Access the last calculated PACF value
Console.WriteLine($"PACF(1): {pacf.Last.Value}");

// Verify key property: PACF(1) should equal ACF(1)
var acf = new Acf(period: 20, lag: 1);
// ... feed same data to both ...
// pacf.Last.Value ≈ acf.Last.Value

// Chained usage
var source = new TSeries();
var pacfChained = new Pacf(source, period: 20, lag: 1);

// Static batch calculation
var output = Pacf.Calculate(source, period: 20, lag: 1);

// Span-based calculation
Span<double> outputSpan = stackalloc double[source.Count];
Pacf.Batch(source.Values, outputSpan, period: 20, lag: 1);

Applications

AR Order Identification

The primary application of PACF is determining the order of AR models:

• PACF cuts off after lag p → suggests AR(p)
• Combined with ACF cutoff after lag q → suggests ARMA(p,q)

Model Validation

After fitting an AR model, residual PACF should show no significant values, indicating all autocorrelation structure has been captured.

Lead-Lag Analysis

In financial markets, PACF can reveal direct lead-lag relationships between assets after controlling for intermediate effects.

Signal Processing

PACF is used in linear prediction and filter design, where the partial correlation structure determines optimal predictor coefficients.

Comparison: ACF vs PACF

Property	ACF	PACF
Measures	Total correlation at lag k	Direct correlation at lag k
AR(p) process	Exponential/sinusoidal decay	Cuts off after lag p
MA(q) process	Cuts off after lag q	Exponential/sinusoidal decay
ARMA(p,q)	Tails off	Tails off
Primary use	MA order identification	AR order identification
Lag 1 value	r_1	\phi_{11} = r_1

References

• Box, G.E.P., Jenkins, G.M. (1970). Time Series Analysis: Forecasting and Control. Holden-Day.
• Durbin, J. (1960). "The fitting of time series models." Review of the International Statistical Institute, 28, 233-243.
• Levinson, N. (1946). "The Wiener RMS error criterion in filter design and prediction." Journal of Mathematics and Physics, 25, 261-278.
• Hamilton, J.D. (1994). Time Series Analysis. Princeton University Press.