Partial Autocorrelation (PACF)

In the previous lesson, we learned about Autocorrelation (ACF), which showed us how the past influences the present.

But ACF has a problem.

It mixes direct influence and indirect influence.

PACF exists to separate those two.

Real-World Analogy (Very Important)

Imagine daily sales of a grocery store.

• Yesterday’s sales influence today → direct effect
• Sales from two days ago influence today through yesterday → indirect effect

ACF measures both.

PACF measures only the direct effect.

Why PACF Is Needed

ACF answers:

“How much does the past affect the present?”

PACF answers:

“How much does a specific past point affect the present, after removing all shorter lags?”

This distinction is critical for building AR models.

Creating a Realistic Time Series

We’ll reuse a realistic dataset with:

• Trend
• Weekly seasonality
• Noise

Python: Time Series Data

import numpy as np
import matplotlib.pyplot as plt

np.random.seed(7)

time = np.arange(300)
trend = time * 0.15
seasonal = 8 * np.sin(2 * np.pi * time / 7)
noise = np.random.normal(0, 3, size=300)

series = trend + seasonal + noise

This is the time series we are working with:

ACF vs PACF – Visual Difference

Before computing PACF, let’s remember what ACF looks like.

ACF shows correlation for all lags, but each lag includes influence from previous lags.

ACF Plot

Notice:

• Correlation decays slowly
• Multiple lags look important
• Hard to tell which lag truly matters

What PACF Actually Does

PACF removes the effect of shorter lags.

So when we look at lag 3:

• ACF includes influence of lag 1 and lag 2
• PACF removes lag 1 and lag 2 effects

What remains is the true direct influence.

Python: Calculating PACF

def pacf(series, max_lag):
    pacf_vals = []
    for lag in range(1, max_lag+1):
        X = []
        y = series[lag:]
        for i in range(lag):
            X.append(series[lag-i-1:-i-1])
        X = np.column_stack(X)
        coef = np.linalg.lstsq(X, y, rcond=None)[0]
        pacf_vals.append(coef[-1])
    return pacf_vals

pacf_values = pacf(series, 20)

Here is the actual PACF plot:

How to Read a PACF Plot

Each bar shows direct dependency.

• High bar → strong direct influence
• Bars drop suddenly → AR order cutoff

In this plot:

• Lag 1 is very strong
• Lag 2 is weak
• Most later lags are near zero

This tells us the series behaves like an AR(1) process.

Why PACF Is Crucial for AR Models

Real-World Interpretation

If PACF shows:

• Strong lag 1 → yesterday matters
• Weak lag 2 → day-before-yesterday doesn’t add new info

Then your system has short memory.

This is common in:

• Sales data
• Traffic counts
• Sensor readings

Practice Questions

Key Takeaways

• ACF mixes direct and indirect effects
• PACF isolates direct dependency
• PACF determines AR order
• Essential for ARIMA modeling

Next Lesson

In the next lesson, we will move into Moving Averages and understand MA behavior visually.