Autocorrelation Calculator

Compute ACF and PACF for time series analysis. Identify lag patterns, seasonality, and determine ARIMA model orders with interactive correlograms.

Data Input

Time Series (comma-separated)

Max Lag:

Summary

Observations

257.5

Mean

Significant Lags

ACF: lag 4: 0.728lag 8: 0.475

PACF: lag 4: 0.718lag 5: -0.474

Blue bars - Significant ACF values (exceed confidence bounds)

Purple bars - Significant PACF values (exceed confidence bounds)

Gray bars - Non-significant autocorrelation at each lag

Dashed lines - 95% confidence bounds (±1.96/√n)

Shaded band - Confidence region (values inside are not significant)

How to Read the Chart

Significant lags: Bars extending beyond the dashed confidence bounds indicate statistically significant autocorrelation at that lag.
ACF pattern: If ACF decays slowly, the series is non-stationary and needs differencing. If it cuts off sharply after lag q, consider an MA(q) model.
PACF pattern: If PACF cuts off after lag p while ACF tails off, this suggests an AR(p) model. Use both plots together to determine ARIMA orders.
Seasonal spikes: Significant bars at regular intervals (e.g., lag 4, 8, 12) indicate seasonal patterns in your data.

Python Code

import numpy as np
import pandas as pd
from statsmodels.tsa.stattools import acf, pacf
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf
from statsmodels.stats.diagnostic import acorr_ljungbox
import matplotlib.pyplot as plt

# Data
data = np.array([200, 220, 250, 280, 210, 230, 260, 290, 220, 240, 270, 300, 230, 250, 280, 310, 240, 260, 290, 320])

# Compute ACF and PACF
acf_values = acf(data, nlags=10)
pacf_values = pacf(data, nlags=10, method='ywm')

print("ACF values:")
for lag, val in enumerate(acf_values):
    sig = "*" if abs(val) > 1.96 / np.sqrt(len(data)) else ""
    print(f"  Lag {lag}: {val:.4f} {sig}")

print("\nPACF values:")
for lag, val in enumerate(pacf_values):
    sig = "*" if abs(val) > 1.96 / np.sqrt(len(data)) else ""
    print(f"  Lag {lag}: {val:.4f} {sig}")

# Ljung-Box test for serial correlation
lb_test = acorr_ljungbox(data, lags=10, return_df=True)
print("\nLjung-Box Test:")
print(lb_test)

# Plot ACF and PACF
fig, axes = plt.subplots(2, 1, figsize=(12, 8))
plot_acf(data, lags=10, ax=axes[0], title='Autocorrelation Function (ACF)')
plot_pacf(data, lags=10, ax=axes[1], title='Partial Autocorrelation Function (PACF)')
plt.tight_layout()
plt.show()

Frequently Asked Questions

What is the difference between ACF and PACF?

ACF (Autocorrelation Function) shows the total correlation between a time series and its lagged values, including indirect correlations through intermediate lags. PACF (Partial ACF) shows only the direct correlation at each lag, removing the effect of shorter lags. PACF is essential for identifying the order of AR models.

How do I use ACF/PACF to choose ARIMA orders?

For ARIMA(p,d,q): If ACF cuts off after lag q and PACF tails off → MA(q) model. If PACF cuts off after lag p and ACF tails off → AR(p) model. If both tail off → ARMA(p,q). Differencing d times should make the series stationary before checking ACF/PACF.

What do the confidence bands mean?

The dashed blue lines are 95% confidence bands at ±1.96/√n. Any bar extending beyond these bands is statistically significant - the autocorrelation at that lag is unlikely to be zero. Bars within the bands are consistent with white noise.

How many lags should I examine?

A common rule of thumb is n/4 or 10·log₁₀(n) lags, where n is the series length. For seasonal data, examine at least 2 full seasonal cycles (e.g., 24 lags for monthly data with yearly seasonality).

What does it mean if ACF decays slowly?

Slowly decaying ACF (many significant lags) indicates a non-stationary series - it needs differencing. After differencing (computing the first differences), re-check the ACF. A stationary series should show ACF that drops off quickly.

Related Tools

ARIMA Model

AR, I, MA time series forecasting

Seasonal Decomposition

Trend, seasonal & residual

Moving Average

SMA, EMA & WMA smoothing

import numpy as np import pandas as pd from statsmodels.tsa.stattools import acf, pacf from statsmodels.graphics.tsaplots import plot_acf, plot_pacf from statsmodels.stats.diagnostic import acorr_ljungbox import matplotlib.pyplot as plt # Data data = np.array([200, 220, 250, 280, 210, 230, 260, 290, 220, 240, 270, 300, 230, 250, 280, 310, 240, 260, 290, 320]) # Compute ACF and PACF acf_values = acf(data, nlags=10) pacf_values = pacf(data, nlags=10, method='ywm') print("ACF values:") for lag, val in enumerate(acf_values): sig = "*" if abs(val) > 1.96 / np.sqrt(len(data)) else "" print(f" Lag {lag}: {val:.4f} {sig}") print("\nPACF values:") for lag, val in enumerate(pacf_values): sig = "*" if abs(val) > 1.96 / np.sqrt(len(data)) else "" print(f" Lag {lag}: {val:.4f} {sig}") # Ljung-Box test for serial correlation lb_test = acorr_ljungbox(data, lags=10, return_df=True) print("\nLjung-Box Test:") print(lb_test) # Plot ACF and PACF fig, axes = plt.subplots(2, 1, figsize=(12, 8)) plot_acf(data, lags=10, ax=axes[0], title='Autocorrelation Function (ACF)') plot_pacf(data, lags=10, ax=axes[1], title='Partial Autocorrelation Function (PACF)') plt.tight_layout() plt.show()