Deciphering ACF and PACF Plots: A Guide to Time Series Forecasting
When working with time series data, it is often useful to analyze the autocorrelation of the data to understand the patterns and dependencies between time steps. Two common tools for this analysis are the Autocorrelation Function (ACF) and the Partial Autocorrelation Function (PACF).
ACF plots show the correlation between a time series and lagged versions of itself. The ACF plot can be used to identify the number of lags that are needed for a time series model. For example, if the ACF plot shows a strong correlation between the time series and its lag-1, lag-2, and lag-3 values, then a time series model that includes these lags would likely be a good fit.
PACF plots, on the other hand, show the correlation between a time series and its lagged values, after accounting for the correlations at shorter lags. This can be useful for identifying the most important lags to include in a time series model, as well as for identifying the presence of seasonal patterns in the data.
To interpret ACF and PACF plots, we can look for the following patterns:
- A sharp cutoff in the ACF plot at a particular lag suggests that a time series model with a corresponding number of lags would be a good fit.
- A slowly decaying ACF plot may indicate the presence of a trend in the data.
- A slowly decaying PACF plot may indicate the presence of seasonality in the data.
- A sharp cutoff in the PACF plot at a particular lag may indicate the presence of a seasonal pattern in the data.
It is worth noting that ACF and PACF plots are only one tool among many that can be used for time series analysis and forecasting. Other techniques, such as decomposition and power spectral density analysis, can also be useful for understanding the patterns in time series data.
In conclusion, ACF and PACF plots are useful tools for analyzing autocorrelation in time series data. By identifying the lags and patterns in the data, we can better understand the dependencies between time steps and build more accurate time series models.
To demonstrate the use of ACF and PACF plots in practice, let’s consider an example time series dataset. For this example, we will use the monthly ridership data for the Toronto Transit Commission (TTC) from January 1973 to September 2018. This dataset contains the number of riders on the TTC’s bus, streetcar, and subway systems each month.
First, we will load and plot the data to get a sense of its overall trend and seasonality:
import pandas as pd
import matplotlib.pyplot as plt# Load the TTC ridership data
ttc = pd.read_csv("ttc_ridership.csv")# Plot the time series data
plt.plot(ttc["time"], ttc["riders"])
plt.show()
[Insert plot of TTC ridership data]
From the plot, we can see that the ridership data exhibits both a long-term trend and seasonal fluctuations.
Next, we will use the plot_acf and plot_pacf functions from the statsmodels library to plot the ACF and PACF of the data:
from statsmodels.graphics.tsaplots import plot_acf, plot_pacf# Plot the ACF and PACF of the data
plot_acf(ttc["riders"], lags=30)
plot_pacf(ttc["riders"], lags=30)
plt.show()From the ACF plot, we can see a strong correlation at lags 1 and 12, which corresponds to the monthly seasonality in the data. The PACF plot shows a sharp cutoff at lag 12, further indicating the presence of a seasonal pattern in the data.
Based on these plots, we could build a time series model that includes lags 1 and 12 to capture the trend and seasonality in the data. We could also consider including additional lags if the ACF plot shows a strong correlation at those lags.
In summary, ACF and PACF plots are useful tools for understanding the patterns in time series data. By analyzing the correlations at different lags, we can identify the most important lags to include in a time series model and build more accurate forecasts.
