Partial Autocorrelation
Partial autocorrelation removes the influence of intermediate lags, providing a clearer picture of the direct relationship between a variable and its past values. Unlike autocorrelation, partial autocorrelation focuses on the direct correlation at each lag.
Mathematical Representation
The partial autocorrelation function (PACF) at lag k for a time series.
Here:
- is the value of the time series at time.
- : is the value of the time series at time (t-k)
- is the conditional covariance between and given the values of the intermediate lags.
- is the conditional variance of given the values of the intermediate lags.
- is the conditional variance of given the values of the intermediate lags.
Interpretation
- Direct Relationship: PACF isolates the direct correlation between the current observation and the observation at lag k, controlling for the influence of lags in between.
- AR Process Identification: Peaks or significant values in PACF at specific lags can indicate potential orders for autoregressive (AR) terms in time series models.
- Modeling Considerations: Analysts often examine PACF to guide the selection of lag orders in autoregressive integrated moving average (ARIMA) models.
R
# Load necessary librarieslibrary(forecast)# Load AirPassengers datasetdata("AirPassengers")# Plot PACFpacf_result <- pacf(AirPassengers, main = "Partial Autocorrelation Function (PACF) for AirPassengers") |
Output:
Partial Autocorrelation
we use the same “AirPassengers” dataset and plot the PACF. The PACF plot shows the direct correlation at each lag, helping identify the order of autoregressive terms.
- The PACF plot helps identify the direct correlation at each lag. Peaks at lags 1 and 12 suggest potential autoregressive terms related to the monthly and yearly patterns in the data.
- The PACF plot, on the other hand, focuses on the direct correlation at each lag, providing insights into the order of autoregressive terms.
- By comparing the two plots, you can observe how they complement each other in revealing the temporal dependencies within the time series. The ACF helps identify overall patterns, while the PACF refines the analysis by highlighting direct correlations.
Perform both on a Time series dataset to compare
R
# Load necessary librarieslibrary(fpp2)# Load the "ausbeer" dataset from fpp2 packagedata("ausbeer")# Plot the time seriesautoplot(ausbeer, main = "Monthly Australian Beer Production") |
Output:
Time Series Plot
Plot ACF
R
# Plot ACFacf(ausbeer, main = "Autocorrelation Function (ACF) for Australian Beer Production") |
Output:
Autocorrelation Plot
Plot PACF for differenced time series
R
# Load PACF from the forecast packagelibrary(forecast)# Plot PACF for differenced time seriesdiff_ausbeer <- diff(ausbeer)pacf_result <- pacf(diff_ausbeer,main = "Partial Autocorrelation Function (PACF) for Differenced Australian Beer Production") |
Output:
Partial Autocorrelation Plot
In this example, we use the “ausbeer” dataset from the fpp2 package, which represents monthly Australian beer production. The ACF plot can provide insights into the potential seasonality and trends in beer production.
- Autocorrelation (ACF): The ACF plot for Australian beer production may reveal patterns related to seasonality, trends, or cyclic behavior. Peaks at certain lags could indicate recurring patterns in beer production.
- Partial Autocorrelation (PACF): Differencing the time series and examining the PACF helps identify potential autoregressive terms that capture the direct correlation at each lag, after removing the influence of trends.
Additional Considerations
- Seasonal Differencing: In some cases, it might be beneficial to apply seasonal differencing (e.g., differencing by 12 for monthly data) to handle seasonality properly.
- Seasonal Differencing: In some cases, it might be beneficial to apply seasonal differencing (e.g., differencing by 12 for monthly data) to handle seasonality properly.
- Model Selection: The combination of ACF and PACF analysis can guide the selection of parameters in time series models, such as autoregressive integrated moving average (ARIMA) models.
- Interpretation: Understanding the patterns revealed by ACF and PACF is crucial for interpreting the underlying dynamics of a time series and building accurate forecasting models.
Autocorrelation and Partial Autocorrelation
Autocorrelation and partial autocorrelation are statistical measures that help analyze the relationship between a time series and its lagged values. In R Programming Language, the acf() and pacf() functions can be used to compute and visualize autocorrelation and partial autocorrelation, respectively.