9.2 Identifying seasonal patterns in ACF and PACF

Seasonal patterns in time series data repeat at fixed intervals, like yearly temperature changes. ACF and PACF plots help spot these patterns by showing significant spikes at lags that are multiples of the seasonal period.

To identify seasonal components, look for spikes at regular intervals in ACF and PACF plots. The lag of the first significant spike indicates the seasonal period. This helps distinguish between short-term fluctuations and long-term, recurring patterns in the data.

Identifying Seasonal Patterns in ACF and PACF

Seasonal patterns in ACF and PACF

• ACF and PACF plots help identify seasonal patterns in time series data which are regular, repeating patterns occurring at fixed intervals (temperature data showing yearly seasonality)
• ACF plot shows significant spikes at lags that are multiples of the seasonal period (lags 12, 24, 36 for monthly data with annual seasonality) with spikes gradually decreasing in magnitude as lag increases
• PACF plot shows significant spikes at lags that are multiples of the seasonal period with spikes cutting off abruptly after the seasonal lag (significant spike at lag 12 followed by insignificant spikes at lags 24, 36 for monthly data with annual seasonality)

Determining seasonal periods

• Seasonal period is the number of time steps between each repetition of the seasonal pattern (12 for monthly data with annual seasonality, 4 for quarterly data with annual seasonality)
• In ACF plots, the lag at which the first significant spike appears represents the seasonal period
• In PACF plots, the lag at which a significant spike appears followed by a cutoff in subsequent lags represents the seasonal period

Order of seasonal AR and MA terms

• Seasonal autoregressive (SAR) terms are identified using the PACF plot
  1. The order of the SAR term is determined by the number of significant spikes at lags that are multiples of the seasonal period
  2. A significant spike at lag 12 in the PACF plot indicates an SAR term of order 1, denoted as SAR(1)
• Seasonal moving average (SMA) terms are identified using the ACF plot
  1. The order of the SMA term is determined by the number of significant spikes at lags that are multiples of the seasonal period
  2. Significant spikes at lags 12 and 24 in the ACF plot indicate an SMA term of order 2, denoted as SMA(2)

Seasonal vs non-seasonal components

• Non-seasonal components are represented by significant spikes at short lags (lags 1, 2, 3) in both ACF and PACF plots indicating the presence of short-term dependencies in the time series data (daily stock price fluctuations)
• Seasonal components are represented by significant spikes at lags that are multiples of the seasonal period in both ACF and PACF plots indicating the presence of long-term, recurring patterns in the time series data (monthly retail sales data showing holiday seasonality)
• To distinguish between seasonal and non-seasonal components, analyze the ACF and PACF plots simultaneously
  • Identify significant spikes at short lags (non-seasonal) and at lags that are multiples of the seasonal period (seasonal)
  • Consider the magnitude and pattern of the spikes to determine the relative importance of seasonal and non-seasonal components (strong seasonal spikes with weak non-seasonal spikes suggest a predominantly seasonal time series)