Activity19

Mastering ACF & PACF Diagnostics

Objective: Identify AR(p)/MA(q) signatures through autocorrelation patterns

The ACF reveals how auto correlations persist over lags, indicating the need for differencing (if autocorrelations decay slowly) or the presence of seasonality (periodic spikes). The PACF pinpoints the order of AR terms by showing where partial correlations become negligible. Together, ACF and PACF help diagnose appropriate ARIMA or seasonal ARIMA structures.

1.1 ACF Decay Patterns with gtemp_both

# Trend-dominated series
gtemp_both |> 
  as_tsibble() |>
  ACF(lag_max = 50) |> 
  autoplot() + 
  labs(subtitle = "Slow ACF decay indicates non-stationarity")

# After differencing
gtemp_both |> 
  as_tsibble() |>
  mutate(dTemp = difference(value)) |> 
  ACF(dTemp) |> 
  autoplot() +
  labs(subtitle = "ACF cuts off after lag 1 -> MA(1) signature")

Key Concept: Persistent ACF decay suggests differencing needed. Sharp cutoff at lag 1 after differencing implies MA(1) component.

1.2 PACF for AR Order Identification with pelt

# Lynx population analysis
pelt |> 
  as_tsibble() |>
  PACF(Lynx, lag_max = 5) |> 
  autoplot() +
  labs(title = "PACF cuts off at lag 2 -> AR(2) process")

Equation: AR(p): \(X_t = \phi_1X_{t-1} + ... + \phi_pX_{t-p} + \epsilon_t\)

PACF spikes within first p lags indicate AR order.

1.3 Seasonal ACF in vic_elec

vic_elec |> 
  filter(year(Time)==2013) |> 
  ACF(Demand, lag_max=200) |> 
  autoplot() +
  labs(subtitle = "Spikes at 48 lags = daily seasonality")

Interpretation: ACF peaks at multiples of fundamental period (48 half-hours = 1 day) reveal seasonality.

Lab Activity:

  1. Load the aus_production dataset from fpp3.
  1. Choose the ‘Gas’ time series. Plot its ACF and PACF to identify any AR/MA components or seasonal effects.
  1. Apply differencing if needed, and re-check the ACF/PACF to refine your model choice.