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L14 Seasonality and SARIMA Model

  • Types of seasonality

    1. Seasonality is deterministic
      • Last period of seasonality is \(s\)
      • \(S_{t}^{(s)} = S_{t+ks}^{(s)}\), \(k = \pm_{1}, \pm 2 \pm 3,\dots\)
      • We can determine the trend in the future as it just repeats.
      • \(S_{t}^{12} = S_{t+12}^{(12)}\)
    2. Seasonality evolves over time as a stationary process
      • \(S_{t}^{(s)} = \mu^{(s)}+v_{t}\) where \(E(v_{t})= 0\)
      • stationary factor is oscillating around \(\mu^{(s)}\), the deterministic effect depending on \(s\)
      • while \(v_{t}\) is a stationary process bringing variability in the stationary factor \(S_{t}^{(s)}\) and is time dependent and not season dependent.
    3. Seasonality evolves over time as a stationary process
      • \(S_{t}^{(s)}\) may follow a non-stationary process.
      • E.g. random walk, \(S_{t}^{(s)} = S_{t-s}^{(s)}+v_{t}\) where \(E(v_{t})= 0\)

  • Applying a seasonal difference corrects for seasonality in all cases.

\(\text{SARIMA}\) model

  • We use seasonal differencing \(Y_{t} - Y_{t-s}\), with lag \(s\).
\[ Z_{t} = (1-B^S)^D (1-B)^d Y_{t} \]
  • \(D\) is number of seasonal differences (usually 0 or 1) → \((1-B^S)\)
  • \(d\) is number of times needed to take the regular difference → \((1-B)\)

The SARIMA model takes the following structure

\[ \Phi_{P}(B^S)\phi_{p}(B)(1-B^S)^D(1-B)^dY_{t} = \Theta_{Q}(B^S)\theta_{q}(B)e_{t} \]

there are four parts

  • \(\Phi_{P}(B^S)\): seasonal AR op. of order \(P\)
  • \(\phi_{p}(B)\): regular AR op. of order \(p\)
  • \(\Theta_{Q}(B^S)\): seasonal MA op. of order \(Q\)
  • \(\theta_{q}(B)\): regular MA op. of order \(q\)
\[ \text{SARIMA}(p,d,q) \times (P,D,Q)_{S} \]
  • Two individual triplets of orders.

Pros & Cons of SARIMA

  • Pros
    • Easy to understand: simplicity and interpretibility
    • Limited variables: fewer parameters to estimate
  • Cons
    • Exponential time complexity: as \(p\) and \(q\) increases
    • Complex data: no optional solution for \(p\) and \(q\)
    • Amount of data required is considerable.

Use Cases of ARIMA and SARIMA Models

  1. Forecast the Dynamics of COVID-19 Epidemic in India
  2. Prediction of Daily and Monthly Average Global Solar Radiation (Seoul, South Korea)
  3. Disease management (spread of a disease, requirement of beds in Singapore)

  4. Forecasting of demand (food company)

  5. In ARIMA, you can capture the trend part.

  6. In case seasonality is also present, use SARIMA.