Seasonality and Trends in Time Series

To handle seasonality and trends in time series, you usually need to make the data stationary (Stationary Time Series) (constant mean and variance).

Trends (long-term upward or downward movement)

• Differencing: Subtract the previous value from the current value: $y_t^{\prime }=y_t-y_{t-1}$ Removes linear trends. Can apply higher-order differencing if needed.
• Detrending with regression: Fit a line or curve to the data and use residuals.
• Transformation: Log transformation or square root to stabilize growth rates.

Seasonality (repeating patterns over fixed periods)

• Seasonal differencing: Subtract the value from the same season in the past: $y_t^{\prime }=y_t-y_{t-s}$ where $s$ = seasonal period (e.g., 12 for monthly data with yearly seasonality).

• Decomposition methods:

  • Additive: $y_t=T_t+S_t+R_t$
  • Multiplicative: $y_t=T_t\times S_t\times R_t$ Use STL (Seasonal-Trend decomposition) or classical decomposition to separate and model components.

• Seasonal dummy variables: Add month/quarter indicators as features (especially for machine learning models).

Combined Trend + Seasonality

• Apply seasonal differencing + regular differencing if both are present (this is what SARIMA does).
• For machine learning approaches (e.g., Random Forest, XGBoost), create lagged features, rolling averages, and seasonal indicators.

Practical considerations

• Check stationarity: Use plots, ACF/PACF, or statistical tests (ADF test).
• Avoid over-differencing: Too many differences can destroy useful structure.
• Model choice:
  • Classical: SARIMA, Exponential Smoothing Forecasting
  • Decomposition + ML: Random Forest for Time Series, Gradient Boosting