5.2 Simple exponential smoothing

Simple exponential smoothing is a forecasting technique that gives more weight to recent data points. It's great for short-term predictions when there's no clear trend or seasonality in your data. Think of it as a way to smooth out the bumps in your time series.

The method uses a single parameter, alpha, to balance the importance of new versus old data. By adjusting alpha, you can make your forecasts more responsive to recent changes or more stable over time. It's a simple yet powerful tool in your time series toolbox.

Simple Exponential Smoothing

Concept of exponential smoothing

• Assigns exponentially decreasing weights to past observations in a time series
  • Gives higher weights to more recent observations compared to older ones
  • Weights decrease exponentially as observations become older (e.g., 0.9, 0.81, 0.729)
• Suitable for time series data lacking clear trends or seasonal patterns
• Simple and effective method for generating short-term forecasts (e.g., daily sales, weekly website traffic)

Application of simple exponential smoothing

• Single-parameter smoothing method denoted as SES
• Forecast for the next period is a weighted average of the current observation and the previous forecast
  • $\hat{y}_{t+1} = \alpha y_t + (1 - \alpha) \hat{y}_t$
    • $\hat{y}_{t+1}$: forecast for the next period
    • $y_t$: actual value at time $t$
    • $\hat{y}_t$: forecast for time $t$
    • $\alpha$: smoothing parameter between 0 and 1 (e.g., 0.2, 0.5, 0.8)
• Initialization: set the first forecast equal to the first observed value in the series
• Appropriate for time series without clear trends or seasonality (e.g., stationary data)

Impact of smoothing parameter

• Smoothing parameter $\alpha$ determines the weight given to the most recent observation
  • Higher $\alpha$ values closer to 1 give more weight to recent observations
    • Results in faster adaptation to changes in the level of the series (e.g., sudden shifts)
  • Lower $\alpha$ values closer to 0 give more weight to past observations
    • Leads to slower adaptation and smoother forecasts (e.g., less sensitive to noise)
• Optimal $\alpha$ value can be estimated by minimizing accuracy measures like sum of squared forecast errors (SSE)
• Choice of $\alpha$ depends on the characteristics of the time series and desired forecast responsiveness (e.g., stable vs. volatile data)

Accuracy of exponential smoothing forecasts

• Forecast accuracy assessed using various measures:
  1. Mean Absolute Error (MAE): average absolute difference between actual values and forecasts
  2. Mean Squared Error (MSE): average squared difference between actual values and forecasts
  3. Mean Absolute Percentage Error (MAPE): average absolute percentage difference between actual values and forecasts
• These measures quantify the average magnitude of forecast errors
  • Lower values indicate better forecast accuracy (e.g., MAE of 10 vs. 20)
• Comparing accuracy measures of SES with other forecasting methods helps determine the most suitable method for a given time series (e.g., SES vs. moving average)