How Smoothing works in Exponential Smoothing? (explained with calculations)

To understand how exponential smoothing works in the context of the Croston method, let's break down the formula and explain the smoothing process.

Exponential Smoothing Formula

In the Croston method, exponential smoothing is applied separately to the non-zero demand sizes and the inter-demand intervals. The general formula for exponential smoothing is:

{X_hat}_t = alpha * X_t + (1-alpha) * {X_hat}_{t-1}        

where:

- "{X_hat}_t" is the smoothed value at time "t".

- "X_t" is the actual observed value at time "t" (demand size or inter-demand interval).

- "{X_hat}_{t-1}" is the smoothed value from the previous period.

- "alpha" is the smoothing parameter, which lies between 0 and 1.

How Smoothing Happens

The essence of exponential smoothing is to give more weight to recent observations while still considering past data, albeit with decreasing importance.

Breaking Down the Formula

1. Weight Assignment:

- The parameter "alpha" determines the weight given to the most recent observation "X_t".

- The term (1-alpha) determines the weight given to the previous smoothed value "{X_hat}_{t-1}".

2. Combining Recent and Past Data:

- By multiplying "alpha" with the current period's observed value "X_t", you emphasize the latest data.

- By multiplying "(1-alpha)" with the previous smoothed value "{X_hat}_{t-1}", you retain some influence from the past data, which smooths out fluctuations over time.

3. Recursive Nature:

- This process is recursive, meaning that each smoothed value depends on the previous smoothed value. As a result, the smoothed value "{X_hat}_t" is a weighted average of all past observations, but with exponentially decreasing weights for older observations.

Example of Smoothing Process in Croston

Let's consider a simplified example with "alpha" = 0.5.

- Step-by-Step Calculation:

- Week 2 (First Non-Zero Demand):

- Observed demand: "X_2" = 3

- Initial smoothed demand "{D_hat}_2" = 3 (since it's the first observation).

- Week 5 (Second Non-Zero Demand):

- Observed demand: "X_5" = 5

- Smoothed demand: "{D_hat}_5" = 0.5 * 5 + (1-0.5) * 3 = 4

- Week 8 (Third Non-Zero Demand):

- Observed demand: "X_8" = 2

- Smoothed demand: "{D_hat}_8" = 0.5 * 2 + (1-0.5) * 4 = 3

Explanation of Smoothing

- Week 5 Calculation:

- The observed demand is 5, which is more than the previous smoothed value (3).

- The new smoothed value "{D_hat}_5" is calculated as the average of the current demand (5) and the previous smoothed value (3), weighted by "alpha".

- Since "alpha" = 0.5, it gives equal weight to the current observation and the previous smoothed value, resulting in a new smoothed value of 4.

- Week 8 Calculation:

- The observed demand is 2, which is less than the previous smoothed value (4).

- The smoothing process takes half of the current observation (2) and half of the previous smoothed value (4), resulting in a new smoothed value of 3.

Why Smoothing Happens

Smoothing happens because the exponential smoothing formula balances between the latest observed value and the historical smoothed value. The weight "alpha" controls this balance:

- If "alpha" is closer to 1, more weight is given to the most recent observation, making the smoothed value respond quickly to changes but also more volatile.

- If "alpha" is closer to 0, more weight is given to the previous smoothed value, making the smoothed value more stable and less sensitive to recent changes.

Thus, exponential smoothing allows you to create a smoothed series that retains recent information while filtering out some of the noise from random fluctuations. This method is particularly useful in forecasting because it helps maintain a balance between sensitivity to recent data and overall trend stability.

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