Time Series Analysis > Exponential Smoothing

## What is Exponential Smoothing?

**Exponential smoothing **of time series data assigns exponentially decreasing weights for newest to oldest observations. In other words, the older the data, the less priority (“weight”) the data is given; newer data is seen as more relevant and is assigned more weight. Smoothing parameters (smoothing constants) — usually denoted by α — determine the weights for observations.

Exponential smoothing is usually used to make short term forecasts, as longer term forecasts using this technique can be quite unreliable.

**Simple (single) exponential smoothing**uses a weighted moving average with exponentially decreasing weights.**Holt’s trend-corrected double exponential smoothing**is usually more reliable for handling data that shows trends, compared to the single procedure.**Triple exponential smoothing**(also called the Multiplicative Holt-Winters) is usually more reliable for parabolic trends or data that shows trends*and*seasonality..

## 1. Simple Exponential Smoothing

The **basic formula** is:

S_{t} = αy_{t-1} + (1 – α) S_{t-1}

*Where*:

- α = the smoothing constant, a value from 0 to 1. When α is close to zero, smoothing happens more slowly. The best value for α is the one that results in the smallest mean squared error (MSE). There are various ways you can do this, but a popular method is the Levenberg–Marquardt algorithm.
- t = time period.

There are **alternative formulas**. For example, Roberts (1959) replaced y_{t-1} with the current observation, y_{t}. Another formula uses the forecast for the previous period and current period:

*Where*:

- Ft-1 = forecast for the previous period,
- At-1 = Actual demand for the period,
- a = weight (must be between 0 and 1). The closer to zero, the smaller the weight.

**Which formula to use **is usually a moot point, as most exponential smoothing is performed using software. Whichever formula you use though, you’ll have to set an initial observation. This is a judgment call. You could use an average of the first few observations, or you could set the second smoothed value equal to the original observation value to get the ball rolling.

## 2. Double Exponential Smoothing

This method is deemed more reliable for analyzing data that shows a trend. This is a more complicated method which adds a second equation to the procedure:

b

_{t}= γ(S_{t}– S_{t-1}) + (1 – γ)b_{t-1}

*Where*:

- γ is a constant that is chosen with reference to α. Like α it can be chosen through the Levenberg–Marquardt algorithm.

## 3. Triple Exponential Smoothing

If your data shows a trend *and *seasonality, use triple exponential smoothing. In addition to the equations for single and double smoothing, a third equation is used to handle the seasonality aspect:

I_{t} = Β y_{t}/S_{t} + (1-Β)I_{t-L+m}

*Where*:

- y = observation,
- S = smoothed observation,
- b = trend factor,
- I = seasonal index,
- F = forecast
*m*periods ahead, - t = time period.

Like α and γ, the optimal Β minimizes the MSE.

**Next**: Exponential Smoothing in Excel > >

**Reference**:

Roberts, S.W. (1959). “Control chart tests based on geometric moving averages.” Technometrics, I. 239-250.

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