A Tutorial on Exponential Smoothing and its Types

Introduction 

 

A widely preferred class of statistical techniques and procedures for discrete time series data, exponential smoothing is used to forecast the immediate future.

 

This method supports time series data with seasonal components, or say, systematic trends where it used past observations to make anticipations. 

 

Exponential smoothing is also considered as the peers, or alternate to the famous Box-Jenkins ARIMA class of methods for time series forecasting. Combinedly, the methods are indicated as ETS models, relating to the explicit modelling of Errors, Trend and Seasonality. 

 

Let us explore blog deeply with; 

 

1. What is exponential smoothing?

2. 3 Types of Exponential Smoothing

3. Key Takeaways

 

 

What is exponential smoothing?

 

Exponential Smoothing is an elementary and pragmatic technique used for forecasting where the forecast is made through the exponentially weighted average of prior observations. 

 

While the largest weight is provided to present observations, less weighted are projected to immediately preceding observations, more less weighted to the observation earlier to that, and so on such that weighted values follow/ reflect exponential decay in terms of influence of past data. 

 

“Forecasts produced using exponential smoothing methods are weighted averages of past observations, with the weights decaying exponentially as the observations get older. In other words, the more recent the observation the higher the associated weight.” - Forecasting: principles and practice

 

In its simplest form, an exponential smoothing of time series data allocates the exponentially decaying weights from newest to oldest observations, ie. analyzing data from a specific period of time via providing more importance to recent data and less importance to former data. This method produces “smoothed data”, the data that has a noise removed, and allows trends and patterns to be more clearly visible.   

 

The essential aim of exponential smoothing is to make original series smooth in the same ways as the moving average does and make use of smoothed data for forecasting futures values of the variable of interest. 

 

Over the half of a century, exponential smoothing is widely used in forecasting applications at the strategic, tactical and operational level, such that;

 

• At strategic level: the forecasting method is deployed for planning on investment & growth and the impact of innovations

• At tactical level: the forecasting method is implemented for deriving expenditure, inventory concern and customer satisfaction.

• At operational level: The forecasting method is used to make targets, predicting qualities and affirmation with standards.  

 

3 Types of Exponential Smoothing

 

Broadly, there are three types of exponential smoothing techniques that rely on trends and seasonality. They are;

 

1. Simple Exponential Smoothing (SES)

 

SES is used for time series anticipation when the data, in particular, doesn’t support any of the following;

 

1. Trend: A slope either inclined upward or downward.

2. Seasonality: Exhibiting a specific pattern due to seasonal factors like hours, days, years, etc.

 

Single exponential smoothing only estimates the level component.

 

SES works on weighted averages (the average of previous level and current observation). Again, largest weights are connected with recent observations and smallest weights are connected with former observations.

 

The weight of each parameter, or decrease in weight is always determined by smoothing parameter, called as  𝜶 (alpha - single parameter/hyperparameter).

 

The value of 𝜶(alpha) lies between 0 to 1 such that;

 

• 𝜶(alpha)=0: signifies that future forecasted values are the average of historical data (giving more weights to historical data) 

• 𝜶(alpha)=1: signifies that future forecast values are the results of the recent observation (giving  more weights to recent observations).

 

A value close to 1 indicates fast learning (that is, only the most recent values influence the forecasts), whereas a value close to 0 indicates slow learning (past observations have a large influence on forecasts).- Source: Practical Time Series Forecasting in R.

 

Hyperparamter for single exponential smoothing is

 

• Alpha: Smoothing factor for the level

 

The formula for the single exponential smoothing is given by:

Where 𝜶 is smoothing parameter, and t is given time period.

 

 

2. Double Exponential Smoothing (DES)

 

DES adds support particularly for trends in the univariate time series. Combined with the additive trends, it is conventionally referred to as Holt’s linear trend model. The name is derived from the name of developer of the method Charles Holt.

 

This method aids changing trends with time in different ways, either additively or multiplicatively, counted on if the trend is linear or exponential correspondingly, i.e, 

 

• Additive Trend: DES with a linear trend

• Multiplicative Trend: DES with an exponential trend

 

To control/determine smoothing factor for a particular level, alpha parameter is used, also an additional smoothing factor is introduced to control the decay of the impact of trends changes, known as beta.

 

When the forecasts have longer range or multi forecasting steps, the trend approaches to become unrealistic such that assisting in dampening the trend over time.

 

Here dampening implies the decreasing the size of the trend over future time steps down to a straight line (no trend).  

 

Dampening the trend takes place the same as modelling the trend, in particular with additively or multiplicatively for  linear or exponential dampening effect respectively. In this case, a damping coefficient phi is used to determine and control the dampening rate, i.e,

 

• Additive Dampening: Make a damped trend linear.

• Multiplicative Dampening: Make a dampened trend exponential.

 

Hyperparameters are following;

 

• Alpha: Level smoothing factor 

• Beta: Trend smoothing factor 

• Trend type: Multiplicative or Additive 

• Dampening type: Multiplicative or Additive

• Phi: Damping coefficient

 

The formulas for double exponential smoothing are given by:

Where,

 

• S_t= smoothed statistic, it is the simple weighted average of recent observation x_t

• S_(t-1) = previous smoothed statistic

• α = smoothing factor of data; 0 < α < 1

• t = time period

• b_t = best estimate of trend at time t

• β= trend smoothing factor; 0 < β <1

 

 

3. Triple Exponential Smoothing (TES)

 

TES explicitly adds support to the univariate time series for seasonality; it is also referred to as Holt-Winters Exponential Smoothing on the name of two contributors Charles Holt and Peter Winters.  

 

The Holt-Winters exponential smoothing model permits the level, trend and seasonality patterns to change over time as it is an adaptive method.

 

Beside the two smoothing factors, alpha and beta, an additional new factor is introduced, called gamma in order to control/determine the impact on the seasonal element.

 

In correspondence with the trend, seasonality can be modeled in the particular of additive or multiplicative process for the linear and exponential variation in the seasonality. 

 

• Additive Seasonality: TES with a linear seasonality.

• Multiplicative Seasonality: TES with an exponential seasonality.

 

Being an advanced variation of exponential smoothing, TES can construct single and double exponential smoothing models through configuration. In addition to that, in order to make and ensure the accuracy of seasonality models, one should specify the number of time steps in a seasonal period. 

 

Hyperparameters are following:

 

• Alpha: Smoothing factor for the level.

• Beta: Smoothing factor for the trend.

• Gamma: Smoothing factor for the seasonality.

• Trend Type: Additive or multiplicative.

• Dampen Type: Additive or multiplicative.

• Phi: Damping coefficient.

• Seasonality Type: Additive or multiplicative.

• Period: Time steps in seasonal period

 

The formulas for triple exponential smoothing are given by:

where,

• S_t= smoothed statistic, it is the simple weighted average of recent observation x₁

• S_(t-1) = previous smoothed statistic

• α = smoothing factor of data; 0 < α < 1

• t = time period

• b_t = best estimate of trend at time t

• β = trend smoothing factor; 0 < β <1

• c_t= sequence of seasonal correction factor at the time t

• γ = seasonal change smoothing factor:  0 < γ < 1 (Source)

 

 

Key Takeaways

 

Below are some key points to be considered for exponential smoothing;

 

1. These methods are the family of traditional forecasting algorithms that work efficiently when time series data exhibit a clear and seasonal pattern.

2. Exponential smoothing has the elements as Error, Trend, and season that can be either additively or multiplicatively.

3. In order to find the trend, season or error, time series decomposition is an accurate approach as it makes the plot of each component as a distinct subplot.

4. As the trends reflect upward or downward behavior (tendency), applied as additively, and if varies exponentially, it is multiplicative.

5. Similarly, if the magnitude of seasonal trends changes linearly, it is additive, and if varies exponentially, it is multiplicatively.