Exponential Smoothing

Exponential Smoothing Equations

Exponential smoothing was initially introduced in the statistical literature without considering the past work done by Robert Goodell Brown in 1956 and then further expanded by Charles C. Holt in 1957. Exponential smoothing is a reliable principle for smoothing time series data through the exponential window function. The controlling input of the exponential smoothing calculation is stated as the smoothing factor or the smoothing constant.

Forecast of the weighted averages of past observations are introduced using exponential smoothing methods, with the weights breaking down exponentially as the observations get formed. In other words, the more the latest the observation the higher the corresponding weight. 

As we are aware of the simple moving average the past observations are weighted equally, exponential functions are used to assign exponentially decreasing weights over time. It can be easily applied for making determinations on the basis of prior assumptions by the user, such as seasonality. Exponential smoothing is primarily used for time-series data analysis.

Exponential Smoothing Formula

The exponential smoothing formula is derived by:

st = θxt+(1 – θ)st-1= st-1+ θ(xt – st-1)

Here,

st is a former smoothed statistic, it is the simple weighted average of present observation xt

st-1 is former smoothed statistic

θ is smoothing factor of data; 0 < θ < 1

t is time period

If the value of the smoothing factor is greater, then the level of smoothing will be minimized. Value of α nearer to 1 minimum smoothing effect and offer higher weights to recent changes in the data, while the value of θ nearer to zero has higher smoothing effect and is less responsive to recent changes.

There is no precise method of choosing θ. Accurate factors are selected on the basis of the statistician's judgments or else a statistical technique may be used to optimize the value of θ. For example, the method of least squares can be used to estimate the value of θ for which the sum of the quantities is diminished.

Exponential Smoothing Methods

The three exponential smoothing methods to determine exponential smoothing. They are:

1. Simple or single exponential smoothing

2. Double exponential smoothing

3. Triple exponential smoothing

Single Exponential Smoothing

If the data which is obtained has no trend and no seasonal pattern, then the single exponential smoothing method for forecasting the time series is primarily used. This method makes use of weighted moving averages with exponentially decreasing weights.

The single exponential smoothing method formula is given by:

st = θxt+(1 – θ)st-1 = st-1 + θ(xt – st-1)

Double Exponential Smoothing

The double exponential smoothing method is also known as Holt's trend corrected or second-order exponential smoothing. This method is primarily used to forecast the time series when the data has a linear trend and no seasonal pattern. The motive of double exponential smoothing is to introduce a term considering the possibility of a series indicating some form of trend. This slope component is itself reformed through exponential smoothing.

The double exponential smoothing formula is derived by:

S1 = y1

B1 = y1-y0

For t>1,

st = θyt + (1 – θ)(st-1 + bt-1)

βt = β(st – st-1) + (1 – β)bt-1

Here,

St is smoothed statistic, it is the simple weighted average of present observation yt

st-1 = former smoothed statistic

θ = smoothing factor of data; 0 < θ < 1

t = time period

bt = accurate estimation of trend at time t

β = trend smoothing factor; 0 < β <1

Triple Exponential Smoothing

In the triple exponential smoothing method, exponential smoothing is used thrice. This method is primarily used to forecast the time series when the data has both linear trend and seasonal patterns.This method is also known as holt-Winters exponential smoothing.

The triple exponential smoothing formula is derived by:

s\[_{0}\] = x\[_{0}\]

s\[_{t}\] = α\[\frac{x_{t}}{c_{t-L}}\] + (1 - α)(s\[_{t-1}\] + b\[_{t-1}\])

b\[_{t}\] = β(s\[_{t}\] - s\[_{t-1}\] + (1 - β)b\[_{t-1}\]

c\[_{t}\] =  γ\[\frac{x_{t}}{s_{t}}\] + (1 - γ)c\[_{t-L}\]

Here,

st = smoothed statistic, it is the simple weighted average of present observation xt

st-1 = previous smoothed statistic

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

t = time period

bt = accurate estimation  of trend at time t

β = trend smoothing factor; 0 < β <1

ct = sequence of seasonal error-free factors at time t

γ = seasonal variation smoothing factor; 0 < γ < 1

Solved Examples

1. The Sales of Books in a Bookstall for the Last 10 Months is Given Below in Tabulated Form. Calculate the Simple Exponential Smoothing Estimating α =0.3 for the Below Data.

Solution

Quiz Time

1. The Use of Smoothing Technique is Accurate When

1. The primary source of variation is random behavior

2. Seasonality is includes

3. Data exhibit a strong trend

4. All the above are accurate

2.  Times Series Data is Classified in 

1. Two components

2. Three components

3. Four components

4. Five components