Exponential Smoothing

Exponential smoothing was first suggested in the statistical literature without reference to previous work by Robert Goodell Brown in 1956 and then expanded by Charles C. Holt in 1957. Exponential smoothing is a broadly accurate principle for smoothing time series data using the exponential window function. The controlling input of the exponential smoothing calculation is defined as the smoothing factor or the smoothing constant.

As we know that, in the simple moving average the past observations are weighted equally, exponential functions are used to assign exponentially decreasing weights over time. It is an easily learned and easily applied method for making some determination based on prior assumptions by the user, such as seasonality. Exponential smoothing is generally used for analysis of time-series data.

Exponential Smoothing Formula

The simplest form of an exponential smoothing formula is given by:

st = αxt+(1 – α)st-1= st-1+ α(xt – st-1)

Here,

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

st-1 = previous smoothed statistic

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

t = time period

If the value of smoothing factor is larger, then the level of smoothing will reduce.Value of α close to 1 has less of a smoothing effect and give greater weight to recent changes in the data, while value of α closer to zero has greater smoothing effect and are less responsive to recent changes.

There is no official accurate procedure for choosing α.The statistician’s judgment is used to choose an appropriate factor sometimes. Otherwise, a statistical technique may be used to optimize the value of α. For example, the method of least squares can be used to determine the value of α for which the sum of the quantities is minimized.

Exponential Smoothing Forecasting

Exponential smoothing is generally used to make short term forecasts but longer term forecasts using this technique can be quite unreliable. More recent observations given larger weights byeExponential smoothing methods, and the weights decrease exponentially as the observations become more distant. When the parameters describing the time series are changing slowly over time then these methods are most effective.

Exponential Smoothing Methods

There are three main methods to estimate exponential smoothing. They are:

  • Simple or single exponential smoothing
  • Double exponential smoothing
  • Triple exponential smoothing

Simple or single exponential smoothing

If the data has no trend and no seational pattern, then this method of forecasting the time series is essentially used. This method uses weighted moving averages with exponentially decreasing weights.

The single exponential smoothing formula is given by:

st = αxt+(1 – α)st-1 = st-1 + α(xt – st-1)

Double exponential smoothing

This method is also called as Holt’s trend corrected or second order exponential smoothing. This method is used for forecasting the time series when the data has linear trend and no seasonal pattern. The primary idea behind double exponential smoothing is to introduce a term to take into account the possibility of a series showing some form of trend. This slope component is itself updated through exponential smoothing.

The double exponential smoothing formulas are given by:

S1 = x1

B1 = x1-x0

For t>1,

st = αxt + (1 – α)(st-1 + bt-1)

βt = β(st – st-1) + (1 – β)bt-1

Here,

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

st-1 = previous smoothed statistic

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

t = time period

bt = best estimate of trend at time t

β = trend smoothing factor; 0 < β <1

Triple exponential smoothing

In this method, exponential smoothing applied three times. This method is used for forecasting the time series when the data has both linear trend and seasonal pattern. This method is also called Holt-Winters exponential smoothing.

The triple exponential smoothing formulas are given by:

Triple exponential formula

Here,

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

st-1 = previous smoothed statistic

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

t = time period

bt = best estimate of trend at time t

β = trend smoothing factor; 0 < β <1

ct = sequence of seasonal correction factor at time t

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

Example problem

The sales of a magazine in a stall for the previous 10 months is given below.

Month

Sales

January

30

February

25

March

35

April

25

May

20

June

30

July

35

August

40

September

30

October

45

Calculate the simple exponential smoothing taking α =0.3 for the above data.

Solution:

Month

Sales

Exponential smooth

α =0.3

January

30

30.00

February

25

30.00

March

35

28.50

April

25

30.45

May

20

14.1

June

30

15.87

July

35

20.109

August

40

24.5763

September

30

29.20341

October

45

29.442387

November

34.1096709

Read more:

Exponential Function

Exponents And Powers Class 7

Exponents And Powers For Class 8

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