Exponential smoothing

The exponential smoothing ( English exponential smoothing ) is a method of time series analysis for short-term forecast from a sample with periodic historical data. Due to exponential smoothing, these are given a higher weighting with increasing topicality. The aging of the measured values is compensated for, the reliability of the prediction is improved, especially when calculating requirements, stocks and orders. A suitable database with measured values from market analyzes is essential.

The exponential smoothing is mainly used if the time series does not reveal any systematic pattern such as linear increase or the like. The procedure is used in warehousing , for example , when it comes to determining the requirements for an item to be ordered in the coming year. With exponential smoothing, for example, the Swiss Army had good success in determining the rifles required in the following year.

Exponential smoothing is used to determine forecast values . It is based on the assumption that the current time series value is always influenced by the past values, the influence weakening the further the value is in the past. By weighting the time series values with a smoothing factor, strong fluctuations of individual observed values are distributed over the estimated time series.

Formal model

A time series with the observations at the points in time is given . At the point in time , a smooth estimated value is calculated, which results as a weighted average from the current time series value and the estimated value of the previous period . The weighting is determined by the smoothing factor , where must be. You get ${\ displaystyle y_ {1}, y_ {2}, \ dotsc, y_ {t}, \ dotsc}$ ${\ displaystyle t \ in \ mathbb {N}}$ ${\ displaystyle t}$ ${\ displaystyle y_ {t}}$ ${\ displaystyle y_ {t} ^ {*}}$ ${\ displaystyle y_ {t}}$ ${\ displaystyle y_ {t-1} ^ {*}}$ ${\ displaystyle \ alpha}$ ${\ displaystyle 0 \ leq \ alpha \ leq 1}$

{\ displaystyle y_ {t} ^ {*} = \ alpha \ cdot y_ {t} + (1- \ alpha) \ cdot y_ {t-1} ^ {*}}

.

For the forecast value is the same as the measured value (no smoothing), for the forecast remains unchanged (smoothing to a parallel to the x-axis). ${\ displaystyle \ alpha = 1}$ ${\ displaystyle \ alpha = 0}$

The time series is built up recursively . Theoretically the current time series is already infinitely long at the point in time . For the practical determination of the smoothed value, however, a start value will be specified and from then on the smoothed time series will be determined. ${\ displaystyle t}$ ${\ displaystyle y_ {0} ^ {*}}$

If one now builds up the smoothed time series starting at , ${\ displaystyle y_ {0} ^ {*}}$

{\ displaystyle y_ {1} ^ {*} = \ alpha y_ {1} + (1- \ alpha) y_ {0} ^ {*} \ ;,}

{\ displaystyle y_ {2} ^ {*} = \ alpha y_ {2} + (1- \ alpha) y_ {1} ^ {*} \ ;,}

{\ displaystyle y_ {3} ^ {*} = \ alpha y_ {3} + (1- \ alpha) y_ {2} ^ {*} \ ;,}

{\ displaystyle \ ldots}

one gets, if one resolves the recursion,

{\ displaystyle y_ {t} ^ {*} = \ alpha y_ {t} + \ alpha (1- \ alpha) y_ {t-1} + \ alpha (1- \ alpha) ^ {2} y_ {t- 2} + \ dotsb + \ alpha (1- \ alpha) ^ {t-1} y_ {1} + (1- \ alpha) ^ {t} y_ {0} ^ {*}}

.

You can see how the influences of the past disappear more and more. ${\ displaystyle \ alpha <1}$

α is therefore also called the present factor. The larger the calculation, the stronger the reference to the more recent values. ${\ displaystyle \ alpha}$

The estimated value then provides the forecast value for the point in time . If the observed time series value is currently available, the prognosis for the next period can be made. ${\ displaystyle y_ {t} ^ {*}}$ ${\ displaystyle t + 1}$ ${\ displaystyle y_ {t}}$

Example for the exponentially smoothed DAX

Graph of the smoothed DAX values

An exponential smoothing is to be calculated with the monthly averages of the DAX share index for the months January 1977 to August 1978. The data are available along with the smoothed time series values:

DAX-Werte und ihre exponentielle Glättung (α = 0,3)
Monat Zeitpunkt t DAX Vt Glättung y*t
1977 Jan 0 512,3 512,3
1977 Feb 1 496,2 507,5
1977 Mrz 2 509,8 508,2
1977 Apr 3 551,9 521,3
1977 Mai 4 539,9 526,9
1977 Jun 5 524,9 526,3
1977 Jul 6 530,3 527,5
1977 Aug 7 540,9 531,5
1977 Sep 8 541,3 534,4
1977 Okt 9 554,2 540,4
1977 Nov 10 557,5 545,5
1977 Dez 11 549,3 546,7
1978 Jan 12 549,4 547,5
1978 Feb 13 552,9 549,1
1978 Mrz 14 549,7 549,3
1978 Apr 15 532,1 544,1
1978 Mai 16 545,5 544,5
1978 Jun 17 553,0 547,1
1978 Jul 18 582,1 557,6
1978 Aug 19 583,1 565,2

The first value is taken as the starting value with 512.3 . We use a smoothing factor α = 0.3. ${\ displaystyle y_ {0} ^ {*}}$

The result is the smoothed values

{\ displaystyle y_ {1} ^ {*} = 0 {,} 3 \ cdot 496 {,} 2 + 0 {,} 7 \ ​​cdot 512 {,} 3 = 507 {,} 5 \ ;,}

{\ displaystyle y_ {2} ^ {*} = 0 {,} 3 \ cdot 509 {,} 8 + 0 {,} 7 \ ​​cdot 507 {,} 5 = 508 {,} 2 \ ;,}

{\ displaystyle y_ {3} ^ {*} = 0 {,} 3 \ cdot 551 {,} 9 + 0 {,} 7 \ ​​cdot 508 {,} 2 = 521 {,} 3 \ ;,}

...

The estimate is now the forecast value for period 2 and so on. ${\ displaystyle y_ {1} ^ {*}}$

The graphic shows the smoothing for α = 0.3 and α = 0.7. You can see that the smaller smoothing factor smooths the time series more, because the current value is now only included with a weight of 0.3, whereas the “mean” past values are still taken into account with 0.7.

Remarks

Exponential smoothing is a recommended procedure if the time series values make a chaotic impression and do not reveal any systematic nature. However, if there are observations that contain a trend , i. H. which are constantly rising or falling, the smoothed values "drag" behind, as can also be seen in the graphic. You can clearly see how between t = 7 and t = 12 the estimated values are always systematically below the observed values. One can remedy this problem with the so-called "double exponential smoothing".

One advantage of exponential smoothing is that the calculation is in the form

{\ displaystyle y_ {t} ^ {*} = \ alpha \ cdot (y_ {t} -y_ {t-1} ^ {*}) + y_ {t-1} ^ {*}}

Only one multiplication, addition and subtraction and requires only a stored value: . This filtering is therefore of interest for embedded systems with little memory and computing power. ${\ displaystyle y_ {t-1} ^ {*}}$

In the case of a Gaussian-distributed input variable, the scatter decreases with a simple moving average over values . The same attenuation of the scatter is obtained with exponential smoothing with . ${\ displaystyle N}$ ${\ displaystyle {\ frac {1} {\ sqrt {N}}}}$ ${\ displaystyle \ alpha = 2 / (N + 1)}$

Smoothing process

A distinction is made between first-order exponential smoothing and second-order exponential smoothing. The exponential smoothing of the 1st order is described here. The variant of the 2nd order takes a trend in the time series into account.