Exponential smoothing

The exponential smoothing (English exponential smoothing ) is a method of time series analysis for short-term forecasting from a sample with periodic historical data. These obtained by exponential smoothing with increasing actuality a higher weighting. The aging of the measured values ​​is balanced, the safety of the prediction improved, particularly in the demand, inventory and order invoice. Fundamental is a suitable data base with measuring values ​​from market analysis.

The exponential smoothing is mainly used when the time series reveals no systematic pattern as linear increase or the like. The method is used for example in storage, if it comes about, to identify the needs of an article to be ordered in the coming year. Thus, the Swiss Army has done with the exponential smoothing good success in the determination of the required guns in the following year.

So you determined using the exponential smoothing forecast values. It starts from the recognition that the current time series value is always influenced by the past values, weakens the influence, the further the value is in the past. By weighting the time series values ​​with a smoothing factor of strong eruptions of individual observed values ​​on the estimated time series are distributed.

Formal model

Given a time series with the observations of the time points. At the time calculated for a smoothed estimate of the weighted average results from the current time series value and the estimated value of the previous period. The weighting is determined by the smoothing factor must be said. obtained

For the predictive value is equal to the measured value ( no smoothing ) for the prediction remains unchanged (smoothing to a line parallel to the x -axis).

The series builds so on recursively. Theoretically, the current time series at time is already infinite. For the practical calculation of the smoothed value However, you will specify a start value and determine from that point on the smoothed time series.

It Establishes now starting to at the smoothed time series,

Obtained when dissolving the recursion,

One sees how are disappearing because of the influences of the past.

α is therefore also called the presence factor. The bigger, the stronger is the calculation of the reference to the more recent values ​​.

The estimate then provides the forecast value for the date. Thus, if before the observed time series value of the present, the forecast for the next period can be made.

Example of the exponentially smoothed DAX

It is intended to August 1978, an exponential smoothing are calculated using the average monthly values ​​of the DAX index for January 1977. The data are stored together with the smoothed time series values ​​before:

DAX values ​​and their exponential smoothing ( α = 0.3 ) Month time t DAX Vt smoothing y * t 1977 January 0 512.3 512.3 1977 Jan 1 496.2 507.5 1977 Mar 2 509.8 508.2 1977 April 3 551.9 521.3 1977 May 4 539.9 526.9 1977 June 5 524.9 526.3 Jul 6, 1977 530.3 527.5 Aug 7, 1977 540.9 531.5 Sep 8, 1977 541.3 534.4 1977 October 9 554.2 540.4 November 10 1977 557.5 545.5 December 11 1977 549.34 546.7 January 12 1978 549.4 547.5 1978 February 13 552.9 549.1 1978 Mar 14 549.7 549.3 1978 April 15 532.1 544.1 May 16 1978 545.5 544.5 1978 June 17 553.0 547.1 1978 July 18 582.1 557.6 1978 August 19 583.1 565.2 The first value is taken with 512.3 as the starting value. We use a smoothing factor α = 0.3.

It results in the smoothed values

The estimate is now the forecast value for period 2, and so on.

The graph shows the smoothing for α = 0.3 and α = 0.7. It is seen that the smaller smoothing factor smooths the time series more, because here the current value is now only one with a weight of 0.3, while the "medium" are past values ​​continue to be included with 0.7.

Note

The exponential smoothing is then a recommended procedure when the time series values ​​look chaotic and show no systematics. However, there are observations before that include a trend, ie the rise or fall continuously " drag ", the smoothed values ​​" behind ", as you can read in the image even partially. So you can clearly see how = 7 and t = 12, the estimates always lie between t systematically the observed values. You can remedy this problem with the so -called "double exponential smoothing ".

Smoothing methods

Differences are the exponential smoothing first -order and second -order exponential smoothing. Described here is the exponential smoothing 1st order. The variant of the second scheme, in a trend in the time series.

Exponential Smoothing ( Materials Management)

The special feature of exponential smoothing is the fact that the calculated predictive value compared with the actual costs of fuel consumption and the resulting deviation in a desired and customized way is taken into account by using the smoothing factor alpha:

• 100% with factor 1
• Not with factor 0

Formula for the exponential smoothing

Vn = forecast demand for new period

Va = forecast demand for old period

Vt = actual consumption for old period

α = smoothing factor

Example: Vn = 100 0.5 * ( 110-100 ) = 100 0.5 * 10 = 100 5 = 105