Prediction algorithm--exponential smoothing method

Directory

? 1. Exponential smoothing definition and formula

? 2. One-time exponential smoothing

? 32-Times Exponential smoothing

? 4. Three-times exponential smoothing

? 5 determination of exponential smoothing coefficient α

1. Definition and formula of exponential smoothing

Background: Exponential smoothing is proposed by Brown, and he believes that the time series is stable or regular, so the time series can be reasonably delayed; he believes that the recent past has, to some extent, continued in the future, so that the larger weights are placed on the most recent data.

The basic principle: exponential smoothing is one of the moving averages, which is characterized by the different weights of the past observations, that is, the weights of the more recent observations are larger than the weights of the long-term observation values. The exponential smoothing method is divided into one exponential smoothing method, two exponential smoothing method and three exponential smoothing method according to the different smoothing times. But the basic idea is that the predicted value is the weighted sum of the previous observations, and different weights are given to different data, the new data is given a larger weighting, and the old data is given a smaller weighting.

method Application: Exponential smoothing method is a common method in production prediction. Also used in short and medium-term economic development trend prediction, in all forecasting methods, exponential smoothing is the most used one.

The basic formula of exponential smoothing method: st=ayt+ (1-a) St-1 type,

st--the smoothed value of time t;

yt--the actual value of the time t;

st-1--the smoothed value of time t-1;

a--smoothing constant with a value range of [0,1]

According to the number of smoothing, exponential smoothing is divided into: one exponential smoothing method, two exponential smoothing and three exponential smoothing method.

2. One-time exponential smoothing prediction

When there is no obvious trend change in the time series, the exponential smoothing prediction can be used. Its prediction formula is:

yt+1 ' =ayt+ (1-a) yt ' style,

? The predicted value of yt+1 '--t+1 period, i.e. the smoothed value St of this period (t phase);

? The actual value of the yt--t period;

? The predicted value of the YT '--t period, that is, the smoothed value of the previous period St-1.

Example: It is known that the sales of a product in the last 15 months are shown in the following table:

Predict the next month's sales Y16 with a single exponential smoothing value.

In order to analyze the characteristics of the different values of the weighted coefficients a , take a =0.1, a =0.3, a =0.5 to calculate an exponential smoothing value, and set the initial value as the average of the first three data: a = 0.5 of an exponential smoothing value is calculated as an example, there are

The following tables are calculated:

According to the table available time 1 May the corresponding 19.9 26.2 28.1 can predict the sales volume of the 16th month respectively according to the forecast formula.

Take a = 0.5 For example: y16=0.5*29+ (1-0.5) *28.1=28.55 (million units)

The conclusions can be drawn from the above examples.

1) The exponential smoothing method has a smooth effect on the actual sequence, the smaller the weight coefficient (smoothing coefficient) , the stronger the smoothing effect, but the slower response to the actual data.

2) in the linear change part of the real sequence, the degree of lag deviation in the exponential smoothing value sequence decreases with the increase of the weight coefficient (smoothing coefficient) , but when the linear trend of the time series changes, there will still be obvious lag deviation with the one-time exponential smoothing method. Therefore, corrections are also required. The modified method also makes two exponential smoothing on the basis of an exponential smoothing, uses the rule of lag deviation to find the development direction and development trend of the curve, and then establishes the linear trend prediction model, so called two times exponential smoothing method.

3, two exponential smoothing predictions

1) A is a weighted coefficient;

2) The exponential smoothing method has a smooth effect on the actual sequence, the smaller the weight coefficient (smoothing coefficient), the stronger the smoothing effect, but the slower the change of the actual data.

3) in the linear variation of the actual sequence, the degree of lag deviation in the exponential smoothing value sequence decreases with the increase of the weight coefficient (smoothing coefficient); But when the time series changes in the linear trend, there will still be obvious lag deviation with the one-time exponential smoothing method. Therefore, corrections are also required.

4) The modified method is also on the basis of an exponential smoothing and then two times the exponential smoothing, using the law of lag deviation to find the direction and development trend of the curve, and then establish a linear trend prediction model, it is called two exponential smoothing method.

On the basis of an exponential smoothing, the formula for two exponential smoothing is:

? In the formula: St (2)--two times exponential smoothing value of the T -period;

? St (1)--an exponential smoothing value of the T -period;

? St-1 (2)--Two exponential smoothing values for the T-1 period;

? a --weighted coefficients (also known as smoothing coefficients).

Two times exponential smoothing is a method of re-exponential smoothing of an exponential smoothing value. It cannot be predicted separately, and it must be combined with an exponential smoothing method to establish a mathematical model of the prediction and then use the mathematical model to determine the predicted value.

Two exponential smoothing mathematical model:

Example 2: The financial information from 1983 to 1993 is as follows, using exponential smoothing to solve the trend linear equation and predicting the fiscal revenue in 1996

Example 3: It is known that the steel output of a factory for 1978-1998 years is shown in the following table, and the steel output of the plant is forecasted in 1999 and 2000. ( How to achieve smoothing index with Excel )

The following steps are made using the exponential smoothing tool:
Select the data-data Analysis command in the Tools menu, and the Data Analysis dialog box appears.

In the Analysis Tools list box, select the Exponential smoothing tool.

The Exponential Smoothing dialog box appears:

One exponential smoothing setting and output

Two exponential smoothing settings and outputs

Output and calculation of final result

Calculates A and B values based on the two-time smoothed exponential mathematical model.

Get the trend line Prediction model: Y=3994.9+141.2T, which can be calculated as:

y1999=3994.9+141.2*1=4136.14

Y2000=3994.9+141.2*2= 4277.34

4, three exponential smoothing predictions

If the change of time series shows a trend of two curves, it is necessary to use three times exponential smoothing method to forecast. Three times the exponential smoothing is smoothed on the basis of two exponential smoothing, which is calculated as:

the prediction model of three-times exponential smoothing method to be :

Example 4: Sales of some durable goods in China from 1996 to 2006 as shown in the table, try to forecast sales volume in 2007 and 2008.

Three-time Exponential smoothing calculation table:

Solution: The nonlinear increment trend is presented by the actual data series, and the three exponential smoothing prediction method is adopted. The steps to solve are as follows. Determine the initial and weight coefficients of the exponential smoothing (smoothing factor)a. The initial value of the first and two exponential smoothing is the average of the earliest three data, i.e.

the tendency of the actual data series is more obvious, the weight coefficient (smoothing coefficient) a Should not take too small, so take a  = 0.3.

The exponential smoothing value is calculated by calculating the formula once, two times, three times the exponential smoothing value:

Calculate the coefficients of nonlinear predictive models at, BT,CT. The current number of cycles t = 11, the relevant data in table 1.6 into the formula (1-19), formula (1-20), formula (1-21) after the

A nonlinear predictive model is established. The coefficients are put into the formula (1-18) to

Forecast product sales for 2007 and 2008. In 2007, its forecast lead period was T = 1; In 2008, its forecast lead period was T = 2. In the model, we have to forecast the sales of products in 2007 and 2008. In 2007, its forecast lead period was t= 1; In 2008, its forecast lead period was t= 2. Substituting the model to

So get 2007 years of product sales forecast value of 8.09 million units, 2008 product sales forecast value of 9.2 million units. Forecasters can evaluate and revise the above forecast results according to the changes of market demand factors.

5. Selection of weighting factor a

In exponential smoothing, the key to a successful prediction is the choice of a . The size of a specifies the proportion of the new data and the original predicted value in the new predicted value. The greater the value of a , the greater the proportion of new data, the smaller the proportion of the original forecast, and vice versa.

Disadvantages of exponential smoothing:

? (1) There is a lack of identification of the turning point of the data, but this can be remedied by either the survey or the expert forecast method.

? (2) The effect of long-term prediction is poor, so it is used for short-term prediction.

the advantages of exponential smoothing method:

? (1) The non-equal right processing of data at different times is more in accord with the actual situation.

? (2) Practical only need to select a model parameter a can be predicted, simple and easy.

? (3) Adaptability, that is to say, predictive models can automatically identify changes in data patterns to be adjusted.

Prediction algorithm--exponential smoothing method