Statistics 8th Edition Jia Junping Chapter XIII Summary of knowledge points of time series analysis and prediction and answers to exercises after class

Catalog

One 、 Knowledge framework

Two 、 Exercises

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One 、 Knowledge framework

Two 、 Exercises

1 yes 1991～2008 China's fiscal revenue data in . Use exponential curve to predict 2009 Annual revenue , And compare the actual value with the predicted value .

Explain ： Let the trend equation of the exponential curve be Yt＝b0b1t, Take logarithm at both ends to get ln（Yt）＝ln（b0）＋tln（b1）. According to the principle of least squares , Get ln（b1）＝0.1709,ln（b0）＝7.8445, The corresponding exponential curve equation is Yt＝2551.6615×1.1864t.
take t＝1,2,…,18 Substitute into the trend equation to get the predicted value of each period , take t＝19 Substitute into the trend equation to get 2009 The forecast value of annual fiscal revenue . The calculation results are shown in

2 surface 13-6 It's a hotel 18 Monthly turnover data .

requirement ：
（1） use 3 The period moving average method predicts the 19 Monthly turnover .
（2） Use exponential smoothing , Use the smoothing coefficient α＝0.3,α＝0.4 and α＝0.5 Forecast monthly turnover , Analyze the prediction error , Explain which smoothing coefficient is more suitable for prediction .
（3） Establish a trend equation to predict the turnover of each month , Calculate the standard error of estimation .

Explain ：（1） The first 19 Months 3 The period moving average forecast value is ：
F19＝（587＋644＋660）/3＝630.33
（2） from Excel Output exponential smoothing prediction value , As shown in the table .

α＝0.3 The predicted value is ：F19＝0.3×660＋（1－0.3）×567.9＝595.5, Square of error ＝87524.82
α＝0.4 The predicted value is ：F19＝0.4×660＋（1－0.4）×591.1＝618.7, Square of error ＝50952.1
α＝0.5 The predicted value is ：F19＝0.5×660＋（1－0.5）×606.5＝633.3, Square of error ＝50235.49 Compare the square of each error ,α＝0.5 More appropriate .
（3） According to the least square method , utilize Excel The output regression results are shown in the table .

So the linear trend equation is ：Yt＝239.73＋21.9288t; Estimate the standard error SY＝31.6628.

3 In our country 1964～1999 Annual yarn production data , As shown in the table 13-11 Shown （ Company ： Ten thousand tons of ）.

（1） Draw a time series diagram to describe its trend .
（2） Choose a suitable trend line to fit the data , And predict according to the trend line 2000 Annual production .

Explain ：（1） Drawing time series , As shown in the figure .

（2） As you can see from the diagram , Yarn output has an obvious linear trend . use Excel The obtained linear trend equation is ：Yt＝69.5202＋13.9495t
therefore 2000 The predicted value is ：Y37＝69.5202＋13.9495×37＝585.65（ Ten thousand tons of ）

4 Counter table 13-12 The data of are respectively fitted to the linear trend line Yt＝b0＋b1t、 Second order curve Yt＝b0＋b1t＋b2t2 And third-order curve Yt＝b0＋b1t＋b2t2＋b3t3, And compare the results .

Explain ： When finding the second-order curve and the third-order curve , First, linearize it , Then use the least square method to solve according to linear regression . use Excel The obtained trend line 、 Coefficients of second-order curve and third-order curve , As shown in the table .

So the trend equations are ：
Linear trend ：Yt＝374.1613－0.6137t
Second order curve ：Yt＝381.6442－1.8272t＋0.0337t2 Third order curve ：Yt＝372.5617＋1.0030t－0.1601t2＋0.0036t3
The prediction value and prediction error obtained from the trend equation , As shown in the table .

The standard errors of different trend line predictions are ：
A straight line ：

Second order curve ：

Third order curve ：

By comparing the prediction errors , The error of the straight line is the largest , The error of the third-order curve is the smallest .
From the prediction diagram of different trend equations （ As shown in the figure ） It can also be seen that , The fitting between the third-order curve and the original sequence is the best .

5 A trading company is mainly engaged in the export business of products , In order to reasonably organize the supply , Need to know the changes of export orders . Table is 2011～2015 The amount of export orders in each month of the year （ Company ： Ten thousand yuan ）.

requirement ：
（1） Draw a trend chart according to the monthly data of each year , Explain the characteristics of the time series .
（2） Calculate the forecast value of each month , What do you think should be done ？
（3） Choose the appropriate method to predict 2016 year 1 The export order amount of the month .

Explain ：（1） Draw a trend chart , As shown in the figure .

As can be seen from the trend chart , There is no trend in the data of each month of each year , But from 2011～2015 Look at the changes in , There is a certain linear trend in the order amount .
（2） Because it predicts the order amount of each month , Therefore, moving average method or exponential smoothing method is more appropriate .
（3） use Excel use 12 The result predicted by the term moving average method is ：F2016/1＝71.4（ Ten thousand yuan ）.
use Excel Use exponential smoothing （α＝0.4） The prediction result is ：F2016/1＝72.5（ Ten thousand yuan ）.

6 surface 13-16 It is the quarterly sales data of a large department store in recent years （ Company ： Ten thousand yuan ）. Decompose the constituent elements of this time series , Calculate the seasonal index , Excluding seasonal changes , Calculate the trend equation after excluding seasonal changes .

Explain ： Calculate the seasonal index by the seasonal average method , Take the moving average number of items equal to the cycle length , namely k＝4, Because the number of moving items is even , So we need to do two moving averages .
for example ：2006 The result of the first moving average in is ：
T2006,2.5＝（y2006,1＋y2006,2＋y2006,3＋y2006,4）/4
T2006,3.5＝（y2006,2＋y2006,3＋y2006,4＋y2006,1）/4
……
Then the result of the second moving average is ：

……
namely 2006 In the first 3 The quarterly moving average is ：

so 2006 In the first 3 The seasonal ratio of the quarter is ：
y2006,3/T2006,3＝2264.1/1627.9875＝1.3907
Empathy 2006 In the first 4 The quarterly moving average is ：

so 2006 In the first 4 The seasonal ratio of the quarter is ：
y2006,4/T2006,4＝1943.3/1833.0875＝1.0601
Similarly, we can get the moving average of other months , Then we can get the corresponding seasonal ratio , Finally, we can get the seasonal index . After calculating the seasonal index , Divide each actual observation by the corresponding seasonal index , So as to eliminate seasonal changes , The formula is ：y/S＝（T×S×I）/S＝T×I. The calculation results are shown in the table .

Draw a seasonal chart , As shown in the figure , It can be seen from the picture that the peak season is 3 quarter , Off season is the first 1 quarter .

Draw the sales volume and its trend chart after excluding seasonal changes , As shown in the figure .

It can be seen from the picture that , One variable linear model can be used to predict the sales of each quarter , Let the trend equation be ：Yt＝b0＋b1t, from Excel Available ：b0＝2043.92,b1＝163.7064. Therefore, the trend equation of sales after excluding seasonal changes is ：Yt＝2043.92＋163.7064t