How to use ARIMA model for prediction?

1、 effect

ARIMA The full name of the model is called autoregressive moving average model , It is the most common statistical model used for time series prediction .

2、 Input / output description

Input ： The characteristic sequence is １ Quantitative variables of time series data
Output ： future Ｎ Forecast value of days

3、 Learning Websites

SPSSPRO- Free professional online data analysis platform

4、 Case example

Case study ： be based on 1985-2021 The sales volume of a magazine in , Forecast the sales volume of a commodity in the next five years .

5、 Case data

ARIMA Case data

6、 Case operation

Step1： New analysis ;
Step2： Upload data ;
Step3： Select the corresponding data to open and preview , Click start analysis after confirmation ;

step4： choice 【 Time series analysis （ARIMA）】;
step5： View the corresponding data format ,【 Time series analysis （ARIMA）】 Request input １ Quantitative variables of time series data .
step6： Select the number of periods to forecast backwards .
step7： Click on 【 To analyze 】, Complete the operation .

7、 Output result analysis

Output results 1：ADF Check list

*p<0.05,**p<0.01,***p<0.001
Intelligent analysis ： The results of this sequence test show that , Based on field annual sales :
The difference is 0 Step time , Significance P The value is 0.998, Don't show significance on the level , The original hypothesis cannot be rejected , This series is an unstable time series . The difference is 1 Step time , Significance P The value is 0.023*, The level is significant , Rejection of null hypothesis , This series is a stationary time series .
The difference is 2 Step time , Significance P The value is 0.000***, The level is significant , Rejection of null hypothesis , This series is a stationary time series .
（ Be careful ： In theory , Sufficient difference operations can fully extract the non-stationary deterministic information in the original time series . But what we need to pay attention to when doing the difference operation is , The order of difference operation is not the more the better . Difference is the extraction of information 、 The process of processing , Every time there is a difference, there will be a loss of information , So the order of the difference needs to be appropriate , To avoid excessive difference .）

Output results 2： Optimal differential sequence diagram

Chart description ： Because the unit root test of the sequence after the first-order difference P Less than 0.05, It shows that the sequence after the first-order difference is stationary data , The figure above shows the raw data 1 Sequence diagram after order difference .

Output results 3： Final difference data autocorrelation diagram （ACF）

Chart description ： According to the autocorrelation diagram , The first-order autocorrelation coefficient is obviously greater than 2 Multiple standard deviation range , After the first-order autocorrelation coefficient , The other autocorrelation coefficients are all in 2 Within times the standard deviation , We can judge that the autocorrelation graph is truncated .

Output results 4： Partial autocorrelation diagram of final difference data （PACF）

Chart description ： From the partial autocorrelation diagram , The first-order partial autocorrelation coefficient is obviously greater than 2 Multiple standard deviation range , After the first order partial autocorrelation coefficient , The other autocorrelation coefficients are all in 2 Within times the standard deviation , We can judge that the partial autocorrelation graph is truncated .

Output results 5： Model parameter table

*p<0.05,**p<0.01,***p<0.001
Chart description ： It is judged by autocorrelation analysis and partial autocorrelation analysis ARIMA There is artificial subjectivity in the parameters of ,SPSSPRO be based on AIC The information criterion automatically finds the optimal parameters , The result of the model is ARIMA Model （0,1,1） Check list , Based on field ： The annual sales , from Q The result of statistical analysis can be ：Q6 There is no significant difference in the level , The assumption that the residual of the model is a white noise sequence cannot be rejected , At the same time, the goodness of fit of the model R2 by 0.981, The model performs well , The model basically meets the requirements .（ Be careful ： Generally speaking , Only before inspection 6 Period and before 12 The delay Q statistic （ namely Q6 and Q12） It can be concluded whether the residual is a random sequence . This is because stationary series usually have short-term correlations , If there is no significant correlation between the values of a short-term delay sequence , There is usually no significant correlation between delays .）

Output results 6： Model residual autocorrelation diagram （ACF

Chart description ： The figure above shows the residual autocorrelation diagram of the model ,（ACF） If the correlation coefficients are all in the dotted line （2 Times the standard deviation ） Inside , Autoregressive model （AR） The residual is a white noise sequence , The time series requires that the model residuals be white noise series . Obviously , The autocorrelation coefficients of the residuals are all in the dotted line .

Output results 7： Partial autocorrelation of model residuals （PACF）

Chart description ： The figure above shows the residual partial autocorrelation of the model （PACF）, If the correlation coefficients are all within the dotted line , Moving average model （MA） The residual is a white noise sequence , The time series requires that the model residuals be white noise series . Obviously , Most of the partial autocorrelation coefficients of the residuals are within the dotted line , Even if the second 9 Order and order 14 The order exceeds 2 Times the standard deviation , This may be caused by accidental factors .

Output results 8： Model check table

*p<0.05,**p<0.01,***p<0.001
Chart description ： Based on field annual sales ,SPSSPRO be based on AIC The information criterion automatically finds the optimal parameters , The result of the model is ARIMA Model （0,1,1） The checklist is based on 1 Differential data , The model formula is as follows ： y(t)=4.996+0.671*ε(t-1)

Output results 9： Time series diagram

Chart description ： The figure above shows the original data graph of the time series model 、 Model fitting value 、 Model predictions . It can be seen from the picture that , There is great similarity between the fitted sequence trend and the real sequence trend , It shows that the fitting effect is good .

Output results 10： Time series prediction table

Chart description ： The table above shows the most recent time series models 5 Forecast of the current data .

8、 matters needing attention

• There are the following , Usually regarded as （ partial ） Autocorrelation coefficient d Order truncation ：
  • In the initial d The order is significantly greater than 2 Multiple standard deviation range
  • Then almost 95% Of （ partial ） The autocorrelation coefficients fall in 2 Within times the standard deviation
  • And the process of attenuation from non-zero autocorrelation coefficient to small value fluctuation near zero is very sudden
  • There are the following , Usually regarded as （ partial ） Autocorrelation coefficient tailing ：
  • If there is more than 5% The sample of （ partial ） The autocorrelation coefficients fall outside the range of double standard deviation
  • Or by significant non 0 Of （ partial ） The attenuation of autocorrelation coefficient to small value fluctuation is slow or very continuous
• After analyzing the autocorrelation diagram and partial autocorrelation diagram , Can be established ARMA Model ：
  • Partial autocorrelation （PACF） The picture is p Step to truncate , Autocorrelation （ACF） Figure trailing ,ARMA The model can be simplified to AR（p） Model ;
  • Autocorrelation （PACF） The picture is q Step to truncate , Partial autocorrelation （ACF） Figure trailing ,ARMA The model can be simplified to MA（q） Model ;
  • If both autocorrelation and partial autocorrelation are tailed , Can be combined with PACF、ACF The most significant order in the graph ( minimum value ) As p、q value ;
  • If both autocorrelation and partial autocorrelation are truncated , You can choose to change to a higher differential , Or not suitable for establishing ARMA Model ;
• SPSSPRO By default AIC The rule is right q And p Carry out optimization and order determination , use adf test + Differential analysis selects the optimal differential hierarchy d

9、 Model theory

ARIMA Model is a method widely used to analyze and model all kinds of time series data . The model is based on the following concepts ： The time series to be predicted is generated by a random process ． If the random process that generates the sequence does not change with time , Then the structure of the stochastic process can be To be accurately characterized and described . Using past observations of the sequence , The future value of the sequence can be extrapolated . stay ARIMA In the model , The future value of the sequence is expressed as a linear function of the current period and lag period of the lag term and random interference term , The general form of the model is shown in the following formula ：

ARIMA The modeling process of the model can be divided into the following four steps ：

step 1 Stationary test of time series ． Usually used ADF or PP Inspection method , Perform unit root test on the original sequence ． If the sequence does not Satisfy the stationarity condition , It can be transformed by difference or logarithmic difference , Transforming non-stationary time series into stationary time series , Then level Stable time series construction ARIMA Model ;

step 2 Determine the order of the model ． With the help of some statistics that can describe the characteristics of the sequence , Such as autocorrelation (AC) Coefficient and partial autocorrelation (PAC) coefficient , Preliminarily identify the possible forms of the model , And then according to AIC Equal order criterion , Select the best model from the available models ;

step 3 Parameter estimation and diagnostic test ． Including testing the significance of model parameters , The validity of the model itself and whether the residual sequence is white noise Sound sequence ． If the model passes the test , Then the model setting is basically correct , otherwise , The form of the model must be redefined , And diagnostic tests , Until we get the setting Determine the correct model form ;

step 4 Use the established ARIMA The model predicts ．

10、 reference

[1] Wang Yan ． Using time series analysis [M]． Beijing ： Renmin University Press of China 2005．
[2] Zheng Li , Duandongmei , Lufengbin , etc. . Integrated forecast of pork consumption demand in China —— be based on ARIMA、VAR and VEC Demonstration of the model [J]. Theory and practice of system engineering ,2013,33(4):918-925.