Why do we do autocorrelation analysis? Explain application scenarios and specific operations

1、 effect

Autocorrelation （ACF） It means that there is a certain degree of correlation between the sequence and the sequence formed by some order lag , Partial autocorrelation function （PACF） It is a measure function of the conditional correlation of two sequences given by other sequences . Generally speaking （ partial ） Autocorrelation is used in time series analysis AR、MA Of p、q Perform order determination .

2、 Input / output description

Input ：１ Quantitative variables of sequence data
Output ：pacf/acf chart , be used for AR、MA Of p、q Perform order determination

3、 Learning Websites

SPSSPRO- Free professional online data analysis platform

4、 Case example

Case study ： be based on 5 Annual monthly sales of goods , Forecast the sales volume of a commodity in the next five months .

5、 Case data

（ partial ） Autocorrelation analysis 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 【（ partial ） Autocorrelation analysis （pcaf/acf)】;
step5： View the corresponding data format ,【（ partial ） Autocorrelation analysis （pcaf/acf)】 Request input １ Quantitative variables of time series data .
step6： Set the differential hierarchy （ Be careful ： In general ,（ partial ） Autocorrelation analysis requires the sequence to be stable or several order difference sequence to be stable . If the original sequence is a stationary sequence , Then select the difference class as 0; If the original sequence is a nonstationary sequence , The first-order difference sequence is a stationary sequence , Then select the difference class as 1; If the original sequence is a nonstationary sequence , The first-order difference sequence is a nonstationary sequence , The second-order difference sequence is a stationary sequence , Then select the difference class as 2.（ In this case, it is found that the original sequence is unstable in the unit root test , The first-order difference sequence is stable , So set the differential hierarchy to 1）
step7： Click on 【 To analyze 】, Complete the operation .

7、 Output result analysis

Output results 1： Model residual autocorrelation diagram （ACF）

Chart description ： The figure above shows the autocorrelation diagram （ACF）, Inclusion factor 、 Upper and lower confidence limits . It can be seen from the picture that , The first and third order autocorrelation coefficients are obviously larger than 2 Multiple standard deviation range , And the process that the autocorrelation coefficient decays into a small value fluctuation is relatively slow or very continuous , We can judge that the autocorrelation graph is trailing .

Output results 2： Partial autocorrelation of model residuals （PACF）

Chart description ： The above figure shows the partial autocorrelation diagram （PACF）, Inclusion factor , Upper and lower confidence limits . It can be seen from the picture that , The first and second order partial autocorrelation coefficients are obviously larger than 2 Multiple standard deviation range , After the first order partial autocorrelation coefficient , The other partial autocorrelation coefficients are all in 2 Within times the standard deviation , And the second-order post biased autocorrelation coefficient decays into a small value fluctuation near zero, which is very sudden . We can judge that the partial autocorrelation graph is truncated .

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 ;
• We recommend that you use ARIMA Of AIC（ Arrange and combine all parameters , solve AIC The combination parameter corresponding to the minimum value ） Automatic optimization mode , adopt PACF And ACF The method of determining the order of the graph is not accurate , It is also difficult to be persuasive .

9、 Model theory

Autocorrelation coefficient and partial autocorrelation coefficient are defined in statistics , It is a statistical index used to reflect the degree of correlation between variables , However, the relationship between the two specific variables is different .
Autocorrelation coefficient measures the degree of correlation between two different periods of the same event , Speaking figuratively is to measure the impact of your past behavior on your present .
Mathematical expression ：

according to ACF Find the lag k Autocorrelation coefficient ACF(k) when , In fact, it's not Z(t) And Z(t-k) A simple correlation between .
because Z(t) At the same time, it will suffer from the middle k-1 Random variables Z(t-1)、Z(t-2)、……、Z(t-k+1) Influence , And this k-1 All random variables and z(t-k) There is a correlation , So the autocorrelation coefficient is actually doped with other variable pairs Z(t) And Z(t-k) Influence .
In order to measure simply Z(t-k) Yes Z(t) Influence , Partial autocorrelation coefficient is introduced （PACF） The concept of . For stationary time series {Z(t)}, The so-called lag k Partial autocorrelation coefficient means that in a given middle k-1 Random variables Z(t-1)、Z(t-2)、……、Z(t-k+1) Under the condition of , Or say , In the middle k-1 Random variables Z(t-1)、Z(t-2)、……、Z(t-k+1) After the interference of ,Z(t-k) Yes Z(t) The relevance of the impact .
Mathematical expression ：

10、 reference

[1] Yu Ningli , Yi Dongyun , Tu Xianqin . Analysis of autocorrelation and partial correlation functions in time series [J]. Mathematical theory and application ,2007(01):54-57.