CN114036736A - Cause and effect network learning method based on local granger cause and effect analysis - Google Patents

Cause and effect network learning method based on local granger cause and effect analysis Download PDF

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CN114036736A
CN114036736A CN202111304900.1A CN202111304900A CN114036736A CN 114036736 A CN114036736 A CN 114036736A CN 202111304900 A CN202111304900 A CN 202111304900A CN 114036736 A CN114036736 A CN 114036736A
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马德伟
韩敏
秦晓梅
许侃
王钧
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Abstract

The invention belongs to the field of data mining, and provides a causal network learning method based on local Glanberg causal analysis. The method comprises the steps of preprocessing acquired data and completing missing data by adopting an average interpolation method. And then carrying out stability inspection and processing on the completed data so as to meet the assumption of establishing a model. And then, normalizing the data to eliminate the influence brought by different variable dimensions. And finally, establishing a cause and effect network learning algorithm based on local Glange cause and effect analysis, realizing the aim of accurately exploring the cause and effect relationship among the variables, and simultaneously displaying a dynamic cause and effect relationship curve among different variables so as to quantitatively and definitely analyze the cause and effect relationship among the variables among the systems.

Description

Cause and effect network learning method based on local granger cause and effect analysis
Technical Field
The invention belongs to the technical field of data mining, and relates to a cause and effect network learning method based on local Glange cause and effect analysis, aiming at researching the relation between variables in high-dimensional data in the fields of weather and the like.
Background
The multivariate time series is a group of discrete observation sets of a plurality of variables distributed according to time, and is widely applied to a plurality of fields such as finance, industry, traffic, meteorology and the like. For example, in the field of air pollution research, with the rapid development of our country industry and the increase of transportation means in recent years, the concentration of harmful substances generated by combustion of coal, oil and the like in the atmosphere is obviously increased, and phenomena such as air quality reduction and haze weather occur. Haze weather is an atmospheric pollution state, and is a general expression of exceeding the standard of various suspended particulate matters in the atmosphere, and especially PM2.5 (particulate matters with aerodynamic equivalent diameter less than or equal to 2.5 micrometers) is considered as the sourcer of the haze weather. Air pollutants such as PM2.5 cause poor atmospheric visibility, so that social problems such as traffic jam and the like are caused, and the air pollutants can be inhaled into human bodies to cause harm to human health. Therefore, modeling is carried out on air pollutants such as PM2.5, causal relations among Air Quality Indexes (AQI) are described, and possible phenomena in the next step are predicted according to the current and historical time states of all variables, so that theoretical support is provided for air pollution prevention and control. Therefore, the establishment of an effective multivariate causal analysis model has important practical significance.
The causal relationship analysis method explains the influence of the causal variable on the effect variable by researching the drive response relationship among all variables of the system, and can effectively infer the internal structure and the operation mechanism of the system, thereby overcoming the defect that the traditional correlation analysis method is difficult to process indirect relationship and asymmetric relationship. The granger causal analysis method is a common method to explore causal relationships between variables. It estimates the time dependence of variables in the model by means of a vector autoregressive model and is based on a predictable idea: for two time series X and Y, there is a Granger causal relationship of X → Y, so to speak, if the addition of time series X history information helps to reduce the prediction error of Y. The directed graph shown in fig. 1 is a granger causal graph, which can visually show the causal relationship existing between variables. Nodes represent variables and arrows represent causal relationships between two variables. For example, x3The cause of the grande causality of (A) is x5。
At present, the improved method based on the granger causal analysis has been widely applied to problems such as multivariable and nonlinearity. However, the above-mentioned improvement method only quantitatively considers the causal relationship between the variables, and neglects the dynamic characteristics between the variables. Stramaglia Sebastiano et al in the article "Stramaglia Sebastiano, Tomas Scagliarini, Yuri Antonacci, et al. local Granger Causality [ J ]. Physical Review E,2021,103 (2)" show that local Glanberg causal analysis can reveal transient states of information transfer between variables, which a quantitative value cannot. The method is more effective especially when the interaction between variables in a certain period of time is explored. In addition, the conventional method based on the granger causal analysis is modeled based on a vector autoregressive model, which results in poor performance when the data dimensions are too high. Runge Jakob et al, in the article "Runge Jakob, Sebastian Bathiany, Erik Bollt, et al. Inferring catalysis from Time Series in Earth System Sciences [ J ]. Nature Communications,2019,10(1):2553 + 2553.", analyzed the current situation of causal analysis methods and indicated that causal network learning methods are applicable to the causal relationship exploration of high dimensional data.
The invention provides a causal network learning method based on local Glange causal analysis, which aims at the problem of causal relationship analysis of a complex high-dimensional system and is used for modeling causal analysis among variables in the research fields of meteorological pollution and the like.
Disclosure of Invention
The invention aims to solve the technical problems that the traditional Glan's cause and effect analysis method cannot be applied to a high-dimensional time sequence due to structural limitation, and cannot search real-time dynamics among variables in a process, an original Glan's cause and effect analysis model is expanded, a cause and effect network learning algorithm based on local Glan's cause and effect analysis is provided, and accurate cause and effect relation search of high-dimensional data and real-time display of information in the process are realized. The method aims to expand the application of the original method to high-dimensional display and display of more dynamic information.
The method provides a causal network learning method for researching the relationship between each variable and the main pollutant PM2.5 among systems in the face of complex systems such as AQI. The acquired data is preprocessed, and missing data is completed by adopting an average interpolation method. And then carrying out stability inspection and processing on the completed data so as to meet the assumption of establishing a model. And then, normalizing the data to eliminate the influence brought by different variable dimensions. And finally, establishing a causal network learning algorithm based on local Glange causal analysis, realizing the purpose of accurately exploring causal relationships among variables, and simultaneously displaying a dynamic causal relationship curve among different variables so as to quantitatively and definitely analyze the relation between each variable and PM2.5 among systems and realize the analysis of PM2.5 influence factors.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows: a cause and effect network learning method based on local Glanberg cause and effect analysis comprises the following specific steps:
step 1: acquiring air quality index AQI and meteorological observation data; preprocessing multidimensional AQI and meteorological time sequence data, performing interpolation on missing samples by an average interpolation method, and analyzing outliers to obtain smooth time sequence data; performing stationarity test on the time sequence data by using a unit root test method, if a unit root exists in a test result, indicating that the data is not stable, performing stationarity processing by using a difference method, otherwise, if the unit root is not obtained, indicating that the data is stable, and not needing to perform stationarity processing; then, normalizing the time series data;
step 2: determining the maximum lag order tau of the time series data obtained by the processing of the step 1 through the Chichi information criterionmaxGenerating a historical variable matrix X-；
Figure BDA0003339827040000031
Wherein the content of the first and second substances,
Figure BDA0003339827040000032
data representing the current time t, τmaxIs the maximum hysteresis order, d represents the number of variables, and n represents the number of samples;
and step 3: using PC algorithm, with each variable
Figure BDA0003339827040000041
Performing feature selection processing on the target to obtain each variable
Figure BDA0003339827040000042
Is related to the feature subset of
Figure BDA0003339827040000043
Wherein, variable
Figure BDA0003339827040000044
For the data collected at the time of t sampling, i is 1,2, …, d; each variable
Figure BDA0003339827040000045
By its associated feature subset
Figure BDA0003339827040000046
The following formula (2) shows:
Figure BDA0003339827040000047
wherein f isi(. cndot.) represents a mapping function,
Figure BDA0003339827040000048
indicating a deviation;
and 4, step 4: subset of relevant features
Figure BDA0003339827040000049
Sending the data as a condition set to a local Greenger causal analysis model to obtain a quantitative causal relationship value, specifically
Step 4.1: selecting a driving variable
Figure BDA00033398270400000410
And target variable
Figure BDA00033398270400000411
Historical variable matrix X from time series by feature selection method-Respectively finding out the most relevant feature subsets:
Figure BDA00033398270400000412
and
Figure BDA00033398270400000413
will change the variables
Figure BDA00033398270400000414
Is backward smoothed by one bit to obtain the corresponding feature subset
Figure BDA00033398270400000415
A subset of features of (a);
step 4.2: to drive variable
Figure BDA00033398270400000416
And target variable
Figure BDA00033398270400000417
The related feature subset is used as a condition set and sent into a local Glanberg causal analysis model, and the formula (3) is as follows:
Figure BDA00033398270400000418
step 4.3: weighted summation to obtain driving variable
Figure BDA00033398270400000419
For target variable
Figure BDA00033398270400000420
The result of the causal analysis of (a);
and 5: averaging the results obtained by the local granger causal analysis to obtain quantitative causal relationship values among the variables, and drawing the results of the local granger causal analysis to obtain dynamic causal relationship curves among the variables;
step 6: obtaining the causal relationship between any variable and other variables, and selecting a driving variable as a related influence factor of a target variable according to the causal relationship result; and establishing a prediction model by using the echo state network for analysis to obtain a prediction analysis result of the target variable.
The local Glanker causal analysis model further explores dynamic characteristics among variables on the basis of a causal analysis result of the obtained quantitative value; the local granger causal analysis model is represented by formula (4):
Lgc(ut,wt,yt)＝GC+γ(ut,wt,yt) (4)
wherein GC is standard glandor causal analysis; gamma (u)t,wt,yt) Is about ut,wt,ytThe function of (2) for revealing the dynamic characteristics between variables; u. oftRepresenting a set of conditional variables, wtRepresenting the driving variable, ytRepresenting a response variable; second order statistics using a unified procedure to represent GC and γ (u)t,wt,yt)：
Figure BDA0003339827040000051
Figure BDA0003339827040000052
Where | represents the determinant of the matrix,
Figure BDA0003339827040000053
is a historical state and a present state observation;
Figure BDA0003339827040000054
Figure BDA0003339827040000055
and
Figure BDA0003339827040000056
is a covariance matrix; gamma (u)t,wt,yt) Has an average value of 0, represents<Lgc(t)>＝GC；<·>Representing a time series average.
Compared with the prior art, the invention has the following obvious advantages:
the present invention is directed to quantitative and dynamic real-time analysis of causal analysis among high-dimensional data variables. Firstly, the original data of the method are expanded through the AIC criterion, and then a causal network learning algorithm is established based on a feature selection method, and the algorithm can be different from a general causal analysis method based on a vector autoregressive framework, so that causal analysis of high-dimensional data is realized. And then, sending the characteristic selection result into a local Glanberg causal analysis model to obtain a quantitative causal relationship value among variables. In addition, the local granger causal analysis method also obtains a dynamic response curve among variables, so that more information is provided for exploring the causal relationship among the variables. Namely, the method not only obtains quantitative causal relationship measurement aiming at high-dimensional data, but also can obtain dynamic causal relationship among variables.
Drawings
FIG. 1 is a Glankey causal graph showing nodes representing vectors and arrows representing causal relationships that exist between the vectors;
FIG. 2 is a flow chart of the present invention;
FIG. 3 is a diagram showing the dynamic relationship between a portion of the variables and the PM2.5 variable.
FIG. 4(a) is a graph comparing GC to PM2.5 target and predicted values;
FIG. 4(b) is a graph of the error of the prediction curve for GC versus PM 2.5;
FIG. 5(a) is a plot of PCMCI versus PM2.5 target and predicted values;
FIG. 5(b) is a plot of the error of the prediction curve for PCMCI versus PM 2.5;
FIG. 6(a) is a comparison graph of the target value and predicted value for PM2.5 according to the present invention;
FIG. 6(b) is a graph of the prediction curve error for PM2.5 according to the present invention.
Detailed Description
The present invention is further described in detail below with reference to specific examples and simulation diagrams.
The hardware equipment used by the invention comprises a PC machine.
Fig. 2 is a flow chart of a cause and effect network learning method based on local granger cause and effect analysis, which specifically includes the following steps:
step 1: acquiring 5088 groups of data which are totally 10-dimensional and are acquired once per hour from 1 month and 1 day of 2021 to 7 months and 31 days of 2021 in Shanghai Xuhui region, wherein the meaning numbers of all variables are shown in table 1, and then interpolating missing values and abnormal values of a multidimensional AQI and meteorological data set to analyze; carrying out stationarity test on the data by adopting a unit root test method, and carrying out primary difference stationarity processing on the data according to a test result; normalizing the time series data;
TABLE 1 Shanghai AQI and Meteorological data number correspondence table
Figure BDA0003339827040000061
Step 2: determining the maximum lag coefficient tau according to the Chi information criterionmaxAt 8, carrying out lag term expansion on the original data according to the formula (1) to obtain a historical variable matrix X-；
And step 3: calculating the most relevant feature subset for each variable by utilizing a PC feature selection method, taking a variable PM2.5 as an example, firstly obtaining the relevant feature subset from a history set, and obtaining the results of PM10(t-1), NO2(t-1), CO (t-1), PM10(t-2) and AQI (t-5);
and 4, step 4: and (3) sending the feature subset obtained in the step (5) as a condition to a local Greenwich causal analysis model, and carrying out causal relationship exploration on each variable in the historical variable matrix and a PM2.5 variable, wherein the causal relationship exploration specifically comprises the following steps:
step 4.1: take an example of exploring the relationship between PM10 and PM 2.5. First find the relevant subset of the driving variable PM10 and the response variable PM 2.5; according to a PC algorithm, obtaining a relevant subset containing n variables after n iterations to meet the conditions;
step 4.2: bringing the time series of the driving variables and the response variables and the time series of the corresponding relevant subsets into a local granger causal analysis model, and calculating a quantitative causal analysis result through a formula (3);
step 4.3: weighted summation to obtain driving variable
Figure BDA0003339827040000072
For target variable
Figure BDA0003339827040000073
The result of the causal analysis of (a);
and 5: steps 4.1 and 4.2 are repeated until each variable in the historical set of variables is calculated as described above for PM 2.5. Through the process, a cause and effect value matrix is obtained
Figure BDA0003339827040000071
10 represents the number of variables, 8 represents the maximum lag coefficient, and then the weighted summation is carried out to obtain the causal analysis result among the variables;
the results of the causal analysis are shown in table 2: the larger the number in the table, the greater the causal relationship between the two variables. Because of the interplay between variables, there are no two variables that are absolutely independent. As can be seen from table 2, the variables in which shanghai xuhui has a causal relationship with PM2.5 are denoted by SO2, O3, air temperature, etc.;
TABLE 2 causal indices for PM2.5
GC PCMCI The method of the invention
PM2.5 0.13 0.37 0.10
PM10 0.02 0.39 0.07
SO2 0.00 0.13 0.59
NO2 0.03 0.33 0.03
O3 0.01 0.22 0.47
CO 0.04 0.25 0.04
AQI 0.06 0.20 0.02
Air temperature 0.01 0.10 1.25
Dew point 0.00 0.14 0.06
Wind speed 0.00 0.10 0.01
Step 6: dynamic causal relationships between some variables and PM2.5 as shown in fig. 3 are obtained by plotting the output time series of the glange causal analysis, from which dynamic information on the time distribution between the variables is obtained. In addition, the time sequence results are averaged to obtain quantitative causal values among the variables, and quantitative indexes are provided for subsequent determination of causal relationships among the variables. Fig. 3 shows the dynamic curve of PM2.5 versus air temperature over time, from which the dynamic relationship between the two variables over time is seen. If a specific period of research needs to be carried out, more information needs to be obtained from the curve in detail; and obtaining the most relevant characteristic subset of the target variable according to the result of the causal analysis, and then establishing a prediction model by using an echo state network to test the result of the causal relationship. The reserve pool parameters of the echo state network are set as follows: reservoir dimension 100, sparsity 0.05, spectral radius 0.5, and connection input weight 0.05. A total of 10 independent experiments were performed, and the average of 10 experiments was taken as the final result to eliminate the effect of error. The prediction index has a Root Mean Square Error (RMSE) and a Symmetric Mean Absolute Percentage Error (SMAPE). Both are defined as follows:
Figure BDA0003339827040000081
Figure BDA0003339827040000082
in the formula, xiAnd
Figure BDA0003339827040000083
respectively representing the true value and the predicted value, and n representing the number of samples.
The predicted results are shown in fig. 4, 5 and 6. It can be seen from the figure that the subset of variables selected by the method of the present invention can reflect the true trend of the predicted PM2.5 result. The error of 10 independent replicates is shown in table 3. In continuing comparison with the standard granger causal analysis method (GC) and a two-stage causal network learning method (PCMCI), the root mean square error and the symmetric mean absolute percentage error obtained by the present invention both gave the best results, which further demonstrates the effectiveness of the present invention.
TABLE 3 prediction results of various methods
GC PCMCI The method of the invention
RMSE 0.0189 0.0184 0.0183
SMAPE 0.1132 0.1120 0.1018
The above examples are only for showing the embodiments of the present invention, and should not be understood as limiting the scope of the present invention, it should be noted that those skilled in the art can make corresponding modifications without departing from the concept of the present invention, and these are all within the protection scope of the present invention.

Claims (2)

1. A cause and effect network learning method based on local Glanberg cause and effect analysis is characterized by comprising the following specific steps:
step 1: acquiring air quality index AQI and meteorological observation data; preprocessing the multidimensional AQI and the meteorological time sequence data; performing stationarity test on the time sequence data by adopting a unit root test method; then, normalizing the time series data;
step 2: determining the maximum lag order tau of the time series data obtained by the processing of the step 1 through the Chichi information criterionmaxGenerating a historical variable matrix X-；
Figure FDA0003339827030000011
Wherein the content of the first and second substances,
Figure FDA0003339827030000012
data representing the current time t, τmaxIs the maximum hysteresis order, d represents the number of variables, and n represents the number of samples;
and step 3: using PC algorithm, with each variable
Figure FDA0003339827030000013
Performing feature selection processing on the target to obtain each variable
Figure FDA0003339827030000014
Is related to the feature subset of
Figure FDA0003339827030000015
Wherein, variable
Figure FDA0003339827030000016
For the data collected at the time of t sampling, i is 1,2, …, d; each variable
Figure FDA0003339827030000017
By its associated feature subset
Figure FDA0003339827030000018
The following formula (2) shows:
Figure FDA0003339827030000019
wherein f isi(. cndot.) represents a mapping function,
Figure FDA00033398270300000110
indicating a deviation;
and 4, step 4: subset of relevant features
Figure FDA00033398270300000111
Sending the data as a condition set to a local granger causal analysis model to obtain a quantitative causal relationship value, which specifically comprises the following steps:
step 4.1: selecting a driving variable
Figure FDA00033398270300000112
And target variable
Figure FDA00033398270300000113
Historical variable matrix X from time series by feature selection method-Respectively finding out the most relevant feature subsets:
Figure FDA00033398270300000114
and
Figure FDA00033398270300000115
will change the variables
Figure FDA00033398270300000116
Is backward smoothed by one bit to obtain the corresponding feature subset
Figure FDA00033398270300000117
A subset of features of (a);
step 4.2: to drive variable
Figure FDA00033398270300000118
And target variable
Figure FDA00033398270300000119
The related feature subset is used as a condition set and sent into a local Glanberg causal analysis model, and the formula (3) is as follows:
Figure FDA00033398270300000120
step 4.3: weighted summation to obtain driving variable
Figure FDA0003339827030000021
For target variable
Figure FDA0003339827030000022
The result of the causal analysis of (a);
and 5: averaging the results obtained by the local granger causal analysis to obtain quantitative causal relationship values among the variables, and drawing the results of the local granger causal analysis to obtain dynamic causal relationship curves among the variables;
step 6: obtaining the causal relationship between any variable and other variables, and selecting a driving variable as a related influence factor of a target variable according to the causal relationship result; and establishing a prediction model by using the echo state network for analysis to obtain a prediction analysis result of the target variable.
2. The cause and effect network learning method based on the local granger cause and effect analysis as claimed in claim 1, wherein the local granger cause and effect analysis model further explores dynamic characteristics between variables on the basis of the cause and effect analysis result of the obtained quantitative value; the local granger causal analysis model is represented by formula (4):
Lgc(ut,wt,yt)＝GC+γ(ut,wt,yt) (4)
wherein GC is standard glandor causal analysis; gamma (u)t,wt,yt) Is about ut,wt,ytThe function of (2) for revealing the dynamic characteristics between variables; u. oftRepresenting a set of conditional variables, wtRepresenting the driving variable, ytRepresenting a response variable; second order statistics using a unified procedure to represent GC and γ (u)t,wt,yt)：
Figure FDA0003339827030000023
Figure FDA0003339827030000024
Where | represents the determinant of the matrix,
Figure FDA0003339827030000025
is a historical state and a present state observation;
Figure FDA0003339827030000026
Figure FDA0003339827030000027
and
Figure FDA0003339827030000028
is a covariance matrix; gamma (u)t,wt,yt) Has an average value of 0, represents<Lgc(t)>＝GC；<·>Representing a time series average.
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