CN103440368B - A kind of multi-model dynamic soft measuring modeling method - Google Patents

Abstract

A kind of multi-model dynamic soft measuring modeling method, multiple submodels are set up using adaptive fuzzy kernel clustering method and least square method supporting vector machine；Then, submodel output is carried out merging the output for obtaining multi-model as weight using the probability distribution function of combining evidences rule construct；Finally, dynamic estimation is carried out to the predicated error of multi-model with reference to autoregressive moving-average model.

Description

A kind of multi-model dynamic soft measuring modeling method

Technical field

The present invention relates to the flexible measurement method of esterification yield in polyester industrial production process, and in particular to a kind of based on evidence reason By composition rule and the multi-model dynamic soft-measuring method of autoregressive moving-average model.

Background technology

Shown in Fig. 1 be esterification basic process, esterification as whole polyester production process key link, Stabilized polyester production is played a decisive role.And Key Quality Indicator --- the esterification yield that the first esterifying kettle is exported in reaction unit Height directly affect the carrying out of subsequent reactions and the crystal property of polyester product, therefore controlled often through control esterification yield Whole production process.But different polycondensating process has different requirements to esterification yield, so must be by adjusting in production process Whole reaction pressure and material quantity than etc. operating condition reach required esterification yield.But the suddenly change of operating condition can cause The quality fluctuation of esterification yield, is unfavorable for the real-time control of whole production process.On the other hand, esterification process uses two esters mostly Change reactor to reach the esterification yield of technological requirement, and the nonlinearity of reaction system, time variation and uncertainty are increased The difficulty of esterification yield on-line measurement.

Field assay instrument is not only expensive, maintaining is complicated, and esterification yield is surveyed using analysis meter During amount, generally there is the delayed of some time, this will cause to control the hydraulic performance decline of quality, it is difficult to meet production requirement.Soft survey The basic skills of amount is that Theory of Automatic Control is combined with production process knowledge, Applied Computer Techniques, for difficulty With measurement or temporary transient immeasurable leading variable, the auxiliary variable that selection other is easily measured, by constituting certain number Relation infers and estimates, replaces the function of field assay instrument with software.Flexible measurement method is rapid because of response, Neng Goulian It is continuous to provide leading variable information, and invest the advantages of low, maintaining is simple and be widely studied and apply in each field.But Be, in nearest decades, with science and technology progress, modern industry produce for production process requirement increasingly Height, data volume is increased dramatically, and data type becomes increasingly complex, and operating mode is complicated and changeable, and on the other hand, industrial process is typically all It is dynamic, static flexible measurement method cannot generally reflect the multidate information and global property of industrial process, causes the suitable of model Answering property is poor, it is impossible to long-term use.So in the past simple, conventional flexible measurement method can not meet the need of modern production process Will, easily there is process characteristic and match that not good, precision of prediction is low and the problems such as bad adaptability.

It is many in order to obtain being applied to the flexible measurement method that esterification yield data are predicted and are analyzed in more typically meaning Improved method is suggested, and forms plentiful and substantial achievement in research, mainly there is the following aspects：Using various modeling methods, such as It is leading to predict that Analysis on Mechanism, artificial neural network, least square method supporting vector machine and Gaussian process set up model to sample set Variable is exported；Using various intelligent optimization methods, such as：The parameter to model such as particle cluster algorithm, genetic algorithm and evolution algorithm Carry out preferably；Using various clustering methods such as：K- mean clusters, Fuzzy C-Means Clustering, manifold cluster and affine propagation clustering Sample is agglomerated into several subclasses by method, builds several submodels to improve model prediction performance etc..

The content of the invention

The present invention is directed to deficiencies of the prior art, there is provided a kind of multi-model dynamic soft measuring modeling method, Based on Combination Rules of Evidence Theory (D-S rule) and autoregressive moving-average model (ARMA), have compared with prior art more preferable Adaptability, precision is higher in the hard measurement to esterification yield.

The present invention is achieved through the following technical solutions：

A kind of multi-model dynamic soft measuring modeling method, comprises the following steps：

S1, data prediction：Selection training sample data collection X_m*n, m is sample dimension, and n is number of samples, rejecting abnormalities Data are simultaneously normalized to data；

S2, the analysis of adaptive fuzzy kernel clustering：Using adaptive fuzzy kernel clustering method to training sample data collection X_m*nEnter Row cluster, obtains the fuzzy class degree of membership and each cluster centre of each sample, and automatically determine out preferable clustering number mesh c；

S3, set up submodel：Is trained to the training sample set of c cluster using least square method supporting vector machine Practise, c son is set up and determined to selection gaussian kernel function as the kernel function of least square method supporting vector machine by cross-validation method The parameter of model：Penalty factor and nuclear parameter σ, and obtain the output of each submodel

S4, the submodel output fusion based on Combination Rules of Evidence Theory：Calculate the evidential probability distribution letter of each submodel Numerical value, as the weight of submodel, the then output to each submodel carries out evidence fusion, obtains static multiple mode Output

S5, the mobilism of model output：The multi-model of current time t is exported using autoregressive moving-average model, i.e., It is rightEnter Mobile state adjustment, first determine whetherWhether it is stationary sequence, if it is not, willBe converted to stationary sequence；Otherwise directly willSubtract each other with true measurement y, obtain a time series on output valve error delta y, then slided using autoregression flat Equal model (p, q) is modeled to the time series, obtains the autoregressive moving-average model on predicated error, finally, will The model of the above two is combined and carries out model prediction, then final dynamic multi-model is output as

Include preferably, in step S2, the step of adaptive fuzzy kernel clustering method：

S21：Cluster object function：To training sample set X={ x_i| i=1,2...n }, adaptive fuzzy kernel clustering method Object function be defined as

In formula,M is fuzzy control index, μ_ij It is subordinate to angle value, v for what i-th sample corresponded to j-th cluster_jIt is j-th cluster centre, K (x_i, v_j) it is gaussian kernel function；

S22：Degree of membership updates：

S23：Cluster centre updates：

S23：Cluster result is evaluated：After cluster terminates, Validity Index is evaluated the result for clustering

Preferably, step S4 includes：

S41：First evidential probability partition function of submodel：Using all c submodels obtained by cluster as evidence Framework of identification in theory, and any submodel is considered as Jiao unit C_j(j=1,2...c), for sample x₁, it is calculated for One submodel, i.e., first Jiao unit C₁Fuzzy class degree of membership, and according to evidence theory, as an evidence, note should The probability distribution function of evidence is m ({ C₁}|x₁)=μ₁₁；

And for all of n test sample data X={ x_i| i=1,2...n }, it is similarly obtained n bar evidences, its probability Partition function is designated as m ({ C₁}|x_i)=μ_i1, i=1,2...n；

Then, these probability distribution functions are merged using Combination Rules of Evidence Theory, by the probability after fusion point With function as first probability distribution function of submodel：

Wherein, the contradiction factorIt is used to reflect the conflict of evidence Degree；

S44：The evidential probability partition function of all submodels：The rest may be inferred, for all of c submodel, according to step Rapid S43, obtains c evidential probability partition function m ({ C₁}|X)...m({C_c}|X)；

S45：Multi-model is exported：Son outputs of the X for each submodel is calculated respectivelyC in S44 is general Rate partition function as each submodel weight, the submodel output result to gained is weighted fusion, then trains sample The multi-model output of notebook data collection is expressed as：

Preferably, step S5 includes：

S51：Static multi-model is exported using autoregressive moving-average modelDynamic calibration is carried out, autoregression is slided The response of averaging model descriptive system current time tI.e.It is not only relevant with the observation before it in time, also be There is certain dependence in the present worth and lagged value of system disturbance, autoregressive moving-average model (p, q) is represented by

Wherein, p is autoregression；Q is rolling average item number；

S52：Linear shift operator B is introduced, is then hadTherefore the formula in S51 can transform to

In formula, ε_tTo meet N (0, σ²) white noise sequence, Φ (B) and θ (B) for Shift operators B m ranks and n ranks it is multinomial Formula,

According to the basic theories of Hilbert space Linear Operators, the random time to meeting steady, normal state, zero-mean SequenceCan be approached with arbitrary accuracy with an autoregressive moving-average model (p, q).

The method first with combining evidences rule process uncertain information focusing advantage, for affine propagation clustering side Each submodel that method is obtained establishes multiple evidential probability partition functions, as the weight of each submodel, to each son The output of model is weighted the multi-model output that fusion obtains test sample, it is to avoid the concussion that switching mode causes, and eliminates Sample mistake divides the influence to model output accuracy, effectively increases the predictive ability of model；Slided then in conjunction with autoregression Dynamic averaging model carries out dynamic calibration to the static multiple mode output error information for obtaining, and significantly improves the dynamic response of system Characteristic.

Brief description of the drawings

Shown in Fig. 1 be esterification process schematic；

Shown in Fig. 2 be multi-model dynamic soft measuring modeling method of the present invention flow chart；

Shown in Fig. 3 be submodel C and σ parameter list；

Shown in Fig. 4 is that LSSVM measuring methods are shown the predicted value of esterification yield test sample and the comparing result of artificial value It is intended to；

Shown in Fig. 5 is contrast of the SFKCM-LSSVM measuring methods to the predicted value and artificial value of esterification yield test sample Result schematic diagram；

Shown in Fig. 6 is AP-LS-SVM measuring methods to the predicted value of esterification yield test sample and the contrast knot of artificial value Fruit schematic diagram；

Shown in Fig. 7 is predicted value and the comparing result schematic diagram of artificial value of the present invention to esterification yield test sample；

Shown in Fig. 8 is the Performance comparision schematic diagram with existing measuring method of the invention.

Specific embodiment

Below with reference to accompanying drawing of the invention, clear, complete description is carried out to the technical scheme in the embodiment of the present invention And discussion, it is clear that as described herein is only a part of example of the invention, is not whole examples, based on the present invention In embodiment, the every other implementation that those of ordinary skill in the art are obtained on the premise of creative work is not made Example, belongs to protection scope of the present invention.

For the ease of the understanding to the embodiment of the present invention, make further by taking specific embodiment as an example below in conjunction with accompanying drawing Illustrate, and each embodiment does not constitute the restriction to the embodiment of the present invention.

First with the focusing advantage of combining evidences rule process uncertain information, obtained for affine propagation clustering method Each submodel establish multiple evidential probability partition functions, as the weight of each submodel, to each submodel Output is weighted the multi-model output that fusion obtains test sample, it is to avoid the concussion that switching mode causes, and eliminates sample Mistake divides the influence to model output accuracy, effectively increases the predictive ability of model；Then in conjunction with autoregressive moving average Model (ARMA, Auto-Regressive Moving Average Model) is to the static multiple mode output error information that obtains Dynamic calibration is carried out, the dynamic response characteristic of system is significantly improved.

This method technical solution adopted for solving the technical problem is：

Fig. 2 is refer to, a kind of multi-model based on Combination Rules of Evidence Theory and autoregressive moving-average model is dynamically soft Measurement modeling method, comprises the following steps：

S1：Data prediction：Selection training sample data collection X_m*n, m is sample dimension, and n is number of samples, rejecting abnormalities Data are simultaneously normalized to data；

S2：Adaptive fuzzy kernel clustering is analyzed：Using adaptive fuzzy kernel clustering method to sample set X_m*nClustered, The fuzzy class degree of membership and each cluster centre of each sample are obtained, and automatically determines out preferable clustering number mesh c；

S3：Set up submodel：To each sub- training sample set, using least square method supporting vector machine (LS-SVM, below with LS-SVM replacements) study is trained to it, and determine the parameter of each submodel.Gaussian kernel function is selected as the core of LS-SVM Function, the parameter of each submodel is determined by cross-validation method：Penalty factor and nuclear parameter σ, as shown in Figure 3；

S4：Submodel output fusion based on D-S：Method according to formula (6) obtains the evidential probability distribution of each submodel Functional value, as the weight of submodel, the then output using formula (7) to each submodelCarry out evidence Fusion, obtains the output of multi-model

S5：The mobilism of model output：Exported in the multi-model that sample is obtained using static models aboveAfterwards, use Arma modeling is exported to the multi-model of current time tIt is i.e. rightEnter Mobile state adjustment.First determine whetherWhether it is stationary sequence, If it is not, willBe converted to stationary sequence；Otherwise directly willSubtract each other with true measurement y, obtain one on output valve error The time series of Δ y, is then modeled using arma modeling (p, q) to the time series, is obtained on predicated error Arma modeling.Finally, two models are combined and carry out model prediction by more than, then the final output of sample is

It is as follows the step of " adaptive fuzzy kernel clustering method " in step S2：

S21：Cluster object function：To training sample set X={ x_i| i=1,2...n }, adaptive fuzzy kernel clustering method Object function be defined as

In formula,M is fuzzy control index, μ_ij It is subordinate to angle value, v for what i-th sample corresponded to j-th cluster_jIt is j-th cluster centre, K (x, y) is gaussian kernel function.

S22：Degree of membership updates：

S23：Cluster centre updates：

S23：Cluster result is evaluated：After cluster terminates, the result for clustering is evaluated using following Validity Index

In step S4, and " based on Combination Rules of Evidence Theory model prediction export " comprise the following steps that：

S41：First evidential probability partition function of submodel：Using all c submodels obtained by cluster as evidence Framework of identification in theory, and any submodel is considered as Jiao unit C_j(j=1,2...c).So, for sample x₁, basis first Formula (3) obtains it for first submodel, namely first Jiao unit C₁Fuzzy class degree of membership.And according to evidence theory, will It is m ({ C as an evidence, the probability distribution function for remembering the evidence₁}|x_i)=μ₁₁。

And for all of n test sample data X={ x_i| i=1,2...n }, similarly, n bar evidences are can obtain, its is general Rate partition function is designated as m ({ C₁}|x_i)=μ_i1, i=1,2...n.

Then, these probability distribution functions are merged using Combination Rules of Evidence Theory, by the probability after fusion point With function as first probability distribution function of submodel, as shown in formula (6)：

Wherein, the contradiction factorIts size reflects evidence Conflict spectrum.

S44：The evidential probability partition function of all submodels：The rest may be inferred, for all of c submodel, according to step Rapid S43, can obtain c evidential probability partition function m ({ C₁}|X)...m({C_c}|X)。

S45：Multi-model is exported：X is calculated respectively for each submodel LS-SVM1 ... the son output of LS-SVMcUsing c probability distribution function obtained above as each submodel weight, to the output of the submodel of gained Result is weighted fusion, then the multi-model output of test sample collection can be expressed as

In step S5, the mobilism of output " model " is concretely comprised the following steps：

S51：The upper static multiple mode for obtaining that saves is exported using autoregressive moving-average model (ARMA)Enter Mobile state school Just.The response of arma modeling descriptive system current time tI.e.It is not only relevant with the observation before it in time, also with There is certain dependence in the present worth and lagged value of system disturbance.Arma modeling (p, q) is represented by

Wherein, AR is autoregression, and p is autoregression；MA is rolling average, and q is rolling average item number.

S52：Linear shift operator B is introduced, is then hadTherefore formula (8) can transform to

In formula, ε_tTo meet N (0, σ²) white noise sequence, Φ (B) and θ (B) for Shift operators B m ranks and n ranks it is multinomial Formula,

According to the basic theories of Hilbert space Linear Operators, the random time to meeting steady, normal state, zero-mean SequenceCan be approached with arbitrary accuracy with an arma modeling (p, q).

Below according to real data for an embodiment：

The first step：To collection in worksite to data process, obtain 1000 groups of normal datas.By 900 groups of numbers therein According to as training dataset X, for the foundation of model；Remaining 100 groups as test data set, for the prediction of testing model Ability.

Second step：Training dataset is clustered using affine propagation clustering method, obtains optimum cluster number for c= 4, corresponding cluster centre v.

3rd step：Four sub- training sample sets obtained by cluster, four submodels are set up using LS-SVM methods, and Training study, the parameter of LS-SVM is determined through cross-validation method, as shown in Figure 3.

4th step：The probability distribution function of each submodel of test sample set pair is calculated according to formula (6), as each The weight of individual submodel, then calculates test sample X^testRelative to the output of each submodelThen it is sharp Output with formula (7) to each submodel is merged, and obtains the output of test sample

5th step：By the predicted value of the test sample of current time tSubtract each other with manual analysis value y, to output error Δ y Time series carry out ARMA modelings.Show that the estimated performance of algorithm is best as optimal factor p=4.

Fig. 4-7 is the estimated performance curve of three kinds of different measuring methods and measuring method of the invention.From simulation result As can be seen that the multi-model dynamic soft measuring based on Combination Rules of Evidence Theory and autoregressive moving-average model of the invention is built Mould method has larger improvement compared to single model and traditional multi-model process to the estimated performance of esterification yield.This be because For esterification has the characteristics of compared with high non-linearity and multi-state, and one model needs to consider all to instruct during single model modeling Practice sample, which has limited the precision of model；And conventional multi-mode type method modeling when although being clustered to training dataset Divide, establish different submodels respectively, but the prediction test sample output stage do not deeply consider test sample with The influence of the difference and dividing condition of training sample and the dynamic change of process to multi-model output result, therefore estimated performance do not have It is significantly improved.The identical sample similar with characteristic of operating mode is first clustered and drawn by the method for the present invention using affine propagation clustering method Point, the output stage in prediction test sample is then taken into full account, each submodel exports the influence to sample final output result, Output using Combination Rules of Evidence Theory construction weight to submodel carries out multi-model fusion, obtains test sample most Output eventually, it is to avoid the prediction deviation that the concussion and sample that switching mode causes have been separated by mistake；In addition, it is contemplated that actual industrial mistake The dynamic characteristic of journey, dynamic calibration is carried out using autoregressive moving-average model to multi-model output, improves the dynamic of system Response characteristic, thus with more preferable adaptability, preferable fitting effect is achieved in the hard measurement to esterification yield.

The performance parameter of different flexible measurement methods is listed in Fig. 8.As can be seen from Figure 8, using provided by the present invention Multi-model dynamic soft measuring modeling method, for more existing modeling method, root-mean-square error and maximum relative error are all Improvement is arrived.

The above, the only present invention preferably specific embodiment, but protection scope of the present invention is not limited thereto, Any one skilled in the art the invention discloses technical scope in, the change or replacement that can be readily occurred in, Should all be included within the scope of the present invention.Therefore, protection scope of the present invention should be with scope of the claims It is defined.

Claims (1)

1. a kind of multi-model dynamic soft measuring modeling method, it is characterised in that comprise the following steps：

S1, data prediction：Selection training sample data collection X_m*n, m is sample dimension, and n is number of samples, rejecting abnormalities data And data are normalized；

S2, the analysis of adaptive fuzzy kernel clustering：Using adaptive fuzzy kernel clustering method to training sample data collection X_m*nGathered Class, obtains the fuzzy class degree of membership and each cluster centre of each sample, and automatically determines out preferable clustering number mesh c；

S3, set up submodel：Study is trained to the training sample set of c cluster using least square method supporting vector machine, is selected Kernel function of the gaussian kernel function as least square method supporting vector machine is selected, c submodel is set up and determined by cross-validation method Parameter：Penalty factor and nuclear parameter σ, and obtain the output of each submodel

S4, the submodel output fusion based on Combination Rules of Evidence Theory：The evidential probability partition function value of each submodel is calculated, As the weight of submodel, the then output to each submodel carries out evidence fusion, obtains static multiple mode output

S5, the mobilism of model output：The multi-model of current time t is exported using autoregressive moving-average model, i.e., it is rightEnter Mobile state is adjusted, and is first determined whetherWhether it is stationary sequence, if it is not, willBe converted to stationary sequence；Otherwise directly willWith it is true Actual measurement value y subtracts each other, and a time series on output valve error delta y is obtained, then using autoregressive moving-average model (p, q) is modeled to the time series, obtains the autoregressive moving-average model on predicated error, finally, by the step The output of static multiple mode in rapid S4Being combined with the output of autoregressive moving-average model in S5 carries out model prediction, then most Whole dynamic multi-model is output as

Include in step S2, the step of the adaptive fuzzy kernel clustering method：

S21：Cluster object function：To training sample set X={ x_i| i=1,2...n }, the target of adaptive fuzzy kernel clustering method Function is defined as

$\left\{\begin{array}{c}{J}_{\mathrm{\Φ}}(U,c)=\underset{i=1}{\overset{n}{\Σ}}\underset{j=1}{\overset{c}{\Σ}}{\mathrm{\μ}}_{ij}^m\left|\right|\[1-K(x_i,v_j)\]|{|}^2\\ \begin{array}{cc}s.t.& U\∈M_{fc}\end{array}\end{array}\right.$

In formula,M is fuzzy control index, μ_ijIt is What i sample corresponded to j-th cluster is subordinate to angle value, v_jIt is j-th cluster centre, K (x_i, v_j) it is gaussian kernel function；

S22：Degree of membership updates：

${\mathrm{\μ}}_{ij}={(1-K(x_i,v_j\left)\right)}^{-1/(m-1)}/\underset{j=1}{\overset{c}{\Σ}}{(1-K(x_i,v_j\left)\right)}^{-1/(m-1)};$

S23：Cluster centre updates：

$v_i=\underset{i=1}{\overset{n}{\Σ}}{{\mathrm{\μ}}_{ij}}^{m}K(x_i,v_j)x_i/\underset{i=1}{\overset{n}{\Σ}}{{\mathrm{\μ}}_{ij}}^{m}K(x_i,v_j);$

S24：Cluster result is evaluated：After cluster terminates, Validity Index is evaluated the result for clustering

$V_{GX}\left(c\right)=\frac{\underset{i=1}{\overset{c}{\Σ}}\underset{k=1}{\overset{n}{\Σ}}{\mathrm{\μ}}_{ik}^m(1-K(v_k,x_i\left)\right)+\frac{1}{c}\underset{i=1}{\overset{c}{\Σ}}(1-K(v_i,\stackrel{\‾}{v}\left)\right)}{\underset{i\≠k}{min}(1-K(v_k,v_i\left)\right)};$

Step S4 includes：

S41：First evidential probability partition function of submodel：Using all c submodels obtained by cluster as evidence theory In framework of identification, and any submodel is considered as Jiao unit C_j, wherein, j=1,2...c, for sample x₁, it is calculated for One submodel, i.e., first Jiao unit C₁Fuzzy class degree of membership, and according to evidence theory, as an evidence, note should The probability distribution function of evidence is m ({ C₁}|x₁)=μ₁₁；

And for all of n test sample data X={ x_i| i=1,2...n }, n bar evidences are similarly obtained, its probability assignments letter Number scale is m ({ C₁}|x_i)=μ_i1, i=1,2...n；

Then, these probability distribution functions are merged using Combination Rules of Evidence Theory, by the probability assignments letter after fusion Number is used as first probability distribution function of submodel：

$\left\{\begin{array}{c}m\left(\right\{C_1\left\}\right|X)=\frac{\underset{\left\{C_1\right\}|x_1\∩\mathrm{...}\∩\{C_1\left\}\right|x_n=\left\{C_1\right\}|X}{\Σ}{m}_1\left(\right\{C_1\left\}\right|x_1)...m_n\left(\right\{C_1\left\}\right|x_n)}{1-k}\\ m\left(\mathrm{\Φ}\right)=0\end{array}\right.$

Wherein, the contradiction factorIt is used to reflect the conflict journey of evidence Degree；

S44：The evidential probability partition function of all submodels：The rest may be inferred, for all of c submodel, obtains c evidence Probability distribution function m ({ C₁}|X)...m({C_c}|X)；

S45：Multi-model is exported：Son outputs of the X for each submodel is calculated respectivelyBy c probability in S44 point With function as each submodel weight, the submodel output result to gained is weighted fusion, then number of training It is expressed as according to the multi-model output of collection：

$\hat{y}=m\left(\right\{C_1\left\}\right|X){\hat{y}}_1+m\left(\right\{C_2\left\}\right|X){\hat{y}}_2+...m\left(\right\{C_c\left\}\right|X){\hat{y}}_c;$

Step S5 includes：

S51：The static multi-model is exported using autoregressive moving-average modelDynamic calibration is carried out, autoregression is slided The response of averaging model descriptive system current time tI.e.It is not only relevant with the observation before it in time, also be There is certain dependence in the present worth and lagged value of system disturbance, autoregressive moving-average model (p, q) is represented by

Wherein, p is autoregression；Q is rolling average item number；

S52：Linear shift operator B is introduced, is then hadTherefore the formula in S51 can transform to

$\mathrm{\Φ}\left(B\right){\hat{y}}_t=\mathrm{\θ}\left(B\right){\mathrm{\ϵ}}_t$

In formula, ε_tTo meet N (0, σ²) white noise sequence, Φ (B) and θ (B) for Shift operators B m ranks and n-order polynomial,

$\left\{\begin{array}{c}\mathrm{\Φ}\left(B\right)=(1-{\mathrm{\φ}}_{1}B-...{\mathrm{\φ}}_m{B}^m)\\ \mathrm{\θ}\left(B\right)=(1-{\mathrm{\θ}}_{1}B-...{\mathrm{\θ}}_m{B}^m)\end{array}\right.$

According to the basic theories of Hilbert space Linear Operators, the Random time sequence to meeting steady, normal state, zero-meanCan be approached with arbitrary accuracy with an autoregressive moving-average model (p, q).