CN103440368A - A multi-model dynamic soft sensor modeling method - Google Patents

Abstract

A multi-model dynamic soft measurement modeling method is characterized in that a plurality of sub-models are established by utilizing a self-adaptive fuzzy kernel clustering method and a least square support vector machine; then, a probability distribution function constructed by an evidence synthesis rule is used as a weight factor to fuse the sub-model outputs to obtain multi-model outputs; and finally, dynamically estimating the prediction error of the multiple models by combining an 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 the polyester industrial production run, be specifically related to a kind of multi-model dynamic soft-measuring method based on Combination Rules of Evidence Theory and autoregressive moving-average model.

Background technology

Shown in Fig. 1 is the basic process of esterification, and esterification is as the key link of whole polyester production process, and to stabilized polyester, production plays a decisive role.And in reaction unit the first esterifying kettle outlet Key Quality Indicator---the height of esterification yield directly affects the carrying out of subsequent reactions and the crystal property of polyester product, therefore usually by controlling esterification yield, control whole production run.But different polycondensating process has different requirements to esterification yield, so must reach required esterification yield by adjusting the operating conditionss such as reaction pressure and material quantity compare in production run.The flip-flop of operating conditions can cause the quality fluctuation of esterification yield, is unfavorable for the real-time control of whole production run.On the other hand, esterification process adopts two esterifiers to reach the esterification yield of technological requirement mostly, and the nonlinearity of reactive system, time variation and the uncertain difficulty of esterification yield on-line measurement that strengthened.

The on-the site analysis instrument not only expensive, maintaining is complicated, and while using analysis meter to be measured esterification yield, usually has hysteresis for a long time, the hydraulic performance decline that this will cause controlling quality, be difficult to meet production requirement.The basic skills of soft measurement is that Theory of Automatic Control and production run knowledge are combined, Applied Computer Techniques, for being difficult to, measure or temporary transient immeasurable leading variable, the auxiliary variable of selecting other easily to measure, infer and estimate by forming certain mathematical relation, replacing the function of on-the site analysis instrument with software.Flexible measurement method, because responding rapidly, can provide leading variable information continuously, and invests the advantage such as low, that maintaining is simple and obtain broad research and application in each field.But, in nearest decades, along with scientific and technical progress, modern industry production is more and more higher for the requirement of production run, data volume sharply increases, data type becomes increasingly complex, and operating mode is complicated and changeable, on the other hand, industrial process is all generally dynamic, static flexible measurement method can't reflect multidate information and the global property of industrial process usually, causes the bad adaptability of model, can't use for a long time.So simple, conventional flexible measurement method can not meet the needs of modern production technique in the past, be prone to that process characteristic coupling is not good, precision of prediction is low and the problem such as bad adaptability.

In order to obtain being applicable to more in general sense the flexible measurement method that the esterification yield data are predicted and analyzed, many improving one's methods is suggested, and formed plentiful and substantial achievement in research, mainly contain the following aspects: utilize various modeling methods, as Analysis on Mechanism, artificial neural network, least square method supporting vector machine and Gaussian process, sample set is set up to model and predict leading variable output; Utilize various intelligent optimization methods, as: particle cluster algorithm, genetic algorithm and evolution algorithm etc. carry out preferably the parameter of model; Utilize various clustering methods as the K-mean cluster, Fuzzy C-Means Clustering, stream shape cluster and affine propagation clustering method are polymerized to several subclasses by sample set, build several submodels and improve model prediction performance etc.

Summary of the invention

The present invention is directed to the prior art above shortcomings, a kind of multi-model dynamic soft measuring modeling method is provided, based on Combination Rules of Evidence Theory (D-S rule) and autoregressive moving-average model (ARMA), have better adaptability than prior art, in the soft measurement to esterification yield, precision is higher.

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 pre-service: select training sample data collection X_m*n, m is the sample dimension, and n is number of samples, and the rejecting abnormalities data are also carried out normalized to data;

S2, adaptive fuzzy kernel clustering are analyzed: adopt adaptive fuzzy kernel clustering method to training sample data collection X_m*ncarry out cluster, obtain fuzzy class degree of membership and each cluster centre of each sample, and automatically determine best clusters number c;

S3, set up submodel: adopt least square method supporting vector machine to carry out training study to the training sample set of c cluster, select the kernel function of gaussian kernel function as least square method supporting vector machine, set up and determine the parameter of c submodel by cross-validation method: penalty factor and nuclear parameter σ, and obtain the output of each submodel

S4, the output of the submodel based on Combination Rules of Evidence Theory are merged: calculate the evidential probability partition function value of each submodel, the weight using it as submodel, then carry out evidence fusion to the output of each submodel, obtains static multiple mode output

The mobilism of S5, model output: use the multi-model output of autoregressive moving-average model to current time t, right

dynamically adjust, at first judgement

whether be stationary sequence, if not, will

be converted to stationary sequence; Otherwise directly will

with true measurement, y subtracts each other, obtain a time series about output valve error delta y, then utilize autoregressive moving-average model (p, q) this time series is carried out to modeling, obtain the autoregressive moving-average model about predicated error, finally, above two models are combined and carry out model prediction, final dynamic multi-model is output as

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

S21: cluster objective function: to training sample set X={x_i| i=1,2...n}, the objective function of adaptive fuzzy kernel clustering method is defined as

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

In formula, μ_ij∈ [0,1];

j=1,2...c, m is the fuzzy control index, μ_ijbe the degree of membership value of i sample corresponding to j cluster, v_jbe j cluster centre, K(x_i, v_j) be gaussian kernel function;

S22: degree of membership is upgraded:

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

S23: cluster centre upgrades:

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

S23: cluster result evaluation: cluster is estimated the result of cluster Validity Index after finishing

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

Preferably, step S4 comprises:

S41: the evidential probability partition function of first submodel: using the framework of identification of all c submodels in evidence theory of cluster gained, and arbitrary submodel is considered as to burnt first C_j(j=1,2...c), for sample x₁, calculate it for first submodel, i.e. first burnt first C₁fuzzy class degree of membership, and, according to evidence theory, using it as an evidence, the probability assignments function of remembering this evidence is m ({ C₁| x₁)=μ₁₁;

And for all n test sample book data X={x_i| i=1,2...n}, obtain n bar evidence equally, and its probability assignments function is designated as m ({ C₁| x_i)=μ_i1(i=1,2...n);

Then, use Combination Rules of Evidence Theory to be merged these probability assignments functions, the probability assignments function using the probability assignments function after fusion as first submodel:

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

Wherein, the contradiction factor$k=\underset{\left\{C_1\right\}|{\mathrm{<figure-callout\; label="x"\; filenames="130812165453.png,130812165457.png"\; state="\{\{state\}\}">x</figure-callout>}}_1\&cap;...\&cap;\{C_1\left\}\right|x_n=\left\{C_1\right\}|X}{\mathrm{\&Sigma;}}{m}_1\left(\right\{C_1\left\}\right|x_1)...m_n\left\{C_1\right\}|x_n,$In order to reflect the conflict spectrum of evidence;

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

S45: multi-model output: calculate respectively the son output of X for each submodel

weight using the c in S44 probability assignments function as each submodel, be weighted fusion to the submodel Output rusults of gained, and the output of the multi-model of training sample data collection is expressed as:

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

Preferably, step S5 comprises:

S51: adopt autoregressive moving-average model to static multi-model output

carry out dynamic calibration, the response of autoregressive moving-average model descriptive system current time t

?

not only relevant with the observed reading before it in time, also with present worth and the lagged value of system disturbance, there is certain dependence, autoregressive moving-average model (p, q) can be expressed as

Wherein, p is the autoregression item; Q is the moving average item number;

S52: introduce linear shift operator B, have

therefore the formula in S51 is variable, be changed to

$\mathrm{\&Phi;}\left(B\right){\hat{y}}_t=\mathrm{\&theta;}\left(B\right){\mathrm{\&epsiv;}}_t$

In formula, ε_tfor meeting N (0, σ²) white noise sequence, the m rank that Φ (B) and θ (B) are Shift operators B and n rank polynomial expression.

$\left\{\begin{array}{c}\mathrm{\&Phi;}\left(B\right)=(1-{\mathrm{\&phi;}}_{1}B-...{\mathrm{\&phi;}}_m{B}^m)\\ \mathrm{\&theta;}\left(B\right)=(1-{\mathrm{\&theta;}}_{1}B-...{\mathrm{\&theta;}}_m{B}^m)\end{array}\right.$

According to the basic theories of Hilbert space Linear Operators, to meeting steadily, the random time sequence of normal state, zero-mean

an available autoregressive moving-average model (p, q) approaches with arbitrary accuracy.

At first the method utilizes the evidence composition rule to process the focusing advantage of uncertain information, each submodel obtained for the affine propagation clustering method has been set up a plurality of evidential probability partition functions, weight using it as each submodel, output to each submodel is weighted the multi-model output that fusion obtains test sample book, the concussion of having avoided switching mode to cause, eliminate the sample mistake and divided the impact on the model output accuracy, effectively improved the predictive ability of model; Then in conjunction with autoregressive moving-average model, the static multiple mode output error information obtained is carried out to dynamic calibration, significantly improved the dynamic response characteristic of system.

The accompanying drawing explanation

Shown in Fig. 1 is the process schematic diagram of esterification;

Shown in Fig. 2 is the process flow diagram of multi-model dynamic soft measuring modeling method of the present invention;

Shown in Fig. 3 is the parameter list of submodel C and σ;

Shown in Fig. 4 is the comparing result schematic diagram of LSSVM measuring method to predicted value and the artificial value of esterification yield test sample book;

Shown in Fig. 5 is the comparing result schematic diagram of SFKCM-LSSVM measuring method to predicted value and the artificial value of esterification yield test sample book;

Shown in Fig. 6 is the comparing result schematic diagram of AP-LS-SVM measuring method to predicted value and the artificial value of esterification yield test sample book;

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

Shown in Fig. 8 is that the Performance Ratio of the present invention and existing measuring method is than schematic diagram.

Embodiment

Below with reference to accompanying drawing of the present invention; technical scheme in the embodiment of the present invention is carried out to clear, complete description and discussion; obviously; as described herein is only a part of example of the present invention; it is not whole examples; embodiment based in the present invention, the every other embodiment that those of ordinary skills obtain under the prerequisite of not making creative work, belong to protection scope of the present invention.

For the ease of the understanding to the embodiment of the present invention, take specific embodiment below in conjunction with accompanying drawing and be further explained as example, and each embodiment does not form the restriction to the embodiment of the present invention.

At first utilize the evidence composition rule to process the focusing advantage of uncertain information, each submodel obtained for the affine propagation clustering method has been set up a plurality of evidential probability partition functions, weight using it as each submodel, output to each submodel is weighted the multi-model output that fusion obtains test sample book, the concussion of having avoided switching mode to cause, eliminate the sample mistake and divided the impact on the model output accuracy, effectively improved the predictive ability of model; Then in conjunction with autoregressive moving-average model (ARMA, Auto-Regressive Moving Average Model), the static multiple mode output error information obtained is carried out to dynamic calibration, significantly improved the dynamic response characteristic of system.

The technical scheme that this method technical solution problem is taked is:

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

S1: data pre-service: select training sample data collection X_m*n, m is the sample dimension, and n is number of samples, and the rejecting abnormalities data are also carried out normalized to data;

S2: the adaptive fuzzy kernel clustering is analyzed: adopt adaptive fuzzy kernel clustering method to sample set X_m*ncarry out cluster, obtain fuzzy class degree of membership and each cluster centre of each sample, and automatically determine best clusters number c;

S3: set up submodel: to each sub-training sample set, adopt least square method supporting vector machine (LS-SVM, below replace with LS-SVM) to carry out training study to it, and determine the parameter of each submodel.Select the kernel function of gaussian kernel function as LS-SVM, determine the parameter of each submodel by cross-validation method: penalty factor and nuclear parameter σ, as shown in Figure 3;

S4: the submodel output based on D-S is merged: obtain the evidential probability partition function value of each submodel according to the method for formula (6), the weight using it as submodel, then utilize the output of formula (7) to each submodel

carry out evidence fusion, obtain the output of multi-model

S5: the mobilism of model output: the static model on utilize obtain the multi-model output of sample

after, use the multi-model output of arma modeling to current time t

rightdynamically adjust.At first judgementwhether be stationary sequence, if not, will

be converted to stationary sequence; Otherwise directly will

with true measurement, y subtracts each other, and obtains a time series about output valve error delta y, then utilizes arma modeling (p, q) to carry out modeling to this time series, obtains the arma modeling about predicated error.Finally, above two models are combined and carry out model prediction, sample finally is output as

In step S2, the step of " adaptive fuzzy kernel clustering method " is as follows:

S21: cluster objective function: to training sample set X={x_i| i=1,2...n}, the objective function of adaptive fuzzy kernel clustering method is defined as

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

In formula, μ_ij∈ [0,1];

j=1,2...c, m is the fuzzy control index, μ_ijbe the degree of membership value of i sample corresponding to j cluster, v_jbe j cluster centre, K (x, y) is gaussian kernel function.

S22: degree of membership is upgraded:

${\mathrm{\&mu;}}_{\mathrm{ij}}={(1-K(x_i,v_j))}^{-1/(m-1)}/\underset{j-1}{\overset{c}{\mathrm{\&Sigma;}}}{(1-K(x_i,v_j))}^{-1/(m-1)}---\left(3\right)$

S23: cluster centre upgrades:

$v_i=\underset{i-1}{\overset{n}{\mathrm{\&Sigma;}}}{{\mathrm{\&mu;}}_{\mathrm{ij}}}^{m}K(x_i,v_j)x_i/\underset{i-1}{\overset{n}{\mathrm{\&Sigma;}}}{{\mathrm{\&mu;}}_{\mathrm{ij}}}^{m}K(x_i,v_j)x_i---\left(4\right)$

S23: cluster result evaluation: cluster adopts following Validity Index to be estimated the result of cluster after finishing

$V_{\mathrm{GX}}\left(c\right)=\frac{\underset{i-1}{\overset{c}{\mathrm{\&Sigma;}}}\underset{k-1}{\overset{n}{\mathrm{\&Sigma;}}}{\mathrm{\&mu;}}_{\mathrm{ik}}^m(1-K(v_k,x_i))+\frac{1}{c}\underset{i-1}{\overset{c}{\mathrm{\&Sigma;}}}(1-K(v_i,\stackrel{\&OverBar;}{v}))}{(1-K(v_k,v_i))_{i\&NotEqual;k}^{\min }}---\left(5\right)$

In step S4, the concrete steps of " the model prediction output based on Combination Rules of Evidence Theory " are as follows:

S41: the evidential probability partition function of first submodel: using the framework of identification of all c submodels in evidence theory of cluster gained, and arbitrary submodel is considered as to burnt first C_j(j=1,2...c).So, for sample x₁, at first according to formula (3), obtain it for first submodel, be also first burnt first C₁fuzzy class degree of membership.And, according to evidence theory, using it as an evidence, the probability assignments function of remembering this evidence is m ({ C₁| x_i)=μ₁₁.

And for all n test sample book data X={x_i| i=1,2...n}, in like manner, can obtain n bar evidence, and its probability assignments function is designated as m ({ C₁| x_i)=μ_i1(i=1,2...n).

Then, use Combination Rules of Evidence Theory to be merged these probability assignments functions, the probability assignments function after merging is as the probability assignments function of first submodel, shown in (6):

$\left\{\begin{array}{c}m\left(\right\{C_1\left\}\right|X)=\frac{\underset{\left\{C_1\right\}|x_1\&cap;...\&cap;\{C_1\left\}\right|x_n=\left\{C_1\right\}|X}{\mathrm{\&Sigma;}}{m}_1\left(\right\{C_1\left\}\right|x_1)...m_n\left\{C_1\right\}|x_n}{1-k}---\left(6\right)\\ m\left(\mathrm{\&Phi;}\right)=0\end{array}\right.$

Wherein, the contradiction factor$k=\underset{\left\{C_1\right\}|x_1\&cap;...\&cap;\{C_1\left\}\right|x_n=\left\{C_1\right\}|X}{\mathrm{\&Sigma;}}m\left(\right\{C_1\left\}\right|X)...m\left\{C_c\right\}|X,$Its size has reflected the conflict spectrum of evidence.

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

S45: multi-model output: calculate respectively X for each submodel LS-SVM1 ... the son output of LS-SVMc

weight using c obtained above probability assignments function as each submodel, be weighted fusion to the submodel Output rusults of gained, and the output of the multi-model of test sample book collection can be expressed as

$\hat{y}=m\left(\right\{C_1\left\}\right|X){\hat{y}}_1+m\left(\right\{C_2\left\}\right|X){\hat{y}}_2+...m\left(\right\{C_c\left\}\right|X){\hat{y}}_c---\left(7\right)$

In step S5, the concrete steps of " mobilism of model output " are:

S51: the static multiple mode output that adopts autoregressive moving-average model (ARMA) to obtain upper joint

carry out dynamic calibration.The response of arma modeling descriptive system current time t

?not only relevant with the observed reading before it in time, also with present worth and the lagged value of system disturbance, there is certain dependence.Arma modeling (p, q) can be expressed as

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

S52: introduce linear shift operator B, have

therefore variable being changed to of formula (8)

$\mathrm{\&phi;}\left(B\right){\hat{y}}_t=\mathrm{\&theta;}\left(B\right){\mathrm{\&epsiv;}}_t---\left(9\right)$

In formula, ε_tfor meeting N (0, σ²) white noise sequence, the m rank that Φ (B) and θ (B) are Shift operators B and n rank polynomial expression.

$\left\{\begin{array}{c}\mathrm{\&Phi;}\left(B\right)=(1-{\mathrm{\&phi;}}_{1}B-...{\mathrm{\&phi;}}_m{B}^m)\\ \mathrm{\&theta;}\left(B\right)=(1-{\mathrm{\&theta;}}_{1}B-...{\mathrm{\&theta;}}_m{B}^m)\end{array}\right.---\left(10\right)$

According to the basic theories of Hilbert space Linear Operators, to meeting steadily, the random time sequence of normal state, zero-mean

an available arma modeling (p, q) approaches with arbitrary accuracy.

Following according to real data for an embodiment:

The first step: to collection in worksite to data processed, obtain 1000 groups of normal datas.Using 900 groups of data wherein as training dataset X, for the foundation of model; Remaining 100 groups as test data set, for the predictive ability of testing model.

Second step: utilize the affine propagation clustering method to carry out cluster to training dataset, obtaining the optimum cluster number is c=4, corresponding cluster centre v.

The 3rd step: to resulting four the sub-training sample sets of cluster, utilize the LS-SVM method to set up four submodels, and training study, determine the parameter of LS-SVM through cross-validation method, as shown in Figure 3.

The 4th step: calculate the probability assignments function of each submodel of test sample book set pair according to formula (6), the weight using it as each submodel, then calculate test sample book X^testoutput with respect to each submodel

then utilize formula (7) to be merged the output of each submodel, obtain the output of test sample book

The 5th step: by the predicted value of the test sample book of current time t

with the manual analysis value, y subtracts each other, and the time series of output error Δ y is carried out to the ARMA modeling.Show that the estimated performance of algorithm is best when optimal factor p=4.

The estimated performance curve that Fig. 4-7 are three kinds of different measuring methods and measuring method of the present invention.From simulation result, can find out, the multi-model dynamic soft measuring modeling method based on Combination Rules of Evidence Theory and autoregressive moving-average model of the present invention is than single model and traditional multi-model process, and the estimated performance of esterification yield has been had to larger improvement.This is because esterification has than the characteristics of high non-linearity and multi-state, and a model need to be considered whole training samples during the single model modeling, and this has limited the precision of model; And although traditional multi-model process has carried out clustering to training dataset when modeling, set up respectively different submodels, but deeply do not consider the impact of the dynamic change of the difference of test sample book and training sample and dividing condition and process on the multi-model Output rusults at prediction test sample book output stage, so not significantly improvement of estimated performance.Method of the present invention utilizes the affine propagation clustering method by the first clustering of the identical sample similar with characteristic of operating mode, then take into full account the output stage in the prediction test sample book, the impact of each submodel output on the final Output rusults of sample, utilize Combination Rules of Evidence Theory structure weight to carry out the multi-model fusion to the output of submodel, obtain the final output of test sample book, the prediction deviation that the concussion of having avoided switching mode to cause and sample mistake minute cause; In addition, consider the dynamic perfromance of actual industrial process, utilize autoregressive moving-average model to export and carry out dynamic calibration multi-model, improved the dynamic response characteristic of system, thereby there is better adaptability, obtained fitting effect preferably in the soft measurement to esterification yield.

Listed the performance parameter of different flexible measurement methods in Fig. 8.As can be seen from Figure 8, adopt multi-model dynamic soft measuring modeling method provided by the present invention, more existing modeling method, root-mean-square error and maximum relative error all are improved.

The above; be only the present invention's embodiment preferably, but protection scope of the present invention is not limited to this, anyly is familiar with in technical scope that those skilled in the art disclose in the present invention; the variation that can expect easily or replacement, within all should being encompassed in protection scope of the present invention.Therefore, protection scope of the present invention should be as the criterion with the protection domain of claim.

Claims (4)

Translated fromChinese

1.一种多模型动态软测量建模方法，其特征在于，包括以下步骤：1. A multi-model dynamic soft sensor modeling method, is characterized in that, comprises the following steps:S1、数据预处理：选择训练样本数据集X_m*n，m为样本维数，n为样本个数，剔除异常数据并对数据进行归一化处理；S1. Data preprocessing: select the training sample data set X_m*n , m is the sample dimension, n is the number of samples, remove abnormal data and normalize the data;S2、自适应模糊核聚类分析：采用自适应模糊核聚类方法对训练样本数据集X_m*n进行聚类，得到每个样本的模糊类隶属度和各聚类中心，并自动确定出最佳聚类数目c；S2. Adaptive fuzzy kernel clustering analysis: use the adaptive fuzzy kernel clustering method to cluster the training sample data set X_m*n , obtain the fuzzy class membership degree and each cluster center of each sample, and automatically determine the The optimal number of clusters c;S3、建立子模型：采用最小二乘支持向量机对c个聚类的训练样本集进行训练学习，选择高斯核函数作为最小二乘支持向量机的核函数，通过交叉验证法建立并确定c个子模型的参数：惩罚因子C和核参数σ，并得到各个子模型的输出

S3. Establish sub-models: use the least squares support vector machine to train and learn the training sample sets of c clusters, select the Gaussian kernel function as the kernel function of the least squares support vector machine, and establish and determine c sub-models by cross-validation method Model parameters: penalty factor C and kernel parameter σ, and get the output of each sub-model

S4、基于证据理论合成规则的子模型输出融合：计算各子模型的证据概率分配函数值，将其作为子模型的权值因子，然后对各子模型的输出进行证据融合，得到静态多模型输出

S4. Sub-model output fusion based on evidence theory synthesis rules: Calculate the evidence probability distribution function value of each sub-model, use it as the weight factor of the sub-model, and then perform evidence fusion on the output of each sub-model to obtain static multi-model output

S5、模型输出的动态化：使用自回归滑动平均模型对当前时刻t的多模型输出，即对

进行动态调整，首先判断

是否是平稳序列，若不是，将

转换为平稳序列；否则直接将

和真实测量值y相减，得到一个关于输出值误差Δy的时间序列，然后利用自回归滑动平均模型(p,q)对该时间序列进行建模，得到关于预测误差的自回归滑动平均模型，最后，将以上两模型相结合进行模型预测，则最终的动态多模型输出为

S5. Dynamic model output: use the autoregressive moving average model to output multiple models at the current moment t, that is, to

For dynamic adjustment, first judge

Whether it is a stationary sequence, if not, will

converted to a stationary sequence; otherwise, the

Subtract the real measurement value y to get a time series about the output value error Δy, and then use the autoregressive moving average model (p, q) to model the time series to get the autoregressive moving average model about the forecast error, Finally, combining the above two models for model prediction, the final dynamic multi-model output is

2.根据权利要求1所述的多模型动态软测量建模方法，其特征在于，步骤S2中，所述自适应模糊核聚类方法的步骤包括：2. multi-model dynamic soft sensor modeling method according to claim 1, is characterized in that, in step S2, the step of described adaptive fuzzy kernel clustering method comprises:S21：聚类目标函数：对训练样本集X={x_i|i=1,2...n}，自适应模糊核聚类方法的目标函数定义为S21: Clustering objective function: For the training sample set X={_xi |i=1,2...n}, the objective function of the adaptive fuzzy kernel clustering method is defined as$\left\{\begin{array}{c}{\mathrm{<span><span>J</span>J</span>}}_{\mathrm{<span><span>\&Phi;</span>\&Phi;</span>}}<span><span>(</span>(</span>\mathrm{<span><span>U</span>u</span>}<span><span>,</span>,</span>\mathrm{<span><span>c</span>c</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\underset{\mathrm{<span><span>i</span>i</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\overset{\mathrm{<span><span>n</span>no</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}\underset{\mathrm{<span><span>j</span>j</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\overset{\mathrm{<span><span>c</span>c</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{\mathrm{<span><span>\&mu;</span>\&mu;</span>}}_{\mathrm{<span><span>ij</span>ij</span>}}^{\mathrm{<span><span>m</span>m</span>}}{<span><span>|</span>|</span><span><span>|</span>|</span><span><span>[</span>[</span>\mathrm{<span><span>1</span>1</span>}<span><span>-</span>-</span>\mathrm{<span><span>K</span>K</span>}<span><span>(</span>(</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>v</span>v</span>}}_{\mathrm{<span><span>j</span>j</span>}}<span><span>)</span>)</span><span><span>]</span>]</span><span><span>|</span>|</span><span><span>|</span>|</span>}^{\mathrm{<span><span>2</span>2</span>}}\\ \mathrm{<span><span>s</span>the\; s</span>}<span><span>.</span>.</span>\mathrm{<span><span>t</span>t</span>}<span><span>.</span>.</span>\mathrm{<span><span>U</span>u</span>}<span><span>\&Element;</span>\&Element;</span>{\mathrm{<span><span>M</span>m</span>}}_{\mathrm{<span><span>fc</span>fc</span>}}\end{array}\right.$式中，μ_ij∈[0,1];

j=1,2...c，m为模糊控制指数，μ_ij为第i个样本对应于第j个聚类的隶属度值，v_j为第j个聚类中心，K（x_i，v_j）为高斯核函数；In the formula, μ_ij ∈ [0,1];

j=1,2...c, m is the fuzzy control index, μ_ij is the membership value of the i-th sample corresponding to the j-th cluster, v_j is the j-th cluster center, K(xi_, v_j ) is a Gaussian kernel function;S22：隶属度更新：S22: Membership update:${\mathrm{<span><span>\&mu;</span>\&mu;</span>}}_{\mathrm{<span><span>ij</span>ij</span>}}<span><span>=</span>=</span>{<span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>-</span>-</span>\mathrm{<span><span>K</span>K</span>}<span><span>(</span>(</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>v</span>v</span>}}_{\mathrm{<span><span>j</span>j</span>}}<span><span>)</span>)</span><span><span>)</span>)</span>}^{<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span><span><span>(</span>(</span>\mathrm{<span><span>m</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}<span><span>)</span>)</span>}<span><span>/</span>/</span>\underset{\mathrm{<span><span>j</span>j</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\overset{\mathrm{<span><span>c</span>c</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{<span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>-</span>-</span>\mathrm{<span><span>K</span>K</span>}<span><span>(</span>(</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>v</span>v</span>}}_{\mathrm{<span><span>j</span>j</span>}}<span><span>)</span>)</span><span><span>)</span>)</span>}^{<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}<span><span>/</span>/</span><span><span>(</span>(</span>\mathrm{<span><span>m</span>m</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}<span><span>)</span>)</span>}<span><span>;</span>;</span>$S23：聚类中心更新：S23: Cluster center update:${\mathrm{<span><span>v</span>v</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>=</span>=</span>\underset{\mathrm{<span><span>i</span>i</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\overset{\mathrm{<span><span>n</span>no</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{{\mathrm{<span><span>\&mu;</span>\&mu;</span>}}_{\mathrm{<span><span>ij</span>ij</span>}}}^{\mathrm{<span><span>m</span>m</span>}}\mathrm{<span><span>K</span>K</span>}<span><span>(</span>(</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>v</span>v</span>}}_{\mathrm{<span><span>j</span>j</span>}}<span><span>)</span>)</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>/</span>/</span>\underset{\mathrm{<span><span>i</span>i</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\overset{\mathrm{<span><span>n</span>no</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{{\mathrm{<span><span>\&mu;</span>\&mu;</span>}}_{\mathrm{<span><span>ij</span>ij</span>}}}^{\mathrm{<span><span>m</span>m</span>}}\mathrm{<span><span>K</span>K</span>}<span><span>(</span>(</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>v</span>v</span>}}_{\mathrm{<span><span>j</span>j</span>}}<span><span>)</span>)</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>;</span>;</span>$S23：聚类结果评价：聚类结束后，对有效性指标对聚类的结果进行评价S23: Evaluation of clustering results: after the clustering is completed, evaluate the clustering results of the effectiveness indicators${\mathrm{<span><span>V</span>V</span>}}_{\mathrm{<span><span>GX</span>GX</span>}}<span><span>(</span>(</span>\mathrm{<span><span>c</span>c</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\frac{\underset{\mathrm{<span><span>i</span>i</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\overset{\mathrm{<span><span>c</span>c</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}\underset{\mathrm{<span><span>k</span>k</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\overset{\mathrm{<span><span>n</span>no</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}{\mathrm{<span><span>\&mu;</span>\&mu;</span>}}_{\mathrm{<span><span>ik</span>ik</span>}}^{\mathrm{<span><span>m</span>m</span>}}<span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>-</span>-</span>\mathrm{<span><span>K</span>K</span>}<span><span>(</span>(</span>{\mathrm{<span><span>v</span>v</span>}}_{\mathrm{<span><span>k</span>k</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>)</span>)</span><span><span>)</span>)</span><span><span>+</span>+</span>\frac{\mathrm{<span><span>1</span>1</span>}}{\mathrm{<span><span>c</span>c</span>}}\underset{\mathrm{<span><span>i</span>i</span>}<span><span>-</span>-</span>\mathrm{<span><span>1</span>1</span>}}{\overset{\mathrm{<span><span>c</span>c</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}}<span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>-</span>-</span>\mathrm{<span><span>K</span>K</span>}<span><span>(</span>(</span>{\mathrm{<span><span>v</span>v</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>,</span>,</span>\stackrel{<span><span>\&OverBar;</span>\&OverBar;</span>}{\mathrm{<span><span>v</span>v</span>}}<span><span>)</span>)</span><span><span>)</span>)</span>}{<span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>-</span>-</span>\mathrm{<span><span>K</span>K</span>}<span><span>(</span>(</span>{\mathrm{<span><span>v</span>v</span>}}_{\mathrm{<span><span>k</span>k</span>}}<span><span>,</span>,</span>{\mathrm{<span><span>v</span>v</span>}}_{\mathrm{<span><span>i</span>i</span>}}<span><span>)</span>)</span><span><span>)</span>)</span>_{\mathrm{<span><span>i</span>i</span>}<span><span>\&NotEqual;</span>\&NotEqual;</span>\mathrm{<span><span>k</span>k</span>}}^{\mathrm{<span><span>min</span>min</span>}}}<span><span>.</span>.</span>$3.根据权利要求1所述的多模型动态软测量建模方法，其特征在于，步骤S4包括：3. multi-model dynamic soft sensor modeling method according to claim 1, is characterized in that, step S4 comprises:S41：第一个子模型的证据概率分配函数：将聚类所得的所有c个子模型作为证据理论中的辨识框架，并将任一子模型视为焦元C_j(j=1,2...c)，对于样本x₁，计算其对于第一个子模型，即第一个焦元C₁的模糊类隶属度，并根据证据理论，将其作为一条证据，记该证据的概率分配函数为m({C₁}|x₁)=μ₁₁；S41: Evidence probability distribution function of the first sub-model: use all the c sub-models obtained by clustering as the identification framework in the evidence theory, and regard any sub-model as the focal element C_j (j=1,2.. .c), for the sample x₁ , calculate its fuzzy class membership degree for the first sub-model, that is, the first focal element C₁ , and use it as a piece of evidence according to the evidence theory, and record the probability distribution function of the evidence is m({C₁ }|x₁ )=μ₁₁ ;而对于所有的n个测试样本数据X={x_i|i=1,2...n}，同样得到n条证据，其概率分配函数记为m({C₁}|x_i)=μ_i1(i=1,2...n)；And for all n test sample data X={x_i |i=1,2...n}, also get n pieces of evidence, and its probability distribution function is recorded as m({C₁ }|xi₎ =μ_i1 (i=1,2...n);然后，使用证据理论合成规则对这些概率分配函数进行融合，将融合后的概率分配函数作为第一个子模型的概率分配函数：Then, these probability distribution functions are fused using evidence-theoretic composition rules, and the fused probability distribution function is used as the probability distribution function of the first sub-model:$\left\{\begin{array}{c}\mathrm{<span><span>m</span>m</span>}<span><span>(</span>(</span><span><span>\{</span>\{</span>{\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>\}</span>\}</span><span><span>|</span>|</span>\mathrm{<span><span>X</span>x</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\frac{\underset{<span><span>\{</span>\{</span>{\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>\}</span>\}</span><span><span>|</span>|</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>\&cap;</span>\&cap;</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>\&cap;</span>\&cap;</span><span><span>\{</span>\{</span>{\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>\}</span>\}</span><span><span>|</span>|</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>n</span>no</span>}}<span><span>=</span>=</span><span><span>\{</span>\{</span>{\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>\}</span>\}</span><span><span>|</span>|</span>\mathrm{<span><span>X</span>x</span>}}{\mathrm{<span><span>\&Sigma;</span>\&Sigma;</span>}}{\mathrm{<span><span>m</span>m</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>(</span>(</span><span><span>\{</span>\{</span>{\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>\}</span>\}</span><span><span>|</span>|</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>)</span>)</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span>{\mathrm{<span><span>m</span>m</span>}}_{\mathrm{<span><span>n</span>no</span>}}<span><span>\{</span>\{</span>{\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>\}</span>\}</span><span><span>|</span>|</span>{\mathrm{<span><span>x</span>x</span>}}_{\mathrm{<span><span>n</span>no</span>}}}{\mathrm{<span><span>1</span>1</span>}<span><span>-</span>-</span>\mathrm{<span><span>k</span>k</span>}}\\ \mathrm{<span><span>m</span>m</span>}<span><span>(</span>(</span>\mathrm{<span><span>\&Phi;</span>\&Phi;</span>}<span><span>)</span>)</span><span><span>=</span>=</span>\mathrm{<span><span>0</span>0</span>}\end{array}\right.$其中，矛盾因子$k=\underset{\left\{C_1\right\}|x_1\&cap;...\&cap;\{C_1\left\}\right|x_n=\left\{C_1\right\}|X}{\mathrm{\&Sigma;}}{m}_1\left(\right\{C_1\left\}\right|x_1)...m_n\left\{C_1\right\}|x_n,$用以反映证据的冲突程度；Among them, the contradiction factor$k=\underset{\left\{C_1\right\}|x_1\&cap;...\&cap;\{C_1\left\}\right|x_{\mathrm{no}}=\left\{C_1\right\}|x}{\mathrm{\&Sigma;}}{m}_1\left(\right\{C_1\left\}\right|x_1)...m_{\mathrm{no}}\left\{C_1\right\}|x_{\mathrm{no}},$ used to reflect the degree of conflict in the evidence;S44：所有子模型的证据概率分配函数：依此类推，对于所有的c个子模型，按照步骤S43，得到c个证据概率分配函数m({C₁}|X)...m{C_c}|X；S44: Evidence probability distribution functions of all sub-models: by analogy, for all c sub-models, follow step S43 to obtain c evidence probability distribution functions m({C₁ }|X)...m{C_c } |X;S45：多模型输出：分别计算出X对于各子模型的子输出

将S44中的c个概率分配函数作为各子模型的权值因子，对所得的子模型输出结果进行加权融合，则训练样本数据集的多模型输出表示为：S45: Multi-model output: separately calculate the sub-output of X for each sub-model

The c probability distribution functions in S44 are used as the weight factors of each sub-model, and the output results of the obtained sub-models are weighted and fused, then the multi-model output of the training sample data set is expressed as:$\stackrel{<span><span>^</span>^</span>}{\mathrm{<span><span>y</span>the\; y</span>}}<span><span>=</span>=</span>\mathrm{<span><span>m</span>m</span>}<span><span>(</span>(</span><span><span>\{</span>\{</span>{\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>\}</span>\}</span><span><span>|</span>|</span>\mathrm{<span><span>X</span>x</span>}<span><span>)</span>)</span>{\stackrel{<span><span>^</span>^</span>}{\mathrm{<span><span>y</span>the\; y</span>}}}_{\mathrm{<span><span>1</span>1</span>}}<span><span>+</span>+</span>\mathrm{<span><span>m</span>m</span>}<span><span>(</span>(</span><span><span>\{</span>\{</span>{\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>\}</span>\}</span><span><span>|</span>|</span>\mathrm{<span><span>X</span>x</span>}<span><span>)</span>)</span>{\stackrel{<span><span>^</span>^</span>}{\mathrm{<span><span>y</span>the\; y</span>}}}_{\mathrm{<span><span>2</span>2</span>}}<span><span>+</span>+</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span>\mathrm{<span><span>m</span>m</span>}<span><span>(</span>(</span><span><span>\{</span>\{</span>{\mathrm{<span><span>C</span>C</span>}}_{\mathrm{<span><span>c</span>c</span>}}<span><span>\}</span>\}</span><span><span>|</span>|</span>\mathrm{<span><span>X</span>x</span>}<span><span>)</span>)</span>{\stackrel{<span><span>^</span>^</span>}{\mathrm{<span><span>y</span>the\; y</span>}}}_{\mathrm{<span><span>c</span>c</span>}}<span><span>.</span>.</span>$4.根据权利要求1所述的多模型动态软测量建模方法，其特征在于，步骤S5包括：4. multi-model dynamic soft sensor modeling method according to claim 1, is characterized in that, step S5 comprises:S51：采用自回归滑动平均模型对所述静态的多模型输出

进行动态校正，自回归滑动平均模型描述系统当前时刻t的响应

即

不仅在时间上同它以前的观测值有关，还与系统扰动的现值和滞后值存在一定的依存关系，自回归滑动平均模型(p,q)可表示为S51: Using an autoregressive moving average model to output the static multi-model

For dynamic correction, the autoregressive moving average model describes the response of the system at the current moment t

Right now

It is not only related to its previous observations in time, but also has a certain dependence relationship with the present value and lag value of the system disturbance. The autoregressive moving average model (p,q) can be expressed as

其中，p为自回归项；q为移动平均项数；Among them, p is the autoregressive item; q is the number of moving average items;S52：引入线性推移算子B，则有

故S51中的公式可变换为S52: Introducing the linear transition operator B, then there is

Therefore, the formula in S51 can be transformed into$\mathrm{<span><span>\&phi;</span>\&phi;</span>}<span><span>(</span>(</span>\mathrm{<span><span>B</span>B</span>}<span><span>)</span>)</span>{\stackrel{<span><span>^</span>^</span>}{\mathrm{<span><span>y</span>the\; y</span>}}}_{\mathrm{<span><span>t</span>t</span>}}<span><span>=</span>=</span>\mathrm{<span><span>\&theta;</span>\&theta;</span>}<span><span>(</span>(</span>\mathrm{<span><span>B</span>B</span>}<span><span>)</span>)</span>{\mathrm{<span><span>\&epsiv;</span>\&epsiv;</span>}}_{\mathrm{<span><span>t</span>t</span>}}$式中，ε_t为满足N(0,σ²)的白噪声序列，Φ(B)和θ(B)为推移算子B的m阶和n阶多项式。In the formula, ε_t is a white noise sequence satisfying N(0,σ² ), and Φ(B) and θ(B) are m-order and n-order polynomials of shift operator B.$\left\{\begin{array}{c}\mathrm{<span><span>\&Phi;</span>\&Phi;</span>}<span><span>(</span>(</span>\mathrm{<span><span>B</span>B</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>-</span>-</span>{\mathrm{<span><span>\&phi;</span>\&phi;</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>B</span>B</span>}<span><span>-</span>-</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span>{\mathrm{<span><span>\&phi;</span>\&phi;</span>}}_{\mathrm{<span><span>m</span>m</span>}}{\mathrm{<span><span>B</span>B</span>}}^{\mathrm{<span><span>m</span>m</span>}}<span><span>)</span>)</span>\\ \mathrm{<span><span>\&theta;</span>\&theta;</span>}<span><span>(</span>(</span>\mathrm{<span><span>B</span>B</span>}<span><span>)</span>)</span><span><span>=</span>=</span><span><span>(</span>(</span>\mathrm{<span><span>1</span>1</span>}<span><span>-</span>-</span>{\mathrm{<span><span>\&theta;</span>\&theta;</span>}}_{\mathrm{<span><span>1</span>1</span>}}\mathrm{<span><span>B</span>B</span>}<span><span>-</span>-</span><span><span>.</span>.</span><span><span>.</span>.</span><span><span>.</span>.</span>{\mathrm{<span><span>\&theta;</span>\&theta;</span>}}_{\mathrm{<span><span>m</span>m</span>}}{\mathrm{<span><span>B</span>B</span>}}^{\mathrm{<span><span>m</span>m</span>}}<span><span>)</span>)</span>\end{array}\right.$根据希尔伯特空间上线性算子的基本理论，对满足平稳、正态、零均值的随机时间序列

可用一自回归滑动平均模型(p,q)以任意精度逼近。According to the basic theory of linear operators on Hilbert space, for random time series that satisfy stationary, normal, zero mean

It can be approximated with arbitrary precision by an autoregressive moving average model (p,q).

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