CN101713983B - Semiconductor process monitoring method based on independent component analysis and Bayesian inference - Google Patents

Abstract

The invention discloses a semiconductor process monitoring method based on independent component analysis and Bayesian inference, comprising the following step of: firstly, dividing working conditions according to the mixed data of the semiconductor process, conducting the independent component analysis for each working condition data, and establishing a corresponding independent component analysis model; and then integrating and combining the monitoring information under the different working conditions by a Bayesian inference method to obtain a final monitoring result. In addition, the invention can also acquire the working condition information of current monitoring data by a posterior probability analysis method, that is to say, the invention can judge that the current monitoring data is in what process operation working condition; compared with the present other methods, the invention can not only greatly enhance the monitoring effect of the semiconductor process, but also largely improve the dependence of the monitoring method on process knowledge and enhance the comprehensive ability and the operating confidence of process operators on the process, thereby being more beneficial to the implementation of the automation of the semiconductor process.

Description

Semiconductor processes monitoring method based on independent component analysis and Bayesian inference

Technical field

The invention belongs to semi-conductor industry process control field, particularly a kind of process monitoring method based on independent component analysis and Bayesian inference.

Background technology

In recent years, the monitoring problem of semi-conductor industry production run more and more obtains the extensive attention of industry member and academia.On the one hand and since semi-conductor industry process itself to product quality require highly, the process that how to prevent effectively produces poor quality or underproof product is the problem that presses for solution.On the other hand, if process is not well monitored, work accident might take place, the lighter influences the quality of product, and weight person will cause the loss of life and property.In addition, the result that semiconductor processes is monitored acquisition can also instruct the improvement of production run and production technology conversely.Therefore, process monitoring has become the research focus of semi-conductor industry production run and has pressed for one of problem of solution.

As a kind of typical batch production process, traditional semiconductor processes monitoring method is except the method based on mechanism model, adopt the multivariate statistical analysis method of multidirectional form mostly, such as multidirectional pca method (MPCA) and multidirectional offset minimum binary method (MPLS) etc.Under the situation that mechanism model is difficult to obtain, become the main stream approach of semiconductor processes monitoring based on the multivariate statistical analysis method of data-driven.But the second-order statistic information that traditional multivariate statistical analysis method can only the leaching process data can not be handled high-order statistic information effectively.By contrast, independent component analysis method (ICA) has himself advantage, the present invention to adopt this method to substitute original multivariate statistical analysis method at the high-order statistic message context of leaching process data process data is carried out information extraction.In addition, because the variation of semiconductor processes product, this process also will operate under the different operation operating modes.Traditional monitoring method hypothesis process operation can't satisfy the monitoring requirement of semiconductor processes under single operating mode.Even the different operating operating mode to process is carried out modeling respectively, also can't reach satisfied monitoring effect.Because when new process data is monitored, need cohesive process knowledge that the operation operating mode of these data is judged, and choose corresponding monitoring model, this has just strengthened the dependence of monitoring method to procedural knowledge greatly, is unfavorable for that the robotization of semiconductor processes is implemented.

Summary of the invention

The objective of the invention is at the deficiencies in the prior art, a kind of semiconductor processes monitoring method based on independent component analysis and Bayesian inference is provided.

The objective of the invention is to be achieved through the following technical solutions:

A kind of semiconductor processes monitoring method based on independent component analysis and Bayesian inference may further comprise the steps:

(1) utilize the data of each nominal situation of Distributed Control System (DCS) collection semiconductor process to form the three-dimensional training sample set that modeling is used: X=[X ₁X ₂, X _C].Wherein, $X_c\∈R^{I_c\×J\×K},$ C=1,2 ..., C is the data matrix corresponding to process operating mode c, I _cBe batch number under this operating mode, J is the variable number, and K counts for the sampled data of each batch.Deposit these data in historical data base respectively.

(2) respectively the data under the different operating modes are expanded into I along separately a batch direction _c* JK two-dimensional data matrix is carried out pre-service and normalization to it, makes that promptly the average of each process variable is zero, and variance is 1, obtains new data matrix collection to be ${\stackrel{\‾}{X}}_c\∈R^{I_c\×JK}.$

(3) along the time point direction each data matrix is arranged again, obtained new data matrix and concentrate and to be ${\stackrel{=}{X}}_c\∈R^{{\mathrm{KI}}_c\×J}.$

(4) adopt the independent component analysis method that each new data matrix is handled, set up the independent component analysis model, obtain residual matrix ${\stackrel{=}{E}}_c\∈R^{{\mathrm{KI}}_c\×J},$ And structure I ²The monitoring and statistics limit of statistic.

(5) at residual matrix The monitoring and statistics limit of structure SPE statistic.

(6) deposit in historical data base and the real-time data base modeling data and each model parameter standby.

(7) collect new process data, and it is carried out pre-service and normalization.

(8) adopt different operating mode models that it is monitored respectively, promptly set up statistic I ²And SPE.

(9) calculate the posterior probability values of current Monitoring Data under each operation operating mode by the Bayesian inference method, and the probability of malfunction value of computational data.Then, the new statistic FI of structure ²With the monitoring result under the integrated different operating modes of FSPE, and provide corresponding performance analysis and positioning result.

The invention has the beneficial effects as follows: the present invention is by carrying out independent component analysis and modeling respectively to each floor data.Then, introducing Bayesian inference method is carried out integrated and comprehensive to the monitoring information under the different operating modes, obtain last monitoring result.In addition, by the posterior probability analytical technology, the present invention can also obtain the work information of current Monitoring Data.Compare other present semiconductor processes monitoring method, the present invention not only can improve the monitoring effect of semiconductor processes greatly, and improved the dependence of monitoring method to a great extent to procedural knowledge, strengthened understandability and the operation confidence of process operator to process, the robotization that helps semiconductor processes is more implemented.

Description of drawings

Fig. 1 is the data scatter diagram of semiconductor processes;

Fig. 2 is the monitoring result of the inventive method to the normal lot data of semiconductor processes;

Fig. 3 is the monitoring result of MPCA to the normal lot data of semiconductor processes;

Fig. 4 is the monitoring result of the inventive method to semiconductor procedure fault lot data;

Fig. 5 is the monitoring result of MPCA method to semiconductor procedure fault lot data;

Fig. 6 is that the inventive method is to normal batch of performance analysis of semiconductor processes and positioning result.

Embodiment

The present invention is directed to the monitoring problem of semiconductor processes, at first utilize the lot data under the Distributed Control System (DCS) collection different operating operating mode, and it is carried out operating mode divide.Respectively at different operation operating modes, set up corresponding independent component analysis model then, and set up two monitoring and statistics amount I ²With SPE and corresponding statistics limit I thereof _Lim ²And SPE _LimDeposit in all process model parameters in the database standby.When new lot data is monitored, at first utilize the monitoring model under the different operating operating mode that it is monitored, obtain corresponding monitoring result.Obtain the operating mode posterior probability of these data then by the Bayesian inference method,, be integrated into last monitoring result in conjunction with its fault probability of happening under each operating mode.In addition, by current data being carried out the analysis of operating mode posterior probability, the present invention can also obtain the work information of these data and it is located accordingly, has strengthened the understanding of operation slip-stick artist to process greatly, has improved their operation confidence.

The key step of the technical solution used in the present invention is as follows respectively:

The first step utilizes the data of each nominal situation of Distributed Control System (DCS) collection semiconductor process to form the three-dimensional training sample set that modeling is used: X=[X ₁X ₂, X _C].Wherein, $X_c\∈R^{I_c\×J\×K},$ C=1,2 ..., C is the data matrix corresponding to process operating mode c, I _cBe batch number under this operating mode, J is the variable number, and K counts for the sampled data of each batch.Deposit these data in historical data base respectively;

Second step expanded into I with the data under the different operating modes along separately a batch direction respectively _c* JK two-dimensional data matrix is carried out pre-service and normalization to it, makes that promptly the average of each process variable is zero, and variance is 1, obtains new data matrix collection to be ${\stackrel{\‾}{X}}_c\∈R^{I_c\×JK};$

In historical data base, the process data that collects is carried out pre-service, reject outlier and tangible coarse error information, in order to make the yardstick of process data can not have influence on the result of monitoring, data to different variablees are carried out normalized respectively, the average that is each variable is zero, and variance is 1.Like this, the data of various process variable just are under the identical yardstick, can not have influence on follow-up monitoring effect afterwards.

The 3rd step arranged each data matrix along the time point direction again, obtained new data matrix and concentrated and to be ${\stackrel{=}{X}}_c\∈R^{{\mathrm{KI}}_c\×J};$

Traditional semiconductor processes monitoring method need be predicted unknown-value when monitoring new lot data.For fear of this problem, we rearrange the data matrix.Like this, monitor sample just becomes single sampled data points by original whole lot data, has avoided the forecasting problem to batch unknown-value preferably.

The 4th step adopted the independent component analysis method that each new data matrix is handled, and set up the independent component analysis model, obtained residual matrix ${\stackrel{=}{E}}_c\∈R^{{\mathrm{KI}}_c\×J},$ And structure I ²The monitoring and statistics limit of statistic;

Adopt independent component analysis (ICA) method to each new data matrix

Handle, extract high-order statistic information, set up the independent component analysis model.Analyze by ICA, can obtain the independent component matrix of this data matrix $S_c\∈R^{r\×K{I}_c},$ Hybrid matrix A _c∈ R ^{J * r}, separation matrix W _c∈ R ^{R * J}And residual matrix ${\stackrel{=}{E}}_c\∈R^{{\mathrm{KI}}_c\×J}$ As follows:

${\stackrel{=}{X}}_c=A_c{S}_c+{\stackrel{=}{E}}_c$

$S_c=W_c{\stackrel{=}{X}}_c$

${\stackrel{=}{E}}_c={\stackrel{=}{X}}_c-A_c{S}_c$

Wherein, c=1,2 ..., C, r are the independent component number of choosing.Then, structure I ²Statistic also utilizes the Density Estimator method to provide its corresponding monitoring and statistics limit I _{Lim, c} ², that is:

$\hat{f}(I_c^2,H)=\frac{1}{{\mathrm{KI}}_c}\underset{i=1}{\overset{{\mathrm{KI}}_c}{\mathrm{\Σ}}}K\left(H^{-1/2}(I_c^2-I_{c,i}^2)\right).---\left(1.16\right)$

Wherein, K () is a kernel function, is chosen for the gaussian kernel form usually, and H is the bandwidth parameter matrix of kernel function, can easy choice be the form at diagonal angle, I _{C, i} ²I for corresponding data under the operating mode c ²The statistics value.Like this, we just can obtain I ²The probability density distribution information of statistic, thus can ask for its statistics limit I under certain degree of confidence easily _{Lim, c} ²Value.

The 5th step is at residual matrix The monitoring and statistics limit of structure SPE statistic;

On the basis of previous step to residual matrix

Set up the SPE statistic and calculate its corresponding monitoring and statistics limit SPE _{Lim, c}, that is:

${\mathrm{SPE}}_{c,\mathrm{tr},i}={\stackrel{=}{E}}_{c,i}{\stackrel{=}{E}}_{c,i}^T(i=\mathrm{1,2},\·\·\·,{\mathrm{KI}}_c)---\left(1.17\right)$

Wherein, SPE _{C, tr, i}Obeying parameter is the χ of g and h ²Distribute,

g·h＝mean(SPE _c，tr)

(1.18)

2g ²h＝var(SPE _c，tr)

Therefore, the monitoring and statistics of SPE statistic statistic limit also can be obtained easily, i.e. SPE _Lim～g χ _h ²

The 6th step deposited in historical data base and the real-time data base modeling data and each model parameter standby;

The 7th step was collected new process data, and it is carried out pre-service and normalization;

For the data sample of newly collecting in the process, except it is carried out the pre-service, the model parameter when adopting modeling is in addition carried out normalization to this data point, promptly deducts the modeling average and divided by the modeling standard deviation.

The 8th step adopted different operating mode models that it is monitored respectively, promptly set up statistic I ²And SPE;

Adopt the model under the different operating modes that new data are monitored, promptly calculate its corresponding I ²As follows with the value of SPE statistic:

$s_c=W_c{\stackrel{=}{x}}_{\mathrm{new}}$

${\stackrel{=}{e}}_{\mathrm{new},c}={\stackrel{=}{x}}_{\mathrm{new}}-A_c{s}_c---\left(1.19\right)$

$I_{\mathrm{new},c}^2=s_c^T{s}_c$

${\mathrm{SPE}}_{\mathrm{new},c}={\stackrel{=}{e}}_{\mathrm{new},c}{\stackrel{=}{e}}_{\mathrm{new},c}^T---\left(1.20\right)$

Wherein, c=1,2 ..., C, s _cBe the independent component vector that corresponding data extracted,

Be residual vector.

The 9th step was calculated the posterior probability values of current Monitoring Data under each operation operating mode by the Bayesian inference method, and the probability of malfunction value of computational data.Then, the new statistic FI of structure ²With the monitoring result under the integrated different operating modes of FSPE, and provide corresponding performance analysis and positioning result.

At first calculate the posterior probability values of current Monitoring Data under each process operation operating mode, that is: by the Bayesian inference method

$P_{I^2}\left(c\right|{\stackrel{=}{x}}_{\mathrm{new}})=\frac{P_{I^2}(c,{\stackrel{=}{x}}_{\mathrm{new}})}{P_{I^2}\left({\stackrel{=}{x}}_{\mathrm{new}}\right)}=\frac{P_{I^2}\left({\stackrel{=}{x}}_{\mathrm{new}}\right|c)P\left(c\right)}{\underset{c=1}{\overset{C}{\mathrm{\Σ}}}\left[P_{I^2}\left({\stackrel{=}{x}}_{\mathrm{new}}\right|c)P\left(c\right)\right]}$

$P_{\mathrm{SPE}}\left(c\right|{\stackrel{=}{x}}_{\mathrm{new}})=\frac{P_{\mathrm{SPE}}(c,{\stackrel{=}{x}}_{\mathrm{new}})}{P_{\mathrm{SPE}}\left({\stackrel{=}{x}}_{\mathrm{new}}\right)}=\frac{P_{\mathrm{SPE}}\left({\stackrel{=}{x}}_{\mathrm{new}}\right|c)P\left(c\right)}{\underset{c=1}{\overset{C}{\mathrm{\Σ}}}\left[P_{\mathrm{SPE}}\left({\stackrel{=}{x}}_{\mathrm{new}}\right|c)P\left(c\right)\right]}$

Wherein,

With

Be the marginal distribution probability, be defined as follows here:

$P_{I^2}\left({\stackrel{=}{x}}_{\mathrm{new}}\right|c)=\exp \{-\frac{I_{\mathrm{new},c}^2}{I_{\lim ,c}^2}\}$

$P_{\mathrm{SPE}}\left({\stackrel{=}{x}}_{\mathrm{new}}\right|c)=\exp \{-\frac{{\mathrm{SPE}}_{\mathrm{new},c}}{{\mathrm{SPE}}_{\lim ,c}}\}$

P (c) is a prior probability, but simple computation is $P\left(c\right)=\frac{{\mathrm{KI}}_c}{\mathrm{KI}}.$

Then, the probability of malfunction of calculating current data under each operating mode is as follows:

$P_{f,I^2}^c\left({\stackrel{=}{x}}_{\mathrm{new}}\right)=\mathrm{Pr}\{I_c^2\left({\stackrel{=}{x}}_{\mathrm{tr},c}\right)\≤I_{\mathrm{new},c}^2\}$

$P_{f,\mathrm{SPE}}^c\left({\stackrel{=}{x}}_{\mathrm{new}}\right)=\mathrm{Pr}\{{\mathrm{SPE}}_c\left({\stackrel{=}{x}}_{\mathrm{tr},c}\right)\≤{\mathrm{SPE}}_{\mathrm{new},c}\}$

Wherein,

Be the modeling data under the corresponding operating mode, Pr{} represents probable value.

Calculate last monitoring result, the monitoring result under the promptly integrated different operating operating mode is as follows:

${\mathrm{FI}}_{\mathrm{new}}^2=\underset{c=1}{\overset{C}{\mathrm{\Σ}}}\left[P_{I^2}\left(c\right|{\stackrel{=}{x}}_{\mathrm{new}})P_{f,I^2}^c\left({\stackrel{=}{x}}_{\mathrm{new}}\right)\right]$

${\mathrm{FSPE}}_{\mathrm{new}}=\underset{c=1}{\overset{C}{\mathrm{\Σ}}}\left[P_{\mathrm{SPE}}\left(c\right|{\stackrel{=}{x}}_{\mathrm{new}})P_{f,\mathrm{SPE}}^c\left({\stackrel{=}{x}}_{\mathrm{new}}\right)\right]$

By analyzing the posterior probability values of current Monitoring Data and each operation operating mode, it is carried out performance analysis and is positioned such that:

Wherein, the value of posterior probability is big more, illustrates that the degree of association of current Monitoring Data and corresponding operating mode is big more.Otherwise the possibility that these data of the more little explanation of value are in corresponding operating mode is just more little.

Below in conjunction with a concrete semiconductor processes example validity of the present invention is described.The data of this process are altogether 129 batch datas from three groups of experiments of TIX, comprising the data under 108 batches of nominal situations and 21 batches of fault datas.By the initial analysis to data, finding has the data deletion condition of two batches of processes more serious, removes after these two batches of deficiencies of data, and one has 107 batches of normal data and 20 batches of fault datas.The source of fault mainly is that the variation of each power and pressure causes, for this process is monitored, has chosen 17 process variable altogether, and is as shown in table 1.In addition, the sampling time of each batch point is 85.Fig. 1 has provided through after the pivot analysis, and all preceding two-dimensional data features of normal batch can obviously be found out, the data of this process are produced by three operation operating modes.Next in conjunction with this detailed process implementation step of the present invention is at length set forth:

1. gatherer process nominal situation data, data pre-service, normalization and rearranging

127 batches of effective process data samples collecting are carried out the data pre-service, reject the outlier and the coarse error point of process.96 batches of normal data choosing are then wherein formed the modeling data matrix, and it is divided into three sub-data matrix X _c∈ R ^{32 * 17 * 85}, c=1,2,3.Each subdata matrix is launched into two-dimensional data matrix and it is carried out normalization according to a batch direction, obtain X _c∈ R ^{32 * 1445}, c=1,2,3.Then, along the sampling instant direction two-dimensional data matrix is arranged again, obtain new data matrix and be ${\stackrel{=}{X}}_c\∈R^{2720\×17},$ c＝1，2，3

2. at each process operation operating mode, set up the confidence limit of independent component analysis model and definite corresponding statistic respectively

Respectively to new data matrix ${\stackrel{=}{X}}_c\∈R^{2720\×17},$ C=1,2,3 carry out the ICA analysis and modeling, choose 3 independent components, obtain detailed ICA model parameter information, i.e. independent component information S _c∈ R ^{3 * 2720}, hybrid matrix A _c∈ R ^{17 * 3}, separation matrix W _c∈ R ^{3 * 17}And residual matrix ${\stackrel{=}{E}}_c\∈R^{2720\×17}.$ Construct I then ²Statistic is also determined its corresponding monitoring and statistics limit by the Density Estimator method.In like manner, can determine the monitoring confidence limit of SPE statistic.Here, our degree of confidence of choosing two statistics is 99%.

3. obtain current Monitoring Data information, and it is carried out pre-service and normalization

In order to test the validity of new method, respectively the data of normal batch and fault batch are tested.The data that picked at random is a certain normal batch, and utilize the normalized parameter under each operating mode that it is handled.Choose a kind of typical fault and test, equally it is carried out normalized.

4. online process monitoring

At first normal batch process data is monitored, the monitoring result that new method and MPCA method obtain respectively as shown in Figures 2 and 3.As can be seen from the figure, new method and MPCA method all can be made preferably monitoring to this batch, promptly do not report phenomenon by mistake and take place, and illustrate that new method do not lose its monitoring effect under nominal situation.Then, fault batch is monitored, the monitoring effect of new method and MPCA method as shown in Figure 4 and Figure 5.Can find out obviously that new method has successfully monitored the fault of process.By contrast, the effect of MPCA method is just far short of what is expected.

5. performance analysis and location

Normal batch data are carried out performance analysis and location, and the result who obtains as shown in Figure 6.As can be seen from the figure, this lot data comes from the possibility maximum of second kind of operation operating mode.

Table 1: monitored variable explanation

Sequence number	Variable	Sequence number	Variable
					1	BCl 3Flow	10	RF power
2	Cl 2Flow	11	The RF impedance
				3	The RF bottom power	12	TCP is tuning
4	A detects end points	13	The TCP phase error
				5	Helium pressure	14	The TCP impedance
6	Constant pressure	15	The TCP top power
				7	RF is tuning	16	The TCP load
8	The RF load	17	The Vat valve
				9	Phase error

Claims (1)

1. the semiconductor processes monitoring method based on independent component analysis and Bayesian inference is characterized in that, may further comprise the steps:

(1) utilize the data of each nominal situation of Distributed Control System (DCS) collection semiconductor process to form the three-dimensional training sample set that modeling is used: X=[X ₁X ₂, X _C]; Wherein,

C=1,2 ..., C is the data matrix corresponding to process operating mode c, I _cBe batch number under this operating mode, J is the variable number, and K counts for the sampled data of each batch; Deposit these data in historical data base respectively;

(2) respectively the data under the different operating modes are expanded into I along separately a batch direction _c* JK two-dimensional data matrix is carried out pre-service and normalization to it, makes that promptly the average of each process variable is zero, and variance is 1, obtains new data matrix collection to be

(3) along the time point direction each data matrix is arranged again, obtained new data matrix collection and be ${\stackrel{\‾}{\stackrel{\‾}{X}}}_c\∈R^{{\mathrm{KI}}_c\×J};$

(4) adopt the independent component analysis method that each concentrated new data matrix of new data matrix that step (3) obtains is handled, set up the independent component analysis model, obtain residual matrix

And structure I ²The monitoring and statistics limit of statistic;

(5) at residual matrix

The monitoring and statistics limit of structure SPE statistic;

(6) deposit in historical data base and the real-time data base modeling data and each model parameter standby;

(7) collect new process data, and new process data is carried out pre-service and normalization;

(8) process data after adopting different operating mode models to step (7) pre-service and normalization is respectively monitored, and promptly sets up statistic I ²And SPE;

(9) calculate the posterior probability values of current Monitoring Data under each operating mode by the Bayesian inference method, and the probability of malfunction value of computational data; Then, the new statistic FI of structure ²With the monitoring result under the integrated different operating modes of FSPE, and provide corresponding performance analysis and positioning result;

Wherein, described step (4) is specially: adopt the independent component analysis method to each new data matrix

Handle, extract high-order statistic information, set up the independent component analysis model; By independent component analysis, can obtain the independent component matrix of this data matrix

Hybrid matrix A _c∈ R ^{J * r}, separation matrix W _c∈ R ^RxJAnd residual matrix

As follows:

${\stackrel{\‾}{\stackrel{\‾}{X}}}_c=A_c{S}_c+{\stackrel{\‾}{\stackrel{\‾}{E}}}_c$

$S_c=W_c{\stackrel{\‾}{\stackrel{\‾}{X}}}_c;$

${\stackrel{\‾}{\stackrel{\‾}{E}}}_c={\stackrel{\‾}{\stackrel{\‾}{X}}}_c-A_c{S}_c$

Wherein, c=1,2 ..., C, r are the independent component number of choosing; Then, structure I ²Statistic also utilizes the Density Estimator method to provide its corresponding monitoring and statistics limit

Described step (5) is specially: on the basis of previous step to residual matrix

Set up the SPE statistic and calculate its corresponding monitoring and statistics limit SPE _{Lim, c}

Described step (8) is specially: for the new data after the normalization

Adopt the model under the different operating modes that it is monitored respectively, it is as follows promptly to set up corresponding monitoring and statistics amount:

$s_c=W_c{\stackrel{\‾}{\stackrel{\‾}{X}}}_{\mathrm{new}}$

${\stackrel{\‾}{\stackrel{\‾}{e}}}_{\mathrm{new},c}={\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}}-A_c{s}_c;$

$I_{\mathrm{new},c}^2=s_c^T{s}_c$

Wherein, c=1,2 ..., C, s _cBe the independent component vector that corresponding data extracted, continue at residual vector Setting up the SPE statistic is:

${\mathrm{SPE}}_{\mathrm{new},c}={\stackrel{\‾}{\stackrel{\‾}{e}}}_{\mathrm{new},c}{\stackrel{\‾}{\stackrel{\‾}{e}}}_{\mathrm{new},c}^T;$

Described step (9) is specially:

(a) at first calculate the posterior probability values of current Monitoring Data under each process operating mode, that is: by the Bayesian inference method

$P_{I^2}\left(c\right|{\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}})=\frac{P_{I^2}(c,{\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}})}{P_{I^2}\left({\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}}\right)}=\frac{P_{I^2}\left({\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}}\right|c)P\left(c\right)}{\underset{c=1}{\overset{C}{\mathrm{\Σ}}}\left[P_{I^2}\left({\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}}\right|c)P\left(c\right)\right]},$

$P_{\mathrm{SPE}}\left(c\right|{\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}})=\frac{P_{\mathrm{SPE}}(c,{\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}})}{P_{\mathrm{SPE}}\left({\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}}\right)}=\frac{P_{\mathrm{SPE}}\left({\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}}\right|c)P\left(c\right)}{\underset{c=1}{\overset{C}{\mathrm{\Σ}}}\left[P_{\mathrm{SPE}}\left({\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}}\right|c)P\left(c\right)\right]};$

Wherein, With

Be the marginal distribution probability, be defined as follows here:

$P_{I^2}\left({\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}}\right|c)=\exp \{-\frac{I_{\mathrm{new},c}^2}{I_{\lim ,c}^2}\},$

$P_{\mathrm{SPE}}\left({\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}}\right|c)=\exp \{-\frac{{\mathrm{SPE}}_{\mathrm{new},c}}{{\mathrm{SPE}}_{\lim ,c}}\};$

P (c) is a prior probability, but simple computation is

(b) probability of malfunction of calculating current data under each operating mode is as follows:

$P_{f,I^2}^c\left({\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}}\right)=\mathrm{Pr}\{I_c^2\left({\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{tr},c}\right)\≤I_{\mathrm{new},c}^2\},$

$P_{f,\mathrm{SPE}}^c\left({\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}}\right)=\mathrm{Pr}\{{\mathrm{SPE}}_c\left({\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{tr},c}\right)\≤{\mathrm{SPE}}_{\mathrm{new},c}\};$

Wherein, Be the modeling data under the corresponding operating mode, Pr{} represents probable value;

(c) calculate last monitoring result, the monitoring result under the promptly integrated different operating modes is as follows:

${\mathrm{FI}}_{\mathrm{new}}^2=\underset{c=1}{\overset{C}{\mathrm{\Σ}}}\left[P_{I^2}\left(c\right|{\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}})P_{f,I^2}^c\left({\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}}\right)\right],$

${\mathrm{FSPE}}_{\mathrm{new}}=\underset{c=1}{\overset{C}{\mathrm{\Σ}}}\left[P_{\mathrm{SPE}}\left(c\right|{\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}})P_{f,\mathrm{SPE}}^c\left({\stackrel{\‾}{\stackrel{\‾}{x}}}_{\mathrm{new}}\right)\right];$

(d) calculate the posterior probability values of current Monitoring Data and each operating mode, it carried out performance analysis and is positioned such that:

Wherein, the value of posterior probability is big more, illustrates that the degree of association of current Monitoring Data and corresponding operating mode is big more; Otherwise the possibility that these data of the more little explanation of value are in corresponding operating mode is just more little.

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