CN107153409A - A kind of nongausian process monitoring method based on missing variable modeling thinking - Google Patents

A kind of nongausian process monitoring method based on missing variable modeling thinking Download PDF

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CN107153409A
CN107153409A CN201710446398.5A CN201710446398A CN107153409A CN 107153409 A CN107153409 A CN 107153409A CN 201710446398 A CN201710446398 A CN 201710446398A CN 107153409 A CN107153409 A CN 107153409A
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matrix
msub
mrow
vector
data
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CN107153409B (en
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石立康
朱莹
童楚东
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DAQING HUAYU PETROLEUM MACHINERY MANUFACTURING CO LTD
Shenzhen Dragon Totem Technology Achievement Transformation Co ltd
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Ningbo University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/418—Total factory control, i.e. centrally controlling a plurality of machines, e.g. direct or distributed numerical control [DNC], flexible manufacturing systems [FMS], integrated manufacturing systems [IMS], computer integrated manufacturing [CIM]
    • G05B19/41885—Total factory control, i.e. centrally controlling a plurality of machines, e.g. direct or distributed numerical control [DNC], flexible manufacturing systems [FMS], integrated manufacturing systems [IMS], computer integrated manufacturing [CIM] characterised by modeling, simulation of the manufacturing system
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
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    • Y02P—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN THE PRODUCTION OR PROCESSING OF GOODS
    • Y02P90/00—Enabling technologies with a potential contribution to greenhouse gas [GHG] emissions mitigation
    • Y02P90/02—Total factory control, e.g. smart factories, flexible manufacturing systems [FMS] or integrated manufacturing systems [IMS]

Abstract

The present invention disclose a kind of nongausian process monitoring method based on missing variable modeling thinking, it is intended to using the modeling approach of variable is lacked by the data conversion of non-gaussian distribution into after the data of Gaussian Profile, so that the fault detect more reliable to nongausian process implementation.Specifically, the inventive method is first on the basis of traditional independent component analysis (ICA) model, by estimating corresponding independent element after assuming each measurand missing data one by one.Then, with the error between independent element actual value and estimate as monitored target.Although the inventive method is directed to the sampled data of nongausian process, the general Gaussian distributed of evaluated error of independent element.See from this point on, the inventive method is the error information that nongausian process sampled data is cleverly converted to Gaussian Profile by lacking the modeling approach of variable.In addition, the inventive method establishes multiple Fault Models, moreover it is possible to played the strong advantage of multi-model generalization ability.

Description

A kind of nongausian process monitoring method based on missing variable modeling thinking
Technical field
The present invention relates to a kind of Industrial Process Monitoring method, more particularly, to a kind of based on the non-of missing variable modeling thinking Gaussian process monitoring method.
Background technology
In industrial process stream, it is ensured that production process is continuously in nominal situation for the tool that ensures product quality stability It is significant.Therefore, the process monitoring module for possessing fault detection capability is essential in produce reality, and this is also academic Boundary and industrial quarters can be continued for constantly carrying out the main cause of theory and application research for process monitoring.From research progress On see, process monitoring progressively develops into data-driven since the fault detection method based on mechanism model most Fault detection method.Compared to mechanism model is set up, the method for data-driven, which is implemented, to be more prone to, and is more suitable for existing For large scale industry process object.Currently, also the mistake of new types of data driving is constantly being expedited the emergence of under the trend of industrial " big data " Journey monitoring method.In this research field, multivariate statistical analysis algorithm is the modeling method being most widely used.Wherein, With principal component analysis (Principal Component Analysis, PCA) and independent component analysis (Independent Component Analysis, ICA) based on fault detection method emerge in an endless stream, derived it is miscellaneous, adapt to In the process monitoring method of different type industrial object.
In general, the process monitoring model based on PCA assumes that sampled data obeys approximate Gaussian distribution mostly, so that The upper control limit of description normal data excursion can easily be determined.However, the complexity of modern industry process object Sampled data is caused to disobey Gaussian Profile.Therefore, traditional PCA methods are used for the presence of more significant failure leakage during fault detect Report rate or rate of false alarm.By contrast, ICA algorithm can excavate the independent element information of non-gaussian, more suitable for monitoring not high This process data.Moreover, many scientific documents are also through case verification, the method based on ICA is superior to the side based on PCA Method.In existing scientific research and patent document, determine that the method for the control limit of ICA Fault Models depends on cuclear density Estimation.From space geometry structure, the control limits the surface for defining a suprasphere in fact, all to be located inside suprasphere Data point all be nominal situation data.However, the non-Gaussian system due to data, the distribution of normal data in itself is impossible It is stuffed entirely with the suprasphere.Therefore, the data point inside suprasphere differs, and to establish a capital be normal data sample.Cause this to ask The independent element that the factor of topic is mostly derived from ICA model extractions is non-gaussian in itself.Therefore, with regard to being based in this point problem ICA nongausian process monitoring model also needs further improvement.
It is worth noting that, error is typically all obeyed or approximate Gaussian distributed.But, the generation of error is usually required One actual value and an estimate, each monitor sample are converted into after corresponding independent element that the reality of ICA models can be treated as Actual value.So, the estimate for how producing independent element is exactly a problem of most critical.Well imagine, if can produce independent The estimate of composition, then the error between independent element actual value and estimate is obtained with.Moreover, original disobey height The independent element of this distribution becomes after error, also with regard to Gaussian distributed, it is determined that corresponding control limit just can be more accurately The distribution of normal data is described.In existing Research Literature, there are some scholars in traditional PCA Fault Model Propose the processing method of missing variable, it is intended to solve the sampled data in produce reality and be possible to that asking for shortage of data occurs Topic.The method of variable is lacked mainly by existing pca model, missing data is not being needed using obtained data have been measured Under the premise of corresponding principal component information can be estimated.In consideration of it, using lack variable processing thinking perhaps can be The estimate for producing independent element provides a feasible resolution policy.
The content of the invention
Technical problem underlying to be solved by this invention is：How one is set up using the modeling approach for lacking variable more may be used The nongausian process monitoring model leaned on.Therefore, the invention provides a kind of nongausian process based on missing variable modeling thinking Monitoring method.This method is on the basis of traditional IC A models, by being estimated after assuming each measurand missing data one by one Corresponding independent element.Then, with the error between independent element actual value and estimate as monitored target, set up appropriate Statistic implements online process monitoring.
The present invention solve the technical scheme that is used of above-mentioned technical problem for：It is a kind of based on missing variable modeling thinking it is non- Gaussian process monitoring method, comprises the following steps：
(1) sample data under production process normal operating condition is collected using sampling system, constitutes training data matrix： X∈Rn×m, and X is standardized, the average for making each measurand is 0, and variance is 1, obtains new data matrixWherein, n is number of training, and m is process measurement variable number, and R is set of real numbers, Rn×mRepresent n × m The real number matrix of dimension,For matrixIn k-th of variable n measurement data, k=1,2 ..., m.
(2) it is using ICA algorithmICA models are set up, i.e.,：Wherein,For d solely Vertical composition Column vector groups into matrix, W ∈ Rm×dFor separation matrix, A ∈ Rm×dFor hybrid matrix, E ∈ Rn×mRepresent model error, The transposition of upper label T representing matrixs or vector.
(3) matrix is assumedMiddle kth column data missing, remaining available column composition matrix Xk∈Rn×(m-1), and utilize following institute Show that formula calculates the estimate for obtaining independent element matrix
In above formula, Ak∈R(m-1)×dTo remove the matrix obtained by row k in hybrid matrix A.
(4) actual value S and estimate are calculatedBetween errorAnd calculating matrix FkCovariance matrix Ck=Fk TFk/(n-1)。
(5) calculating matrix CkCharacteristic vector α corresponding to eigenvalue of maximumk∈Rd×1, juxtaposition
(6) k ＜ m are judgedIf so, then putting return to step after k=k+1 (3)；If it is not, then performing step (7).
(7) calling model parameter setImplement online process monitoring, specific implementation process It is as follows：
1. the sampled data y at production process object newest moment is gatherednew∈R1×m, it is carried out and X same standards Processing is obtained
2. sample vector is calculatedThe actual value of corresponding independent element vectorAnd initialize k=1.
3. sample vector is assumedIn k-th variable data missing, willThe middle new vector of remainder data composition And utilize formulaCalculate the estimate for obtaining independent element vector
4. the error between the actual value and estimate of independent element vector is calculated
5. monitoring and statisticses amount Q is calculated according to formula as followsk：
Qk=(fkαk)2 (1)
7. k ＜ m are judgedIf so, then putting after k=k+1 return to step 3.；If it is not, then performing step 8..
8. Q=max { Q are put1, Q2..., QmAfter, judge whether to meet condition：If so, then current sample is just Normal sample, 1. production process is in normal operating conditions and returned continues to monitor next new samples；If it is not, then the sample is event Hinder sample, production process enters damage and triggers fault warning.Wherein, symbol max { } represents to take maximum,Table It is that 1, confidence level is numerical value corresponding to δ chi square distribution to show the free degree.
Compared with conventional method, the advantage of the inventive method is mainly reflected in following two aspects：
On the one hand, the object of the inventive method monitoring is the evaluated error of independent element, although independent element is non-gaussian Distribution but error general Gaussian distributed in itself.It can be said that the inventive method is will by lacking the modeling approach of variable Nongausian process sampled data is cleverly converted to the error information of Gaussian Profile, and the control that should determine that is limited also with regard to that can align Regular data carries out more accurate description.On the other hand, the inventive method is by assuming each measurand shortage of data one by one, from And establish the Fault Model equal with monitored parameterses number.With fault detection method of the tradition based on single ICA models Compare, the inventive method has also played the strong advantage of multi-model generalization ability.In summary two aspect advantage, the inventive method is A kind of nongausian process monitoring method being more highly preferred to.
Brief description of the drawings
Fig. 1 is the implementing procedure figure of the inventive method.
Fig. 2 is error F1The Gaussian Profile of middle first row and the first column element in independent element S, which is examined, to be schemed.
Fig. 3 is the inventive method and traditional IC A method mean failure rate rate of failing to report comparison diagrams.
Embodiment
The inventive method is described in detail below in conjunction with the accompanying drawings.
As shown in figure 1, the present invention relates to a kind of nongausian process monitoring method based on missing variable modeling thinking, the party The specific implementation step of method is as follows：
(1) sample data under production process normal operating condition is collected using sampling system, constitutes training data matrix： X∈Rn×m, and X is standardized, the average for making each measurand is 0, and variance is 1, obtains new data matrixWherein, n is number of training, and m is process measurement variable number, and R is set of real numbers, Rn×mRepresent n × m The real number matrix of dimension,For matrixIn k-th of variable n measurement data, k=1,2 ..., m.
(2) it is using ICA algorithmICA models are set up, and initialize k=1.Set up the implementation process of ICA models such as Shown in lower：
1. calculateCovariance matrixWherein C ∈ Rm×m；
2. calculating matrix C all characteristic values and characteristic vector, and reject characteristic value less than 0.0001 and its corresponding Characteristic vector, obtains eigenvectors matrix P=[p1, p2..., pM]∈Rm×MAnd characteristic value diagonal matrix D=diag (λ1, λ2..., λM)∈RM×M；
It is worth noting that, solving obtained characteristic vector p here1, p2..., pMIt all must be the vector of unit length.
3. according to formulaIt is rightWhitening processing is carried out, Z ∈ R are obtainedn×M, and initialize i=1；
4. column vector c is takeniThe i-th row in unit matrix are tieed up for M × M,
5. c is updated according to formula as followsi, i.e.,：
ci←E{Zg(ci TZ)}-E{h(ci TZ)}ci (2)
In above formula (2), E { } represents to ask for desired value (i.e. vectorial average value), function g and the h following institute of concrete form Show：
G (u)=tanh (u) (3)
H (u)=[sech (u)]2 (4)
In above formula (3) and (4), u is function argument, refers herein to ci TElement in Z.
6. to the vectorial c after renewaliCarry out orthogonal standardization according to the following formula successively：
ci←ci/||ci|| (6)
7. repeat step 5.~6. until vector ciConvergence, and preserve vectorial ci；
8. i ＜ M are judgedIf so, put after i=i+1, repeat step 4.~8.；If it is not, performing step 9.；
9. by obtained all M vector c1, c2..., cMConstitute Matrix C=[c1, c2..., cM]∈RM×M, and according to such as Formula calculates separation matrix W shown in lower0∈Rm×MWith hybrid matrix A0∈Rm×M：
A0=PD1/2C (7)
W0=PD-1/2C (8)
10. A is calculated0In each column vector length, L is designated as respectively1, L2..., LM, and by L1, L2..., LMAccording to numerical value Size carries out descending arrangement and obtains l1, l2..., lM, then the independent element number d of reservation is the minimum value for meeting following condition：
By A0The hybrid matrix A ∈ R of d maximum Column vector groups Cheng Xin of middle column vector lengthm×d, while from W0In take Go out Column vector groups corresponding with A into new separation matrix W ∈ Rm×d；
The ICA models finally obtained areWherein,For d independent element arrange to Measure the matrix of composition, E ∈ Rn×mRepresent model error.
(3) matrix is assumedMiddle kth column data missing, remaining available column composition matrix Xk∈Rn×(m-1), and utilize following institute Show that formula calculates the estimate for obtaining independent element matrix
In above formula, Ak∈R(m-1)×dTo remove the matrix obtained by row k in hybrid matrix A.
(4) actual value S and estimate are calculatedBetween errorAnd calculating matrix FkCovariance matrix Ck=Fk TFk/(n-1)。
(5) calculating matrix CkCharacteristic vector α corresponding to eigenvalue of maximumk∈Rd×1, juxtaposition
(6) k ＜ m are judgedIf so, then putting return to step after k=k+1 (3)；If it is not, then performing step (7).
(7) calling model parameter setImplement online process monitoring, specific implementation process It is as follows：
1. the sampled data y at production process object newest moment is gatherednew∈R1×m, it is carried out and X same standards Processing is obtained
2. sample vector is calculatedThe actual value of corresponding independent element vectorAnd initialize k=1.
3. sample vector is assumedIn k-th variable data missing, willThe middle new vector of remainder data composition And utilize formulaCalculate the estimate for obtaining independent element vector
4. the error between the actual value and estimate of independent element vector is calculated
5. monitoring and statisticses amount Q is calculated according to formula as followsk：
Qk=(fkαk)2 (11)
7. k ＜ m are judgedIf so, then putting after k=k+1 return to step 3.；If it is not, then performing step 8..
8. Q=max { Q are put1, Q2..., QmAfter, judge whether to meet condition：If so, then current sample is just Normal sample, 1. production process is in normal operating conditions and returned continues to monitor next new samples；If it is not, then the sample is event Hinder sample, production process enters damage and triggers fault warning.Wherein, symbol max { } represents to take maximum,Table It is that 1, confidence level is numerical value corresponding to δ chi square distribution to show the free degree.
Illustrate that the inventive method is superior relative to existing method with reference to the example of a specific industrial process Property and reliability.The process data comes from the experiment of the U.S. Tennessee-Yi Siman (TE) chemical process, and prototype is Yi Siman chemical industry One actual process flow of workshop.At present, TE processes are because of the complexity of its flow, as a standard test platform It is widely used in fault detect research.Whole TE processes include 22 measurands, 12 performance variables and 19 composition measurements Variable.The data gathered are divided into 22 groups, including the data set under 1 group of nominal situation and 21 groups of fault datas.And at this In a little fault datas, it is known fault type to have 16, the changing of such as cooling water inlet temperature or feed constituents, it is valve viscous, Kinetics drift etc., also 5 fault types are unknown.In order to be monitored to the process, choose as shown in table 1 33 process variables, next specific implementation step of the present invention is explained in detail with reference to the TE processes.
Table 1：TE process monitoring variables.
Sequence number Variable description Sequence number Variable description Sequence number Variable description
1 Material A flow 12 Separator liquid level 23 D material inlet valves position
2 Material D flows 13 Separator pressure 24 E material inlet valves position
3 Material E flows 14 Separator bottom of towe flow 25 A material inlet valves position
4 Combined feed flow 15 Stripper grade 26 A and C material inlet valves position
5 Circular flow 16 Pressure of stripping tower 27 Compressor cycle valve location
6 Reactor feed 17 Stripper bottom rate 28 Empty valve location
7 Reactor pressure 18 Stripper temperature 29 Separator liquid phase valve location
8 Reactor grade 19 Stripper upper steam 30 Stripper liquid phase valve location
9 Temperature of reactor 20 Compressor horsepower 31 Stripper steam valve position
10 Rate of evacuation 21 Reactor cooling water outlet temperature 32 Reactor condensate flow
11 Separator temperature 22 Separator cooling water outlet temperature 33 Condenser cooling water flow
(1) process data under collection TE process object nominal situations, and choose 960 normal data composition matrix X ∈ R960×33, it is standardized and obtained
(2) it is using ICA algorithmICA models are set up, i.e.,：And initialize k=1.
(3) matrix is assumedMiddle kth column data missing, remaining available column composition matrix Xk∈R960×32, and utilize following institute Show that formula calculates the estimate for obtaining independent element matrix
In above formula, Ak∈R32×28To remove the matrix obtained by row k in hybrid matrix A；
(4) actual value S and estimate are calculatedBetween errorAnd calculating matrix FkCovariance matrix Ck=Fk TFk/(n-1)；
Because the object that the inventive method is monitored is error Fk, and that the monitoring of traditional IC A methods is independent element S, spy is to F1 The data of middle first row and first row in independent element matrix S carry out Gaussian Profile inspection, are as a result shown in Fig. 2.If being surveyed It is then straight line to try the scatter diagram in the strict Gaussian distributed of data, Fig. 2.It will be apparent that error F in the inventive method1Closely Like Gaussian distributed, and in traditional IC A methods due to extraction be non-gaussian independent element, Gauss point is not just met certainly Cloth.It can also illustrate from this contrast, the composition of non-gaussian can dexterously be converted into approximately obeying Gauss by the inventive method The error of distribution.So, the control limit that the inventive method is determined can just be obtained according to chi square distribution point.
(5) calculating matrix CkCharacteristic vector α corresponding to eigenvalue of maximumk∈R28×1, juxtaposition
(6) k ＜ m are judgedIf so, then putting return to step after k=k+1 (3)；If it is not, then off-line modeling is completed and implemented Line malfunction monitoring；
To test the superiority in fault detect of the inventive method, to monitor the mean failure rate of 21 kinds of failures of TE processes Detect exemplified by success rate, contrast the inventive method and traditional fault detect effect based on ICA methods.The test data has 960 Individual sample composition, its preceding 160 sample is nominal situation down-sampling, and rear 800 samples are sampled for fault condition.
1. the sampled data y at production process object newest moment is gatherednew∈R1×33, it is carried out and X same standards Processing is obtained
2. sample vector is calculatedThe actual value of corresponding independent element vectorAnd initialize k=1；
3. sample vector is assumedIn k-th variable data missing, willThe middle new vector of remainder data composition And utilize formulaCalculate the estimate for obtaining independent element vector
4. the error between the actual value and estimate of independent element vector is calculated
5. monitoring and statisticses amount Q is calculated according to formula as followsk：
Qk=(fkαk)2 (2)
7. k ＜ 33 are judgedIf so, then putting after k=k+1 return to step 3.；If it is not, then performing step 8.；
8. Q=max { Q are put1, Q2..., Q33After, according to conditionWhether decision-making failure occurs.
Mean failure rate rate of failing to report of the present invention hair with traditional IC A methods for 21 kinds of fault types of TE processes is shown in Fig. 3 In, it is found that the mean failure rate rate of failing to report of the inventive method is significantly lower than traditional IC A methods.Therefore, the inventive method is improved The fault detect performance of nongausian process monitoring method based on ICA.
Above-described embodiment is only used for explaining the present invention, rather than limits the invention, in the spirit and power of the present invention In the protection domain that profit is required, any modifications and changes made to the present invention are both fallen within protection scope of the present invention.

Claims (2)

1. a kind of nongausian process monitoring method based on missing variable modeling thinking, it is characterised in that comprise the following steps：
(1) sample data under production process normal operating condition is collected using sampling system, constitutes training data matrix：X∈Rn ×m, and X is standardized, the average for making each measurand is 0, and variance is 1, obtains new data matrixWherein, n is number of training, and m is process measurement variable number, and R is set of real numbers, Rn×mRepresent n × m The real number matrix of dimension,For matrixIn k-th of variable n measurement data, k=1,2 ..., m；
(2) it is using ICA algorithmICA models are set up, i.e.,：And initialize k=1.Wherein,For d independent element Column vector groups into matrix, W ∈ Rm×dFor separation matrix, A ∈ Rm×dFor hybrid matrix, E∈Rn×mRepresent the transposition of model error, upper label T representing matrixs or vector；
(3) matrix is assumedMiddle kth column data missing, remaining available column composition matrix Xk∈Rn×(m-1), and utilize public affairs as follows Formula calculates the estimate for obtaining independent element matrix
<mrow> <msub> <mover> <mi>S</mi> <mo>^</mo> </mover> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>X</mi> <mi>k</mi> </msub> <msub> <mi>A</mi> <mi>k</mi> </msub> <msup> <mrow> <mo>(</mo> <msup> <msub> <mi>A</mi> <mi>k</mi> </msub> <mi>T</mi> </msup> <msub> <mi>A</mi> <mi>k</mi> </msub> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
In above formula, Ak∈R(m-1)×dTo remove the matrix obtained by row k in hybrid matrix A；
(4) actual value S and estimate are calculatedBetween errorAnd calculating matrix FkCovariance matrix Ck= Fk TFk/(n-1)；
(5) calculating matrix CkCharacteristic vector α corresponding to eigenvalue of maximumk∈Rd×1, juxtaposition
(6) k ＜ m are judgedIf so, then putting return to step after k=k+1 (3)；If it is not, then performing step (7)；
(7) calling model parameter setImplement online process monitoring, specific implementation process is as follows It is shown：
1. the sampled data y at production process object newest moment is gatherednew∈R1×m, it is carried out and the processing of X same standardizations Obtain
2. sample vector is calculatedThe actual value of corresponding independent element vectorAnd initialize k=1；
3. sample vector is assumedIn k-th variable data missing, willThe middle new vector of remainder data compositionAnd profit Use formulaCalculate the estimate for obtaining independent element vector
4. the error between the actual value and estimate of independent element vector is calculated
5. monitoring and statisticses amount Q is calculated according to formula as followsk：
Qk=(fkαk)2 (2)
7. k ＜ m are judgedIf so, then putting after k=k+1 return to step 3.；If it is not, then performing step 8.；
8. Q=max { Q are put1, Q2..., QmAfter, judge whether to meet condition：If so, then current sample is normal sample This, 1. production process is in normal operating conditions and returned continues to monitor next new samples；If it is not, then the sample is failure sample This, production process enters damage and triggers fault warning.Wherein, symbol max { } represents to take maximum,Represent certainly It is numerical value corresponding to 1, confidence level is δ chi square distribution as spending.
2. a kind of nongausian process monitoring method based on missing variable modeling thinking according to claim 1, its feature It is, is using ICA algorithm in the step (2)The implementation process for setting up ICA models is as follows：
1. calculateCovariance matrixWherein C ∈ Rm×m；
2. calculating matrix C all characteristic values and characteristic vector, and reject the characteristic value and its corresponding feature less than 0.0001 Vector, obtains eigenvectors matrix P=[p1, p2..., pM]∈Rm×MAnd characteristic value diagonal matrix D=diag (λ1, λ2..., λM)∈RM×M；
It is worth noting that, solving obtained characteristic vector p here1, p2..., pMIt all must be the vector of unit length.
3. according to formulaIt is rightWhitening processing is carried out, Z ∈ R are obtainedn×M, and initialize i=1；
4. column vector c is takeniThe i-th row in unit matrix are tieed up for M × M,
5. c is updated according to formula as followsi, i.e.,：
ci←E{Zg(ci TZ)}-E{h(ci TZ)}ci (3)
In above formula (3), E { } represents to ask for desired value (i.e. vectorial average value), and function g and h concrete form are as follows：
G (u)=tanh (u) (4)
H (u)=[sech (u)]2 (5)
In above formula (4) and (5), u is function argument, refers herein to ci TElement in Z.
6. to the vectorial c after renewaliCarry out orthogonal standardization according to the following formula successively：
<mrow> <msub> <mi>c</mi> <mi>i</mi> </msub> <mo>&amp;LeftArrow;</mo> <msub> <mi>c</mi> <mi>i</mi> </msub> <mo>-</mo> <msubsup> <mi>&amp;Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <msub> <mi>c</mi> <mi>i</mi> </msub> <mi>T</mi> </msup> <msub> <mi>c</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <msub> <mi>c</mi> <mi>j</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
ci←ci/||ci|| (7)
7. repeat step 5.~6. until vector ciConvergence, and preserve vectorial ci；
8. i ＜ M are judgedIf so, put after i=i+1, repeat step 4. -8.；If it is not, performing step 9.；
9. by obtained all M vector c1, c2..., cMConstitute Matrix C=[c1, c2..., cM]∈RM×M, and according to following institute Show that formula calculates separation matrix W0∈Rm×MWith hybrid matrix A0∈Rm×M：
A0=pD1/2C (8)
W0=PD-1/2C (9)
10. A is calculated0In each column vector length, L is designated as respectively1, L2..., LM, and by L1, L2..., LMEnter according to numerical values recited The arrangement of row descending obtains l1, l2..., lM, then the independent element number d of reservation is the minimum value for meeting following condition：
<mrow> <mfrac> <mrow> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>d</mi> </munderover> <msub> <mi>l</mi> <mi>j</mi> </msub> </mrow> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>L</mi> <mn>2</mn> </msub> <mo>+</mo> <mo>...</mo> <mo>+</mo> <msub> <mi>L</mi> <mi>M</mi> </msub> </mrow> </mfrac> <mo>&amp;GreaterEqual;</mo> <mn>0.9</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow>
By A0The hybrid matrix A ∈ R of d maximum Column vector groups Cheng Xin of middle column vector lengthm×d, while from W0Middle taking-up and A Corresponding Column vector groups are into new separation matrix W ∈ Rm×d；
The ICA models finally obtained areWherein,For d independent element Column vector groups into Matrix, E ∈ Rn×mRepresent model error.
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