Selection of the Number of Principal Components: The Variance of the Reconstruction Error Criterion with a Comparison to Other Methods


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1 Ind. Eng. Che. Res. 1999, 38, Selection of the Nuber of Principal Coponents: The Variance of the Reconstruction Error Criterion with a Coparison to Other Methods Sergio Valle, Weihua Li, and S. Joe Qin* Departent of Cheical Engineering, The University of Texas at Austin, Austin, Texas 7871 One of the ain difficulties in using principal coponent analysis (PCA) is the selection of the nuber of principal coponents (PCs). There exist a plethora of ethods to calculate the nuber of PCs, but ost of the use onotonically increasing or decreasing indices. Therefore, the decision to choose the nuber of principal coponents is very subjective. In this paper, we present a ethod based on the variance of the reconstruction error to select the nuber of PCs. This ethod deonstrates a iniu over the nuber of PCs. Conditions are given under which this iniu corresponds to the true nuber of PCs. Ten other ethods available in the signal processing and cheoetrics literature are overviewed and copared with the proposed ethod. Three data sets are used to test the different ethods for selecting the nuber of PCs: two of the are real process data and the other one is a batch reactor siulation. 1. Introduction Principal coponent analysis (PCA) has wide applications in signal processing, 1 factor analysis, cheoetrics, 3 and cheical processes data analysis. 4,5 Although the original concept of PCA was introduced by Pearson 33 and developed by Hotelling, 3 the vast ajority of applications happened only in the last few decades with the popular use of coputers. In the cheical engineering area. the applications of PCA include the following: (1) onitoring of batch and continuous processes, 6,7 () extraction of active biocheical reactions, 8 (3) product uality control in the principal coponent subspace, 9 (4) issing value replaceent 10 (5) sensor fault identification and reconstruction, 11 (6) process fault identification and reconstruction, 1 and (7) disturbance detection. 13 Early PCA applications to the uality control of cheical processes are docuented in Jackson. 4 A key issue in developing a PCA odel is to choose the adeuate nuber of PCs to represent the syste in an optial way. If fewer PCs are selected than reuired, a poor odel will be obtained and an incoplete representation of the process results. On the contrary, if ore PCs than necessary are selected, the odel will be overparaeterized and will include noise. Zwick and Velicer 0 copared five different approaches available in the literature. Hies et al. 34 provide further coparisons of several approaches using the Tennessee Eastan proble. Different approaches had been proposed in the past to select the optial nuber of PCs: (1) Akaike inforation criterion (AIC), 14 () iniu description length (MDL), 15 (3) ibedded error function (IEF), 16 (4) cuulative percent variance (CPV), (5) scree test on residual percent variance (RPV), 17,18 (6) average eigenvalue (AE), 19 (7) parallel analysis (PA), 0 (8) autocorrelation (AC), 1 (9) cross validation based on the * To who correspondence should be addressed. Tel: (51) Fax: (51) Eail: An earlier version of the paper was presented at the AIChE Annual Meeting, Miai, FL, Nov 160, PRESS and R ratio, 3,,3 and (10) variance of the reconstruction error (VRE). 9 In the signal processing literature, PCA has been used to deterine the nuber of independent source signals fro noisy observations by selecting the nuber of significant principal coponents. 1 Under the assuption that easureent noise corresponds to the sallest eual eigenvalues of the covariance atrix, the Akaike inforation criterion and the iniu description length principle have been applied to selecting the nuber of PCs. 1 These criteria apply to PCA based on the covariance atrix of the data and do not work on the correlationbased PCA of the data, although correlationbased PCA is preferred in cheical processes to eually weight all variables by variance scaling. One advantage of the AIC and MDL criteria is that they have a solid statistical basis and are theoretically shown to have a iniu nuber of PCs. To develop a selection criterion that works with both correlationbased and covariancebased PCA, Qin and Dunia 9 propose a new criterion to deterine the nuber of PCs based on the best reconstruction of the variables. An iportant feature of this new approach is that the proposed index has a iniu (i.e.. nononotonic) corresponding to the best reconstruction. When the PCA odel is used to reconstruct issing values or faulty sensors, the reconstruction error is a function of the nuber of PCs. Qin and Dunia 9 use the variance of the reconstruction error to deterine the nuber of PCs. The VRE can be decoposed into a portion in the principal coponent subspace (PCS) and a portion in the residual subspace (RS). The portion in the RS is shown to decrease onotonically with the nuber of PCs and that, in the PCS in general, increases with the nuber of PCs. As a result, the VRE always has a iniu which points to the optial nuber of PCs for best reconstruction. In the case of issing value reconstruction, the VRE for different issing patterns are cobined with weights based on the freuency of occurrence. In the case of faulty sensor identification and reconstruction, the VREs are weighted based on the variance of each variable. In the special /ie990110i CCC: $ Aerican Cheical Society Published on Web 09/30/1999
2 4390 Ind. Eng. Che. Res., Vol. 38, No. 11, 1999 case where each sensor is reconstructed once, the procedure can be thought of as a cross validation by leaving sensors out, in contrast to the traditional leaving saples out cross validation. The ultiate purpose of this paper is 3fold: (1) Introduce the VRE, AIC, MDL, and other related ethods for the selection of the nuber of principal coponents in cheical process data odeling. () Provide further analysis and coparison of the VRE ethod with other ethods, including the AIC and MDL criteria. (3) Copare and benchark various ethods for selecting the nuber of PCs using data fro a siulated batch reactor and two industrial processes data sets. The effectiveness of these ethods will be tested and bencharked on the basis of these data sets. The organization of the paper is given as follows. Section describes the selection criteria based on AIC, MDL, and IEF ethods. Section 3 describes different selection criteria coonly used in the PCA literature. Section 4 presents the VRE ethod based on the correlation atrix and covariance atrix. Section 5 copares the ethods using three exaples: a siulated batch reactor, a boiler process, and an incineration process. Section 6 concludes the paper.. Criteria Based on AIC, MDL, and IEF The AIC and MDL criteria are popular in signal processing, while the IEF approach is proposed in factor analysis. 16 Although these three ethods are developed in different areas. There are coon attributes about the: (1) they work only with covariancebased PCA; () the variances of easureent noise in each variable are assued to be identical; and (3) they deonstrate a iniu over the nuber of PCs given the above assuptions. In this section, we first present the notation and basic relations for PCA. Then we provide a linkage between the use of PCA for signal processing and the use of PCA for process odeling. Finally, the criteria for AIC, MDL, and IEF are presented..1. PCA Notation. Let x R denote a saple vector of sensors. Assuing that there are N saples for each sensor, a data atrix X R N is coposed with each row representing a saple. The atrix X is scaled to zeroean for covariancebased PCA and, in addition to unit variance for correlationbased PCA, by either the NIPALS 4 or a singularvalue decoposition (SVD) algorith. The atrix X can be decoposed into a score atrix T and a loading atrix P whose coluns are the right singular vectors of X, where X ) T P T is the residual atrix. Since the coluns of T are orthogonal, the covariance atrix is where X ) TP T + X )TP T + T P T ) [T T ][P P ] T (1) S 1 N  1 XT X ) [P P ] Λ [P P ] T () Λ ) 1 N  1 [T T ]T [T T ] ) diag{λ 1, λ,, λ } (3) and t i is the ith colun of T and λ i are the eigenvalues of the covariance atrix. If N is very large, the approxiate euality in e becoes an euality. We assue the euality holds in this paper unless otherwise specified. For variance scaled X, e is the correlation atrix R. A saple vector x can be projected on the PCS (S p ), which is spanned by P R l, and RS (S r ), respectively, Since S p and S r are orthogonal, and λ i ) 1 N  1 t T i t i ) var{t i } (4) The task for deterining the nuber of PCs is to choose l such that xˆ contains ostly inforation and xˆ contains noise... PCA for Signal Processing and Process Modeling. In signal processing, PCA has been used to detect independent signal sources fro observation variables, that is, where x(k) R is a vector of observation variables, s(k) R is a vector of signal sources, n(k) R is a vector of observation noise, and A R is a atrix with appropriate diension. This odel is used for radar signal detection, 1 for instance. In cheical process odeling, the physical principles that govern the process behavior are ainly aterial and energy balances. Assuing there exists a linear (or linearized) relationship aong the process variables, the following steadystate relation can be derived, where xj(k) R are the true process variables free of easureent noise, B R is the odel atrix, and n(k) R is the easureent noise. While es 9 and 11 represent different physical phenoena, we have the following rearks under the condition that )  : (1) Euation 9 can be converted to es 10 and 11 by preultiplying on both sides of e 9, where xˆ ) PP T x S p (5) x ) P P Tx ) (I  PP T ) x S r (6) xˆ Tx ) 0 (7) xˆ + x ) x (8) x(k) ) As(k) + n(k) (9) Bxj(k) ) 0 (10) x(k) ) xj(k) + n(k) (11) B ) [I  A(A T A) 1 A T ] (1) xj(k) ) x(k)  n(k) ) As(k) () Euation 11 can be converted to e 9 by selecting A as a basis in R(B T ), where R(B T ) is the range space of B T. On the basis of the euivalence of es 9 and 11, we can directly apply AIC and MDL criteria originally
3 Ind. Eng. Che. Res., Vol. 38, No. 11, proposed for the detection of signal sources in signal processing 1 to the deterination of the nuber of PCs in process odeling..3. AIC and MDL Criteria. AIC and MDL are respectively proposed by Akaike 14 and Rissanen 15 for odel selection. Assuing that n(k) N(0, σ I) is independently identically distributed, we can describe the covariance atrix of x(k) in the following for, where Sh )E{xjxj T ) has a rank e. Denoting the eigenvalues of S by λ 1 g λ,, gλ,it is easy to verify that the sallest  eigenvalues of S are eual to σ, that is, and S ) E{xx T } ) Sh +σ I (13) λ +1 ) λ + ) ) λ ) σ S ) (λ i  σ )v i v T i + σ I i)1 where λ 1,, λ and v 1,, v are the largest eigenvalues and the associated eigenvectors of S. We denote w ) [λ 1,, λ, σ, v 1 T,, v T ] T R (+1)+1 as the paraeter vector for the PCA odel. Wax and Kailath 1 propose to deterine the nuber of PCs based on AIC and MDL, which can be suarized as follows. Given a set of N observations x(1),, x(n), which are assued to be a seuence of independent and identically distributed zeroean Gaussian vectors, the axiu likelihood estiate of w 1,5 is given by ŵ ) [ d 1,, d l, 1 T]T d i, c T 1,, c l  li)l+1 where d 1,, d are the eigenvalues and c 1,, c l the eigenvectors of the saple covariance atrix, which is N Ŝ ) 1/(N  1) k)1 x(k)x T (k). Wax and Kailath 1 have shown that the AIC and MDL functions will have the following fors, 1/(l) d s)l+1 AIC(l) ) log( s 1 ls)l+1 1/(nl) d s)l+1 MDL(l) ) 1og( s where M is the nuber of independent paraeters in the odel vector ŵ. For coplex valued signals, the eigenvectors c 1, c,, c l are coplex, and the nuber of independent paraeters is 1 1 ls)l+1 M ) l(  l) d s)(l)n d s)(l)n + M (14) + M log N (15) For real valued signals such as those encountered in cheical processes, the eigenvectors c 1, c,, c l are real. Since these vectors are noralized and utually orthogonal, the nuber of independent paraeters in these eigenvectors is M ) l (  l + 1) This forula is used for the application exaples in this paper. The first ter in AIC(l) or MDL(l) tends to decrease and the second ter increases with l. Theoretically, a iniu results for l [1, ]. Wax and Kailath 1 show that, for a large data length N, MDL gives a consistent estiate, while AIC tends to overestiate the nuber of PCs..4. Ibedded Error Function. The IEF ethod 16 is based on the idea that the easureent space can be divided in two subspaces: a principal coponent subspace which represents the iportant signals of the process and a residual subspace that has only noise in its easureents. The only reuireent to calculate this function is to know the eigenvalues. The calculation of this function is done using only the residual eigenvalues: [ l λ j IEF(l) ) (16) N(  l)]1/ where l represents the nuber of PCs used to represent the data. The IEF ethod assues that the data contain signals plus easureent errors siilar to those in e 11. Each PC will contain a portion of the signals and a portion of ibedded errors. Before all signals are extracted, the IEF(l) contains a ixture of signal and ibedded errors, which tends to decrease with l. At the point all signals are extracted, the IEF(l) contains ibedded errors only and should increase with l. The iniu value of the IEF, if exists, corresponds to the nuber of PCs needed to describe the data by leaving out easureent errors only. 3. Selection Criteria in Cheoetrics 3.1. Cuulative Percent Variance. The CPV is a easure of the percent variance captured by the first l PCs: l λ j j)1 CPV(l) ) 100[ (17) λ j]% j)1 With this criterion one selects a desired CPV, e.g., 90%, 95%, or 99%, which is very subjective. While we would like to account for as uch of the variance as possible, we want to retain as few principal coponents as possible. The decision then becoes a balance between the aount of parsiony and coprehensiveness of the odel. We ust decide what balance will serve our purpose. The CPV ethod to select the nuber of PCs is often abiguous because the CPV is onotonically increasing with the nuber of PCs.
4 439 Ind. Eng. Che. Res., Vol. 38, No. 11, Scree Test on Residual Percent Variance. The Scree test on RPV 17,18 is an epirical procedure to select the nuber of PCs using the RPV as a basis. The ethod looks for a knee point in the residual percent variance plotted against the nuber of principal coponents. The ethod is based on the idea that the residual variance should reach a steady state when the factors begin to account for rando errors. When a break point is found or when the plot stabilizes, that point corresponds to the nuber of principal coponents to represent the process. The RPV uses only the residual eigenvalues to do the calculations: λ j RPV(l) ) 100[ λ j j)1 ]% (18) The ipleentation of this ethod is relatively easy, but in soe cases it is difficult to find a knee if the curve decreases soothly. This point is observed in section 5.3 of the paper using the incineration process data Average Eigenvalue. The average eigenvalue approach 19 is a soewhat popular criterion to choose the nuber of PCs. This criterion accepts all eigenvalues with values above the average eigenvalue and rejects those below the average. The reason is that a PC contributing less than an average variable is insignificant. For covariancebased PCA, the average eigenvalue is 1 / trace(s), and for correlationbased PCA the average eigenvalue 1 / trace(r), which is 1. Then all the eigenvalues above 1 will be selected as the principal eigenvalues to for the odel. Therefore, this criterion is also known as the eigenvalueone rule Paralle1 Analysis. The PA ethod basically builds PCA odels for two atrices: one is the original data atrix and the other is an uncorrelated data atrix with the sae size as the original atrix. This ethod was developed originally by Horn 6 to enhance the perforance of the Scree test. When the eigenvalues for each atrix are plotted in the sae figure, all the values above the intersection represent the process inforation and the values under the intersection are considered noise. Because of this intersection, the parallel analysis ethod is not abiguous in the selection of the nuber of PCs. For a large nuber of saples, the eigenvalues for a correlation atrix of uncorrelated variables are 1. In this case, the PA ethod is identical to the AE ethod. However, when the saples are generated with a finite nuber of saples, the initial eigenvalues exceed 1, while the final eigenvalues are under 1. That is why Horn 6 suggested coparing the correlation atrix eigenvalues for uncorrelated variables with those of a real data atrix based on the sae saple size Autocorrelation. According to Shrager and Hendler, 1 it is possible to use an autocorrelation function to separate the noisy eigenvectors fro the sooth ones using either the score atrix or the loading atrix. A siple statistic for detecting noise is the autocorrelation function of order 1 for the PCA scores, N1 AC(l) ) t i,l t i+1,l (19) i)1 where N represents the nuber of saples. A value greater than +0.5 indicates soothness of the scores, while a value saller than 0.5 indicates that the coponent contains ainly noise and should not be included in the odel. 1 Two ajor drawbacks of this ethod are (1) this 0.5 threshold is soewhat arbitrary and () a large variance PC can have little autocorrelation. The second drawback is observed in the incineration data in section 5 of this paper Cross Validation Based on the R Ratio. Cross validation 3,7 is a popular statistical criterion to choose the nuber of factors in PCA. The basis of this ethod is to estiate the values of soe deleted data fro a odel and then copare these estiates with the actual values. Wold 3 proposes a crossvalidation ethod known as the R ratio. This ethod randoly divides the data set X into G groups. Then, for a starting value of l ) 1, a reduced data set is fored by deleting each group in turn and the paraeters in the odel are calculated fro the reduced data set. The predicted error su of suares (PRESS) is calculated fro the predicted and actual values of the deleted objects. In the eantie, the residual su of suares (RSS) the next coponent is extracted is calculated. A ratio, R(l) ) PRESS(l) RSS(l  1) (0) is then calculated starting with l ) 1,,  1. A value of the R ratio less than 1 indicates that the newly added coponent in the odel iproved the prediction and hence the calculation proceeds. A value of this ratio larger than 1 indicates that the new coponent does not iprove the prediction and hence should be deleted. Since the PCA odel calculation involves randoly deleted data, the NIPALS algorith is used which can handle issing values. The brief procedure for calculating the R ratio is suarized as follows. Refer to Wold 3 for the detailed procedure. (1) Scale the data X to zeroean (and unit variance for correlationbased PCA). Set l ) 1 and X 0 ) X. () Calculate: RSS(l  1) ) trace(x T l1 X l1 ) (3) Divide X l1 randoly into G groups. For all but one group calculate the lth PCA factor using the NIPALS algorith that handles issing values. After leaving out each group in turn. calculate PRESS(l) for all groups of data. (4) Calculate the R ratio using e 0. If R < 1, then calculate the lth PCA factor based on all data X l1, set X l ) X l1  t l p T l, set l :) l + 1, and return to step. If R g 1, terinate the progra Cross Validation Based on PRESS. In the R ratio ethod, the PRESS is calculated in a anner that X l1 ay be rerandoized after each increent of l. If a new PC results in a PRESS that is larger than the RSS without this PC, it is discarded. An alternative approach is to use PRESS alone to deterine the nuber of PCs. This procedure is related to Wold 3 and has been successfully used in practice. 8 The procedure to use PRESS to deterine the nuber of PCs is suarized as follows: (1) Scale the data X to zeroean (and unit variance for correlationbased PCA). Set l ) 1.
5 Ind. Eng. Che. Res., Vol. 38, No. 11, () Divide X randoly into G groups. For all but one group calculate all PCA factors using the NIPALS algorith that handles issing values. After leaving out each group in turn, calculate PRESS(l) for all groups of data. (3) A iniu in PRESS(l) corresponds to the best nuber of PCs to choose. Note that the nuber of PCs calculated in this procedure can be different fro that in the ethod of the R ratio. 4. Variance of the Reconstruction Error Criterion The variance of the reconstruction error ethod was developed by Qin and Dunia 9 to select the nuber of principal coponents based on the best reconstruction of the variables. An iportant characteristic of this approach is that the VRE index has a iniu, corresponding to the best reconstruction. When the PCA odel is used to reconstruct issing values or faulty sensors, the reconstruction error is a function of the nuber of PCs. The iniu found in the VRE calculation directly deterines the nuber of PCs. This is because the VRE is decoposed in two subspaces: the principal coponents subspace and a residual subspace. The portion in PCS has a tendency to increase with the nuber of PCs, and that in the RS has a tendency to decrease, resulting in a iniu in VRE. Qin and Dunia 9 consider that the sensor easureent is corrupted with a fault along a direction ξ i, where x* is the noral portion, f the fault agnitude, and ξ i ) 1. For sensor faults, ξ i corresponds to the ith colun of an identity atrix. For process faults, ξ i can be an arbitrary vector with unit nor. The reconstruction of the fault is given by correction along the fault direction, that is, so that x i is ost consistent with the PCA odel. The difference x*  x i is known as the reconstruction error. Qin and Dunia 9 define the variance of the reconstruction error as where x ) x* + fξ i x i ) x  f i ξ i u i var{ξ T i (x  x i )} ) ξ i T Rξ i (1) (ξ i T ξ i) ξ i ) (I  PP T )ξ i ) P P Tξ i () The proble to find the nuber of PCs is to iniize u i with respect to the nuber of PCs. Considering all possible faults, the VRE to be iniized is defined as u i VRE(l) ) ) (3) i)1var{ξ T i x} i)1ξ T i Rξ i The variancebased weighting factors are necessary to eualize the iportance of each variable. Note that the u i Figure 1. Reaction network for the batch reactor. above derivation is for a correlationbased PCA sincethe correlation atrix R is used. For a covariancebased PCA, one can siply replace with S the covariance atrix. The VRE procedure for selecting the nuber of PCs can be suarized as follows: (1) Build a PCA odel using noral data fro all variables. () Reconstruct each variable using other variables and calculate the VRE, u i. (3) The total VRE is calculated using e 3. (4) The nuber of PCs that gives the iniu VRE is selected, which corresponds to the best reconstruction. For a particular variable or fault direction ξ i,itis possible that u i g var{ξ T x}, which eans the odel gives a worse prediction than the ean of the data. In this case, the variable has little correlation with others and should be dropped fro the odel. Therefore, the VRE can be used to achieve the following objectives: (1) deterine the nuber of PCs by leaving out variables; () select variables that can be reliably reconstructed fro the odel; (3) siultaneously leave out a group of variables; and (4) include a particular fault or disturbance direction in selecting the nuber of PCs for the purpose of fault or disturbance detection VRE versus PRESS. Like PRESS, the VRE ethod works for both a correlationbased PCA and covariancebased PCA. Another siilarity between VRE and PRESS is that they do not reuire any fault to occur; they just need to leave out a portion of the data and reconstruct the fro the odels. Furtherore, the issing value treatent in the NIPALS algorith is euivalent to the fault reconstruction as shown in Dunia et al. 11 However, the VRE and PRESS ethods are different in the following ways: (1) The VRE ethod can include process faults with arbitrary direction ξ i to ephasize particular process faults or disturbances, while PRESS cannot. () If each sensor direction (ξ i ) is used once and only once, the VRE ethod is siilar to cross validation by leaving sensors out, while the PRESS leaves saples out randoly. (3) In PRESS, as any as G odels are built in the presence of issing values, while VRE builds one PCA odel only based on all available data. (4) In the VRE ethod, it is possible to check if a group of data can be reliably reconstructed, while the PRESS ethod fails to do this rigorously. (5) The VRE ethod is also suitable for recursive online calculation, as shown in Li et al Consistency of the VRE Method. Wax and Kailath 1 show that the MDL approach gives a consistent estiate of the nuber of PCs, while the AIC ethod
6 4394 Ind. Eng. Che. Res., Vol. 38, No. 11, 1999 Figure. Coparison for the siulated batch reactor free of noise. does not and tends to overestiate the nuber of PCs. The consistency of ost other ethods are not known. For the VRE ethod, Dunia and Qin 1 show that the VRE deonstrates a iniu, but it is not known whether the iniu corresponds to the true nuber of PCs. In this section, we establish the consistency conditions for the VRE ethod. First, we exaine the covariancebased PCA where the noise variances are assued eual. Fro e 13, the eigenvalues of covariance atrix S are λ i ) { λh i + σ σ 1 e i e < i e where λh i (i ) 1,,..., ) are the nonzero eigenvalues of Sh, the covariance of xj. For convenience we reuire λh 1 g λh g g λh. Assuing l PCs are included in the PCA odel, the VRE can be written as follows using e, u il ) ξ il T ξ il (ξ il T ξ il ) ) ξ T i P l(p T l SP l)p l (ξ T i P lp T l ξ i ) where e il P lt ξ i R l and Λ l ) diag{λ l+1, λ l+,, λ }. Given the above analysis, we have the following theore about the consistency of the covariancebased VRE ethod. Theore 1. Assuing the covariance atrix of the data is given in e 13, then (1) the VRE achieves a iniu at l e and () the VRE achieves a global iniu at l ) if λh g e il / e i σ for l <. The proof of this theore is given in Appendix A. We ake the following rearks about Theore 1: (1) The VRE ethod will never over estiate the nuber of PCs. T ξ i e T ilp T l SP le il ) (e T il e il ) e il T Λ l e il (e il T e il ) (4)
7 Ind. Eng. Che. Res., Vol. 38, No. 11, Figure 3. Coparison for the siulated batch reactor with stationary additive Gaussian noise. () The condition for a consistent estiate reuires that the variance of the signal is slightly larger than the variance of the noise, which is uite reasonable. For a correlationbased PCA or the general case of a covariancebased PCA, the easureent errors do not necessarily have the sae variance. In this case, we denote the eigenvalues of the correlation (or covariance) atrix as λ 1, λ,, λ, σ +1, σ +,, σ, where σ i are variance uantities of the noise ters. To guarantee that VRE gives a consistent estiate in this case, we have the following theore. Theore. Assuing the eigenvalues of the correlation or covariance atrix with descending order are λ 1, λ,, λ, σ +1,, σ, the VRE achieves a iniu at l ) if (1) σ g σ +1 e il / e i for l g, and () λ l g (1 + e il / e i ) for l <. σ +1 The proof of this theore is given in Appendix B. The first condition reuires that the noise variances, although different, should not be too different. The second condition reuires that the variance of the signal be larger than the variance of the noise. 5. Coparisons on Three Case Studies 5.1. Siulated Batch Reactor Data. An isotheral batch reactor is siulated with four firstorder reactions taking place. 31 Figure 1 depicts the reaction network for the batch reactor. The aterial balance euation for the batch reactor is dc i dt ) r i (5) where the reaction rates for the five species are
8 4396 Ind. Eng. Che. Res., Vol. 38, No. 11, 1999 Figure 4. Coparison for the siulated batch reactor with nonstationary ultiplicative Gaussian noise. r A )k 1 C A  k C A )(k 1 + k )C A r R ) k 1 C A  k 3 C R  k 4 C R ) k 1 C A  (k 3 + k 4 )C R r T ) k C A r U ) k 4 C R r S ) k 3 C R The reaction rate constants are k 1 ) 10 7 exp { T } [)]h1 k ) exp { 4000 T k 3 ) exp { T } [)]h1 } [)]h1 k 4 ) exp { T } [)]h1 The teperature was kept constant at C. It was assued that initially only the raw aterial A was present and the vessel was wellixed. Three different cases of easureent noise were siulated for the batch reactor: (1) a noisefree case, () stationary additive Gaussian noise with a standard deviation of 0.00, and (3) nonstationary ultiplicative Gaussian noise with a standard deviation of 10% of the variables aplitude. The siulation generated a total of 00 saples for the 5 variables using this kinetic odel. The data were scaled to zeroean and unit variance, and then analyzed with the 11 different ethods to deterine the nuber of PCs to adeuately represent the odel.
9 Ind. Eng. Che. Res., Vol. 38, No. 11, Figure 5. Coparison for a real boiler process. Table 1. Nuber of PCs Deterined fro Each Analyzed Method reactor boiler incinerator CPV RPV abiguous 1 abiguous AE PA IEF no solution no solution AC no solution PRESS R ratio 5 AIC no solution no solution MDL no solution no solution VRE cov 1 VRE cor 1 5 The analysis results are shown in Figure for the noisefree case, Figure 3 for the additive noise, and Figure 4 for the case of ultiplicative noise. Table 1 suarizes the results for the case of additive noise and shows if the ethod has found a solution, an abiguous solution or no solution at all. Observe that, for the additive noise case, the RPV ethod shows a sooth curve, aking it difficult to decide on the nuber of PCs. The CPV and PRESS ethods choose three PCs. AE and PA deterine one PC, and the rest of the seven ethods agree in two PCs. We conclude that to have an adeuate representation of the process, it is necessary to use two PCs. Also observe in Figure (noisefree) and Figure 4 (ultiplicative noise) that the AIC and MDL ethods were unable to find a solution. 5.. Boiler Process Data. The data fro an industrial boiler consist of 7 variables and 630 saples. The variables represent teperatures, pressures, flows, and concentrations. Before the analysis was done the data were centered and scaled to zeroean and unit vari
10 4398 Ind. Eng. Che. Res., Vol. 38, No. 11, 1999 Figure 6. Coparison for a real incineration process. Table. Characteristics for the 11 Methods ( ) no; ) yes; L ) Low; M ) Mediu; H ) High) CPV RPV AE PA AC IEF R PRESS AIC MDL VRE objectiveness uniueness covariancebased correlationbased reliability coputation L L L L L L H M L L L ance. The results fro the 11 ethods are shown in Figure 5, with the selected nuber of PCs shown in Table 1. It is seen fro Figure 5 that, for the CPV ethod, this process is highly correlated because only with one PC that one is able to capture ore than 95% of the variance. Basically, all the ethods that found a solution agree with one PC with the exception of the R ratio ethod. The IEF, AIC, and MDL ethods are unable to find a iniu for this real data set. The autocorrelation ethod is also ineffective Incineration Process Data. The incineration data are collected fro an industrial waste treatent incinerator, where waste aterial fro different reactions in a process are burned before they are released to the atosphere. The total analyzed data are 900 saples and 0 variables. The variables included are teperatures, pressures, flows, and concentrations. Figure 6 depicts the results fro the 11 ethods, and Tables 1 and suarize the selected nuber of PCs for each ethod and the characteristics for the 11 ethods respectively.
11 Ind. Eng. Che. Res., Vol. 38, No. 11, This process is very noisy and, as expected, the nuber of PCs to represent the process is large. The IEF, AIC, and MDL ethods did not find a solution at all. The autocorrelation ethod is soewhat abiguous as the autocorrelation of the 10th and 11th PCs becoe larger than 0.5 after several PCs being saller than 0.5. The RPV index is onotonically decreasing but fails to show a knee location, aking the solution abiguous. The rest of the ethods found a solution. However, the PRESS solution is too high and VRE cov too low for this process. The R ratio and VRE cor agree in the sae solution. 6. Conclusions Aong the 11 ethods presented in the paper, ost of the have onotonically decreasing or increasing indices. The IEF, AC, AIC, and MDL worked well with the siulated exaple, but they failed when used with real data. The ost reliable ethods are CPV, AE, PA, PRESS, and VRE. A coparison of the 11 tested ethods for the selection of the nuber of PCs is suarized in Table. Considering the effectiveness, reliability, and objectiveness, the PRESS and correlationbased VRE ethods are superior to the others. The VRE ethod is preferred to the PRESS ethod in the consistency of the estiate, coputational cost, and ability to include a particular disturbance or fault direction in selecting the nuber of PCs. Acknowledgent This work is supported by the National Science Foundation (under Grant CTS ), AMD, Air Products, ALCOA Foundation, DuPont, Union Carbide, Consejo Nacional de Ciencia y Tecnología (CONACyT), and Instituto Tecnológico de Durango (ITD). Noenclature AC ) autocorrelation index B ) odel atrix C ) odel projection atrix C i ) coponent i concentration CPV ) cuulative percent variance index f ) fault agnitude I ) identity atrix IEF ) ibedded error function index k i ) reaction rate constant for coponent i l ) nuber of principal coponents ) nuber of variables M ) nuber of independent paraeters n ) observation noise vector N ) nuber of saples P ) loadings atrix PRESS ) predicted error su of suares index r i ) reaction rate for coponent i R ) R ratio index R ) correlation atrix RPV ) residual percent variance index RSS ) residual su of suares E ) expectation R ) real s ) signal sources vector S ) covariance atrix S p ) principal coponents subspace S r ) residual subspace t ) score vector T ) teperature T ) score atrix u ) variance of the reconstruction error x ) saple vector X ) data atrix Greek Letters λ ) eigenvalue ξ ) fault direction Λ ) eigenvalues diagonal atrix ) Euclidean nor Superscripts and Subscripts )referred to the residual subspace ˆ )referred to the principal coponents subspace / ) uncorrupted portion T ) transpose Abbreviations AC ) autocorrelation CPV ) cuulative percent variance diag ) diagonal IEF ) ibedded error function PCA ) principal coponents analysis PCS ) principal coponents subspace PRESS ) predicted su of suares RS ) residual subspace RSS ) residual su of suares PCs ) principal coponents RPV ) residual percent variance VRE ) variance of the reconstruction error Appendix A Proof of Theore 1. (1) Denoting e il ) [ɛ il+1, ɛ i,l+,, ɛ i, ] T, it is clear that e i ) [e i,+1,, e i, ] T for > l. Ifl g, the VRE can be written as which is onotonically increasing with l. Therefore, it is ipossible for VRE to have a iniu at l >. () To prove that VRE achieves a global iniu at l ), we just need to show u il g u i for l <. Denoting where Λh l ) diag{λh l+1,, λh,0,, 0}, the VRE in e 4 can be written as follows: and u il ) e T il σ I l e il ) σ (e T il e il ) (e T il e il ) u il ) e T il(λh l + σ I l )e il (e T il e il ) ) e T ilλh l e il e il + σ 4 e il ) Λ l ) Λh l + σ I l λ j ɛ i,j + σ g e il 4 e il ) σ ɛ i,j λh ɛ i,j + σ (6) e il 4 e il
12 4400 Ind. Eng. Che. Res., Vol. 38, No. 11, 1999 To reuire u il g u i, we need to have or Appendix B Proof of Theore. In the case of uneual noise variances e 4 still holds. If l g, and To reuire u il g u i, we need or If I <, σ u i ) e i ɛ i,j σ λh g σ  σ ɛ i,j ) e il 4 e i e il e i e il u il ) u i ) λh g e il e i σ QED u il ) e T ilλ l e il (e T il e il ) g e T ilσ I l e il ) σ e il 4 e il u i ) e T iλ e i (e T i e i ) e σ +1 ɛ i σ e il g σ +1 e i λ j ɛ i,j + (e T il e il ) σ k ɛ i,k k)+1 (e i T e i ) e σ il g σ +1 e i k)+1 σ k ɛ i,k (e il T e il ) λ j ɛ i,j u il  u i ) + σ k ɛ i,k e il 4 k)+1 [ 1 e il 41 e i 4] g ) To reuire u il g u i for l <, we need λ g σ +1 ( ) 1 + e il QED e i Literature Cited λ ɛ i,j  e σ il 4  e i 4 k ɛ i,k e il 4 k)+1 e il 4 e i 4 λ λ j ɛ i,j  e il 4 k)+1 (1) Wax, M.; Kailath, T. Detection of signals by inforation criteria. IEEE Trans. Acoust. Speech Signal Process. ASSP33, 1985, () Malinowski, E. R. Factor Analysis in Cheistry; Wiley Interscience: New York, (3) Wold, S. Cross validatory estiation of the nuber of coponents in factor and principal coponents analysis. Technoetrics 1978, 0, (4) Jackson, J. E. A User s Guide to Principal Coponents; WileyInterscience: New York, (5) MacGregor, J. F. Statistical process control of ultivariate processes. In IFAC ADCHEM Proceedings, Kyoto, Japan, (6) Noikos, P.; MacGregor, J. F. Multivariate SPC charts for onitoring batch process. Technoetrics 1995, 37 (3), (7) MacGregor, J. F.; Kourti, T. Statistical process control of ultivariate processes. Control Eng. Practice 1995, 3 (3), (8) Haron, J. L.; Duboc, Ph.; Bonvin, D. Factor analytical odeling of biocheical data. Coput. Che. 1995, 19, (9) Piovoso, M. J.; Kosanovich, K. A.; Pearson, R. K. Monitoring process perforance in realtie. In Proceedings of ACC, Chicago, 199, (10) Martens, H.; Naes, T. Multivariate Calibration; John Wiley and Sons: New York, (11) Dunia, R.; Qin, J.; Edgar, T. F.; McAvoy, T. J. Sensor fault identification and reconstruction using principal coponent analysis. In 13th IFAC World Congress, San Francisco, 1996; N: (1) Dunia, R.; Qin, S. J. A unified geoetric approach to process and sensor fault identification. Coput. Che. 1998,, ɛ i,j ( e il + e i ) σ k ɛ i,k e il 4 e i 4 ɛ i,j σ k ɛ i,k ( e il + e i ] ) k)+1 ) [λ  e il 4 e i 4 ɛ i,j σ +1 ɛ i,k ] k)+1 g [λ  e il 4 e i 4 ɛ i,j e il + e i ] ) [λ  σ +1 e il 4 e i
13 Ind. Eng. Che. Res., Vol. 38, No. 11, (13) Ku, W.; Storer, R. H.; Georgakis, C. Disturbance detection and isolation by dynaic principal coponent analysis. Cheo. Intell. Lab. Syst. 1995, 30, (14) Akaike, H. Inforation theory and an extension of the axiu likelihood principle. In Proceedings nd International Syposiu on Inforation Theory; Petrov and Caski, Eds., 1974; pp (15) Rissanen, J. Modeling by shortest data description. Autoatica 1978, 14, (16) Malinowski, E. R. Deterination of the nuber of factors and the experiental error in a data atrix. Anal. Che. 1977, 49 (4), (17) Cattell, R. B. The scree test for the nuber of factors. Multivariate Behav. Res. 1966, April, (18) Rozett, R. W.; Petersen, E. M. Methods of factor analysis of ass spectra. Anal. Che. 1975, 47 (8), (19) Kaiser, H. F. The application of electronic coputers to factor analysis. Educ. Psychol. Meas. 1960, 0 (1), (0) Zwick, W. R.; Velicer, W. F. Coparison of five rules for deterining the nuber of coponents to retain. Psychol. Bull. 1986, 99 (3), (1) Shrager, R. I.; Hendler, R. W. Titration of individual coponents in a ixture with resolution of difference spectra, pks, and redox transitions. Anal. Che. 198, 54 (7), () Carey, R. N.; Wold, N. S.; Westgard, J. O. Principal coponent analysis: an alternative to referee ethods in ethod coparison studies. Anal. Che. 1975, 47 (11), (3) Osten, D. W. Selection of optial regression odels via crossvalidation. J. Cheo. 1988,, (4) Wold, S.; Esbensen, K.; Geladi, P. Principal Coponent Analysis. Cheo. Intell. Lab. Syst. 1987,, (5) Anderson, T. W. Asyptotic theory for principal coponent analysis. Ann. Math. Stat. 1963, 34, (6) Horn, J. L. A rationale and test for the nuber of factors in factor analysis. Psychoetrica 1965, 30 (), (7) Eastent, H. T.; Krzanowski, W. Crossvalidatory choice of the nuber of coponents fro a principal coponent analysis. Technoetrics 198, 4 (1), (8) MacGregor, J. F. Personal counication, (9) Qin, S. J.; Dunia, R. Deterining the nuber of principal coponents for best reconstruction. In IFAC DYCOPS 98, Greece, June (30) Li, W.; Yue, H.; Valle, S.; Qin, J. Recursive PCA for adaptive process onitoring. Subited to J. Process Control (31) Kraer, M. A. Nonlinear principal coponent analysis using autoassociative neural networks. AIChE J. 1991, 37, (3) Hotelling, H. Analysis of a coplex of statistical variables into principal coponents. J. Educ. Psychol. 1933, 4 (6), (33) Pearson, K. On lines and planes of closest fit to systes of points in space. Phil. Mag. 1901, series 6, (11), (34) Hies, D. M.; Storer, R. H.; Georgakis, C. Deterination of the nuber of principal coponents for disturbance detection and isolation. Proceedings of the Aerican Control Conference, Baltiore, MD, June 1994; pp Received for review February 15, 1999 Revised anuscript received July 8, 1999 Accepted August, 1999 IE990110I
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