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


 Bruce Walsh
 1 years ago
 Views:
Transcription
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
ExtendedHorizon Analysis of Pressure Sensitivities for Leak Detection in Water Distribution Networks: Application to the Barcelona Network
2013 European Control Conference (ECC) July 1719, 2013, Zürich, Switzerland. ExtendedHorizon Analysis of Pressure Sensitivities for Leak Detection in Water Distribution Networks: Application to the Barcelona
More informationOnline Bagging and Boosting
Abstract Bagging and boosting are two of the ost wellknown enseble learning ethods due to their theoretical perforance guarantees and strong experiental results. However, these algoriths have been used
More informationCRM FACTORS ASSESSMENT USING ANALYTIC HIERARCHY PROCESS
641 CRM FACTORS ASSESSMENT USING ANALYTIC HIERARCHY PROCESS Marketa Zajarosova 1* *Ph.D. VSB  Technical University of Ostrava, THE CZECH REPUBLIC arketa.zajarosova@vsb.cz Abstract Custoer relationship
More information6. Time (or Space) Series Analysis
ATM 55 otes: Tie Series Analysis  Section 6a Page 8 6. Tie (or Space) Series Analysis In this chapter we will consider soe coon aspects of tie series analysis including autocorrelation, statistical prediction,
More informationA CHAOS MODEL OF SUBHARMONIC OSCILLATIONS IN CURRENT MODE PWM BOOST CONVERTERS
A CHAOS MODEL OF SUBHARMONIC OSCILLATIONS IN CURRENT MODE PWM BOOST CONVERTERS Isaac Zafrany and Sa BenYaakov Departent of Electrical and Coputer Engineering BenGurion University of the Negev P. O. Box
More informationarxiv:0805.1434v1 [math.pr] 9 May 2008
Degreedistribution stability of scalefree networs Zhenting Hou, Xiangxing Kong, Dinghua Shi,2, and Guanrong Chen 3 School of Matheatics, Central South University, Changsha 40083, China 2 Departent of
More informationEvaluating Inventory Management Performance: a Preliminary DeskSimulation Study Based on IOC Model
Evaluating Inventory Manageent Perforance: a Preliinary DeskSiulation Study Based on IOC Model Flora Bernardel, Roberto Panizzolo, and Davide Martinazzo Abstract The focus of this study is on preliinary
More informationAn Innovate Dynamic Load Balancing Algorithm Based on Task
An Innovate Dynaic Load Balancing Algorith Based on Task Classification Hongbin Wang,,a, Zhiyi Fang, b, Guannan Qu,*,c, Xiaodan Ren,d College of Coputer Science and Technology, Jilin University, Changchun
More informationAdaptive Modulation and Coding for Unmanned Aerial Vehicle (UAV) Radio Channel
Recent Advances in Counications Adaptive odulation and Coding for Unanned Aerial Vehicle (UAV) Radio Channel Airhossein Fereidountabar,Gian Carlo Cardarilli, Rocco Fazzolari,Luca Di Nunzio Abstract In
More informationASIC Design Project Management Supported by Multi Agent Simulation
ASIC Design Project Manageent Supported by Multi Agent Siulation Jana Blaschke, Christian Sebeke, Wolfgang Rosenstiel Abstract The coplexity of Application Specific Integrated Circuits (ASICs) is continuously
More informationImage restoration for a rectangular poorpixels detector
Iage restoration for a rectangular poorpixels detector Pengcheng Wen 1, Xiangjun Wang 1, Hong Wei 2 1 State Key Laboratory of Precision Measuring Technology and Instruents, Tianjin University, China 2
More informationAnalyzing Spatiotemporal Characteristics of Education Network Traffic with Flexible Multiscale Entropy
Vol. 9, No. 5 (2016), pp.303312 http://dx.doi.org/10.14257/ijgdc.2016.9.5.26 Analyzing Spatioteporal Characteristics of Education Network Traffic with Flexible Multiscale Entropy Chen Yang, Renjie Zhou
More informationSearching strategy for multitarget discovery in wireless networks
Searching strategy for ultitarget discovery in wireless networks Zhao Cheng, Wendi B. Heinzelan Departent of Electrical and Coputer Engineering University of Rochester Rochester, NY 467 (585) 75{878,
More informationMachine Learning Applications in Grid Computing
Machine Learning Applications in Grid Coputing George Cybenko, Guofei Jiang and Daniel Bilar Thayer School of Engineering Dartouth College Hanover, NH 03755, USA gvc@dartouth.edu, guofei.jiang@dartouth.edu
More informationAN ALGORITHM FOR REDUCING THE DIMENSION AND SIZE OF A SAMPLE FOR DATA EXPLORATION PROCEDURES
Int. J. Appl. Math. Coput. Sci., 2014, Vol. 24, No. 1, 133 149 DOI: 10.2478/acs20140011 AN ALGORITHM FOR REDUCING THE DIMENSION AND SIZE OF A SAMPLE FOR DATA EXPLORATION PROCEDURES PIOTR KULCZYCKI,,
More informationManaging Complex Network Operation with Predictive Analytics
Managing Coplex Network Operation with Predictive Analytics Zhenyu Huang, Pak Chung Wong, Patrick Mackey, Yousu Chen, Jian Ma, Kevin Schneider, and Frank L. Greitzer Pacific Northwest National Laboratory
More informationUse of extrapolation to forecast the working capital in the mechanical engineering companies
ECONTECHMOD. AN INTERNATIONAL QUARTERLY JOURNAL 2014. Vol. 1. No. 1. 23 28 Use of extrapolation to forecast the working capital in the echanical engineering copanies A. Cherep, Y. Shvets Departent of finance
More informationApplying Multiple Neural Networks on Large Scale Data
0 International Conference on Inforation and Electronics Engineering IPCSIT vol6 (0) (0) IACSIT Press, Singapore Applying Multiple Neural Networks on Large Scale Data Kritsanatt Boonkiatpong and Sukree
More informationThis paper studies a rental firm that offers reusable products to price and qualityofservice sensitive
MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol., No. 3, Suer 28, pp. 429 447 issn 523464 eissn 5265498 8 3 429 infors doi.287/so.7.8 28 INFORMS INFORMS holds copyright to this article and distributed
More informationComment on On Discriminative vs. Generative Classifiers: A Comparison of Logistic Regression and Naive Bayes
Coent on On Discriinative vs. Generative Classifiers: A Coparison of Logistic Regression and Naive Bayes JingHao Xue (jinghao@stats.gla.ac.uk) and D. Michael Titterington (ike@stats.gla.ac.uk) Departent
More informationPERFORMANCE METRICS FOR THE IT SERVICES PORTFOLIO
Bulletin of the Transilvania University of Braşov Series I: Engineering Sciences Vol. 4 (53) No.  0 PERFORMANCE METRICS FOR THE IT SERVICES PORTFOLIO V. CAZACU I. SZÉKELY F. SANDU 3 T. BĂLAN Abstract:
More informationReliability Constrained Packetsizing for Linear Multihop Wireless Networks
Reliability Constrained acketsizing for inear Multihop Wireless Networks Ning Wen, and Randall A. Berry Departent of Electrical Engineering and Coputer Science Northwestern University, Evanston, Illinois
More information( C) CLASS 10. TEMPERATURE AND ATOMS
CLASS 10. EMPERAURE AND AOMS 10.1. INRODUCION Boyle s understanding of the pressurevolue relationship for gases occurred in the late 1600 s. he relationships between volue and teperature, and between
More informationCapacity of MultipleAntenna Systems With Both Receiver and Transmitter Channel State Information
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO., OCTOBER 23 2697 Capacity of MultipleAntenna Systes With Both Receiver and Transitter Channel State Inforation Sudharan K. Jayaweera, Student Meber,
More informationMedia Adaptation Framework in Biofeedback System for Stroke Patient Rehabilitation
Media Adaptation Fraework in Biofeedback Syste for Stroke Patient Rehabilitation Yinpeng Chen, Weiwei Xu, Hari Sundara, Thanassis Rikakis, ShengMin Liu Arts, Media and Engineering Progra Arizona State
More informationPhysics 211: Lab Oscillations. Simple Harmonic Motion.
Physics 11: Lab Oscillations. Siple Haronic Motion. Reading Assignent: Chapter 15 Introduction: As we learned in class, physical systes will undergo an oscillatory otion, when displaced fro a stable equilibriu.
More informationData Set Generation for Rectangular Placement Problems
Data Set Generation for Rectangular Placeent Probles Christine L. Valenzuela (Muford) Pearl Y. Wang School of Coputer Science & Inforatics Departent of Coputer Science MS 4A5 Cardiff University George
More informationInformation Processing Letters
Inforation Processing Letters 111 2011) 178 183 Contents lists available at ScienceDirect Inforation Processing Letters www.elsevier.co/locate/ipl Offline file assignents for online load balancing Paul
More informationReal Time Target Tracking with Binary Sensor Networks and Parallel Computing
Real Tie Target Tracking with Binary Sensor Networks and Parallel Coputing Hong Lin, John Rushing, Sara J. Graves, Steve Tanner, and Evans Criswell Abstract A parallel real tie data fusion and target tracking
More informationAudio Engineering Society. Convention Paper. Presented at the 119th Convention 2005 October 7 10 New York, New York USA
Audio Engineering Society Convention Paper Presented at the 119th Convention 2005 October 7 10 New York, New York USA This convention paper has been reproduced fro the authors advance anuscript, without
More informationPerformance Evaluation of Machine Learning Techniques using Software Cost Drivers
Perforance Evaluation of Machine Learning Techniques using Software Cost Drivers Manas Gaur Departent of Coputer Engineering, Delhi Technological University Delhi, India ABSTRACT There is a treendous rise
More informationAn Approach to Combating Freeriding in PeertoPeer Networks
An Approach to Cobating Freeriding in PeertoPeer Networks Victor Ponce, Jie Wu, and Xiuqi Li Departent of Coputer Science and Engineering Florida Atlantic University Boca Raton, FL 33431 April 7, 2008
More informationData Streaming Algorithms for Estimating Entropy of Network Traffic
Data Streaing Algoriths for Estiating Entropy of Network Traffic Ashwin Lall University of Rochester Vyas Sekar Carnegie Mellon University Mitsunori Ogihara University of Rochester Jun (Ji) Xu Georgia
More informationEquivalent Tapped Delay Line Channel Responses with Reduced Taps
Equivalent Tapped Delay Line Channel Responses with Reduced Taps Shweta Sagari, Wade Trappe, Larry Greenstein {shsagari, trappe, ljg}@winlab.rutgers.edu WINLAB, Rutgers University, North Brunswick, NJ
More informationQuality evaluation of the modelbased forecasts of implied volatility index
Quality evaluation of the odelbased forecasts of iplied volatility index Katarzyna Łęczycka 1 Abstract Influence of volatility on financial arket forecasts is very high. It appears as a specific factor
More informationPricing Asian Options using Monte Carlo Methods
U.U.D.M. Project Report 9:7 Pricing Asian Options using Monte Carlo Methods Hongbin Zhang Exaensarbete i ateatik, 3 hp Handledare och exainator: Johan Tysk Juni 9 Departent of Matheatics Uppsala University
More informationMarkovian inventory policy with application to the paper industry
Coputers and Cheical Engineering 26 (2002) 1399 1413 www.elsevier.co/locate/copcheeng Markovian inventory policy with application to the paper industry K. Karen Yin a, *, Hu Liu a,1, Neil E. Johnson b,2
More informationExercise 4 INVESTIGATION OF THE ONEDEGREEOFFREEDOM SYSTEM
Eercise 4 IVESTIGATIO OF THE OEDEGREEOFFREEDOM SYSTEM 1. Ai of the eercise Identification of paraeters of the euation describing a onedegreeof freedo (1 DOF) atheatical odel of the real vibrating
More informationINTEGRATED ENVIRONMENT FOR STORING AND HANDLING INFORMATION IN TASKS OF INDUCTIVE MODELLING FOR BUSINESS INTELLIGENCE SYSTEMS
Artificial Intelligence Methods and Techniques for Business and Engineering Applications 210 INTEGRATED ENVIRONMENT FOR STORING AND HANDLING INFORMATION IN TASKS OF INDUCTIVE MODELLING FOR BUSINESS INTELLIGENCE
More informationREQUIREMENTS FOR A COMPUTER SCIENCE CURRICULUM EMPHASIZING INFORMATION TECHNOLOGY SUBJECT AREA: CURRICULUM ISSUES
REQUIREMENTS FOR A COMPUTER SCIENCE CURRICULUM EMPHASIZING INFORMATION TECHNOLOGY SUBJECT AREA: CURRICULUM ISSUES Charles Reynolds Christopher Fox reynolds @cs.ju.edu fox@cs.ju.edu Departent of Coputer
More informationModelling of Thermal Behavior NDoped Silicon Resistor
Journal of Sensor Technology, 0,, 337 http://dx.doi.org/0.436/jst.0.309 Published Online Septeber 0 (http://www.scirp.org/journal/jst) Modelling of Theral Behavior NDoped Silicon Resistor Fouad Kerrour,
More informationInternational Journal of Management & Information Systems First Quarter 2012 Volume 16, Number 1
International Journal of Manageent & Inforation Systes First Quarter 2012 Volue 16, Nuber 1 Proposal And Effectiveness Of A Highly Copelling Direct Mail Method  Establishent And Deployent Of PMOSDM Hisatoshi
More informationSoftware Quality Characteristics Tested For Mobile Application Development
Thesis no: MGSE201502 Software Quality Characteristics Tested For Mobile Application Developent Literature Review and Epirical Survey WALEED ANWAR Faculty of Coputing Blekinge Institute of Technology
More informationAn Optimal Task Allocation Model for System Cost Analysis in Heterogeneous Distributed Computing Systems: A Heuristic Approach
An Optial Tas Allocation Model for Syste Cost Analysis in Heterogeneous Distributed Coputing Systes: A Heuristic Approach P. K. Yadav Central Building Research Institute, Rooree 247667, Uttarahand (INDIA)
More informationExact Matrix Completion via Convex Optimization
Exact Matrix Copletion via Convex Optiization Eanuel J. Candès and Benjain Recht Applied and Coputational Matheatics, Caltech, Pasadena, CA 91125 Center for the Matheatics of Inforation, Caltech, Pasadena,
More informationThe AGA Evaluating Model of Customer Loyalty Based on Ecommerce Environment
6 JOURNAL OF SOFTWARE, VOL. 4, NO. 3, MAY 009 The AGA Evaluating Model of Custoer Loyalty Based on Ecoerce Environent Shaoei Yang Econoics and Manageent Departent, North China Electric Power University,
More informationResource Allocation in Wireless Networks with Multiple Relays
Resource Allocation in Wireless Networks with Multiple Relays Kağan Bakanoğlu, Stefano Toasin, Elza Erkip Departent of Electrical and Coputer Engineering, Polytechnic Institute of NYU, Brooklyn, NY, 0
More informationPure Bending Determination of StressStrain Curves for an Aluminum Alloy
Proceedings of the World Congress on Engineering 0 Vol III WCE 0, July 68, 0, London, U.K. Pure Bending Deterination of StressStrain Curves for an Aluinu Alloy D. TorresFranco, G. UrriolagoitiaSosa,
More informationModified Latin Hypercube Sampling Monte Carlo (MLHSMC) Estimation for Average Quality Index
Analog Integrated Circuits and Signal Processing, vol. 9, no., April 999. Abstract Modified Latin Hypercube Sapling Monte Carlo (MLHSMC) Estiation for Average Quality Index Mansour Keraat and Richard Kielbasa
More informationThe Virtual Spring Mass System
The Virtual Spring Mass Syste J. S. Freudenberg EECS 6 Ebedded Control Systes Huan Coputer Interaction A force feedbac syste, such as the haptic heel used in the EECS 6 lab, is capable of exhibiting a
More informationThe Research of Measuring Approach and Energy Efficiency for Hadoop Periodic Jobs
Send Orders for Reprints to reprints@benthascience.ae 206 The Open Fuels & Energy Science Journal, 2015, 8, 206210 Open Access The Research of Measuring Approach and Energy Efficiency for Hadoop Periodic
More informationBudgetoptimal Crowdsourcing using Lowrank Matrix Approximations
Budgetoptial Crowdsourcing using Lowrank Matrix Approxiations David R. Karger, Sewoong Oh, and Devavrat Shah Departent of EECS, Massachusetts Institute of Technology Eail: {karger, swoh, devavrat}@it.edu
More informationFuzzy Sets in HR Management
Acta Polytechnica Hungarica Vol. 8, No. 3, 2011 Fuzzy Sets in HR Manageent Blanka Zeková AXIOM SW, s.r.o., 760 01 Zlín, Czech Republic blanka.zekova@sezna.cz Jana Talašová Faculty of Science, Palacký Univerzity,
More informationImplementation of Active Queue Management in a Combined Input and Output Queued Switch
pleentation of Active Queue Manageent in a obined nput and Output Queued Switch Bartek Wydrowski and Moshe Zukeran AR Special Research entre for UltraBroadband nforation Networks, EEE Departent, The University
More informationEvaluating the Effectiveness of Task Overlapping as a Risk Response Strategy in Engineering Projects
Evaluating the Effectiveness of Task Overlapping as a Risk Response Strategy in Engineering Projects Lucas Grèze Robert Pellerin Nathalie Perrier Patrice Leclaire February 2011 CIRRELT201111 Bureaux
More informationDesign of Model Reference Self Tuning Mechanism for PID like Fuzzy Controller
Research Article International Journal of Current Engineering and Technology EISSN 77 46, PISSN 347 56 4 INPRESSCO, All Rights Reserved Available at http://inpressco.co/category/ijcet Design of Model Reference
More informationResearch Article Performance Evaluation of Human Resource Outsourcing in Food Processing Enterprises
Advance Journal of Food Science and Technology 9(2): 964969, 205 ISSN: 20424868; eissn: 20424876 205 Maxwell Scientific Publication Corp. Subitted: August 0, 205 Accepted: Septeber 3, 205 Published:
More informationENZYME KINETICS: THEORY. A. Introduction
ENZYME INETICS: THEORY A. Introduction Enzyes are protein olecules coposed of aino acids and are anufactured by the living cell. These olecules provide energy for the organis by catalyzing various biocheical
More informationThe Velocities of Gas Molecules
he Velocities of Gas Molecules by Flick Colean Departent of Cheistry Wellesley College Wellesley MA 8 Copyright Flick Colean 996 All rights reserved You are welcoe to use this docuent in your own classes
More informationRECURSIVE DYNAMIC PROGRAMMING: HEURISTIC RULES, BOUNDING AND STATE SPACE REDUCTION. Henrik Kure
RECURSIVE DYNAMIC PROGRAMMING: HEURISTIC RULES, BOUNDING AND STATE SPACE REDUCTION Henrik Kure Dina, Danish Inforatics Network In the Agricultural Sciences Royal Veterinary and Agricultural University
More informationEfficient Key Management for Secure Group Communications with Bursty Behavior
Efficient Key Manageent for Secure Group Counications with Bursty Behavior Xukai Zou, Byrav Raaurthy Departent of Coputer Science and Engineering University of NebraskaLincoln Lincoln, NE68588, USA Eail:
More informationDynamic Placement for Clustered Web Applications
Dynaic laceent for Clustered Web Applications A. Karve, T. Kibrel, G. acifici, M. Spreitzer, M. Steinder, M. Sviridenko, and A. Tantawi IBM T.J. Watson Research Center {karve,kibrel,giovanni,spreitz,steinder,sviri,tantawi}@us.ib.co
More informationABSTRACT KEYWORDS. Comonotonicity, dependence, correlation, concordance, copula, multivariate. 1. INTRODUCTION
MEASURING COMONOTONICITY IN MDIMENSIONAL VECTORS BY INGE KOCH AND ANN DE SCHEPPER ABSTRACT In this contribution, a new easure of coonotonicity for diensional vectors is introduced, with values between
More informationOptimal ResourceConstraint Project Scheduling with Overlapping Modes
Optial ResourceConstraint Proect Scheduling with Overlapping Modes François Berthaut Lucas Grèze Robert Pellerin Nathalie Perrier Adnène Hai February 20 CIRRELT2009 Bureaux de Montréal : Bureaux de
More informationLecture L263D Rigid Body Dynamics: The Inertia Tensor
J. Peraire, S. Widnall 16.07 Dynaics Fall 008 Lecture L63D Rigid Body Dynaics: The Inertia Tensor Version.1 In this lecture, we will derive an expression for the angular oentu of a 3D rigid body. We shall
More informationIEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1. Secure Wireless Multicast for DelaySensitive Data via Network Coding
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1 Secure Wireless Multicast for DelaySensitive Data via Network Coding Tuan T. Tran, Meber, IEEE, Hongxiang Li, Senior Meber, IEEE,
More informationConstruction Economics & Finance. Module 3 Lecture1
Depreciation: Construction Econoics & Finance Module 3 Lecture It represents the reduction in arket value of an asset due to age, wear and tear and obsolescence. The physical deterioration of the asset
More informationWORK TRAVEL MODE CHOICE MODELING USING DATA MINING: DECISION TREES AND NEURAL NETWORKS
TRB Paper No. 034348 WORK TRAVEL MODE CHOICE MODELING USING DATA MINING: DECISION TREES AND NEURAL NETWORKS Chi Xie Research Assistant Departent of Civil and Environental Engineering University of Massachusetts,
More informationAUC Optimization vs. Error Rate Minimization
AUC Optiization vs. Error Rate Miniization Corinna Cortes and Mehryar Mohri AT&T Labs Research 180 Park Avenue, Florha Park, NJ 0793, USA {corinna, ohri}@research.att.co Abstract The area under an ROC
More informationAn Integrated Approach for Monitoring Service Level Parameters of SoftwareDefined Networking
International Journal of Future Generation Counication and Networking Vol. 8, No. 6 (15), pp. 1974 http://d.doi.org/1.1457/ijfgcn.15.8.6.19 An Integrated Approach for Monitoring Service Level Paraeters
More informationPresentation Safety Legislation and Standards
levels in different discrete levels corresponding for each one to a probability of dangerous failure per hour: > > The table below gives the relationship between the perforance level (PL) and the Safety
More informationA framework for performance monitoring, load balancing, adaptive timeouts and quality of service in digital libraries
Int J Digit Libr (2000) 3: 9 35 INTERNATIONAL JOURNAL ON Digital Libraries SpringerVerlag 2000 A fraework for perforance onitoring, load balancing, adaptive tieouts and quality of service in digital libraries
More informationCLOSEDLOOP SUPPLY CHAIN NETWORK OPTIMIZATION FOR HONG KONG CARTRIDGE RECYCLING INDUSTRY
CLOSEDLOOP SUPPLY CHAIN NETWORK OPTIMIZATION FOR HONG KONG CARTRIDGE RECYCLING INDUSTRY Y. T. Chen Departent of Industrial and Systes Engineering Hong Kong Polytechnic University, Hong Kong yongtong.chen@connect.polyu.hk
More informationModeling operational risk data reported above a timevarying threshold
Modeling operational risk data reported above a tievarying threshold Pavel V. Shevchenko CSIRO Matheatical and Inforation Sciences, Sydney, Locked bag 7, North Ryde, NSW, 670, Australia. eail: Pavel.Shevchenko@csiro.au
More informationCooperative Caching for Adaptive Bit Rate Streaming in Content Delivery Networks
Cooperative Caching for Adaptive Bit Rate Streaing in Content Delivery Networs Phuong Luu Vo Departent of Coputer Science and Engineering, International University  VNUHCM, Vietna vtlphuong@hciu.edu.vn
More informationThe Fundamentals of Modal Testing
The Fundaentals of Modal Testing Application Note 2433 Η(ω) = Σ n r=1 φ φ i j / 2 2 2 2 ( ω n  ω ) + (2ξωωn) Preface Modal analysis is defined as the study of the dynaic characteristics of a echanical
More information2. FINDING A SOLUTION
The 7 th Balan Conference on Operational Research BACOR 5 Constanta, May 5, Roania OPTIMAL TIME AND SPACE COMPLEXITY ALGORITHM FOR CONSTRUCTION OF ALL BINARY TREES FROM PREORDER AND POSTORDER TRAVERSALS
More informationLeak detection in open water channels
Proceedings of the 17th World Congress The International Federation of Autoatic Control Seoul, Korea, July 611, 28 Leak detection in open water channels Erik Weyer Georges Bastin Departent of Electrical
More informationON SELFROUTING IN CLOS CONNECTION NETWORKS. BARRY G. DOUGLASS Electrical Engineering Department Texas A&M University College Station, TX 778433128
ON SELFROUTING IN CLOS CONNECTION NETWORKS BARRY G. DOUGLASS Electrical Engineering Departent Texas A&M University College Station, TX 7788 A. YAVUZ ORUÇ Electrical Engineering Departent and Institute
More informationThe individual neurons are complicated. They have a myriad of parts, subsystems and control mechanisms. They convey information via a host of
CHAPTER 4 ARTIFICIAL NEURAL NETWORKS 4. INTRODUCTION Artificial Neural Networks (ANNs) are relatively crude electronic odels based on the neural structure of the brain. The brain learns fro experience.
More informationFactored Models for Probabilistic Modal Logic
Proceedings of the TwentyThird AAAI Conference on Artificial Intelligence (2008 Factored Models for Probabilistic Modal Logic Afsaneh Shirazi and Eyal Air Coputer Science Departent, University of Illinois
More informationModeling Parallel Applications Performance on Heterogeneous Systems
Modeling Parallel Applications Perforance on Heterogeneous Systes Jaeela AlJaroodi, Nader Mohaed, Hong Jiang and David Swanson Departent of Coputer Science and Engineering University of Nebraska Lincoln
More informationMethod of supply chain optimization in Ecommerce
MPRA Munich Personal RePEc Archive Method of supply chain optiization in Ecoerce Petr Suchánek and Robert Bucki Silesian University  School of Business Adinistration, The College of Inforatics and Manageent
More informationModeling Cooperative Gene Regulation Using Fast Orthogonal Search
8 The Open Bioinforatics Journal, 28, 2, 889 Open Access odeling Cooperative Gene Regulation Using Fast Orthogonal Search Ian inz* and ichael J. Korenberg* Departent of Electrical and Coputer Engineering,
More informationMatching and decision for Vehicle tracking in road situation
Matching and decision for Vehicle tracking in road situation Doinique GRUYER, Véronique BERGECHERFAOUI HeudiasycUMR CNRS 6599 Université de Technologie de Copiègne BP 0 59, 6005 Copiègne, France eail
More informationPreferencebased Search and Multicriteria Optimization
Fro: AAAI02 Proceedings. Copyright 2002, AAAI (www.aaai.org). All rights reserved. Preferencebased Search and Multicriteria Optiization Ulrich Junker ILOG 1681, route des Dolines F06560 Valbonne ujunker@ilog.fr
More informationBinary Embedding: Fundamental Limits and Fast Algorithm
Binary Ebedding: Fundaental Liits and Fast Algorith Xinyang Yi The University of Texas at Austin yixy@utexas.edu Eric Price The University of Texas at Austin ecprice@cs.utexas.edu Constantine Caraanis
More informationLecture L9  Linear Impulse and Momentum. Collisions
J. Peraire, S. Widnall 16.07 Dynaics Fall 009 Version.0 Lecture L9  Linear Ipulse and Moentu. Collisions In this lecture, we will consider the equations that result fro integrating Newton s second law,
More informationAn Improved Decisionmaking Model of Human Resource Outsourcing Based on Internet Collaboration
International Journal of Hybrid Inforation Technology, pp. 339350 http://dx.doi.org/10.14257/hit.2016.9.4.28 An Iproved Decisionaking Model of Huan Resource Outsourcing Based on Internet Collaboration
More informationModel Predictive Control Approach to Improve the Control Performance in Pasteurized Milk Process
Paper code: pc IChE International Conference Noveber, at Hatyai, Songkhla HAIAND odel Predictive Control Approach to Iprove the Control Perforance in Pasteurized ilk Process Sathit Niasuwan, Paisan Kittisupakorn
More informationCrossDomain Metric Learning Based on Information Theory
Proceedings of the TwentyEighth AAAI Conference on Artificial Intelligence CrossDoain Metric Learning Based on Inforation Theory Hao Wang,2, Wei Wang 2,3, Chen Zhang 2, Fanjiang Xu 2. State Key Laboratory
More informationA ComputerAided System for Melt Quality and Shrinkage Propensity Evaluation Based on the Solidification Process of Ductile Iron
Paper 08008(05).pdf, Page of 5 AFS Transactions 2008 Aerican Foundry Society, Schauburg, IL USA A CoputerAided Syste for Melt Quality and Shrinkage Propensity Evaluation Based on the Solidification Process
More informationMultiClass Deep Boosting
MultiClass Deep Boosting Vitaly Kuznetsov Courant Institute 25 Mercer Street New York, NY 002 vitaly@cis.nyu.edu Mehryar Mohri Courant Institute & Google Research 25 Mercer Street New York, NY 002 ohri@cis.nyu.edu
More informationCalculation Method for evaluating Solar Assisted Heat Pump Systems in SAP 2009. 15 July 2013
Calculation Method for evaluating Solar Assisted Heat Pup Systes in SAP 2009 15 July 2013 Page 1 of 17 1 Introduction This docuent describes how Solar Assisted Heat Pup Systes are recognised in the National
More informationA Scalable Application Placement Controller for Enterprise Data Centers
W WWW 7 / Track: Perforance and Scalability A Scalable Application Placeent Controller for Enterprise Data Centers Chunqiang Tang, Malgorzata Steinder, Michael Spreitzer, and Giovanni Pacifici IBM T.J.
More informationStandards and Protocols for the Collection and Dissemination of Graduating Student Initial Career Outcomes Information For Undergraduates
National Association of Colleges and Eployers Standards and Protocols for the Collection and Disseination of Graduating Student Initial Career Outcoes Inforation For Undergraduates Developed by the NACE
More informationADJUSTING FOR QUALITY CHANGE
ADJUSTING FOR QUALITY CHANGE 7 Introduction 7.1 The easureent of changes in the level of consuer prices is coplicated by the appearance and disappearance of new and old goods and services, as well as changes
More informationEnergy Efficient VM Scheduling for Cloud Data Centers: Exact allocation and migration algorithms
Energy Efficient VM Scheduling for Cloud Data Centers: Exact allocation and igration algoriths Chaia Ghribi, Makhlouf Hadji and Djaal Zeghlache Institut MinesTéléco, Téléco SudParis UMR CNRS 5157 9, Rue
More informationMarkov Models and Their Use for Calculations of Important Traffic Parameters of Contact Center
Markov Models and Their Use for Calculations of Iportant Traffic Paraeters of Contact Center ERIK CHROMY, JAN DIEZKA, MATEJ KAVACKY Institute of Telecounications Slovak University of Technology Bratislava
More information 265  Part C. Property and Casualty Insurance Companies
Part C. Property and Casualty Insurance Copanies This Part discusses proposals to curtail favorable tax rules for property and casualty ("P&C") insurance copanies. The syste of reserves for unpaid losses
More information