Inverse-Variance Weighting PCA-based VRE criterion to select the optimal number of PCs

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1 Preprints of the 18th IFAC Word Congress Miano (Itay) August 28 - September 2, 211 Inverse-Variance Weighting PCA-based VRE criterion to seect the optima number of PCs Baigh Mnassri E Mostafa E Ade Mustapha Ouadsine Laboratory of Sciences of Information s and Systems (LSIS - UMR CNRS 6168); University of Pau Cézanne (Aix-Marseie III) Domaine Universitaire de Saint-Jérôme, Avenue Escadrie Normandie-Niemen; Marseie Cedex 2, France (Te: +33()491563; e-mai: {baigh.mnassri, mostafa.eade, mustapha.ouadsine}@sis.org). Abstract: In earier works, we have shown that the cassica VRE criterion of the SPE index is optimaintheseectionofthenumberofpcsexceptionaybetweenthecorreatedvariabes.the perfection of this criterion is independent of the measurement noises. In this paper, we propose a newinverse-varianceweightingpca-basedvrecriterionmorerobustandgeneraizedforpca modeing task. It enabes to found easiy the optima number of PCs in data by considering the uncorreated variabes. By considering numerica exampes, it has deduced that the sensitivities to the measurement noises of the new VRE and the cassica VRE of SPE index are identica. Keywords: PCA; SPE index; Inverse-Variance Weighting PCA; β index; VRE. 1. INTRODUCTION The use of principa components anaysis (PCA), in many different industrias processes monitoring and anaysis, has been intensivey studied. It has been more recenty proposed as a technique in mutivariate statistica process contro for on-ine detection and isoation of fauts (Qin [23]). For the purpose of detection, PCA is used to generate an empirica static and inear mode based on norma operating data. Moreover, the PCA mode is buit using the measured data during norma operation conditions. A key issue in deveoping a PCA mode is to choose the adequate number of principa components (PCs) in order to represent the system in an optima way. Athough the PCA method has been widey used, the seection of the optimum number of PCs is rather subjective. There are two disadvantages which can occur with a bad choice. If fewer PCs are seected than required, a poor mode wi be obtained and an incompete representation of the process resuts. On the contrary, if more PCs than necessary are retained, the mode wi be overparameterized and wi incude noise (Vae et a. [1999]). As a resut, severa criteria for seecting the optimum number of PCs were proposed in the iterature such as the scree pot, eigenvaues imits, cumuative percent variance, cross-vaidation method, variance of reconstruction error (VRE) (Vae et a. [1999], Qin and Dunia [2], Dunia et a. [1996], Dunia and Qin [1998a,b]). The scree pot and eigenvaues imits approaches consider that components with sma eigenvaues are not important for modeing. This work was supported in part by the Laboratory of Sciences of Information s and Systems (LSIS - UMR CNRS 6168). The cumuative percent variance method consists of the seection of the minimum mode dimension that can express a substantia part of the tota variance of data. The cross-vaidation method uses part of the training sampes for mode construction. The remaining sampes are compared with the predicted ones using the mode. When the prediction residua sum of squares becomes smaer than the residua sum of squares of the previous mode, the new component is added to the mode. Most of these existing approaches use an index that is monotonicay decreasing. They are caed subjective methods because there can be more than one ocation which satisfies the criterion. The ast and important criterion is the variance of reconstruction error (VRE) (Qin [23], Yue and Qin [21], Dunia et a. [1996], Dunia and Qin [1998a,b]). An important feature of this approach is that it has a minimum corresponding to the best reconstruction of variabes. When the PCA-mode is used to reconstruct missing vaues, the reconstruction error is a function on the number of PCs. As a resut, the VRE has aways a minimum which points to the optima number of PCs for best reconstruction. In PCA-based process monitoring, severa detection indices have been widey used for faut detection (Nomikos and MacGregor [1995], Qin and Dunia [2], Mnassri et a. [28]). The Hoteing s T 2 statistic is expressed in the Principa Components Subspace (PCS). It gives a measure of variation with the PCA-mode. The Squared Prediction Error (SPE) and Hawkins s TH 2 (or Squared Weighted Error: SWE) are cacuated into the Residua Subspace (RS). They indicate how much each sampe deviates from the mode. The combined index uses simutaneousy the T 2 and SPE indices that are weighted by their contro imits respectivey. It is we noted that a detection indices are quadratic forms (Qin [23]). Copyright by the Internationa Federation of Automatic Contro (IFAC) 2851

2 Preprints of the 18th IFAC Word Congress Miano (Itay) August 28 - September 2, 211 In the iterature, it has been proposed one VRE criterion that depends on the SPE index. The minimum of this criterion points on the number of redundancies in data without considering the uncorreated variabes (Mnassri et a. [21a]). Given that the reconstruction task depends on the used detection index (Tharraut et a. [28]), we have estabished a generaized or unified VRE criterion vaid for any quadratic detection index (Mnassri et a. [21a]). We have verified that the VRE of the combined index can determine moderatey the optimum PCA-mode in the presence of the independent variabes according unusua significance eve vaues. For this, we have aso proposed a new bidimensiona VRE criterion (Mnassri eta.[21b]).itseectstheoptimapca-modeaccording to usua vaues of significance eve. However, the perfection of these proposed VRE criteria is biased by the measurement noises in variabes. In this paper, we propose a new VRE criterion based on new Inverse-Variance Weighting PCA (IVWPCA) approach for modeing task. Its sensitivity to the measurement noises in the seection of the optimum number of PCs is the same as the VRE criterion of the SPE index in the seection of the number of redundancies in data. The organization of the paper is as foows: Section 2 is a background of the inear PCA as we as known faut detectionindices(spe,t 2 andϕ)andtheircontroimits. Faut reconstruction approach and the unified VRE criterion (Mnassri et a. [21a,b]) are detaied in section 3 and section 4 respectivey. The fifth section deas with the theoretica way of our proposed Inverse-Variance Weighting PCA approach and the reated new VRE criterion. From simuation exampes, the rates of the correct seection of our proposed VRE and the cassica VRE reated to the SPE are compared in section 6. Section 7 concudes the paper. 2. FAULT MODELLING AND DETECTION INDICES Statistica process monitoring reies on the use of norma process data to buid process modes. PCA is one of the most popuar statistica methods for extracting information from measured data. It finds the significant variabiity directions by forming inear combinations of variabes. PCA modes are predominanty used to extract variabes correation from data. 2.1 Linear PCA Let x R m denotes a sampe vector of m variabes. Assuming that there are N sampes for each variabe, data matrixx R N m iscomposedwitheachrowrepresenting a sampe. To avoid the scaing probem, we consider that the data are scaed to zero mean and unit variance. PCA determines an optima inear transformation of the data X (Mnassri et a. [28, 29]) which can be decomposed into a score matrix T and a oading matrix P as foows: X = TP T (1) with T = [t 1 t a t m ] R N m and a = {1,,m}, where the vectors t a are caed scores or PCs. The matrix P = [p 1 p a p m ] R m m, where the orthogona vectors p a caed oading or principa vectors are the eigenvectors associated to the eigenvaues λ a of the correation matrix Σ of X such that: Σ = PΛP T with PP T = P T P = I m (2) Λ = diag(λ 1,λ a λ m ) is a diagona eigenvaues matrix with eements in the decreasing order. The partition of eigenvectors and principa components matrices gives respectivey: P = [ˆP Pm ] = [ˆP P], T = [ˆT T m ] = [ˆT T](3) where represents the number of the more significant PCs which are sufficient to expain the variabiity of the process through their data X. The first eigenvectors constitute the representation space or the PCS defined by S p = span{ˆp}, whereas the RS is: S r = span{ P}. A sampe vector x can be projected on the PCS and RS respectivey by: ˆx = ˆP ˆP T x = Ĉx S p (4) x = P P T x = Cx S r (5) where Ĉ and C = (I Ĉ) represent respectivey the projection matrices on the PCS and RS with ( < m). When the seected number of PCs is optima, the diagona eements of these two above projection matrices shoud indicate correcty the inear correation for each variabe. Since S p and S r are orthogona, ˆx T x = x Tˆx = and x = ˆx+ x (6) The vectors ˆx and x represent respectivey the modeed and unmodeed variations of x based on PCs. Thus, represents the dimension of the PCA-mode. Its vaue is optima if ˆx and x contain mosty information and noise respectivey. It wi be studied in the foowing sections. 2.2 Faut Detection Indices FautdetectionisusuaythefirststepinMutivariateStatistica Process Contro (MSPC). Many typica statistics for detecting abnorma conditions have been proposed in the iterature. Tabe 1 iustrates the more known indices. SPE (or Q-statistic) and Hoteing s T 2 (or D-statistic) represent the variabiity in the RS and PCS respectivey. The combined index ϕ combines simutaneousy these two statistics that are pondered by their contro imits. In genera, a detection indices are quadratic forms that are unified according to the foowing expression: γ(k) = x T (k)υ γ x(k) (7) where Υ γ is a positive-definite matrix which characterizes the studied detection index γ. By considering one of these detection indices for monitoring task, the process is considered norma if (Box [1954]): γ(k) γ 2 α = g γ χ 2 (h γ,α) (8) γ 2 α is the contro imit of the statistic γ with h γ and α are the degree of freedom and the significance eve of the Chi2 distribution respectivey. g γ and h γ are determined as foows (Box [1954]): g γ = tr[(συ γ ) 2 ]/tr[συ γ ] (9) 2852

3 Preprints of the 18th IFAC Word Congress Miano (Itay) August 28 - September 2, 211 Tabe 1. Different detection indices Index γ SPE T 2 Combined index ϕ P P T SPE 2 α Υ γ C ˆP ˆΛ 1 ˆP T + ˆP ˆΛ 1 ˆP T h γ = (tr[συ γ ]) 2 /tr[(συ γ ) 2 ] (1) where Σ is the correation matrix of the data X and tr[ ] impies the trace of the matrix. 3. FAULT RECONSTRUCTION APPROACH The reconstruction method assumes that a group of variabesmaybefautyandsuggestsreconstructingthemfrom the remaining variabes using a PCA mode. In the unidimensiona and singe fauts cases, the best reconstruction is obtained by considering an optima PCA mode. The reconstruction of variabes consists in estimating the reconstructed vector by eiminating the fauts effect aong fauts directions. The sampe vector for norma operating conditions is denoted by x that is unknown when a faut hasoccurred.inthepresenceofamutidimensionaprocess fauts R, the sampe vector x can be written as (Dunia et a. [1996], Dunia and Qin [1998a,b], Qin and Dunia [2], Tharraut et a. [28]): T 2 α x = x +Ξ R f (11) where R represents a subset of r among m variabes which are assumed in fauty and shoud be reconstructed. R is a subset containing the indices of the faut directions. f represents the magnitude of the faut. Note that f may change depending on how the actua faut deveops over time. The orthonorma matrix Ξ R R m r indicates the reconstruction directions. This matrix is buit with and 1 where 1 indicates the reconstructed variabes. To find the optima PCA-mode, the unidimensiona and singe fauts cases are considered. For this, Ξ R is the Rth coumn of the identity matrix. Let x R is the reconstructed vector of x aong the Rth direction: x R = x Ξ R f R (12) then the reconstructed detection index γ R that corresponds to the reconstructed vector x R is: γ R = x T RΥ γ x R (13) The estimated faut magnitude is determined by minimizing the above reconstructed detection index as foows: f γ R =argmin {γ }{{} R } f γ R { =argmin (x ΞR f γ }{{} R )T Υ γ (x Ξ R f γ R )} (14) f γ R After derivation of γ R with respect to f γ R, the magnitude of the estimated faut and its corresponding reconstructed vector are respectivey given by: f γ R = (ΞT RΥ γ Ξ R ) 1 Ξ T RΥ γ x (15) x R = (I Ξ R (Ξ T RΥ γ Ξ R ) 1 Ξ T RΥ γ )x (16) where I R m m is the identity matrix. When considering one detection index, PCA-mode which corresponds to the minimum of the variance of the reconstructed error can be retained as the optima mode. On the other hand, It is cear that faut reconstruction approach depends impicity on the used detection index γ and PCA-mode dimension. Therefore, the VRE criterion is not uniquey reated to the SPE statistic as it has been proposed in the iterature. In (Mnassri et a. [21a]), we have estabished a unified VRE criterion expression vaid for any quadratic detection index. 4. UNIFIED VRE CRITERION According to the VRE method, the PC index that corresponds to the minimum of the VRE criterion is considered as the optimum number for modeing. If faut is oriented towards the Rth direction (or variabe), the corresponding unified individua VRE which depends on the number of PCs and the detection index γ is given by: u γ R = var{ Ξ T R(x x R ) } = ΞT R Υ γσυ γ Ξ R (Ξ T R Υ γξ R ) 2 (17) Considering a possibe unidimensiona and singe fauts, the unified goba VRE based on the detection index γ and the number of the PCs is the foowing: m u γ R VRE γ () = Ξ T R ΣΞ (18) R R=1 where Ξ T R ΣΞ R = 1because data X arescaed to zeromean and unit variance. Thus, PC index for which the vaue of the goba VRE criterion attains its minimum vaue represents the optimum number of PCs: m u γ R op = argminvre }{{} γ () = argmin }{{} Ξ T R=1 R ΣΞ (19) R According to the different detection indices proposed in the iterature, the unique VRE criterion which is abe to determine perfecty the optimum number of PCs, under the condition that a variabes are correated (Mnassri et a. [21a]), is the VRE of the SPE index: m Ξ T ν op = argminvre }{{} SPE () = argmin R Σ Ξ R (2) }{{} ( Ξ T R=1 R Ξ R ) 2 On the other hand and to consider the uncorreated variabes, we have shown that the optima PCA-mode can be obtained based on the VRE ϕ criterion but with an appropriate and deicate choice of the significance eve α (Mnassrieta.[21a]).Therefore,wehaveproposedanew bidimensiona VRE criterion reated to a new combined detectionindex(mnassrieta.[21b]).however,therates of these two VRE criteria in the seection of the true optima number of PCs are biased by the measurement noises in variabes. The VRE reated to a remaining detection indices are unabe to achieve the modeing task. 2853

4 Preprints of the 18th IFAC Word Congress Miano (Itay) August 28 - September 2, NEW VRE CRITERION BASED ON IVWPCA Inthissection,wedefinefirsttheobjectiveofourproposed IVWPCA method as foows: Suppose that m measured variabes x 1,x 2,...,x m can be expressed as inear combinations of m other correated variabes y 1,y 2,...,y m. The reationships between them, by considering a mixing symmetric and invertibe matrix A R m m, is given by: y = Ax (21) In order to find A, we suppose that each variabe y i shoud depend ineary ony with the variabe x i and their corresponding covariance coefficient σ yix i = E{y i x i } = 1. In other words, we want to eiminate the inear effects of a variabes x j (j = 1,...,m and j i) on the variabe x i. Based on these arguments, the covariance matrix of data X compared to data Y shoud equa to the identity matrix: E{yx T } = E{xy T } = AΣ = ΣA = I (22) where I is the identity matrix. This impies that: A = Σ 1 (23) which represents the pseudo-inverse matrix of the correation matrix Σ of data X. From (1), the sampe PCs vector tandthevariance-covariancematrixs T ofthepcsmatrix T of data X are given respectivey by: t = P T x (24) S T = E{tt T } = Λ (25) On the other hand, the variance-covariance matrix of data Y is the foowing: S Y = E{yy T } = Σ 1 = A (26) SinceP isorthonorma,thisimpiesthatp 1 = P T.Then, the singuar vaue decomposition of the matrix S Y gives: S Y = Σ 1 = (PΛP T ) 1 = PΛ 1 P T (27) Denoting H and h as the PCs matrix and sampe PCs vector of data Y respectivey. Then, the vector h can be written as: h=p T y = P T Ax =Λ 1 P T x = Λ 1 t (28) and the variance-covariance matrix S H of data H verifies the foowing equaity: S H = E{hh T } = Λ 1 (29) According to the previous equations, two fundamentas properties are concuded: (i) The PCs of data Y keep the same directions of those of data X. (ii) The variance of each PC of data Y is equa to the inverse of its corresponding PC of data X. For this reason, this approach is caed Inverse-Variance Weighting PCA (IVWPCA). Therefore, the PCs of data Y are determined by diagonaization of the pseudo-inverse of the correation matrix Σ of data X. It is cear that the singuar vaue decomposition of the variance-covariance matrix S Y of data Y keeps the same orthonorma matrix P of the correation matrix Σ of data X. However, its eigenvaues matrix is the inverse of those of data X. Contrary to the cassica principe of PCA approach, the eigenvaues diagona eements of our IVWPCA method are considered in increasing order. Based on data Y, our main idea is to propose a new VRE criterion ess sensitive to the measurement noises in order to seect correcty the optimum number of PCs by taking into account the independent variabes. Let consider y as the sampe vector for norma operating conditions which is unknown when a faut has occurred. In the presence of a mutidimensiona process fauts J, y can be written as: y = y +ξ J d (3) where J represents a subset of j among m variabes that are assumed in fauty and shoud be reconstructed. ξ J R m j is an orthonorma matrix. d represents the faut magnitude. To find the optima PCA-mode, the unidimensiona and singe fauts cases must be considered. This means that ξ J is the Jth coumn of the identity matrix. Let y J is the reconstructed vector of y aong the Jth direction: y J = y ξ J d J (31) Defining β as a new unified detection index that corresponds to the sampe vector y as foows: β = y T Z β y (32) Z β is a positive-definite matrix that characterizes the unified detection index β cacuated into the Inverse-Variance Weighting PCS, Inverse-Variance Weighting RS or the both subspaces. Noticeaby, β is a unified statistic that may represent many detection indices. Then, the unified reconstructed detection index noted β J that corresponds to the reconstructed vector y J is given by: β J = y T JZ β y J = (y ξ J d J ) T Z β (y ξ J d J ) (33) Theoptima estimatedfautmagnituded β J,thatminimizes the reconstructed index (33), and the reconstructed vector have respectivey the foowing expressions: d β J = (ξt JZ β ξ J ) 1 ξ T JZ β y (34) y J = ( I ξ J (ξ T JZ β ξ J ) 1 ξ T JZ β ) y (35) Thus,theunifiedandindividuaVREreatedtotheunified detection index β is given by: u β J = var{ ξ T J(y y J ) } = ξt J Z βs Y Z β ξ J (ξ T J Z βξ J ) 2 (36) Considering a possibe unidimensiona and singe fauts, the unified goba VRE criterion of the detection index β based on PCs has the foowing expression: VRE β () = m J=1 u β J ξ T J S Yξ J (37) PC index that corresponds to the minimum of the above VRE criterion represents the optima number of PCs: m u β J op = argminvre }{{} β () = argmin }{{} ξ J=1 J TS (38) Yξ J 2854

5 Preprints of the 18th IFAC Word Congress Miano (Itay) August 28 - September 2, 211 The ony avaiabe VRE criterion that seects perfecty the optima number of PCs under the condition that a variabes are correated is the VRE SPE (2). According to our approach, the VRE criterion that seects perfecty the optima number of PCs in the presence of the independent variabes is reated to an index β cacuated into the Inverse-Variance Weighting PCS (Z β = Ĉ): β = y T Ĉy (39) Therefore, the VRE criterion that wi be considered and the optima number of PCs have respectivey the foowing expressions: m ˆξ J T VRE β () = ˆΛ 1ˆξ J (ξ T Σ 1 ξ)(ˆξ J T ˆξ (4) J ) 2 J=1 m ˆξ T op = argmin ˆΛ 1ˆξ J J }{{} J=1 (ξ T Σ 1 ξ)(ˆξ T ˆξ (41) J J ) 2 The goa of this paper is to compare the effectiveness of the cassica VRE SPE (2) and our proposed VRE criterion(41)intheseectionoftheoptimanumberofpcs ony among the correated variabes and in the presence of independent variabes respectivey. This comparison is achieved according to the measurement noises vaues introduced in variabes. 6. SIMULATION EXAMPLES In this section, we have simuated an academic exampe of data (Tharraut et a. [28]) as shown in tabe 2. ϑ, x 8 andx 9 areuncorreatednormarandomvariabeswithzero mean and.35, 1 and 1 as standard deviation respectivey. To consider the measurement noises, uncorreated random signas normay distributed with zero mean and SD = 1 2 : 1 2 : 1 as standard deviation were superimposed to each generated variabe. The data are constituted of N = 45 sampes generated under norma operating conditions. These data contain inear and noninear anaytica redundancy reations, as we as independent variabes. These reations or characteristics are captioned in the second coumn of tabe 2. The expressions: Ind., S.R.{*} and D.{*} indicate respectivey that the corresponding variabe is independent, source of redundancy and depends on. For exampe, in second row of tabe 2, S.R.{4,5,6} signifies that the variabe x 2 is source of redundancy for the fourth, fifth and sixth variabes. This is equivaent to say that the fourth variabe depends on the second one. This is why, we have introduced D.{1,2} in the fourth row of the same tabe. Consequenty, when these variabes: x 2, x 4, x 5 and x 6 are grouped in the same data sets, they represent together one PC in the PCA-mode. However, each uncorreated or independent variabe represents a fu PC as x 8 and x 9. In this paper, three data sets are considered: D1 = {x 1 to x 7 }, D2 = {x 1 to x 8 } and D3 = {x 1 to x 9 }. In D1, a variabes are correated. However, in the second and third data sets, there are one and two independent variabes respectivey. In third coumn of tabe 2, we attribute the vaue of 1 to each independent or source of redundancy Tabe 2. Simuated Data Variabes Characteristic PC x 1 (k) = 1+ϑ(k) 2 +sin( k 3 ) S.R.{4,5,6,7} 1 x 2 (k) = 2sin( k 6 )cos(k 4 )exp( k N ) S.R.{4,5,6} 1 x 3 (k) = og(x 2 (k) 2 ) S.R.{7} 1 x 4 (k) = x 1 (k)+x 2 (k) D.{1,2} x 5 (k) = x 1 (k) x 2 (k) D.{1,2} x 6 (k) = 2x 1 (k)+x 2 (k) D.{1,2} x 7 (k) = x 1 (k)+x 3 (k) D.{1,3} x 8 (k) N(,1) Ind. 1 x 9 (k) N(,1) Ind. 1 variabe. From this way, we can determine the theoretica number of redundancies (ν op ) and the optimum number of PCs ( op ) constituting such data sets (tabe 3). For exampe, in data D1, the number of redundancies is equa to the optimum number of PCs because there are not independent variabes. In order to assess the effectiveness of the studied criteria, each data set was simuated 1 times for every SD vaue of the measurement noises. The VRE criteria are computed during each simuation and their rates in the seection of the correct number of PCs are determined. Theoreticay, the number of sources of redundancies in data D1 is the same as the optima number of PCs. By investigating Fig.1., the curves rates of the two studied VRE criteria in the correct seection of PCs are practicay simiar. They seect in 1 of cases the correct number of PCs. On the other hand, D2 and D3 shoud be represented by 4 and 5 PCs respectivey. The new proposed criterion shows with highy perfection (1) that the optima PCA-mode for D2 and D3 is spanned by = 4 and = 5 PCs respectivey (Fig.2. and Fig.3.). In both data sets, the cassica VRE of the SPE index has shown that the number of redundancies is aways the same (ν = 3 PCs) except for high noises variance cases. When the noises variance reaches highy vaues, it is more probabe that some correated variabes are transformed to independent variabes. For this reason, the rates curves of the correct seection of the number of PCs, according to these criteria, are decreased at these vaues. From these exampes, the resuts show that the new proposed criterion VRE β is more generaized than the VRE SPE. It determines perfecty the optima number of PCs by taking into account the uncorreated variabes. 7. CONCLUSIONS In this paper, we have proposed a new unified detection index β reated toanewcaed Inverse-VarianceWeighting PCA (IVWPCA) approach. Our study was imited to estabish the VRE criterion of the β index. In future work, we wi investigate the usefuness of this new statistic for faut diagnosis. This new VRE criterion finds perfecty the optima number of PCs in data by considering the uncorreated variabes as we as the number of redundancies when a variabes are correated. From a simuation exampes, we have shown that the sensitivities to the measurement noises of the new VRE and the cassica VRE of the SPE index are identica. This impies that our proposed approach is more robust and generaized for PCA modeing task. 2855

6 Preprints of the 18th IFAC Word Congress Miano (Itay) August 28 - September 2, Tabe 3. Theoretica number of redundancies and optimum number of PCs in the studied data sets Data sets ν op op D1={x 1 to x 7 } 3 PCs 3 PCs D2={x 1 to x 8 } 3 PCs 4 PCs D3={x 1 to x 9 } 3 PCs 5 PCs SPE index SD vaue β index SD vaue Fig.1.VREratesoftheSPE andβ indicesintheseection of the correct number of redundancies (ν op = 3) and optima number of PCs ( op = 3) respectivey by varying the noises standard deviation (SD) in D SPE index SD vaue β index SD vaue Fig.2.VREratesoftheSPE andβ indicesintheseection of the correct number of redundancies (ν op = 3) and optima number of PCs ( op = 4) respectivey by varying the noises standard deviation (SD) in D SPE index SD vaue β index SD vaue REFERENCES G.E.P. Box. Some theorems on quadratic forms appied in the study of anaysis of variance probems, i. effect of inequaity of variance in the one-way cassification. Ann. Math. Statist., 25(2):29 32, R. Dunia and S.J. Qin. A unified geometric approach to process and sensor faut identification and reconstruction: the unidimensiona faut case. Computers & Chemica Engineering, 22(7-8): , 1998a. R. Dunia and S.J. Qin. Subspace approach to mutidimensiona faut identification and reconstruction. Process Systems Engineering, 44(8): , 1998b. R. Dunia, S.J. Qin, T.F. Edgar, and T.J. McAvoy. Identification of fauty sensors using principa component anaysis. Process Systems Engineering, 42(1): , B. Mnassri, E.M. E Ade, and M. Ouadsine. Faut ocaization using principa component anaysis based on a newcontributiontothesquaredpredictionerror. In16th Mediterranean Conference on Contro and Automation, pages 65 7, Ajaccio, France, 28. B. Mnassri, E.M. E Ade, B. Ananou, and M. Ouadsine. Faut detection and diagnosis based on pca and a new contribution pots. In 7th IFAC Symposium on Faut Detection, Supervision and Safety of Technica Processes, pages , Barceona, Spain, 29. B. Mnassri, E.M. E Ade, B. Ananou, and M. Ouadsine. A generaized variance of reconstruction error criterion for determining the optimum number of principa components. In 18th Mediterranean Conference on Contro and Automation, pages , Marrakech, Morocco, 21a. B. Mnassri, E.M. E Ade, M. Ouadsine, and B. Ananou. Seection of the number of principa components based on the faut reconstruction approach appied to a new combined index. In 49th IEEE Conference on Decision and Contro, pages , Atanta, Georgia USA, 21b. P. Nomikos and J.F. MacGregor. Mutivariate spc charts for monitoring bach processes. Technometrics, 37(1): 41 59, S.J. Qin. Statistica process monitoring: basics and beyond. Journa of Chemometrics, 17(8 9):48 52, 23. S.J. Qin and R. Dunia. Determining the number of principa components for best reconstruction. Journa of Process Contro, 1(2):245 25, 2. Y. Tharraut, G. Mourot, J. Ragot, and D. Maquin. Faut detection and isoation with robust principa component anaysis. Int. J. App. Math. Comput. Sci., 18(4): , 28. S. Vae, W. Li, and S.J. Qin. Seection of the number of principa components: The variance of the reconstruction error criterion with a comparison to other methods. Ind. Eng. Chem. Res., 38(11): , H.H. Yue and S.J. Qin. Reconstruction-based faut identification using a combined index. Ind. Eng. Chem. Res., 4(2): , 21. Fig.3.VREratesoftheSPE andβ indicesintheseection of the correct number of redundancies (ν op = 3) and optima number of PCs ( op = 5) respectivey by varying the noises standard deviation (SD) in D3 2856