where x is the total mean vector
and n = YH=i n< denotes the total number of observations in all classes. Eq. 8.33 can be rewritten by adding — Xj +5^ to each term and rearranging the sums so that the total scatter is the sum of the within-class scatter and the between-class scatter as 
ST = y: y>tJ - Xj + x, - x)(xjj - Xj + Xj - x)T i=i j=i
The first FDA vector wi maximizes the scatter between classes (Ss) while minimizing the scatter within classes (Sw) is obtained as
under the assumption of Sw being invertible [139, 99]. The second FDA vector is calculated to maximize the scatter between classes while minimizing the scatter within classes among all axes perpendicular to the first FDA vector (wi). Additional FDA vectors are determined if necessary by using the same maximization objective and orthogonality constraint. These FDA vectors wa form the columns of an optimal W that are the generalized eigenvectors corresponding to the largest eigenvalues in
where the magnitude ordered eigenvalues Aa indicate the degree of overall separability among the classes by linearly transforming the data onto wa [139, 99], The eigenvalues in Eq. 8.37 can be computed as the roots of the characteristic polynomial det(Ss — AaSw) = 0 and then solving (Sb — AaSn/)wa — 0 directly for the eigenvectors wa .
Consider a data set from a fermentation process with g distinct events such as normal process operation and operations under g — 1 different faults. Each operation type (class) is determined on the basis of p measurements xi, ■ ■ • ,x„ that belong to one of the classes 7r», i = 1, • • • ,g with prior probabilities of pj, i — 1, • • • ,g and probability density functions of /¿(x). The objective of the fault diagnosis is to assign the new on-line out-of-control observations (xo) to the most likely fault class.
FDA is used to diagnose faults by modifying the quadratic discrimination score for the 2 th population defined in Eq. 8.16 in the FDA framework such that dtQ(x0) = hi Pi - I(x0 - Xj)TWa (WjSiW,)"1 Wj(x0 - x,)
where Wa contains the first a FDA vectors . The allocation rule is: Allocate xo to ~k if d®(xo) is the largest of all d®(xo), i — 1, ■ • • ,g.
The classification rule in conjunction with Bayes' rule is used [262, 99] so that the posterior probability (Eq. 8.13) assuming -f>(7rfelx) = 1 that the class membership of the observation xo is i. This assumption may lead to a situation where the observation will be classified wrongly to one of the fault cases which were used to develop the FDA discriminant when an unknown fault occurs. Chiang et al.  proposed several screening procedures to detect unknown faults. One of them involves FDA related T2 statistic before applying Eq. 8.38 as
so that it can be used to determine if the observation is associated with fault class i. The threshold for Tfa is defined as where Fa (a, n — a) denotes the F-distribution with a and n — a degrees of freedom , Chiang et al.  introduce another class of data that are collected under NO to allow the class information in the known fault data to improve the ability to detect faults. The first step then becomes the detection of out-of-control situation. A threshold for NO class is developed based on Eq. 8.40 for detection; if Tfa > T2 Q, there is an out-of-control situation. One proceeds with calculation at thresholds for each class i using Eq. 8.40. If T?a > T2 a for alH = 1,... ,g, then the observation xo does not belong to any fault class i, and it is most likely associated with an unknown fault. If Tfa < T2 for some fault class i, then xo belongs to a known fault class. Once this is determined, Fisher's discriminant score in Eq. 8.38 can be used to assign it to a fault class 7Tj with the highest (¿^(xq) of all diQ(x0), i = 1, • • • ,g.
FDA and PC A can also be combined to avoid assigning an unknown fault to one of the known fault classes i246, 99, 5301. PC A is widely used for fault detection as discussed in Chapter 6. Chiang et al.  proposed two algorithms incorporating FDA and PCA. In the first algorithm (PCA/FDA), PCA is used to detect unknown faults and FDA to diagnose faults (by assigning them to fault classes). The NO class and classes with fault conditions are used to develop the PCA model. When a new observation xo becomes available, T2 value is calculated based on PCA as
where Aa is (a x a) diagonal matrix containing eigenvalues and P are the loading vectors. A set of threshold values based on NO and the known fault classes using Eq. 8.40 is calculated. If T2 < T2 , it is concluded that this is a known class (either NO or faulty) and FDA assignment rule is used to diagnose the fault class (or NO class if it is in-control).
The second combined algorithm (FDA/PCA) deploys FDA initially to determine the most probable fault class i. Then it uses PCA T2 statistic to find out if the observation xo is truly associated with fault class i.
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