Overview of Linear Discriminant Analysis in .NET

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1.2 Overview of Linear Discriminant Analysis
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We are given a data set that consists of n samples {(xi , yi )}n , where xi IRd denotes i=1 the d-dimensional input, yi {1, 2, . . . , k} denotes the corresponding class label, n is the sample size, and k is the number of classes. Let X = [x1 , x2 , . . . , xn ] Rd n be the data matrix and let Xj Rd nj be the data matrix of the jth class, where nj is the sample size of the jth class and k nj = n. Classical LDA computes a linear j=1 L transformation G Rd that maps xi in the d-dimensional space to a vector xi in the -dimensional space as follows:
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L xi IRd xi = GT xi R ,
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In LDA, three scatter matrices, called the within-class, between-class, and total scatter matrices are de ned as follows [8]: 1 Sw = n Sb = 1 n 1 n
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(x c(j) )(x c(j) )T ,
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j=1 x Xj k
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nj (c(j) c)(c(j) c)T ,
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j=1 n
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St = where c(j)
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(xi c)(xi c)T ,
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is the centroid of the jth class and c is the global centroid. It can be veri ed from the de nitions that St = Sb + Sw [8]. De ne three matrices Hw , Hb , and Ht as follows: 1 Hw = [X1 c(1) (e(1) )T , . . . , Xk c(k) (e(k) )T ], (1.4) n 1 Hb = [ n1 (c(1) c), . . . , nk (c(k) c)], n 1 Ht = (X ceT ), n (1.5) (1.6)
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where e(j) and e are vectors of all ones of length nj and n, respectively. Then the three scatter matrices, de ned in Eqs. (1.1) (1.3), can be expressed as
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T Sw = Hw Hw , T Sb = Hb Hb ,
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St = Ht HtT .
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It follows from the properties of matrix trace that trace(Sw ) = 1 n
k j=1 x Xj
x c(j) 2 ,
Discriminant Analysis for Dimensionality Reduction
trace(Sb ) =
k j=1
nj c(j) c 2 .
Thus trace(Sw ) measures the distance between the data points and their corresponding class centroid, and trace(Sb ) captures the distance between the class centroids and the global centroid. In the lower-dimensional space resulting from the linear transformation G, the scatter matrices become
L Sw = GT Sw G, L Sb = GT Sb G,
StL = GT St G.
L L An optimal transformation G would maximize trace(Sb ) and minimize trace(Sw ) siL ) and minimizing trace(S L ) multaneously, which is equivalent to maximizing trace(Sb t L L simultaneously, since StL = Sw + Sb . The optimal transformation, GLDA , of LDA is computed by solving the following optimization problem [8, 16]: L GLDA = arg max trace Sb StL G 1
It is known that the optimal solution to the optimization problem in Eq. (1.11) can be obtained by solving the following generalized eigenvalue problem [8]: Sb x = St x. (1.12)
More speci cally, the eigenvectors corresponding to the k 1 largest eigenvalues form columns of GLDA . When St is nonsingular, it reduces to the following regular eigenvalue problem: St 1 Sb x = x. (1.13)
When St is singular, the classical LDA formulation discussed above cannot be applied directly. This is known as the singularity or undersampled problem in LDA. In the following discussion, we consider the more general case when St may be singular. The transformation, GLDA , then consists of the eigenvectors of St+ Sb corresponding to the nonzero eigenvalues, where St+ denotes the pseudo-inverse of St [27]. Note that when St is nonsingular, St+ equals St 1 . The above LDA formulation is an extension of the original Fisher linear discriminant analysis (FLDA) [7], which deals with binary-class problems, that is, k = 2. The optimal transformation, GF , of FLDA is of rank one and is given by [15, 16] GF = St+ (c(1) c(2) ). (1.14)
Note that GF is invariant of scaling. That is, GF , for any = 0, is also a solution / to FLDA. When the dimensionality of data is larger than the sample size, which is the case for many high-dimensional and low sample size data, all of the three scatter matrices are singular. In recent years, many algorithms have been proposed to deal with this singularity problem. We rst review these LDA extensions in the next subsection. To