Overview of Linear Discriminant Analysis in .NET

Development QR Code in .NET Overview of Linear Discriminant Analysis
1.2 Overview of Linear Discriminant Analysis
QR barcode library in .net
Using Barcode Control SDK for visual .net Control to generate, create, read, scan barcode image in visual .net applications.
OVERVIEW OF LINEAR DISCRIMINANT ANALYSIS
Assign qr codes for .net
using visual studio .net toadd qr code 2d barcode for asp.net web,windows application
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:
Visual .net qrcode readerin .net
Using Barcode reader for .net framework Control to read, scan read, scan image in .net framework applications.
L xi IRd xi = GT xi R ,
.net Framework bar code writeron .net
generate, create barcode none in .net projects
< d.
.net Framework barcode scannerwith .net
Using Barcode decoder for VS .NET Control to read, scan read, scan image in VS .NET applications.
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
Control qr barcode image with c#.net
generate, create qr code none with c#.net projects
(x c(j) )(x c(j) )T ,
Qr Bidimensional Barcode barcode library with .net
using asp.net web service toinsert qr-codes for asp.net web,windows application
j=1 x Xj k
Control qr codes data on vb.net
qr code 2d barcode data in vb.net
(1.1)
Qr Barcode generating in .net
use visual studio .net qr code implement todraw qr in .net
nj (c(j) c)(c(j) c)T ,
.net Framework barcode implementationin .net
using barcode generator for .net vs 2010 control to generate, create bar code image in .net vs 2010 applications.
j=1 n
Code 128C barcode library on .net
generate, create barcode 128 none on .net projects
(1.2)
Assign codeabar with .net
generate, create code 2 of 7 none in .net projects
St = where c(j)
UPC A encoding with visual c#
use .net upc symbol creation tointegrate upc symbol in .net c#
(xi c)(xi c)T ,
Incoporate european article number 13 for excel spreadsheets
using barcode implementation for excel control to generate, create ean-13 supplement 2 image in excel applications.
(1.3)
Barcode generator with .net
generate, create barcode none with .net projects
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)
Control pdf417 size in excel
pdf-417 2d barcode size for excel spreadsheets
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
Control ean13 image in office word
using barcode writer for microsoft word control to generate, create ean 13 image in microsoft word applications.
T Sw = Hw Hw , T Sb = Hb Hb ,
Pdf417 2d Barcode encoder with c#
using .net framework todisplay pdf417 in asp.net web,windows application
St = Ht HtT .
Control barcode 39 image with vb.net
use .net vs 2010 barcode code39 printing toencode code 39 full ascii in visual basic.net
(1.7)
Control ean 13 data with word
to incoporate upc - 13 and upc - 13 data, size, image with word documents barcode sdk
It follows from the properties of matrix trace that trace(Sw ) = 1 n
k j=1 x Xj
x c(j) 2 ,
(1.8)
1
Discriminant Analysis for Dimensionality Reduction
trace(Sb ) =
k j=1
nj c(j) c 2 .
(1.9)
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.
(1.10)
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
(1.11)
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