A New Feature Extraction Algorithm Based on Orthogonal Regularized Kernel CCA and Its Application

In this paper, an orthogonal regularized kernel canonical correlation analysis algorithm (ORKCCA) is proposed. ORCCA al-gorithm can deal with the linear relationships between two groups of random variables. But if the linear relationships between two groups of random variables do not exist, the performance of ORCCA algorithm will not work well. Linear orthogonal regularized CCA algorithm is extended to nonlinear space by introducing the kernel method into CCA. Simulation experimental results on both artiﬁcial and handwritten numerals databases show that the proposed method outperforms ORCCA for the nonlinear problems.


Introduction
Canonical correlation analysis (CCA) is a technique of multivariate statistical analysis, which deals with the mutual relationships of two sets of variables [1][2][3].
is method extracts the representative variables which are the linear combination of the variables in each group.e relationships between new variables can reflect the overall relationships between two groups of variables [4].
e orthogonal regularization canonical correlation analysis (ORCCA) algorithm [5] is that the original formula of CCA algorithm with orthogonal constraints is substituted for CCA conjugate orthogonalization [6,7].When the number of samples is less and the sample distribution patterns of different classifications are different, the ORCCA algorithm has the better ability of classification.A suboptimal solution to eigenvalue decomposition problem can be obtained by introducing two regularization parameters [8].So, the complexity of time and space for the quadratic optimization problem should be considered at the same time.ORCCA algorithm is the same as CCA algorithm that both their goals look for the linear combinations of the variables in each group.But when the nonlinear relationships between the variables exist, ORCCA algorithm cannot extract effectively the comprehensive variables.
In this paper, the kernel method [9][10][11] is introduced into ORCCA algorithm, and ORKCCA algorithm is presented.e kernel method maps the linear inseparable data in the low-dimensional space into a higher-dimensional space [12,13].In the higher-dimensional space, the characteristics of the data can be extracted and analyzed through the linear method.By introducing kernel function, the computation of the orthogonal regularization canonical correlation analysis extends to a nonlinear feature space.Experimental results show that the accuracies of classification of our method in the nonlinear space are significantly improved.
e experimental results show ORKCCA is feasible.
directions a and b which satisfy the following optimal problem [5].

Orthogonal Regularized Kernel CCA Algorithm (ORKCCA)
ORCCA algorithm can give the linear relationships between two groups of random variables.But if the linear relationships between two groups of random variables do not exist, the performance of ORCCA will not work well.e kernel method is an effective way to analyze the nonlinear pattern problem.So, the kernel method is introduced into ORCCA algorithm, and ORKCCA algorithm is proposed.Both Φ x and Φ y are nonlinear mappings which map original random variables x i and ORCCA is implemented in higher-dimensional spaces F x and F y .So, Equation ( 7) can be obtained by substituting a, b, Φ x (x i ), and Φ y (y i ) into Equation (1) as follows: Expanding the objective function in Equation ( 7), we get Applying the kernel trick to Equation ( 8), K x and Centralization is exerted on K x and K y .e optimal model in which the kernel method is introduced can be given by using Equation ( 9): where x , and M yy � 1/nK T y K y .According to the Lagrange multiplier method, the Lagrange function is as follows where ζ 1 and ζ 2 are Lagrange multipliers.Taking the partial derivatives of L ′ (α, β) with respect to α and β and letting them zero, we get where M xx and M yy are positive semidefinite matrices and ζ 1 and ζ 2 are positive numbers.

Journal of Electrical and Computer Engineering
So, α and β can be obtained from Equation (11): where I P and I Q are the identity matrices of size P * P and Q * Q, respectively.Equations ( 14) and ( 15) can be obtained through replacing α and β with their expressions in Equations ( 12) and (13), respectively.

Simulation Experiments
In this section, we evaluate our method compared with ORCCA on artificial and handwritten numerals databases.

Experiment on Artifical Databases.
e pairwise samples X and Y are generated from the expressions in Equations ( 16) and (17), respectively.
According to Equations ( 16) and ( 17), 100 pairs of data are randomly generated as the training samples.Canonical variables are calculated from the ORCCA and ORKCCA algorithms for the different values of regularization parameters.e correlation coefficients of canonical variables are sorted by the descending order.Many pairs of canonical variables can be gained from the two algorithms.For the sake of simplicity, the most representative of the former two groups of canonical variables are examined.
e average value of the correlation coefficients of the former two groups of canonical variables is regarded as criterion that judges the regularization parameters is good or not.e larger the average value is, the better the regularization parameters are.
Table 1 lists the average value of the correlation coefficients of the former two groups of canonical variables for the different values of the regularization parameters.
Table 1 shows that the optimal values of the regularization parameters for the ORCCA and ORKCCA algorithms are 10 −3 and 10 −1 , respectively.e optimal regularization parameters are used to perform simulations in the next section.

Simulation Experiment 1.
According to Equations ( 16) and ( 17), 200 pairs of data are randomly generated as the test samples.For the regularization parameters λ � 10 −3 and ζ � 10 −1 in the ORCCA and ORKCCA algorithms, the canonical variables are obtained for test samples, respectively.e correlation coefficients of the canonical variables are sorted in the descending order.
Tables 2 and 3 list the correlation coefficients of the first two groups of canonical variables for ORCCA and ORKCCA algorithms.u 1 and v 1 denote the first group of canonical variables.u 2 and v 2 are the second group of canonical variables.
e experimental results in Tables 2 and 3 show that the correlationships between the same pair of the canonical variables are better than that between the different pairs of canonical variables, especially for nonlinear data.16) and ( 17), 5 pairs of data are randomly generated as the sample data.Each pair of sample data represents the center data of each class.100 pairs of data for each class are given by adding Gaussian noise with standard deviation of 0.05 to each class center data.So we have five class data, which contains 100 samples for each class.100, 175, and 250 pairs of data are chosen from the 500 pairs of the whole data as the training samples, respectively.e rest 400, 325, and 250 pairs of data are the test samples, respectively.

Simulation Experiment 2. According to Equations (
e classification experiments based on Kneighbors algorithm are carried out on the test samples data which are preprocessed in the above way.And, the accuracies of classification are given.For the test samples with 400, 325, and 250 pairs of data, the experiments are performed 15 times, respectively.e accuracies of classification for 400, 325, and 250 pairs of data are the averages of the accuracies of classification for the 15 experiments results, Journal of Electrical and Computer Engineering respectively.Table 4 gives the accuracies of classi cation for ORCCA and ORKCCA for the test samples with the different number.
In Table 4, the rst column is the numbers of the training samples and the second column and the third column are the accuracies of classi cation for ORCCA and ORKCCA for the training samples with the di erent number.
e experimental results show that the accuracies of classi cation for ORKCCA are higher than those for ORCCA.So, the performance of ORKCCA outperforms that of ORCCA for the nonlinear problem.e comparison curves of the accuracies of classi cation for ORCCA and ORKCCA are given in Figure 1.

Experiments on Handwritten Numerals Databases.
e Concordia University CENPARMI database of handwritten Arabic numerals have 10 classes, that is, 10 digits (from 0 to 9), and 600 samples for each.e rst 400 samples are used as the training set, and the remaining samples as the test set in each class.en, the training samples and the test samples are 4000 and 2000, respectively.e handwritten digital images are preprocessed by the method given in [14].Four kinds of features are extracted as follows: X G (256dimensional Gabor transformation feature), X L (121-dimensional Legendre moment feature), X P (36-dimensional Pseudo-Zernike moment feature), and X Z (30-dimensional Zernike moment feature).
For the choice of the regularization parameters, let λ λ 1 λ 2 and ζ ζ 1 ζ 2 .e regularization parameters were chosen from 10 −5 , 10 −3 , and 1. e results of our method are compared with the results of ORCCA in order to verify the e ectiveness of ORKCCA.Table 5 lists the accuracies of classi cation for ORCCA and ORKCCA in di erent feature combinations and regularization parameters.Experimental results show that (1) the classi cation e ect of the two methods is the best as the regularization parameter is 1; (2) the classi cation accuracies of ORKCCA are higher than that of ORCCA for di erent features combinations; (3) the classi cation accuracies of ORKCCA in the regularization parameters 10 −5 and 10 −3 are higher than those of ORCCA in the regularization parameters 1.

Data Availability
e experiments in paper were performed by the author Xi 2 years ago.Some troubles happened to his computer.e data can not be gotten from his computer.I'm sorry that the data is unable to be provided.
and S xy � (1/n) X T Y. e optimal model in Equation (1) can be rewritten as max a,b 2a T S xy b − a T S xx a − b T S yy b, s.t. a T a � 1, b T b � 1.

Figure 1 :
Figure 1: Comparison curves of the accuracies of classi cation for ORCCA and ORKCCA.

Table 5 :
Comparisons of the accuracies of classi cation for ORCCA and ORKCCA in di erent feature combinations and regularization parameters.

Table 1 :
e mean values of the correlation coe cients of the former two groups of canonical variables for the di erent values of the regularization parameters from ORCCA and ORKCCA.

Table 4 :
Comparisons of the accuracies of classi cation for ORCCA and ORKCCA.

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Journal of Electrical and Computer Engineering nonlinear problem.Contrast experiments of ORCCA and ORKCCA are performed on artificial and handwritten numerals databases.Experimental results show that the proposed method outperforms ORCCA for the correlation coefficients of canonical variables and the accuracies of classification on the test data.e experimental results show ORKCCA is feasible.