Abstract
In this paper, an orthogonal regularized kernel canonical correlation analysis algorithm (ORKCCA) is proposed. ORCCA algorithm 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.
1. Introduction
Canonical correlation analysis (CCA) is a technique of multivariate statistical analysis, which deals with the mutual relationships of two sets of variables [1–3]. This method extracts the representative variables which are the linear combination of the variables in each group. The relationships between new variables can reflect the overall relationships between two groups of variables [4].
The 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 [9–11] is introduced into ORCCA algorithm, and ORKCCA algorithm is presented. The 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. The experimental results show ORKCCA is feasible.
2. Orthogonal Regularized CCA Algorithm
Given n pairs of pairwise samples and , where , . We assume that the samples have been centered. ORCCA algorithm aims at finding a pair of projection directions and which satisfy the following optimal problem [5].
The objective function in Equations (1) can be expanded as follows:where , , and .
The optimal model in Equation (1) can be rewritten as
According to the Lagrange multipliers method, Lagrange function is as follows:where both and are Lagrange multipliers.
The solutions to Equation (4) are given as follows:where and denote identity matrices of size p p and q q, respectively.
Both and in Equations (5) and (6) are called regularization parameters. By solving Equation (5), the eigenvalues and their corresponding eigenvectors can be obtained. The eigenvalues and their corresponding eigenvectors can be obtained from Equation (6).
3. 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. The 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 and are nonlinear mappings which map original random variables and into and in P-dimensional space (P > p) and Q-dimensional space (Q > q), . Let , , where , .
ORCCA is implemented in higher-dimensional spaces and . So, Equation (7) can be obtained by substituting , , , and into Equation (1) as follows:
Expanding the objective function in Equation (7), we get
Applying the kernel trick to Equation (8), and can be computed, namely, , where is kernel function. Centralization is exerted on and . The optimal model in which the kernel method is introduced can be given by using Equation (9):where , , and .
According to the Lagrange multiplier method, the Lagrange function is as followswhere and are Lagrange multipliers. Taking the partial derivatives of with respect to and and letting them zero, we getwhere and are positive semidefinite matrices and and are positive numbers.
So, and can be obtained from Equation (11):where and are the identity matrices of size P P and Q Q, respectively.
Equations (14) and (15) can be obtained through replacing and with their expressions in Equations (12) and (13), respectively.
As like before, both and in Equations (14) and (15) are called regularization parameters. By solving Equation (14), the eigenvalues and their corresponding eigenvectors can be obtained. The eigenvalues and their corresponding eigenvectors can be obtained from Equation (15).
4. Simulation Experiments
In this section, we evaluate our method compared with ORCCA on artificial and handwritten numerals databases.
4.1. Experiment on Artifical Databases
The pairwise samples and are generated from the expressions in Equations (16) and (17), respectively.where obeys uniform distribution on and and are Gaussian noise with standard deviation 0.05. The radial basis function is chosen as kernel function, where .
4.1.1. Determining Regularization Parameters
For the selection of the regularization parameters, by far there is no reliable method to determine the optimal values. In this paper, in order to simplify the calculation, let and . The regularization parameters were chosen from 10−5, 10−4, 10−3, 10−2, 10−1, and 1. This method is used in the literature [5].
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. The 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.
The 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. The 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. The optimal regularization parameters are used to perform simulations in the next section.
4.1.2. Simulation Experiment 1
According to Equations (16) and (17), 200 pairs of data are randomly generated as the test samples. For the regularization parameters and in the ORCCA and ORKCCA algorithms, the canonical variables are obtained for test samples, respectively. The 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. and denote the first group of canonical variables. and are the second group of canonical variables.
The 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.
4.1.3. Simulation Experiment 2
According to Equations (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. The rest 400, 325, and 250 pairs of data are the test samples, respectively. The classification experiments based on K-neighbors 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. The accuracies of classification for 400, 325, and 250 pairs of data are the averages of the accuracies of classification for the 15 experiments results, respectively. Table 4 gives the accuracies of classification for ORCCA and ORKCCA for the test samples with the different number.
In Table 4, the first column is the numbers of the training samples and the second column and the third column are the accuracies of classification for ORCCA and ORKCCA for the training samples with the different number. The experimental results show that the accuracies of classification for ORKCCA are higher than those for ORCCA. So, the performance of ORKCCA outperforms that of ORCCA for the nonlinear problem. The comparison curves of the accuracies of classification for ORCCA and ORKCCA are given in Figure 1.
4.2. Experiments on Handwritten Numerals Databases
The Concordia University CENPARMI database of handwritten Arabic numerals have 10 classes, that is, 10 digits (from 0 to 9), and 600 samples for each. The first 400 samples are used as the training set, and the remaining samples as the test set in each class. Then, the training samples and the test samples are 4000 and 2000, respectively. The handwritten digital images are preprocessed by the method given in [14]. Four kinds of features are extracted as follows: XG (256-dimensional Gabor transformation feature), XL (121-dimensional Legendre moment feature), XP (36-dimensional Pseudo-Zernike moment feature), and XZ (30-dimensional Zernike moment feature).
For the choice of the regularization parameters, let and . The regularization parameters were chosen from 10−5, 10−3, and 1. The results of our method are compared with the results of ORCCA in order to verify the effectiveness of ORKCCA. Table 5 lists the accuracies of classification for ORCCA and ORKCCA in different feature combinations and regularization parameters. Experimental results show that (1) the classification effect of the two methods is the best as the regularization parameter is 1; (2) the classification accuracies of ORKCCA are higher than that of ORCCA for different features combinations; (3) the classification accuracies of ORKCCA in the regularization parameters 10−5 and 10−3 are higher than those of ORCCA in the regularization parameters 1.
5. Conclusions
An orthogonal regularized kernel CCA algorithm for nonlinear problem is presented. By introducing the kernel function, our proposed algorithm is more suitable for solving 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. The experimental results show ORKCCA is feasible.
Data Availability
The experiments in paper were performed by the author Xi 2 years ago. Some troubles happened to his computer. The data can not be gotten from his computer. I’m sorry that the data is unable to be provided.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors are grateful to the support of the Hainan Provincial Natural Science Foundation (117150) and the Scientific Research Foundation of Hainan Tropical Ocean University (RHDXB201624).