Nonlinear Principal Component Analysis Method And Its Applications In Medicine

There are multivariate analyses in the medical research, and we often encounter the variables which are nonlinear related among them. In this case, it is inappropriate that to handle the data with a linear method. Thus, many methods which can deal with the nonlinear data are proposed. Among them, nonlinear data processing methods based on kernel trick are very popular to deal with these variables for its excellent property that it is easy to operate.The main idea of kernel methods is original input space data are mapped into high dimension feature spaces through nonlinear mapping, Data is applied to deal with in the feature spaces. Its key is inducted into kernel function, that scalar product operation in high dimension feature is transformed into kernel function compute in input space, and don't need to compute nonlinear mapping, so non-linearization is achieved in input spaces.In this paper we discuss the work principle and mathematics model of kernel principal component analysis method. Principal component analysis is a traditional statistic method and its analysis object is the covariance structure of the multivariate observed value and its purpose is obtain the principal component which can simply describe the relation of the observed value. In detail, PCA method is changing original n dimension observed variable into a set of new feature with the same n number through linear transformation, and each new feature is the linear combination of the original feature. If these new features are unrelated between each other, the small number main features among them which contain the principal information of the original data are called principal components. PCA is a method of feature extraction and dimensionality reduction. While kernel PCA is a nonlinear generalization of PCA in the sense that it is performing PCA in feature spaces and diagonalizing kernel matrix in high dimensional spaces. Kernel PCA can find at most l (the number of observed value) number of nonzero eigenvalues, which can exceed the sample dimensionality. The dimensionality of feature spaces is very high, even to infinite. However, kernel PCA don't need to look for principal components in the full spaces F, but just in the subspace spanned by the observed data. Kernel PCA need only to compute kernel function and rather than to compute nonlinear transformation and scalar product. Thus, the amount of calculation of kernel PCA isn't very complicated compared to PCA. When come to especially complexity problem, we even don't need to compute the whole eigenvalue, only need to compute the largest one or two eigenvalues.Our results demonstrate that kernel principal component analysis shows better results than principal components analysis in dimensions reduction and can deal with the nonlinear relation between the variables. Provide a theoretical basis for the popularized application of these methods in medical studies.We use Matlab software as platform to handle the analysis of our application example.