Analysis And Research Of Principal Component Analysis In Spherical Coordinate

Nowadays,datasets with high dimensional vectors are used for conducting research in various fields of science and engineering.In order to efficiently analyze the vectors in these datasets and implement various machine intelligence algorithms on the vectors,it is required to reduce the dimensions of the vectors but preserve most of their information.Among the many dimension reduction methods,the principal component analysis,which is an advanced data analysis and dimension reduction method,is the most common method used in many science and engineering applications.It has been enriched in theory development and practical applicationThe conventional principal component analysis method decomposes the eigenvalues of the covariance matrix of the original vectors,then obtains a transformation matrix for dimension reduction.However,when the total number of the vectors is very limited,the covariance matrix may drop rank.In this case,the transformation matrix obtained by the eigen decomposition is no longer unique.Also,the matrix with its columns being these eigenvectors does not guarantee to be unitary.Furthermore,the conventional principal component analysis and its enhanced versions,such as kernel principal component analysis based on different kernel functions,require huge computing power.For some engineering applications that require high computational efficiency,it is difficult to perform the principal component analysis.In terms of this issue,the following work is carried outFirstly,the properties of the principal component analysis when the covariance matrix of the vectors drops rank are studied.It is found that the eigen decomposition of the covariance matrix is not uniquely defined and a theoretical proof is given.This implies that different transform matrices could be obtained for performing the principal component analysis.Hence,the generalized form of the eigen decomposition of the covariance matrix is given.Also,the necessary and sufficient condition for the matrix with its columns being the eigenvectors of the covariance matrix to be unitary is derived.Moreover,since the design of the unitary transform matrix for performing the principal component analysis is usually formulated as an optimization problem,the necessary and sufficient condition for the first order derivative of the Lagrange function to be equal to the zero vector is derived.Furthermore,the necessary and sufficient condition for the second order derivative of the Lagrange function to be a positive definite function is also derived.The computer numerical simulation results are given to valid the theoretical analysis resultsSecondly,since the conventional principal component analysis are required huge computational power and its transformation matrix may not be unique,this paper proposes a particular type of principal component analysis,namely the spherical coordinated based principal component analysis.In this method,the vectors represented in the Cartesian coordinate system are expressed as those represented in the spherical coordinate system.Then,certain rotational angles or the radii of the vector are set to their corresponding mean values The principal component of data is consisted of the remaining rotational angles or the radii of the vector.Finally,the processed vectors represented in the spherical coordinate system are expressed back in the Cartesian coordinate system and it can easily to calculate the reconstruction error.As the degrees of the freedoms of the processed vectors represented in the spherical coordinate system are reduced,the dimension of the manifold of the processed vectors represented in the Cartesian coordinate system is also reduced.Compared with the conventional principal component analysis and kernel principal component analysis,the comparative analysis of complexity shows that the proposed method has lower computational complexity.The computer numerical simulation results also show that the spherical coordinated based principal component analysis has lower reconstruction error than the conventional principal component analysis and kernel principal component analysis,and the required computing power is significantly reducedFinally,to illustrate the practicability of the proposed spherical coordinate based principal component analysis on the practical engineering applications,the electrocardiogram denoising problem is considered.The public data from real scenarios are selected in this paper Concretely,using the electrocardiogram signal and different levels of additive Gaussian white noise to build the data set required for the experiment.Then,the proposed method is used for signal denoising.The signal to noise ratio,mean square error,correlation coefficient and smoothness are used to evaluate the effect of signal denoising.At the same time,a method of joint spherical coordinate based principal component analysis and empirical mode decomposition method is also proposed and compared with the classical wavelet analysis method.The computer numerical simulation results show that the proposed method effectively suppresses noise.