Complex-valued neural networks for nonlinear complex principal component analysis

Abstract

Principal component analysis (PCA) has been generalized to complex principal component analysis (CPCA), which has been widely applied to complex-valued data, 2-dimensional vector fields, and complexified real data through the Hilbert transform. Nonlinear PCA (NLPCA) can also be performed using auto-associative feed-forward neural network (NN) models, which allows the extraction of nonlinear features in the data set. This paper introduces a nonlinear complex PCA (NLCPCA) method, which allows nonlinear feature extraction and dimension reduction in complex-valued data sets. The NLCPCA uses the architecture of the NLPCA network, but with complex variables (including complex weight and bias parameters). The application of NLCPCA on test problems confirms its ability to extract nonlinear features missed by the CPCA. For similar number of model parameters, the NLCPCA captures more variance of a data set than the alternative real approach (i.e. replacing each complex variable by 2 real variables and applying NLPCA). The NLCPCA is also used to perform nonlinear Hilbert PCA (NLHPCA) on complexified real data. The NLHPCA applied to the tropical Pacific sea surface temperatures extracts the El Niño-Southern Oscillation signal better than the linear Hilbert PCA.

Nonlinear complex principal component analysis of the tropical Pacific interannual wind variability

“…(see Rattan and Hsieh [2004] for details of the NLCPCA method). It is well known that a feed-forward NN only needs one hidden layer of neurons for it to model any nonlinear continuous function [Bishop, 1995].…”

“…[4] For nonlinear feature extraction in the complex domain, the nonlinear CPCA (NLCPCA) method has recently been proposed using a complex-valued NN and applied to the tropical Pacific sea surface temperatures [Rattan and Hsieh, 2004]. This research letter will be the first application of the NLCPCA to a 2-D vector field, the monthly tropical Pacific wind data.…”

Complex-valued forecasting of the global solar irradiation

“…They can outperform their real counterparts in many ways. 19 CVNNs give also the possibility of the simultaneous modeling and forecasting, 20 where they have been used to forecast the wind's speed and direction simultaneously. The main motivation to use CVNNs is due to faster convergence, reduction in learning parameters, and ability to learn two dimension motion of signal in complex-valued neural network.…”

Software aging estimation and prediction of a real VOD system based on PCA and neural networks

“…We then construct artificial neural network (ANN) [7], [8] models to analyze the extracted data series of systematic parameters and to predict software aging of the VOD system. In order to reduce the complexity of ANN and to improve its efficiency, principal component analysis (PCA) [9], [10] is used to reduce the dimensionality of input variables of ANN. We finally make the sensitivity analysis of the number of principal components.…”