Signal fraction analysis for subspace processing of high dimensional data
Date
2007
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Abstract
A general tool for computing subspaces that decomposes data into potentially useful features is proposed. The technique is called Signal Fraction Analysis (SFA). The row-energy and column-energy optimization problems for signal-to-signal ratios are investigated. A generalized singular value problem is presented. This setting is distinguished from the Singular Value Decomposition (SVD). Preprocessing mappings of the data is used in situations where domain specific knowledge is available as a guide. We suggest an optimization problem where these mapping functions may be adapted using a problem dependent objective function. These ideas are illustrated using Wavelet and Fourier filters applied to EEG data. A self-contained description of the motivating maximum noise fraction method is included and a procedure for estimating the covariance matrix of the noise is described. We extend SFA by introducing novel constraints and propose two new generalized SVD type problems for computing subspace representations. A connection between SFA and Canonical Correlation Analysis is maintained. We implement and investigate a nonlinear extension to SFA based on a kernel method, i.e., Kernel SFA. Moreover, a second algorithm that uses noise adjustment in the data domain prior to kernelization is suggested. We include a detailed derivation of the methodology using kernel principal component analysis as a prototype. These methods are compared using toy examples and the benefits of KSFA are illustrated. This work establishes the potential of a SFA beamforming technique via its merger with a wide band MC-CDMA system. The details of non-overlapping window adaptive realization of SFA are introduced. We discuss the relationship between the SFA and DOA estimation via MUSIC. A novel structure for wide band MC-CDMA systems that utilizes the benefits of path diversity (inherent in direct sequence CDMA) and frequency diversity (inherent in MC-CDMA systems) is introduced. Simulations were performed to study the impact of noise perturbations on the performance of SFA. Simulations confirm that SFA enhances the performance and separability of interfering users. KSFA is applied to the classification of EEG data arising in the Brain Computer Interface Problem. We use Fourier and Wavelet filters to generate signal fractions as well as differencing methods.
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Subject
classification
component analysis
kernel
nonlinear
signal fraction analysis
signal separation
subspaces
mathematics