Most naturally occurring processes are inherently multivariate in their origination. Simultaneous analysis of multiple random variables reveals insights about the relationship among variables. This leads to more compelling analysis than marginal consideration of the components alone. Multivariate analysis is considerably more complicated than univariate analysis, especially when the assumption of multivariate normality does not apply. Methods for reducing the complexity of multivariate observations become essential because of the curse of dimensionality.
Independent component analysis (ICA) is a means for finding a suitable representation of multivariate data. We propose methods for measuring mutual independence, introduce a novel statistical framework under minimal prior assumptions for estimation of independent components from observations, establish consistency, and consider mis-specification testing of the proposed ICA model.
Due to computational constraints, principal component analysis (PCA) is commonly used for dimension reduction prior to ICA (PCA+ICA). We propose likelihood component analysis (LCA), a novel methodology in which dimension reduction and latent variable estimation are achieved simultaneously by maximizing a likelihood with Gaussian and non-Gaussian components. LCA is a useful data visualization and dimension reduction tool that reveals features not apparent from PCA or PCA+ICA.
Risk, B., Matteson, D.S. and Ruppert, D. (2015), “Likelihood Component Analysis.”
Matteson, D.S. and Tsay, R.S. (2015), “Independent Component Analysis via Distance Covariance,” To Appear, Journal of the American Statistical Association.
Risk, B., Matteson, D.S., Ruppert, D., Eloyan, A. and Cao, B. (2014), “An Evaluation of Independent Component Analyses with an Application to Resting State fMRI,” Biometrics, Vol. 70, No. 1: 224-236.
- R Package: steadyICA – ICA and Tests of Independence via Multivariate Distance Covariance (2015)
- Risk, B., James, N.A. and Matteson, D.S.
- R Package: ica4fts – Independent Components for Time Series (2009)
- Ang, E. and Matteson, D.S.