Penalized partially linear models using sparse representations with an application to fMRI time series

• Published in: IEEE transactions on signal processing Vol. 53; no. 9; pp. 3436 - 3448
• Main Authors: Fadili, J.M., Bullmore, E.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.09.2005; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Biological and medical sciences; Biological system modeling; Biomedical engineering; Biomedical imaging; Computer Science; Computer simulation; Computerized, statistical medical data processing and models in biomedicine; Econometrics; Economic models; Engineering Sciences; Estimators; fMRI; Humans; Large-scale systems; Linear regression; Magnetic resonance imaging; Mathematical analysis; Medical management aid. Diagnosis aid; Medical sciences; Neuroimaging; partially linear models; penalized estimation; Polynomials; Redundant; Representations; Signal and Image processing; sparse representations; Studies; Time series; Vectors; wavelets; penalized estimation; Threshold detection; sparse representations; Time series; Non parametric estimation; Functional analysis; Nuclear magnetic resonance imaging; Modeling; Linear model; neuroimaging; Penalty function; fMRI; Simulation; partially linear models; wavelets; Medical imagery; Sparse representation; Stationary signal; Fast algorithm; Functional imaging
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2005.853207

In this paper, we consider modeling the nonparametric component in partially linear models (PLMs) using linear sparse representations, e.g., wavelet expansions. Two types of representations are investigated, namely, orthogonal bases (complete) and redundant overcomplete expansions. For bases, we introduce a regularized estimator of the nonparametric part. The important contribution here is that the nonparametric part can be parsimoniously estimated by choosing an appropriate penalty function for which the hard and soft thresholding estimators are special cases. This allows us to represent in an effective manner a broad class of signals, including stationary and/or nonstationary signals and avoids excessive bias in estimating the parametric component. We also give a fast estimation algorithm. The method is then generalized to handle the case of overcomplete representations. A large-scale simulation study is conducted to illustrate the finite sample properties of the estimator. The estimator is finally applied to real neurophysiological functional magnetic resonance imaging (MRI) data sets that are suspected to contain both smooth and transient drift features.