TY - JOUR
T1 - Penalized Least Squares methods for solving the EEG Inverse Problem
AU - Vega-Hernández, Mayrim
AU - Martínez-Montes, Eduardo
AU - Sánchez-Bornot, José M.
AU - Lage-Castellanos, Agustín
AU - Valdés-Sosa, Pedro A.
PY - 2008/10/1
Y1 - 2008/10/1
N2 - Most of the known solutions (linear and nonlinear) of the ill-posed EEG Inverse Problem can be interpreted as the estimated coefficients in a penalized regression framework. In this work we present a general formulation of this problem as a Multiple Penalized Least Squares model, which encompasses many of the previously known methods as particular cases (e.g., Minimum Norm, LORETA). New types of inverse solutions arise since recent advances in the field of penalized regression have made it possible to deal with non-convex penalty functions, which provide sparse solutions (Fan and Li (2001)). Moreover, a generalization of this approach allows the use of any combination of penalties based on 11 or 12-norms, leading to solutions with combined properties such as smoothness and sparsity. Synthetic data is used to explore the benefits of non-convex penalty functions (e.g., LASSO, SCAD and LASSO Fusion) and mixtures (e.g., Elastic Net and LASSO Fused) by comparing them with known solutions in terms of localization error, blurring and visibility. Real data is used to show that a mixture model (Elastic Net) allows for tuning the spatial resolution of the solution to range from very concentrated to very blurred sources.
AB - Most of the known solutions (linear and nonlinear) of the ill-posed EEG Inverse Problem can be interpreted as the estimated coefficients in a penalized regression framework. In this work we present a general formulation of this problem as a Multiple Penalized Least Squares model, which encompasses many of the previously known methods as particular cases (e.g., Minimum Norm, LORETA). New types of inverse solutions arise since recent advances in the field of penalized regression have made it possible to deal with non-convex penalty functions, which provide sparse solutions (Fan and Li (2001)). Moreover, a generalization of this approach allows the use of any combination of penalties based on 11 or 12-norms, leading to solutions with combined properties such as smoothness and sparsity. Synthetic data is used to explore the benefits of non-convex penalty functions (e.g., LASSO, SCAD and LASSO Fusion) and mixtures (e.g., Elastic Net and LASSO Fused) by comparing them with known solutions in terms of localization error, blurring and visibility. Real data is used to show that a mixture model (Elastic Net) allows for tuning the spatial resolution of the solution to range from very concentrated to very blurred sources.
KW - EEG
KW - Inverse Problem
KW - Least Squares
KW - Penalized regression
UR - http://www.scopus.com/inward/record.url?scp=60149096965&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:60149096965
SN - 1017-0405
VL - 18
SP - 1535
EP - 1551
JO - Statistica Sinica
JF - Statistica Sinica
IS - 4
ER -