Estimation of general linear model coefficients for real-time application

Abstract

An algorithm using an orthogonalization procedure to estimate the coefficients of general linear models (GLM) for functional magnetic resonance imaging (fMRI) calculations is described. The idea is to convert the basis functions or explanatory variables of a GLM into orthogonal functions using the usual Gram–Schmidt orthogonalization procedure. The coefficients associated with the orthogonal functions, henceforth referred to as auxiliary coefficients, are then easily estimated by applying the orthogonality condition. The original GLM coefficients are computed from these estimates. With this formulation, the estimates can be updated when new image data become available, making the approach applicable for real-time estimation. Since the contribution of each image data is immediately incorporated into the estimated values, storing the data in memory during the estimation process becomes unnecessary, minimizing the memory requirements of the estimation process. By employing Cholesky decomposition, the algorithm is a factor of two faster than the standard recursive least-squares approach. Results of the analysis of an fMRI study using this approach showed the algorithm’s potential for real-time application.