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.

Introduction

Functional magnetic resonance imaging (fMRI) offers a noninvasive approach in studying human brain functions by mapping parts of the human brain that are activated in response to various physical stimuli or activities such as sounds, images, or finger movements. This “brain mapping” is achieved by detecting local changes in the blood oxygenation level that can occur in response to an increase in neuronal activities (Ogawa et al., 1990). These hemodynamic changes are measured using an MR scanner and the resulting series of MR images is then assessed for evidence of experimentally induced effects at every intracerebral voxel. Statistical analysis is carried out to check for evidence against the null hypothesis of no effects. This results in an “image” of statistics showing how significantly an intracerebral voxel is affected by the given experimental task.

Several statistics have been employed for hypothesis testing. The general linear model (GLM), however, provides a unified framework in the analysis of fMRI data (Friston et al., 1995). With its ability to model multiple experimental and confounding effects simultaneously, GLM provides for greater flexibility in experimental design and can be used to analyze fMRI time series extensively. However, to obtain reliable estimates of GLM coefficients, gathering several MR images is required in order to minimize the effects of noise, which can include sensor noise, bulk motion, and other physiological fluctuations. The large number of data involved in the analysis often requires considerable processing time and data storage. Thus, GLM is usually used to analyze fMRI time series long after the scanning session is completed.

Real-time analysis of fMRI data offers several advantages both for theoretical studies and clinical applications. With real-time analysis, monitoring the task performance of the subject and the resulting quality of the acquired data can be easily achieved. It could also make functional mapping experiments more interactive by allowing an ongoing paradigm to be altered if a need arises. These advantages and many others have been demonstrated in several studies Cox et al 1995, Gembris et al 2000, Smyser et al 2001, Voyvodic 1999, Yoo et al 1999. For instance, Cox et al. (1995) presented an algorithm that computes the correlation coefficient, in real-time, of a time course of images with a reference time series, with the mean and linear trend projected out. Gembris et al. (2000) reported a real-time algorithm that combines correlation analysis with a detrending, sliding window technique, and reference vector optimization. Most of these approaches employ a single predefined statistical quantity such as correlation coefficient and estimate this quantity while the data are being generated or immediately after all the data become available to produce the activation map. On the other hand, Smyser et al. (2001) reported a real-time implementation of multiple linear regression supported by a time-aware acquisition and processing system. Multiple linear regression analysis was carried out using Gentleman’s algorithm (Gentleman, 1974), which is designed to minimize both computational cost and memory requirement for large regression computations. This approach provides a flexible tool in performing substantial parametric analysis in real time.

Real-time estimation of general linear model coefficients can provide more sophisticated statistical analysis of fMRI time series as exemplified in Smyser et al. (2001). In this paper, we use an orthogonalization procedure to estimate the coefficients of GLMs for the real-time analysis of fMRI data. The algorithm is based on the usual Gram–Schmidt orthogonalization procedure. The basic idea is to convert the basis functions or explanatory variables of a GLM into orthogonal functions. The auxiliary coefficients, i.e., the coefficients associated with the orthogonal functions, are then easily estimated using the orthogonality condition. The values of the original GLM coefficients are computed from these estimates. This formulation offers several advantages, especially in the analysis of fMRI time series. With this approach, the estimates can be immediately updated when new image data become available, making it attractive for real-time estimation. Since the contribution of each image data is immediately incorporated into the estimated values, storing the data in memory during estimation becomes unnecessary. This minimizes the memory requirement, which is an important consideration in fMRI analysis. The contribution of each basis function in reducing the mean square error can also be known from the square of the auxiliary coefficients. Thus, values of the mean square error can be immediately obtained without recomputing the contribution of previous observations.

Several approaches based on orthogonalization procedures (e.g., Billings et al 1989, Korenberg 1988 have been employed in many model estimation problems. These methods have been proven to be very efficient in determining significant terms of the model and providing unbiased parameter estimates. One of these is Korenberg’s fast orthogonal search (FOS) algorithm (Korenberg, 1988), known for its robustness in obtaining a parsimonious model even when the model includes nonlinear terms. FOS has been successfully applied in the estimation of vector autoregressive model parameters (Bagarinao and Sato, 2002), in the detection of nonlinearities in short noisy time series (Barahona et al., 1996), and in the reconstruction of bifurcation diagrams (Bagarinao et al., 1999), among others. In the present study, an implicit orthogonalization procedure similar to the FOS is given. Unlike the other approaches where the estimation is done after all the data become available, the emphasis of the present study is on the real-time estimation of model coefficients.

Under Methods, we describe the algorithm and define the needed quantities relevant to the analysis of fMRI data. We discuss first the univariate case and later consider the multivariate implementation of the approach. The section ends with an outline of the different steps in the estimation process. Under Results, we apply the algorithm to the analysis of actual fMRI time series. Emphasis is given to the algorithm’s attractive features for real-time implementation. Results of a comparison of the computational complexity of the proposed approach to that of the recursive least-squares approach is also given. The discussion of results, together with the conclusion, is presented in the last section.