On the Probability Density Function of the Test Statistic for One Nonlinear GLR Detector Arising from fMRI

Fangyuan Nan

doi:10.3844/jmssp.2007.38.43

Abstract

Recently an important and interesting nonlinear generalized likelihood ratio (GLR) detector emerged in functional magnetic resonance imaging (fMRI) data processing. However, the study of that detector is incomplete: the probability density function (pdf) of the test statistic was draw from numerical simulations without much theoretical support and is therefore, not firmly grounded. This correspondence presents more accurate (asymptotic) closed form of the pdf by resorting to a non-central Wishart matrix and by asymptotic expansion of some integrals. It is then confirmed theoretically that the detector does possess constant false alarm rate (CFAR) property under some practical regimes of signal to noise ratio (SNR) for finite samples and the correct threshold selection method is given, which is very important for real fMRI data processing.

Highlights

In our research on voxelwise detection of functional magnetic resonance imaging (fMRI), we encountered the following model and problem [1]
generalized likelihood ratio (GLR) test statistic: The GLRT statistic is given by l3 (y) =
This study provides an asymptotic, closed form of the pdf of the test statistic of one non linear GLRT detector with important applications to fMRI, thereby a 1

Summary

INTRODUCTION

In our research on voxelwise detection of fMRI, we encountered the following model and problem [1]. The term nc denotes a standard complex Gaussian vector (with mean zero and covariance matrix IN×N ). The detection is with regard to b: under H0, b=0 indicates there is no brain activity in the corresponding voxel; under H1, b ≠ 0 indicates the presence of activity in the voxel We have compared this model to actual fMRI time-series and found that our assumptions are in good agreement with actual data[1]. Because of the fundamental importance of threshold selection in detection problems and the above-mentioned importance of[1] in fMRI data processing, it is worthwhile to do some rigorous theoretical work to fix the defect. We attack the more complex problem with unknown noise variance.

Results on known noise variance

Results on unknown noise variance

DISCUSSION