Noise-cleaning the precision matrix of fMRI time series

TL;DR

This paper compares various noise-cleaning algorithms for estimating precision matrices from small fMRI datasets, demonstrating that the optimal rotationally invariant estimator performs best, especially under severe undersampling conditions.

Contribution

The study introduces a variant of the Optimal Rotationally Invariant Estimator with cross-validation and an iterative likelihood gradient ascent method, improving precision matrix estimation in small, undersampled datasets.

Findings

Optimal Rotationally Invariant Estimator reduces distance to true matrix in synthetic data.

It achieves higher test likelihood on natural fMRI data.

The iterative likelihood gradient ascent accurately estimates weakly correlated datasets.

Abstract

We present a comparison between various algorithms of inference of covariance and precision matrices in small datasets of real vectors, of the typical length and dimension of human brain activity time series retrieved by functional Magnetic Resonance Imaging (fMRI). Assuming a Gaussian model underlying the neural activity, the problem consists in denoising the empirically observed matrices in order to obtain a better estimator of the true precision and covariance matrices. We consider several standard noise-cleaning algorithms and compare them on two types of datasets. The first type are time series of fMRI brain activity of human subjects at rest. The second type are synthetic time series sampled from a generative Gaussian model of which we can vary the fraction of dimensions per sample q = N/T and the strength of off-diagonal correlations. The reliability of each algorithm is assessed in terms of test-set likelihood and, in the case of synthetic data, of the distance from the true precision matrix. We observe that the so called Optimal Rotationally Invariant Estimator, based on Random Matrix Theory, leads to a significantly lower distance from the true precision matrix in synthetic data, and higher test likelihood in natural fMRI data. We propose a variant of the Optimal Rotationally Invariant Estimator in which one of its parameters is optimised by cross-validation. In the severe undersampling regime (large q) typical of fMRI series, it outperforms all the other estimators. We furthermore propose a simple algorithm based on an iterative likelihood gradient ascent, providing an accurate estimation for weakly correlated datasets.