Experimental Design with Circulant Property and Its Application to fMRI Experiments

Abstract

Experimental designs have been widely used for cost-efficiency. Orthogonal arrays are commonly used to study the effects of many factors simultaneously, but they do not exist in any sizes. Recently, orthogonal arrays with circulant property receive great attention and are applied to experiments in many fields, such as functional magnetic resonance imaging (fMRI). fMRI is a pioneering technology for studying brain activity in response to mental stimuli. Efficient fMRI experimental designs are important for rendering precise statistical inference on brain functions, but a systematic construction method for this important class of designs does not exist. In this work, we propose an innovative and unified construction method for efficient, if not optimal, fMRI designs via circulant almost orthogonal arrays (CAOAs). Since circulant Hadamard matrices, that can also be viewed as circulant orthogonal arrays of symbols two and strength two, have been conjectured nonexistence, CAOAs are considered.????? We characterize this new class of efficient designs and propose a systematic construction via a newly invented algebraic tool called complete difference system (CDS). We not only prove the equivalence relation of CDS and CAOA, but also construct many classes of CAOAs with very high efficiency. Finally, we apply these efficient CAOAs to fMRI experiments, demonstrating that our constructed designs have better properties than the traditional designs in terms of cost-efficiency.Key words and phrases: Optimal Designs; Circulant Orthogonal Arrays; Complete Difference System; functional Magnetic Resonance Imaging (fMRI).