Aggregated wavelet estimation and its application to ultra-fast fMRI

Abstract

The methodology of aggregation of known nonparametric regression estimators into a single better estimator has received increasing attention in statistical literature. Traditional aggregation means that a linear or convex combination of several estimators is considered. Wavelet regression estimation, due to its multiresolution nature, presents another opportunity for aggregation – using different estimation procedures on different resolution scales. Such an opportunity becomes attractive if known wavelet estimators have desired complementary properties on different frequencies. The difficulty of such an aggregation is that the assignment of scales depends on an underlying regression function and regression errors. This paper proposes a data-driven aggregation of two wavelet estimators – SureBlock of Cai and Zhou [(2009), ‘A Data-driven Block Thresholding Approach to Wavelet Estimation’, Annals of Statistics, 37, 569–595] and Universal of Efromovich [(1999a,b), Nonparametric Curve Estimation: Methods, Theory and Applications, New York: Springer; ‘Quasi-linear Wavelet Estimation’, Journal of the American Statistical Association, 94, 189–204] – to achieve a better quality of estimation, better data-compression, and better visualisation of functions with different smoothness characteristics on low and high frequencies. The proposed estimator is motivated by an applied problem of denoising and compression of ultra-fast (UF) functional magnetic resonance imaging (fMRI) – the new magnetic resonance technology that screens the activity of brain voxels every 50 ms with the purpose of understanding human brain activity. The proposed aggregated wavelet estimator is supported by the asymptotic theory, tested via intensive numerical simulations and UF fMRI applications, and it is expected to be useful in similar applications.