Nonparametric Matrix Response Regression with Application to Brain Imaging Data Analysis

Abstract

With the rapid growth of neuroimaging technologies, a great effort has been dedicated recently to investigate the dynamic changes in brain activity. Examples include time course calcium imaging and dynamic brain functional connectivity. In this paper, we propose a novel nonparametric matrix response regression model to characterize the nonlinear association between 2D image outcomes and predictors such as time and patient information. Our estimation procedure can be formulated as a nuclear norm regularization problem, which can capture the underlying low-rank structure of the dynamic 2D images. We present a computationally efficient algorithm, derive the asymptotic theory and show that the method outperforms other existing approaches in simulations. We then apply the proposed method to a calcium imaging study for estimating the change of fluorescent intensities of neurons, and an electroencephalography study for a comparison in the dynamic connectivity covariance matrices between alcoholic and control individuals. For both studies, the method leads to a substantial improvement in prediction error.

Broadcasted Nonparametric Tensor Regression

“…One recent statistical development under this setup is tensor regression (Zhou et al, 2013). In this work, we focus on models that involve a tensor covariate X = (X i 1 ,i 2 ,...,i D ) ∈ R p 1 ×p 2 ו••×p D of order D. In passing, it is also worth mentioning that regression of a tensor response on a vector covariate (e.g., Sun and Li, 2017;Li and Zhang, 2017;Hu et al, 2019) is also a popular research direction.…”

Matrix Linear Discriminant Analysis

“…Second, turning a p × q matrix into a pq × 1 vector generates unmanageably high dimensionality. E.g., estimating the population precision matrix for LDA can be troublesome if pq n. Third, covariates, while Hu et al [2019] developed a nonparametric matrix response regression model.…”

Broadcasted nonparametric tensor regression

“…One recent statistical development under this setup is tensor regression. In this work, we focus on models that involve a tensor covariate X = (X i 1 ,i 2 ,...,i D ) ∈ R p 1 ×p 2 ו••×p D of order D. Notice that tensor regression based on vector covariate (e.g., Sun and Li, 2017;Li and Zhang, 2017;Hu et al, 2019) is also a popular research direction.…”