Effective dimension reduction for sparse functional data

Functional sufficient dimension reduction through average Fréchet derivatives

Sufficient dimension reduction (SDR) embodies a family of methods that aim for reduction of dimensionality without loss of information in a regression setting. In this article, we propose a new method for nonparametric function-on-function SDR, where both the response and the predictor are a function. We first develop the notions of functional central mean subspace and functional central subspace, which form the population targets of our functional SDR. We then introduce an average Fréchet derivative estimator, which extends the gradient of the regression function to the operator level and enables us to develop estimators for our functional dimension reduction spaces. We show the resulting functional SDR estimators are unbiased and exhaustive, and more importantly, without imposing any distributional assumptions such as the linearity or the constant variance conditions that are commonly imposed by all existing functional SDR methods. We establish the uniform convergence of the estimators for the functional dimension reduction spaces, while allowing both the number of Karhunen-Loève expansions and the intrinsic dimension to diverge with the sample size. We demonstrate the efficacy of the proposed methods through both simulations and two real data examples.

Nonparametric testing of lack of dependence in functional linear models

An important inferential task in functional linear models is to test the dependence between the response and the functional predictor. The traditional testing theory was constructed based on the functional principle component analysis which requires estimating the covariance operator of the functional predictor. Due to the intrinsic high-dimensionality of functional data, the sample is often not large enough to allow accurate estimation of the covariance operator and hence causes the follow-up test underpowered. To avoid the expensive estimation of the covariance operator, we propose a nonparametric method called Functional Linear models with U-statistics TEsting (FLUTE) to test the dependence assumption. We show that the FLUTE test is more powerful than the current benchmark method (Kokoszka P,2008; Patilea V,2016) in the small or moderate sample case. We further prove the asymptotic normality of our test statistic under both the null hypothesis and a local alternative hypothesis. The merit of our method is demonstrated by both simulation studies and real examples.

Subgroup identification via homogeneity pursuit for dense longitudinal/spatial data

In the clinical trial community, it is usually not easy to find a treatment that benefits all patients since the reaction to treatment may differ substantially across different patient subgroups. The heterogeneity of treatment effect plays an essential role in personalized medicine. To facilitate the development of tailored therapies and improve the treatment efficacy, it is important to identify subgroups that exhibit different treatment effects. We consider a very general framework for subgroup identification via the homogeneity pursuit methods usually employed in econometric time series analysis. The change point detection algorithm in our procedure is most suitable for analyzing dense longitudinal or spatial data which are quite common for biomedical studies these days. We demonstrate that our proposed method is fast and accurate through extensive numerical studies. In particular, our method is illustrated by analyzing a diffusion tensor imaging data set.

Neuroprotective Natural Molecules, From Food to Brain

The prevalence of neurodegenerative disorders is increasing; however, an effective neuroprotective treatment is still remaining. Nutrition plays an important role in neuroprotection as recently shown by epidemiological and biochemical studies which identified food components as promising therapeutic agents. Neuroprotection includes mechanisms such as activation of specific receptors, changes in enzymatic neuronal activity, and synthesis and secretion of different bioactive molecules. All these mechanisms are focused on preventing neuronal damage and alleviating the consequences of massive cell loss. Some neuropathological disorders selectively affect to particular neuronal populations, thus is important to know their neurochemical and anatomical properties in order to design effective therapies. Although the design of such treatments would be specific to neuronal groups sensible to damage, the effect would have an impact in the whole nervous system. The difficult overcoming of the blood brain barrier has hampered the development of efficient therapies for prevention or protection. This structure is a physical, enzymatic, and influx barrier that efficiently protects the brain from exogenous molecules. Therefore, the development of new strategies, like nanocarriers, that help to promote the access of neuroprotective molecules to the brain, is needed for providing more effective therapies for the disorders of the central nervous system (CNS). In order both to trace the success of these nanoplatforms on the release of the bioactive cargo in the CNS and determinate the concentration at trace levels of targets biomolecules by analytical chemistry and concretely separation instrumental techniques, constitute an essential tool. Currently, these techniques are used for the determination and identification of natural neuroprotective molecules in complex matrixes at different concentration levels. Separation techniques such as chromatography and capillary electrophoresis (CE), using optical and/or mass spectrometry (MS) detectors, provide multiples combinations for the quantitative and qualitative analysis at basal levels or higher concentrations of bioactive analytes in biological samples. Bearing this in mind, the development of food neuroprotective molecules as brain therapeutic agents is a complex task that requires the intimate collaboration and engagement of different disciplines for a successful outcome. In this sense, this work reviews the new advances achieved in the area toward a better understanding of the current state of the art and highlights promising approaches for brain neuroprotection.

Optimal Penalized Function-on-Function Regression under a Reproducing Kernel Hilbert Space Framework

Many scientific studies collect data where the response and predictor variables are both functions of time, location, or some other covariate. Understanding the relationship between these functional variables is a common goal in these studies. Motivated from two real-life examples, we present in this paper a function-on-function regression model that can be used to analyze such kind of functional data. Our estimator of the 2D coefficient function is the optimizer of a form of penalized least squares where the penalty enforces a certain level of smoothness on the estimator. Our first result is the Representer Theorem which states that the exact optimizer of the penalized least squares actually resides in a data-adaptive finite dimensional subspace although the optimization problem is defined on a function space of infinite dimensions. This theorem then allows us an easy incorporation of the Gaussian quadrature into the optimization of the penalized least squares, which can be carried out through standard numerical procedures. We also show that our estimator achieves the minimax convergence rate in mean prediction under the framework of function-on-function regression. Extensive simulation studies demonstrate the numerical advantages of our method over the existing ones, where a sparse functional data extension is also introduced. The proposed method is then applied to our motivating examples of the benchmark Canadian weather data and a histone regulation study.