arXiv subjects

Martin Lindquist

Publications and source records attributed to Martin Lindquist.

4 recordsLinked to original sources

arXiv2023

Super Ensemble Learning Using the Highly-Adaptive-Lasso

We introduce the Meta Highly-Adaptive-Lasso Minimum Loss Estimator (M-HAL-MLE), a novel ensemble approach for estimating functional parameters of realistically modeled data distribution from independent and identically distributed observations. Given $J$ initial estimators, candidate ensembles are generated by finite-sectional-variation cadlag functions. Using $V$-fold cross-validation, the M-HAL-MLE selects the optimal cadlag ensemble minimizing the cross-validated empirical risk, with the sectional variation bound as a tuning parameter. The final estimator, M-HAL super-learner, is obtained by averaging ensemble compositions across folds. In contrast, the oracle ensemble and oracle estimator are defined by minimizing the population excess risk relative to the true function. We establish following theoretical properties: 1) the M-HAL super-learner converges to the oracle estimator at rate $n^{-2/3}$ in excess risk, up to log-n factors; 2) by appropriate undersmoothing, target features of the M-HAL super-learner are asymptotically linear for corresponding target features of the oracle estimator; 3) the excess risk between the oracle estimator and true function, along with the difference between their target features, is generally second-order. Simulations validate the theoretical results, demonstrating effectiveness in high-dimensional settings. We further illustrate the method in a real-data application involving mediation analysis of functional MRI from human pain studies.

stat.ME↗

arXiv2022

Graph-constrained Analysis for Multivariate Functional Data

Functional Gaussian graphical models (GGM) used for analyzing multivariate functional data customarily estimate an unknown graphical model representing the conditional relationships between the functional variables. However, in many applications of multivariate functional data, the graph is known and existing functional GGM methods cannot preserve a given graphical constraint. In this manuscript, we demonstrate how to conduct multivariate functional analysis that exactly conforms to a given inter-variable graph. We first show the equivalence between partially separable functional GGM and graphical Gaussian processes (GP), proposed originally for constructing optimal covariance functions for multivariate spatial data that retain the conditional independence relations in a given graphical model. The theoretical connection help design a new algorithm that leverages Dempster's covariance selection to calculate the maximum likelihood estimate of the covariance function for multivariate functional data under graphical constraints. We also show that the finite term truncation of functional GGM basis expansion used in practice is equivalent to a low-rank graphical GP, which is known to oversmooth marginal distributions. To remedy this, we extend our algorithm to better preserve marginal distributions while still respecting the graph and retaining computational scalability. The insights obtained from the new results presented in this manuscript will help practitioners better understand the relationship between these graphical models and in deciding on the appropriate method for their specific multivariate data analysis task. The benefits of the proposed algorithms are illustrated using empirical experiments and an application to functional modeling of neuroimaging data using the connectivity graph among regions of the brain.

stat.ME↗

arXiv2018

Functional Mediation Analysis with an Application to Functional Magnetic Resonance Imaging Data

Causal mediation analysis is widely utilized to separate the causal effect of treatment into its direct effect on the outcome and its indirect effect through an intermediate variable (the mediator). In this study we introduce a functional mediation analysis framework in which the three key variables, the treatment, mediator, and outcome, are all continuous functions. With functional measures, causal assumptions and interpretations are not immediately well-defined. Motivated by a functional magnetic resonance imaging (fMRI) study, we propose two functional mediation models based on the influence of the mediator: (1) a concurrent mediation model and (2) a historical mediation model. We further discuss causal assumptions, and elucidate causal interpretations. Our proposed models enable the estimation of individual causal effect curves, where both the direct and indirect effects vary across time. Applied to a task-based fMRI study, we illustrate how our functional mediation framework provides a new perspective for studying dynamic brain connectivity. The R package cfma is available on CRAN.

stat.AP↗

arXiv2015

An M-Estimator for Reduced-Rank High-Dimensional Linear Dynamical System Identification

High-dimensional time-series data are becoming increasingly abundant across a wide variety of domains, spanning economics, neuroscience, particle physics, and cosmology. Fitting statistical models to such data, to enable parameter estimation and time-series prediction, is an important computational primitive. Existing methods, however, are unable to cope with the high-dimensional nature of these problems, due to both computational and statistical reasons. We mitigate both kinds of issues via proposing an M-estimator for Reduced-rank System IDentification (MR. SID). A combination of low-rank approximations, L-1 and L-2 penalties, and some numerical linear algebra tricks, yields an estimator that is computationally efficient and numerically stable. Simulations and real data examples demonstrate the utility of this approach in a variety of problems. In particular, we demonstrate that MR. SID can estimate spatial filters, connectivity graphs, and time-courses from native resolution functional magnetic resonance imaging data. Other applications and extensions are immediately available, as our approach is a generalization of the classical Kalman Filter-Smoother Expectation-Maximization algorithm.

stat.ME↗