Statistical inference in mechanistic models: time warping for improved gradient matching

Inference in mechanistic models of non-linear differential equations is a challenging problem in current computational statistics. Due to the high computational costs of numerically solving the differential equations in every step of an iterative parameter adaptation scheme, approximate methods based on gradient matching have become popular. However, these methods critically depend on the smoothing scheme for function interpolation. The present article adapts an idea from manifold learning and demonstrates that a time warping approach aiming to homogenize intrinsic length scales can lead to a significant improvement in parameter estimation accuracy. We demonstrate the effectiveness of this scheme on noisy data from two dynamical systems with periodic limit cycle, a biopathway, and an application from soft-tissue mechanics. Our study also provides a comparative evaluation on a wide range of signal-to-noise ratios

Penalized Likelihood and Bayesian Function Selection in Regression Models

Challenging research in various fields has driven a wide range of methodological advances in variable selection for regression models with high-dimensional predictors. In comparison, selection of nonlinear functions in models with additive predictors has been considered only more recently. Several competing suggestions have been developed at about the same time and often do not refer to each other. This article provides a state-of-the-art review on function selection, focusing on penalized likelihood and Bayesian concepts, relating various approaches to each other in a unified framework. In an empirical comparison, also including boosting, we evaluate several methods through applications to simulated and real data, thereby providing some guidance on their performance in practice

Augmented balancing weights as linear regression

We provide a novel characterization of augmented balancing weights, also known as Automatic Debiased Machine Learning (AutoDML). These estimators combine outcome modeling with balancing weights, which estimate inverse propensity score weights directly. When the outcome and weighting models are both linear in some (possibly infinite) basis, we show that the augmented estimator is equivalent to a single linear model with coefficients that combine the original outcome model coefficients and OLS; in many settings, the augmented estimator collapses to OLS alone. We then extend these results to specific choices of outcome and weighting models. We first show that the combined estimator that uses (kernel) ridge regression for both outcome and weighting models is equivalent to a single, undersmoothed (kernel) ridge regression; this also holds when considering asymptotic rates. When the weighting model is instead lasso regression, we give closed-form expressions for special cases and demonstrate a ``double selection'' property. Finally, we generalize these results to linear estimands via the Riesz representer. Our framework ``opens the black box'' on these increasingly popular estimators and provides important insights into estimation choices for augmented balancing weights

A Partially Linear Framework for Massive Heterogeneous Data

We consider a partially linear framework for modelling massive heterogeneous data. The major goal is to extract common features across all sub-populations while exploring heterogeneity of each sub-population. In particular, we propose an aggregation type estimator for the commonality parameter that possesses the (non-asymptotic) minimax optimal bound and asymptotic distribution as if there were no heterogeneity. This oracular result holds when the number of sub-populations does not grow too fast. A plug-in estimator for the heterogeneity parameter is further constructed, and shown to possess the asymptotic distribution as if the commonality information were available. We also test the heterogeneity among a large number of sub-populations. All the above results require to regularize each sub-estimation as though it had the entire sample size. Our general theory applies to the divide-and-conquer approach that is often used to deal with massive homogeneous data. A technical by-product of this paper is the statistical inferences for the general kernel ridge regression. Thorough numerical results are also provided to back up our theory.Comment: 40 pages main text + 40 pages suppl, To appear in Annals of Statistic

Non-asymptotic Optimal Prediction Error for RKHS-based Partially Functional Linear Models

Under the framework of reproducing kernel Hilbert space (RKHS), we consider the penalized least-squares of the partially functional linear models (PFLM), whose predictor contains both functional and traditional multivariate part, and the multivariate part allows a divergent number of parameters. From the non-asymptotic point of view, we focus on the rate-optimal upper and lower bounds of the prediction error. An exact upper bound for the excess prediction risk is shown in a non-asymptotic form under a more general assumption known as the effective dimension to the model, by which we also show the prediction consistency when the number of multivariate covariates $p$ slightly increases with the sample size $n$. Our new finding implies a trade-off between the number of non-functional predictors and the effective dimension of the kernel principal components to ensure the prediction consistency in the increasing-dimensional setting. The analysis in our proof hinges on the spectral condition of the sandwich operator of the covariance operator and the reproducing kernel, and on the concentration inequalities for the random elements in Hilbert space. Finally, we derive the non-asymptotic minimax lower bound under the regularity assumption of Kullback-Leibler divergence of the models.Comment: 24 page

Explicit connections between longitudinal data analysis and kernel machines

Two areas of research – longitudinal data analysis and kernel machines – have large, but mostly distinct, literatures. This article shows explicitly that both fields have much in common with each other. In particular, many popular longitudinal data fitting procedures are special types of kernel machines. These connections have the potential to provide fruitful cross-fertilization between longitudinal data analytic and kernel machine methodology. © 2009, Institute of Mathematical Statistics. All rights reserved