Optimal two-stage procedures for estimating location and size of maximum of multivariate regression function

We propose a two-stage procedure for estimating the location \bolds{\mu} and size M of the maximum of a smooth d-variate regression function f. In the first stage, a preliminary estimator of \bolds{\mu} obtained from a standard nonparametric smoothing method is used. At the second stage, we "zoom-in" near the vicinity of the preliminary estimator and make further observations at some design points in that vicinity. We fit an appropriate polynomial regression model to estimate the location and size of the maximum. We establish that, under suitable smoothness conditions and appropriate choice of the zooming, the second stage estimators have better convergence rates than the corresponding first stage estimators of \bolds{\mu} and M. More specifically, for $\alpha$-smooth regression functions, the optimal nonparametric rates $n^{-(\alpha-1)/(2\alpha+d)}$ and $n^{-\alpha/(2\alpha+d)}$ at the first stage can be improved to $n^{-(\alpha-1)/(2\alpha)}$ and $n^{-1/2}$, respectively, for $\alpha>1+\sqrt{1+d/2}$. These rates are optimal in the class of all possible sequential estimators. Interestingly, the two-stage procedure resolves "the curse of the dimensionality" problem to some extent, as the dimension d does not control the second stage convergence rates, provided that the function class is sufficiently smooth. We consider a multi-stage generalization of our procedure that attains the optimal rate for any smoothness level $\alpha>2$ starting with a preliminary estimator with any power-law rate at the first stage.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1053 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Estimating the minimizer and the minimum value of a regression function under passive design

We propose a new method for estimating the minimizer $\boldsymbol{x}^*$ and the minimum value $f^*$ of a smooth and strongly convex regression function $f$ from the observations contaminated by random noise. Our estimator $\boldsymbol{z}_n$ of the minimizer $\boldsymbol{x}^*$ is based on a version of the projected gradient descent with the gradient estimated by a regularized local polynomial algorithm. Next, we propose a two-stage procedure for estimation of the minimum value $f^*$ of regression function $f$. At the first stage, we construct an accurate enough estimator of $\boldsymbol{x}^*$, which can be, for example, $\boldsymbol{z}_n$. At the second stage, we estimate the function value at the point obtained in the first stage using a rate optimal nonparametric procedure. We derive non-asymptotic upper bounds for the quadratic risk and optimization error of $\boldsymbol{z}_n$, and for the risk of estimating $f^*$. We establish minimax lower bounds showing that, under certain choice of parameters, the proposed algorithms achieve the minimax optimal rates of convergence on the class of smooth and strongly convex functions.Comment: 35 page

Estimation in Dirichlet random effects models

We develop a new Gibbs sampler for a linear mixed model with a Dirichlet process random effect term, which is easily extended to a generalized linear mixed model with a probit link function. Our Gibbs sampler exploits the properties of the multinomial and Dirichlet distributions, and is shown to be an improvement, in terms of operator norm and efficiency, over other commonly used MCMC algorithms. We also investigate methods for the estimation of the precision parameter of the Dirichlet process, finding that maximum likelihood may not be desirable, but a posterior mode is a reasonable approach. Examples are given to show how these models perform on real data. Our results complement both the theoretical basis of the Dirichlet process nonparametric prior and the computational work that has been done to date.Comment: Published in at http://dx.doi.org/10.1214/09-AOS731 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fighting the curse of sparsity: probabilistic sensitivity measures from cumulative distribution functions

Quantitative models support investigators in several risk analysis applications. The calculation of sensitivity measures is an integral part of this analysis. However, it becomes a computationally challenging task, especially when the number of model inputs is large and the model output is spread over orders of magnitude. We introduce and test a new method for the estimation of global sensitivity measures. The new method relies on the intuition of exploiting the empirical cumulative distribution function of the simulator output. This choice allows the estimators of global sensitivity measures to be based on numbers between 0 and 1, thus fighting the curse of sparsity. For density-based sensitivity measures, we devise an approach based on moving averages that bypasses kernel-density estimation. We compare the new method to approaches for calculating popular risk analysis global sensitivity measures as well as to approaches for computing dependence measures gathering increasing interest in the machine learning and statistics literature (the Hilbert–Schmidt independence criterion and distance covariance). The comparison involves also the number of operations needed to obtain the estimates, an aspect often neglected in global sensitivity studies. We let the estimators undergo several tests, first with the wing-weight test case, then with a computationally challenging code with up to k = 30, 000 inputs, and finally with the traditional Level E benchmark code

PREDICTION OF RESPIRATORY MOTION

Radiation therapy is a cancer treatment method that employs high-energy radiation beams to destroy cancer cells by damaging the ability of these cells to reproduce. Thoracic and abdominal tumors may change their positions during respiration by as much as three centimeters during radiation treatment. The prediction of respiratory motion has become an important research area because respiratory motion severely affects precise radiation dose delivery. This study describes recent radiotherapy technologies including tools for measuring target position during radiotherapy and tracking-based delivery systems. In the first part of our study we review three prediction approaches of respiratory motion, i.e., model-based methods, model-free heuristic learning algorithms, and hybrid methods. In the second part of our work we propose respiratory motion estimation with hybrid implementation of extended Kalman filter. The proposed method uses the recurrent neural network as the role of the predictor and the extended Kalman filter as the role of the corrector. In the third part of our work we further extend our research work to present customized prediction of respiratory motion with clustering from multiple patient interactions. For the customized prediction we construct the clustering based on breathing patterns of multiple patients using the feature selection metrics that are composed of a variety of breathing features. In the fourth part of our work we retrospectively categorize breathing data into several classes and propose a new approach to detect irregular breathing patterns using neural networks. We have evaluated the proposed new algorithm by comparing the prediction overshoot and the tracking estimation value. The experimental results of 448 patients’ breathing patterns validated the proposed irregular breathing classifier

Nonlinear Structural Functional Models

A common objective in functional data analyses is the registration of data curves and estimation of the locations of their salient structures, such as spikes or local extrema. Existing methods separate curve modeling and structure estimation into disjoint steps, optimize different criteria for estimation, or recast the problem into the testing framework. Moreover, curve registration is often implemented in a pre-processing step. The aim of this dissertation is to ameliorate the shortcomings of existing methods through the development of unified nonlinear modeling procedures for the analysis of structural functional data. A general model-based framework is proposed to unify registration and estimation of curves and their structures. In particular, this work focuses on three specific research problems. First, a Sparse Semiparametric Nonlinear Model (SSNM) is proposed to jointly register curves, perform model selection, and estimate the features of sparsely-structured functional data. The SSNM is fitted to chromatographic data from a study of the composition of Chinese rhubarb. Next, the SSNM is extended to the nonlinear mixed effects setting to enable the comparison of sparse structures across group-averaged curves. The model is utilized to compare compositions of medicinal herbs collected from two groups of production sites. Finally, a Piecewise Monotonic B-spline Model (PMBM) is proposed to estimate the locations of local extrema in a curve. The PMBM is applied to MRI data from a study of gray matter growth in the brain

Population size estimation via alternative parametrizations for Poisson mixture models

We exploit a suitable moment-based reparametrization of the Poisson mixtures distributions for developing classical and Bayesian inference for the unknown size of a finite population in the presence of count data. Here we put particular emphasis on suitable mappings between ordinary moments and recurrence coefficients that will allow us to implement standard maximization routines and MCMC routines in a more convenient parameter space. We assess the comparative performance of our approach in real data applications and in a simulation study