TY - BOOK

T1 - Model Selection and Risk Estimation with Applications to Nonlinear Ordinary Differential Equation Systems

AU - Mikkelsen, Frederik Vissing

PY - 2017

Y1 - 2017

N2 - Broadly speaking, this thesis is devoted to model selection applied to ordinary dierentialequations and risk estimation under model selection. A model selection framework was developedfor modelling time course data by ordinary dierential equations. The frameworkis accompanied by the R software package, episode. This package incorporates a collectionof sparsity inducing penalties into two types of loss functions: a squared loss function relyingon numerically solving the equations and an approximate loss function based on inversecollocation methods. The goal of this framework is to provide eective computational toolsfor estimating unknown structures in dynamical systems, such as gene regulatory networks,which may be used to predict downstream eects of interventions in the system. A recommendedalgorithm based on the computational tools is presented and thoroughly tested invarious simulation studies and applications.The second part of the thesis also concerns model selection, but focuses on risk estimation,i.e., estimating the error of mean estimators involving model selection. An extension of Stein'sunbiased risk estimate (SURE), which applies to a class of estimators with model selection,is developed. The extension relies on studying the degrees of freedom of the estimator, whichfor a broad class of estimators decomposes into two terms: one ignoring the selection stepand one correcting for it. The classic SURE assumes that the estimator in question is almostdierentiable and it therefore only accounts for the rst term of the decomposition. In order toaccount for the second term the continuum of models arising when the selection procedure hasa tuning parameter is studied. By exploiting the duality between varying the tuning parameterfor xed observations and perturbing the observations for xed tuning parameter, an identityis derived for a class of estimators which support the extension of SURE. The resultingcorrected version of SURE is generally fast to compute and for the lasso-OLS estimator itshows promising results when compared to risk estimation via cross validation.

AB - Broadly speaking, this thesis is devoted to model selection applied to ordinary dierentialequations and risk estimation under model selection. A model selection framework was developedfor modelling time course data by ordinary dierential equations. The frameworkis accompanied by the R software package, episode. This package incorporates a collectionof sparsity inducing penalties into two types of loss functions: a squared loss function relyingon numerically solving the equations and an approximate loss function based on inversecollocation methods. The goal of this framework is to provide eective computational toolsfor estimating unknown structures in dynamical systems, such as gene regulatory networks,which may be used to predict downstream eects of interventions in the system. A recommendedalgorithm based on the computational tools is presented and thoroughly tested invarious simulation studies and applications.The second part of the thesis also concerns model selection, but focuses on risk estimation,i.e., estimating the error of mean estimators involving model selection. An extension of Stein'sunbiased risk estimate (SURE), which applies to a class of estimators with model selection,is developed. The extension relies on studying the degrees of freedom of the estimator, whichfor a broad class of estimators decomposes into two terms: one ignoring the selection stepand one correcting for it. The classic SURE assumes that the estimator in question is almostdierentiable and it therefore only accounts for the rst term of the decomposition. In order toaccount for the second term the continuum of models arising when the selection procedure hasa tuning parameter is studied. By exploiting the duality between varying the tuning parameterfor xed observations and perturbing the observations for xed tuning parameter, an identityis derived for a class of estimators which support the extension of SURE. The resultingcorrected version of SURE is generally fast to compute and for the lasso-OLS estimator itshows promising results when compared to risk estimation via cross validation.

UR - https://soeg.kb.dk/permalink/45KBDK_KGL/fbp0ps/alma99121918119405763

M3 - Ph.D. thesis

BT - Model Selection and Risk Estimation with Applications to Nonlinear Ordinary Differential Equation Systems

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -