High-Dimensional Gaussian and Generalized Linear Mixed Models
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1 Research Collection Doctoral Thesis High-Dimensional Gaussian and Generalized Linear Mixed Models Author(s): Schelldorfer, Jürg Publication Date: 2011 Permanent Link: Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library
2 Diss. ETH No High-Dimensional Gaussian and Generalized Linear Mixed Models A dissertation submitted to ETH ZURICH for the degree of Doctor of Sciences presented by JÜRG SCHELLDORFER Dipl. Math. ETH born September 11, 1981 citizen of Bauma ZH accepted on the recommendation of Prof. Dr. Peter Bühlmann, examiner Prof. Dr. Sara van de Geer, co-examiner 2011
3 Abstract Nowadays, statisticians are faced with new challenges due to the availability of high-dimensional data sets in many scientific fields such as biology and information technology. To be more precise, highdimensional means that we consider situations where the number of observations is much smaller than the number of unknown model parameters. In the simplest example of a linear model, the ordinary least squares estimator is ill-posed in the high-dimensional setup. It is therefore necessary to make additional assumptions in order to deal with this problem. One common approach is based on a sparsity scenario. In other words, it is believed that only a few variables (out of hundreds or thousands) are relevant and hence only a small number of the unknown model parameters is indeed different from zero. Under this sparsity assumption, high-dimensional linear regression problems have been extensively studied over the past years. There has been substantial interest in Lasso-type estimators. The Lasso enforces sparsity by setting some model coefficients exactly to zero. In nearly all these works parameter estimation involves examining convex optimization problems. And therein convexity is a crucial tool for establishing nice statistical theory as well as for developing fast computational algorithms. However, in this thesis, we study a specific class of models involving regularized non-convex optimization problems. We specialise in high-dimensional data which are heterogenous in the sense that the observations are not independent but grouped. More precisely, we assume that observations between units are independent, but measurements within each unit are dependent. This kind of data structure leads to mixed-effects models. The purpose of the present work is to
4 x Abstract study theoretical aspects and to develop efficient computational methods for l 1 -regularized high-dimensional generalized linear mixed models. In the first part of the thesis, included in Chapter 2, we address the issue of high-dimensional Gaussian mixed-effects models. We present theoretical results about our proposed l 1 -penalized maximum likelihood estimator as well as an efficient algorithm for fitting truly high-dimensional Gaussian mixed-effects models. Finally, we apply the proposed methodology to a real data set and thereby we can illustrate the attractiveness of high-dimensional (Gaussian) mixed-effects models. In the second part of the thesis, included in Chapter 3, we consider high-dimensional generalized linear mixed models. We focus on the computational aspects and suggest an algorithm for performing variable selection and parameter estimation. We will see that there are some additional challenges compared to Gaussian mixed-effects models.
5 Zusammenfassung Aufgrund der Verfügbarkeit von hoch-dimensionalen Daten in vielen wissenschaftlichen Bereichen wie Biologie und Informationstechnologie sind Statistiker mit neuen Herausforderungen konfrontiert. Präziser ausgedrückt bedeutet hoch-dimensional, dass wir Situationen betrachten wo die Anzahl der Beobachtungen viel kleiner ist als die Anzahl der unbekannten Modellparameter. Am einfachsten Beispiel eines linearen Modells ist der Kleinste Quadrate Schätzer im hoch-dimensionalen Fall unterbestimmt. Um mit diesem Problem umzugehen müssen deshalb zusätzliche Annahmen gemacht werden. Ein weit verbreiteter Ansatz basiert auf der Annahme von dünn besetzten Schätzern. Es wird dabei angenommen, dass nur wenige Variablen (aus Hunderten oder Tausenden) relevant sind und somit nur eine kleine Anzahl der unbekannten Modellparameter in der Tat von Null verschieden ist. Während den letzten Jahren wurden hoch-dimensionale lineare Regressionsprobleme unter der Annahme der Dünnbesetztheit ausführlich untersucht. Es besteht ein grosses Interesse an Lasso-artigen Schätzern. Der Lasso Schätzer erzwingt die Dünnbesetztheit, indem einige Koeffizienten exakt Null geschätzt werden. In vielen Arbeiten führt das Schätzproblem auf die Untersuchung konvexer Optimierungsprobleme. Dabei ist die Konvexität ein zentrales Element sowohl für die Ausarbeitung schöner statistischer Theorie als auch für die Entwicklung von schnellen Algorithmen. In dieser Arbeit studieren wir hingegen eine spezifische Klasse von Modellen, welche auf das Studium von regularisierten nichtkonvexen Optimisierungsproblemen führt. Wir spezialisieren uns auf hoch-dimensionale Daten, welche in dem Sinne heterogen sind, dass die Beobachtungen nicht unabhängig, sondern
6 xii Zusammenfassung gruppiert sind. Genauer gesagt nehmen wir an, dass die Beobachtungen zwischen verschiedenen Einheiten unabhängig, aber zwischen den Beobachtungen derselben Einheit abhängig sind. Diese Art von Daten führt uns zu gemischten Modellen. Das Ziel der vorliegenden Arbeit ist das Studium von theoretischen Aspekten und die Entwicklung eines effizienten Algorithmus für l 1 -regularisierte, hoch-dimensionale verallgemeinerte lineare gemischte Modelle. Im ersten Teil dieser Arbeit, bestehend aus Kapitel 2, studieren wir hoch-dimensionale Gauss sche gemischte Modelle. Wir präsentieren theoretische Resultate für unseren vorgeschlagenen l 1 -bestraften Maximum Likelihood Schätzer als auch einen effizienten Algorithmus zum Anpassen realer hoch-dimensionaler Gauss scher gemischter Modelle. Schliesslich wenden wir die vorgeschlagene Methodologie auf einen realen Datensatz an und illustrieren dabei die Attraktivität von hoch-dimensionalen (Gauss schen) gemischten Modellen. Im zweiten Teil dieser Arbeit, bestehend aus Kapitel 3, betrachten wir hoch-dimensionale verallgemeinerte lineare gemischte Modelle. Wir fokussieren uns auf numerische Aspekte und schlagen einen Algorithmus zur Variablenselektion und zur Schätzung der Parameter vor. Wir werden sehen, dass es verglichen mit Gauss schen gemischten Modellen zusätzliche Herausforderungen gibt.
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