Com- putational Subset Model Selection Algorithms and Applications

This dissertation develops new computationally e±cient algorithms for identifying the subset of variables that minimizes any desired information criteria in model selection. In recent years, the statistical literature has placed more and more empha- sis on information theoretic model selection criteria. A model selection crite- rion chooses model that \closely approximates the true underlying model. Recent years have also seen many exciting developments in the model se- lection techniques. As demand increases for data mining of massive data sets with many variables, the demand for model selection techniques are be- coming much stronger and needed. To this end, we introduce a new Implicit Enumeration (IE) algorithm and a hybridized IE with the Genetic Algorithm (GA) in this dissertation. The proposed Implicit Enumeration algorithm is the ¯rst algorithm that explicitly uses an information criterion as the objective function. The algo- rithm works with a variety of information criteria including some for which the existing branch and bound algorithms developed by Furnival and Wil- son (1974) and Gatu and Kontoghiorghies (2003) are not applicable. It also ¯nds the \best subset model directly without the need of ¯nding the \best subset of each size as the branch and bound techniques do. The proposed methods are demonstrated in multiple, multivariate, logis- tic regression and discriminant analysis problems. The implicit enumeration algorithm converged to the optimal solution on real and simulated data sets v with up to 80 predictors, thus having 280 = 1; 208; 925; 819; 614; 630; 000; 000; 000 possible subset models in the model portfolio. To our knowledge, none of the existing exact algorithms have the capability of optimally solving such problems of this size

Elliptic genera from multi-centers

I show how elliptic genera for various Calabi-Yau threefolds may be understood from supergravity localization using the quantization of the phase space of certain multi-center configurations. I present a simple procedure that allows for the enumeration of all multi-center configurations contributing to the polar sector of the elliptic genera\textemdash explicitly verifying this in the cases of the quintic in $\mathbb{P}^4$, the sextic in $\mathbb{WP}_{(2,1,1,1,1)}$, the octic in $\mathbb{WP}_{(4,1,1,1,1)}$ and the dectic in $\mathbb{WP}_{(5,2,1,1,1)}$. With an input of the corresponding `single-center' indices (Donaldson-Thomas invariants), the polar terms have been known to determine the elliptic genera completely. I argue that this multi-center approach to the low-lying spectrum of the elliptic genera is a stepping stone towards an understanding of the exact microscopic states that contribute to supersymmetric single center black hole entropy in $\mathcal{N}=2$ supergravity.Comment: 30+1 pages, Published Versio

Identification of Interaction Patterns and Classification with Applications to Microarray Data

Emerging patterns represent a class of interaction structures which has been recently proposed as a tool in data mining. In this paper, a new and more general definition refering to underlying probabilities is proposed. The defined interaction patterns carry information about the relevance of combinations of variables for distinguishing between classes. Since they are formally quite similar to the leaves of a classification tree, we propose a fast and simple method which is based on the CART algorithm to find the corresponding empirical patterns in data sets. In simulations, it can be shown that the method is quite effective in identifying patterns. In addition, the detected patterns can be used to define new variables for classification. Thus, we propose a simple scheme to use the patterns to improve the performance of classification procedures. The method may also be seen as a scheme to improve the performance of CARTs concerning the identification of interaction patterns as well as the accuracy of prediction

An Output-sensitive Algorithm for Computing Projections of Resultant Polytopes

We develop an incremental algorithm to compute the Newton polytope of the resultant, aka resultant polytope, or its projection along a given direction. The resultant is fundamental in algebraic elimination and in implicitization of parametric hypersurfaces. Our algorithm exactly computes vertex- and halfspace-representations of the desired polytope using an oracle producing resultant vertices in a given direction. It is output-sensitive as it uses one oracle call per vertex. We overcome the bottleneck of determinantal predicates by hashing, thus accelerating execution from $18$ to $100$ times. We implement our algorithm using the experimental CGAL package {\tt triangulation}. A variant of the algorithm computes successively tighter inner and outer approximations: when these polytopes have, respectively, 90\% and 105\% of the true volume, runtime is reduced up to $25$ times. Our method computes instances of $5$-, $6$- or $7$-dimensional polytopes with $35$K, $23$K or $500$ vertices, resp., within $2$hr. Compared to tropical geometry software, ours is faster up to dimension $5$ or $6$, and competitive in higher dimensions

Storage management in Ada. Three reports. Volume 1: Storage management in Ada as a risk to the development of reliable software. Volume 2: Relevant aspects of language. Volume 3: Requirements of the language versus manifestations of current implementations

The risk to the development of program reliability is derived from the use of a new language and from the potential use of new storage management techniques. With Ada and associated support software, there is a lack of established guidelines and procedures, drawn from experience and common usage, which assume reliable behavior. The risk is identified and clarified. In order to provide a framework for future consideration of dynamic storage management on Ada, a description of the relevant aspects of the language is presented in two sections: Program data sources, and declaration and allocation in Ada. Storage-management characteristics of the Ada language and storage-management characteristics of Ada implementations are differentiated. Terms that are used are defined in a narrow and precise sense. The storage-management implications of the Ada language are described. The storage-management options available to the Ada implementor and the implications of the implementor's choice for the Ada programmer are also described