A standard approach to model reduction of large-scale higher-order linear dynamical systems is to rewrite the system as an equivalent first-order system and then employ Krylov-subspace techniques for model reduction of first-order systems. This paper presents some results about the structure of the block-Krylov subspaces induced by the matrices of such equivalent first-order formulations of higher-order systems. Two general classes of matrices, which exhibit the key structures of the matrices of first-order formulations of higher-order systems, are introduced. It is proved that for both classes, the block-Krylov subspaces induced by the matrices in these classes can be viewed as multiple copies of certain subspaces of the state space of the original higher-order system.

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## Scholarly Works (127 results)

A major limitation on performing detailed behavioral analysis of honey bee colonies is that there is currently no efficient way to carry it out. Due to the time required in manually analyzing the data, the current approach and small sample sizes limit the statistical power of these analyses. An automated system can provide a breakthrough in the way this research is performed. Waggle dances are an important aspect of understanding the behavior of honey bees as it serves as a way to communicate among themselves. In this thesis, we develop an automated system using computer vision and learning techniques to solve two problems i) Single bee tracking and waggle detection and ii) Multiple bee waggle detection. Our approach shows that it is possible to train learning algorithms to detect when and where a waggle happens in the hive.

The MindReader web app is an online freely accessible version of the “matching pennies” game. Matching pennies is an old game mentioned in the works of Edgar Allen Poe and analyzed by Claude Shannon, but our implementation is very modern. By visiting, “www.mindreaderpro2.appspot.com”, a user can instantly play on their computer, tablet, or phone and can save their game results by signing into Facebook. The website is part of the larger MindReader project, which collects game data from the app and provides a data analysis platform. We present the inner workings of the project so that academic researchers can easily run and contribute to MindReader.

In this dissertation, we explore two fundamental sets of inference problems arising in machine learning and statistics. We present robust, efficient, and straightforward algorithms for both, adapting sensitively to structure in data by viewing these problems as playing games against an adversary representing our uncertainty.

In the problem of classification, there is typically much more unlabeled data than labeled data, but classification algorithms are largely designed to be supervised, only taking advantage of labeled data. We explore how to aggregate the predictions of an ensemble of such classifiers as accurately as possible in a semi-supervised setting, using both types of data. The insight is to formulate the learning problem as playing a game over the unlabeled data against an adversary, who plays the unknown true labels. This formulation uses unlabeled data to improve performance over labeled data alone in an extremely general and efficient way, without model assumptions or tuning parameters. We demonstrate this by devising and evaluating a number of practical, scalable semi-supervised learning algorithms. The theoretical contributions include a proof that the optimal aggregation rules in this semi-supervised setting are artificial neurons for many natural loss functions, with efficient convex algorithms for learning them.

We also provide fundamental results for a second set of problems relating to sequential learning and testing. Random variation in such situations can typically be described by a martingale, a generalization of a random walk that describes any repeated fair game. We describe the concentration behavior of a martingale's sample path, extending to finite times the law of the iterated logarithm, a classic result of probability theory. With this powerful tool, we are able to show how to design simple sequential tests that use as few samples as possible to detect an effect, provably adapting to the unknown effect size. We also apply our results to optimally correct the p-values of many common statistical hypothesis tests, making them robust to the common practice of 'peeking' at test results and waiting for a significant one to report.

This dissertation consists of three papers on a central topic: the application of scale type theory to the Rasch and 2PL models. Each paper uses a different framework or set of frameworks for defining a typology of scales.

In the first paper, I begin the analysis of scale types through the Stevens (1946) typology. I introduce the notion of difference scales using a formalization of this typology. In applying this formalization to the Rasch and 2PL models, I discuss alternate paradigms for conceptualizing the restrictive assumptions of the Rasch model.

In the second paper, I apply the Suppes (1958) typology to the two IRT models. The conclusions here echo and reinforce those of the first chapter. In the second half of this paper, I examine other perspectives on connecting scale type theory and the Rasch model, focusing on the case of additive conjoint measurement theory.

The first two papers consider the scale types of the odds and logit forms of the models separately. In the third paper, I look at typologies in which the model form is not a factor. The first of these is the axiomatic system of Hölder (1901), which identifies two types of quantities: magnitudes, and points on a line. The second is the homogeneity-uniqueness typology of Narens and Luce (1986), which classifies scales by the types of choices that are possible during the process of numeral assignment.

Together, the three papers form an argument for considering psychometric properties behaving as predicted by the Rasch model as having the scale type of ratio scale attributes, while those that behave as predicted by the 2PL model have the scale type of interval scale attributes.