Browsing by Subject "Probability"
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Item Combinatorial and probabilistic techniques in harmonic analysis(2012-05) Lewko, Mark J., 1983-; Vaaler, Jeffrey D.; Beckner, William; Pavlovic, Natasa; Rodriguez-Villegas, Fernando; Zuckerman, DavidWe prove several theorems in the intersection of harmonic analysis, combinatorics, probability and number theory. In the second section we use combinatorial methods to construct various sets with pathological combinatorial properties. In particular, we answer a question of P. Erdos and V. Sos regarding unions of Sidon sets. In the third section we use incidence bounds and bilinear methods to prove several new endpoint restriction estimates for the Paraboloid over finite fields. In the fourth and fifth sections we study a variational maximal operators associated to orthonormal systems. Here we use probabilistic techniques to construct well-behaved rearrangements and base changes. In the sixth section we apply our variational estimates to a problem in sieve theory. In the seventh section, motivated by applications to sieve theory, we disprove a maximal inequality related to multiplicative characters.Item The effects of payoffs and feedback on the disambiguation of relative clauses(2014-12) Chacartegui Quetglas, Luis; Bannard, ColinThis dissertation investigates two facts about language processing. The Good Enough Approach claims that language users do not form a fully detailed representation of the input unless the task at hand requires it. On the other hand it has been shown that language users display internal preferences when they are faced with ambiguous input, as to what direction disambiguation should take. It has been proposed that these preferences are based on previous experience with similar inputs. This thesis investigates these two issues using tools from the fields of decision making and reinforcement learning. Specifically feedback and payoffs associated with sentence interpretations are manipulated to explore reading behavior, understood as a process of information seeking, and disambiguation choices. In four eye-tracking-reading experiments, the experimental stimuli are sentences containing a relative clause attachment ambiguity. Experiment 1 investigates whether the combination of the degree of ambiguity of a sentence and the possible payoffs, affect people’s reading times for the potentially ambiguous parts of a sentence, as well as their disambiguation choices. Experiment 2 investigates the role of feedback in such processes, a combination related to expected utility maximization. Experiment 3 studies how participants learn from feedback under risky or non-risky conditions. The last experiment investigates whether participants adjust their responses to evidence provided by feedback even overriding their internal initial bias towards a default response.Item Greedy structure learning of Markov Random Fields(2011-08) Johnson, Christopher Carroll; Ravikumar, Pradeep; Dhillon, InderjitProbabilistic graphical models are used in a variety of domains to capture and represent general dependencies in joint probability distributions. In this document we examine the problem of learning the structure of an undirected graphical model, also called a Markov Random Field (MRF), given a set of independent and identically distributed (i.i.d.) samples. Specifically, we introduce an adaptive forward-backward greedy algorithm for learning the structure of a discrete, pairwise MRF given a high dimensional set of i.i.d. samples. The algorithm works by greedily estimating the neighborhood of each node independently through a series of forward and backward steps. By imposing a restricted strong convexity condition on the structure of the learned graph we show that the structure can be fully learned with high probability given $n=\Omega(d\log (p))$ samples where $d$ is the dimension of the graph and $p$ is the number of nodes. This is a significant improvement over existing convex-optimization based algorithms that require a sample complexity of $n=\Omega(d^2\log(p))$ and a stronger irrepresentability condition. We further support these claims with an empirical comparison of the greedy algorithm to node-wise $\ell_1$-regularized logistic regression as well as provide a real data analysis of the greedy algorithm using the Audioscrobbler music listener dataset. The results of this document provide an additional representation of work submitted by A. Jalali, C. Johnson, and P. Ravikumar to NIPS 2011.Item On the Poisson Follower Model(2020-08-14) Dragović, Nataša; Baccelli, F. (François), 1954-; De Veciana, Gustavo; Zitkovic, Gordan; Tran, Ngoc; Taillefumier, Thibaud OThis dissertation presents studies of dynamics over the Poisson point process. In particular, we study a special case of Hegselmann-Krause Dynamics [1] over ℝ². Chapter 1 is a brief introduction to the thesis and its structure. Chapter 2 introduces the notation, the definitions and examples of phenomena of interest. In Chapter 3, we go deeper in analyzing the phenomena described by calculating frequency of these phenomena. A system of quadratic inequalities will be introduced to allow one to calculate the probabilities of the events pertaining to this dynamics using methods from integral geometry. Chapter 4 uses percolation arguments to prove the absence of percolation at step 1. In Chapter 5, we provide geometric results of independent interest pertaining to the Follower Dynamics. In Chapter 6, we discuss the limiting behavior of this process and include some more simulations. In Chapter 7 we propose future steps and discuss more general Hegselmann-Krause Dynamics.Item Quantifying and mitigating wind power variability(2015-12) Niu, Yichuan; Santoso, Surya; Arapostathis, Aristotle; Baldick, Ross; Longoria, Raul G.; Tiwari, MohitUnderstanding variability and unpredictability of wind power is essential for improving power system reliability and energy dispatch in transmission and distribution systems. The research presented herein intends to address a major challenge in managing and utilizing wind energy with mitigated fluctuation and intermittency. Caused by the varying wind speed, power variability can be explained as power imbalances. These imbalances create power surplus or deficiency in respect to the desired demand. To ameliorate the aforementioned issue, the fluctuating wind energy needs to be properly quantified, controlled, and re-distributed to the grid. The first major study in this dissertations is to develop accurate wind turbine models and model reductions to generate wind power time-series in a laboratory time-efficient manner. Reliable wind turbine models can also perform power control events and acquire dynamic responses more realistic to a real-world condition. Therefore, a Type 4 direct-drive wind turbine with power electronic converters has been modeled and designed with detailed aerodynamic and electric parameters based on a given generator. Later, using averaging and approximation techniques for power electronic circuits, the order of the original model is lowered to boost the computational efficiency for simulating long-term wind speed data. To quantify the wind power time-series, efforts are made to enhance adaptability and robustness of the original conditional range metric (CRM) algorithm that has been proposed by literatures for quantitatively assessing the power variability within a certain time frame. The improved CRM performs better under scarce and noisy time-series data with a reduced computational complexity. Rather than using a discrete probability model, the improved method implements a continuous gamma distribution with parameters estimated by the maximum likelihood estimators. With the leverage from the aforementioned work, a wind farm level behavior can be revealed by analyzing the data through long-term simulations using individual wind turbine models. Mitigating the power variability by reserved generation sources is attempted and the generation scenarios are generalized using an unsupervised machine learning algorithm regarding power correlations of those individual wind turbines. A systematic blueprint for reducing intra-hour power variations via coordinating a fast- and a slow- response energy storage systems (ESS) has been proposed. Methods for sizing, coordination control, ESS regulation, and power dispatch schemes are illustrated in detail. Applying the real-world data, these methods have been demonstrated desirable for reducing short-term wind power variability to an expected level.Item “Should I switch?” Controversies created by an advice column(2010-08) Lehman, Sandra Elizabeth; Daniels, Mark L.; Armendáriz, Efraim P.In 1990’s, the circumstances of being a contestant on a popular game show were published in a trendy question and answer column in Parade Magazine. If contestant switched from the initial choice to a second choice offer by the host, would the chances of winning the desired prize be increase? The columnist’s response to the reader sparked a good deal of controversy among mathematicians. Shortly after the publication of this answer, articles appeared in various mathematical publications some supporting and some refuting the columnist’s answer. This document reports the results of research into the controversy generated by some of the probability problems used on Let’s Make a Deal game show. Using a variety of approaches and assumption, the author attempts to formulate mathematical proof to explain the correct answer to the contestant’s question, “Should I switch?”Item Side information, robustness, and self supervision in imitation learning(2021-08-12) Memarian, Farzan; Topcu, Ufuk; Ghattas, Omar; Pingali, Keshav; Mueller, Peter; Niekum, ScottImitation learning refers to a family of learning algorithms enabling the learning agents to learn directly from demonstrations provided by experts, practitioners, and users. While imitation learning methods have been successfully applied to many robot learning and autonomous driving problems, existing imitation learning methods still perform poorly for certain types of problems and suffer from lack of robustness. Most existing imitation learning algorithms are designed to learn purely from demonstrations, however, there are several other sources of information that could be used to improve learning from demonstrations. We identify the following shortcomings in imitation learning algorithms pertaining to their lack of use of other available sources of information: First, most existing imitation learning algorithms are oblivious to potentially existing domain knowledge and side information. Second, they are oblivious to the possibility of using the sparse rewards provided by the environment, which might guide learning and ease the requirement of access to informative demonstrations. And third, they are oblivious to the fact that, in physically embodied applications, reward functions and policies have certain underlying structures, such as Lipschitzness. In order to achieve robust and fast imitation learning, in this dissertation, we propose novel imitation learning algorithms that exploit the above-mentioned sources of information and priors which have been ignored in existing imitation learning algorithms.