Probability and Computing: Randomized Algorithms and Probabilistic Analysis

Randomized self-routing algorithms for Clos networks

Random projection (RP) is a common technique for dimensionality reduction under L2 norm for which many significant space embedding results have been demonstrated. In particular, random projection techniques could yield sharp results for Rd under L2 norm in time linear to the number of data points and dimensionality in question only. Inspired by the use of symmetric probability distributions in previous work, we propose a RP algorithm based on the hyper- spherical symmetry and give its probabilistic analyses based on beta and Gaussian distribution. Later, we present evaluations of our algorithm with other RP algorithms as well as the singular value decomposition (SVD). In particular, we benchmark our results via cosine similarity and L2 norm on a image collection.

Probabilistic Look-ahead Contingency Analysis Integration with Commercial Tool and Practical Data

Predictive demand and collapse fragility functions are two essential components of the probabilistic seismic demand analysis that are commonly developed based on statistics with enormous, costly and time consuming data gathering. Although this approach might be justified for research purposes, it is not appealing for practical applications because of its computational cost. Thus, in this paper, Bayesian regression-based demand and collapse models are proposed to eliminate the need of timeconsuming analyses. The demand model developed in the form of linear equation predicts overall maximum inter-story drift of the lowto mid-rise regular steel moment resisting frames (SMRFs), while the collapse model mathematically expressed by lognormal cumulative distribution function provides collapse occurrence probability for a given spectral acceleration at the fundamental period of the structure. Next, as an application, the proposed demand and collapse functions are implemented in a seismic fragility analysis to develop fragility and consequently seismic demand curves of three example buildings. The accuracy provided by utilization of the proposed models, with considering computation reduction, are compared with those directly obtained from Incremental Dynamic analysis, which is a computer-intensive procedure.

Introduction : To perform a cost-effectiveness analysis of follicular lymphoma maintenance treatment after first-line induction with rituximab compared to observation in Portugal. Methods : A Markov follicular lymphoma model was developed with four health states: progression free survival on first and second lines, progression and death. The transition probabilities between health states were obtained from the PRIMA and EORTC20981 clinical trials and the health states utilities from the literature. The analysis considered the Portuguese National Health Service perspective and includes only direct costs. Resource consumption was estimated by a Portuguese clinical expert panel. The unit costs (€ in 2014) were estimated using Portuguese official databases. Deterministic and probabilistic analyses were performed. Results : In the base case, for a time horizon of 10 years, the cost per life year gained and per quality-adjusted life-year were € 10,634 and € 10,664, respectively. The sensitivity analyses confirmed the stability of the base case for time horizons of 20 and 30 years, varying between € 7,430 and € 7,155. Maintenance with rituximab is cost-effective for a time horizon of 5.5 years (cost per quality-adjusted life-year gained of € 23,616). Conclusion : According to the present model rituximab maintenance treatment of follicular lymphoma patients that respond to first line induction therapy in comparison with observation is cost-effective in Portugal.

The present thesis studies some important random walk-based algorithms, which are randomized rumor spreading and balanced allocation protocols on networks. In the rst part of the thesis, we study the Push and the Push-Pull protocols introduced by [DGG+87], which are basic randomized protocols for information dissemination on networks. In Chapter 2, we propose a new model where the number of calls of each node in every round is chosen independently according to a probability distribution R with bounded mean determined at the beginning of the process. In addition to the model being a natural extension of the standard protocols, it also serves as a more realistic model for rumor spreading in a network whose entities are not completely uniform and may have di erent levels of power. We provide both lower and upper bounds on the rumor spreading time depending on statistical properties of R such as the mean or the variance. While it is well-known that the standard protocols need Θ(log n) rounds to spread a rumor on a complete network with n nodes, we show that, if R follows a power law distribution with exponent β ∈ (2, 3), then the Push-Pull protocol spreads a rumor in Θ(log log n) rounds. Moreover, when β = 3, we show a runtime of Θ ( logn log logn ) . In Chapter 3, we analyze the behavior of the standard Push-Pull protocol on a class of random graphs, called random k-trees for every integer k > 2, that are suitable to model poorly connected, small-world and scale free networks. Here, we show that the Push-Pull protocol propagates a rumor from a randomly chosen informed node to almost all nodes of a random k-tree with n nodes in O((log n)1+ck) rounds with high probability, where 0 3 can be any integer. Let l be a given positive integer. In each round t, 1 6 t 6 n, ball t picks a node of G uniformly at random and performs a nonbacktracking random walk of length l from the chosen node. Then it deterministically selects a subset of the visited nodes as the potential choices and allocates itself on one of the choices with minimum load (ties are broken uniformly at random). Provided G has a su ciently large girth, we establish an upper bound for the maximum number of balls at any bin after allocating n balls by the algorithm. We also show that the upper bound is tight up to a O(log log n) factor. In particular, we show that if we set l = b(log n) 1+ 2 c, for any constant ∈ (0, 1], and G has girth at least ω(l), then the maximum load is bounded by O(1/ ) with high probability.

This research investigated the reliability of a newly designed steel roof truss system of an industrial building to be constructed in one of the major cities in Nigeria. The probabilistic analysis technique was done with the aid of CalREL, a general-purpose structural reliability analysis software program. The longest span truss element (consisting of 73 members), is the most critical in the system, was selected and first analysed using SAP2000 Advanced 12.0.0 finite element analysis (FEA) software program in order to obtain the forces in the steel truss members; this forms part of the inputs required in CalREL. Four load variations (referred to as load ratios in the study) were tested on the selected truss. The strengths of the truss members and other properties were determined as specified in BS 5950-1: 2000. Limit state equations were derived for the calculation of the probability of failure of the individual members of the truss system. A reliability index as a measure of structural performance and related to the probability of failure was developed for all the elements of the truss. The results showed that compression members displayed a noticeable violation of the ultimate limit state requirement, while tension members showed a negligible violation. Sensitivity factors that reflect the relative importance of the individual variables in the design of roof trusses were also presented. The estimated reliability indices also revealed structural members that require immediate redesign; though they appear satisfactory in the level of deterministic design. A probabilistic approach for the reappraisal of new and existing civil structures is well supported by the findings of this investigation.

Error handling in multimodal voice-enabled interfaces of tour-guide robots using graphical models

Measurements provide one with results, in the form of both quantitative estimates of measured quantity along with attributed quantitative probabilistic analysis. Measurement is prescribed precisely in order to enable researchers, experts or other measurers to obtain maximum confidence in its results. In that way, the probability of obtaining unpredicted or unwanted consequences is minimised. Yet, owing to a rather large number of degrees of freedom in a typical measurement sequence, its nonlinear character and nonlinear couplings, in general it is not known in what amount a variation in measurement conditions brings about significantly larger variations in measured quantities or its derivatives. In this article we treat in some details the aforementioned influence of variations and argue about possible results. In order to illustrate the treated influences we present results of a rather simple and common measurement of surface roughness of solid state objects. It is argued that there is no significant augmentation of variations in results of initial measurements throughout measurement sequence.

In mathematics and statistics, the desire to eliminate mathematical tedium and facilitate exploration has lead to computer algebra systems. These computer algebra systems allow students and researchers to perform more of their work at a conceptual level. The design of generic algorithms for tedious computations allows modelers to push current modeling boundaries outward more quickly. Probability theory, with its many theorems and symbolic manipulations of random variables is a discipline in which automation of certain processes is highly practical, functional, and efficient. There are many existing statistical software packages, such as SPSS, SAS, and S-Plus, that have numeric tools for statistical applications. There is a potential for a probability package analogous to these statistical packages for manipulation of random variables. The software package being developed as part of this dissertation, referred to as “A Probability Programming Language” (APPL) is a random variable manipulator and is proposed to fill a technology gap that exists in probability theory. My research involves developing algorithms for the manipulation of discrete random variables. By defining data structures for random variables and writing algorithms for implementing common operations, more interesting and mathematically intractable probability problems can be solved, including those not attempted in undergraduate statistics courses because they were deemed too mechanically arduous. Algorithms for calculating the probability density function of order statistics, transformations, convolutions, products, and minimums/maximums of independent discrete random variables are included in this dissertation.

Multi-criteria decision analysis can be a useful tool in routing out and ranking different alternatives. However, many such analyses involve imprecise information, including estimates of utilities, outcome probabilities and criteria weights. This paper presents a general multi-criteria approach, allowing the modelling of multi-criteria and probabilistic problems in the same tree form, which includes a decision tree evaluation method integrated with a framework for analyzing decision situations under risk with a criteria hierarchy. The general method of probabilistic multi-criteria analysis extends the use of additive and multiplicative utility functions for supporting evaluation of imprecise and uncertain facts. Thus, it relaxes the requirement for precise numerical estimates of utilities, probabilities, and weights. The evaluation is done relative to a set of decision rules, generalizing the concept of admissibility and computationally handled through the optimization of aggregated utility functions. The approach required design and development of computationally intensive algorithms for which there was no template

Observing the environment and recoganizing patterns for the purpose of decision making are fundamental to any scientific enquiry. Pattern recognition is a scientific discipline so much so that it enables perception in machines and also it has applications in diverse technology areas. Among the scientific community, statistical pattern recognition has received considerable attention in recent years. The statistical pattern recognition challenges are mostly approached by Hidden Markov Models (HMMs). A Hidden Markov Model (HMM) is a probabilistic mathematical discrete structure with the state emission probabilities apart from consisting the components of a probabilistic finite state automaton (PFA). Over the years, researches have been carried out to study the relations between HMM and PFA. Probabilistic finite state automata are mathematical models constructed to generate distributions over a set of strings. The computation of the probability of generating a string as a total, and a string with given prefix or suffix have important applications in the field of parsing. In this attempt, the Semi Probabilistic Finite State Automata (Semi-PA), the most general class of Probabilistic Automata is discussed in detail. AMS Classification: 68Q10 and 68Q45.

Most previous research done in probabilistic schedulability analysis assumes a known distribution of execution times for each task of a real-time application. This is however not trivial to determine it with a high level of confidence. Methods based on measurements are often biased since not in general exhaustive on all the possible execution paths, whereas methods based on static analysis are mostly worst-case execution time - WCET-oriented. Using static analysis, this work proposes a method to obtain probabilistic distributions of execution times. It assumes that the given real time application is divided into multiple tasks, whose source code is known. Ignoring in this paper hardware considerations and based only on the source code of the tasks, the proposed technique allows designers to associate to any execution path an execution time and a probability to go through this path. A source code example is presented to illustrate the method.

We introduce a fast approach to classification and clustering applicable to high-dimensional continuous data, based on Bayesian mixture models for which explicit computations are available. This permits us to treat classification and clustering in a single framework, and allows calculation of unobserved class probability. The new classifier is robust to adding noise variables as a drawback of the built-in spike-and-slab structure of the proposed Bayesian model. The usefulness of classification using our method is shown on metabololomic example, and on the Iris data with and without noise variables. Agglomerative hierarchical clustering is used to construct a dendrogram based on the posterior probabilities of particular partitions, to provide a dendrogram with a probabilistic interpretation. An extension to variable selection is proposed which summarises the importance of variables for classification or clustering and has probabilistic interpretation. Having a simple model provides estimation of the model parameters using maximum likelihood and therefore yields a fully automatic algorithm. The new clustering method is applied to metabolomic, microarray, and image data and is studied using simulated data motivated by real datasets. The computational difficulties of the new approach are discussed, solutions for algorithm acceleration are proposed, and the written computer code is briefly analysed. Simulations shows that the quality of the estimated model parameters depends on the parametric distribution assumed for effects, but after fixing the model parameters to reasonable values, the distribution of the effects influences clustering very little. Simulations confirms that the clustering algorithm and the proposed variable selection method is reliable when the model assumptions are wrong. The new approach is compared with the popular Bayesian clustering alternative, MCLUST, fitted on the principal components using two loss functions in which our proposed approach is found to be more efficient in almost every situation.

In this paper, the robust H∞-control problem is reported for a class of uncertain discrete-time fuzzy systems with both multiple probabilistic delays and multiple missing measurements and random missing control from the fuzzy controllers to the actuator. A sequence of random variables including accounting for the probabilistic communication delays and the random missing control are thought as mutually independent and obey the Bernoulli distribution. The measurement-missing phenomenon can be assumed to occur stochastically. Assumption that the missing probability for each sensor satisfies a certain probabilistic distribution in the interval [0 1] is given. Much attention is focused on design of H∞ the fuzzy output feedback controllers to ensure that the resulting close-loop Takagi-Sugeno (T-S) system is exponentially stable in the mean square. The developed method makes disturbance rejection attenuation satisfy a given level by means of the H∞-performance index. Intensive analysis is employed to reach the sufficient conditions about the existence of admissible output feedback controllers which satisfies the exponential stability as well as the prescribed H∞ performance. In addition, the cone-complementarity linearization procedure is utilized to transform the controller-design problem into a sequential minimization one which can be solved by the semi-definite program method. Simulation results conform the feasibility as weil as the effectiveness of the proposed design method.