Speaker: Ted Hill, School of Mathematics, Georgia Tech

Title: Some Surprising Statistical Properties of Deterministic Sequences

Abstract: Following recent base-invariance characterizations of Benford's Law (logarithmic probability distribution of significant digits), and a CLT-like theorem for convergence to Benford, two years ago physicists discovered that certain laboratory experiments and numerical simulations of classical dynamical systems and differential equations also seem to follow Benford's law. Motivated by this new empirical evidence, it has now been shown that the significant digits of a surprisingly general class of deterministic systems have a Benford distribution. This class includes the iterates of almost all rational and exponential functions, and even compositions of these. For example, the significant digits of the orbits O_T(x) = (x, T(x), T(T(x)),...) are logarithmically distributed for Lebesgue almost all x>1, whether T is the function x^2, or 3^x, or x^x, and even for orbits of non-autonomous systems formed by alternating application of these functions in any order. These results generalize the well-known facts that classical deterministic sequences such as the Fibonacci numbers F(n) and (n!) are Benford, and the proofs rely on shadowing techniques, theories of uniform distribution mod 1, and classical limit theorems in probability. This lecture will be aimed for the non-specialist, and will include a number of Benford-related open problems in probability, number theory, differential equations and iterated function systems.

Speaker: Amy R. Ward, ISyE, Georgia Tech

Title: Optimal Control of Assemble-to-Order Systems with Delay Guarantees

Abstract: (Joint work with Erica Plambeck) We consider an assemble-to-order (ATO) system with multiple products and components having guaranteed maximum delivery leadtimes. For such a system, the first-order operational challenge is to balance the demand and supply of components through pricing and supply contracts (or capacity planning). However, due to stochastic fluctuations in supply and demand, component shortages occur. Then, the manufacturer must dynamically allocate scarce components to customer orders for various products, and/ or pay to expedite component production. In practice, many firms adopt simple static rules to sequence customer orders for assembly, such as FIFO, and expedite components on an ad hoc basis. Recently, Dell has begun to prioritize customer orders dynamically, based on component availability. However, dynamic control of an ATO system is difficult, and theory is needed to guide business practice. We undertake a holistic analysis of a high-volume ATO system. Our objective is to maximize expected infinite horizon discounted profit subject to assembling orders within a guaranteed maximum delivery leadtime. Our key assumption is that the inter-arrival time between customers is small compared to guaranteed delivery leadtimes, meaning the ATO system is high-volume. (This is certainly true for Dell; its new OptiPlex assembly plant assembles more than 20,000 computers per day whereas a customer's order is guaranteed to ship within 10 days.) We first establish that any revenue-maximizing strategy requires optimal prices and component production rates to be close. In such a regime, diffusion approximations are extremely accurate. Furthermore, in this regime, the system exhibits a reduction in problem dimensionality. This simplifies the dynamic control problem and allows us to provide easily implementable policies for sequencing customer orders and expediting components that are asymptotically optimal in the heavy traffic limit. We finish by characterizing the structure of near-optimal inventory policies. By taking account of important component dependencies, the near-optimal inventory policy will outperform any base stock policy.

Speaker: Wenjiang Jiang, School of Mathematical Sciences, Yunnan Normal University, China

Title: A Class of Non-Gaussian GARCH type models

Abstract: We propose a class of Non-Gaussian GARCH type models, using two classes of new distributional classes as building blocks. The distributions from the first class have closed-form formulae for probability densities and distribution functions, while the distributions from the second class may have extremely unbalanced tails. More importantly, both classes have closed-form quantile functions, which provides a great deal of convenience in Value-at-Risk (VAR) or Conditional-VaR (CVaR) modelling. Having rich tail behaviors, both classes allow realistic modelling in many fields, for instance, the power price modelling and turbulence modelling. In addition, the resulting GARCH type models are very easy to simulate, which enables us to achieve a reasonably fast speed in both parameter estimation and other relevant calculations in VaR and CVaR models.

Speaker: Mark Borodovsky, School of Biology and Biomedical Engineering, Georgia Tech

Title: Gene Finding and Statistical Models of Partially Observed Systems

Abstract: Within the first decade of the 21st century the collection of complete genomes will be measured in hundreds. Yet, these "books of life" will carry many enigmatic "words" and "sentences". The problem of interpreting DNA was encountered as soon as the first DNA sequence was determined. Through the years many algorithms for DNA sequence interpretation and particularly gene finding algorithms were developed. I will talk about ab initio gene finding methods. These methods are capable to predict genes not identifiable by similarity search. The ab initio methods GeneMark and GeneMark.hmm were developed at Georgia Tech. They were used for annotation of several prokaryotic and eukaryotic complete genomes. Both algotihms can be interpreted in terms of Hidden Markov model. The GeneMark algorithms produces approximation to a posterior decoding. It identifies a coding potential (an a posteriori probability of carrying protein code) within a rather short sliding window based on inhomogeneous three-periodic Markov models of coding region and homogeneous models of non-coding region. The GeneMark.hmm algorithm is analyzing a whole DNA sequence at once. This Viterbi like algorithm finds the most likely sequence of coding and non-coding regions using explicit hidden Markov model of gene organization. In the ideal case, when the nucleotide sequence has no errors, the GeneMark.hmm algorithm is more accurate than GeneMark in finding the boundaries between coding and non-coding regions, specifically gene starts and exon-intron junctions. However, GeneMark.hmm is more sensitive to sequence errors that may corrupt the predictions in a rather long region. Both GeneMark and GeneMark.hmm can be used for high and low eukaryotes, bacteria and archaea, eukaryotic viruses and phages: http://opal.biology.gatech.edu/GeneMark/

Speaker: Jan Rosinski, University of Tennessee, Knoxville

Title: Continuity of infinitely divisible processes via Poisson processes

Abstract

Speaker: Andrzej Swiech, School of Mathematics, Georgia Tech

Title: Infinite dimensional PDE coming from mathematical finance

Abstract: I will discuss results on infinite dimensional, second order equations of Hamilton-Jacobi-Bellman type related to option pricing in the Musiela model of interest rates. This model describes the dynamics of the so called forward rates in terms of evolution of infinite dimensional diffusion processes. I will present a viscosity solution approach to such equations that guarantees the existence of unique solutions and will discuss their properties and their connection to the underlying stochastic optimal control/option pricing problem.

Speaker: Anda Gadidov, School of Mathematics, Georgia Tech

Title: Strong Law of Large Numbers for U-statistics

Abstract: Since their introduction into the statistical practice in 1948, U-statistics proved to be useful in a wide range of applications. Many statistics in common use are in fact U-statistics and their relative simple structure makes them ideal for studying general estimations processes like bootstrapping and jackknifing. Unlike the case of sums of independent identically distributed random variables, the finiteness of the first moment is not necessary for the strong law of large numbers for U-statistics. We present some results on the strong law of large numbers from a historical perspective, tools and techniques used, and recent developments.

Speaker: William P. McCormick, Department of Statistics, University of Georgia

Title: Asymptotic Expansions Of Convolutions Of D.F.'s With Regularly Varying Tails

Abstract: In this talk I will obtain second-order tail area approximations for sums of independent random variables having d.f with a regularly varying tail plus a little additional regularity. When further smoothness is imposed on the d.f. of the summands, a higher order expansion is possible. The number of terms in the expansion is dependent on how many moments are finite. An interesting approach through linear algebra is used to acheive the higher-order results.

Speaker: Richard Serfozo, Industrial and Systems Engineering , Georgia Tech

Title: Reversible Markov Processes on General Spaces: Spatial Birth-Death and Queueing Processes

Abstract: Thus study describes the stationary distributions of spatial birth-death and queueing processes that represent systems in which discrete units (customers, particles) move about in an Euclidean or partially ordered space where they are processed. These are reversible Markov jump process with uncountable state spaces (sets of "finite" counting measures). Reversible Markov processes on countable state spaces, introduced by Kolmogorov, have the exceptional property that their stationary distributions have a canonical form: a simple ratio of products of transition rates. We present an analogue of this result for uncountable state spaces. This involves representing two-way communication by certain Radon-Nikodym derivatives for measures on product spaces. Included is a Kolmogorov criterion that establishes the reversibility in the same spirit as one studies -irreducible Markov jump processes (Meyn and Tweedie (1993). Stationary Markov Chains and Stochastic Stability). Related references for "infinite" birth-death processes are: Lopes Garcia, N. (1995). Birth and death processes as projections of higher-dimensional Poisson processes. Adv. in Appl. Probab. Glotzl, E. (1981). Time reversible and Gibbsian point processes. I. Markovian spatial birth and death processes on a general phase space. Math. Nachr.

Speaker: Patricia Reynaud-Bouret, École normale supérieure, Paris and School of Mathematics, Georgia Tech

Title: Adaptive Estimation of the Intensity of an Inhomogeneous Poisson Process by Model Selection

Abstract: We derive oracle type inequalities for penalized projection estimators of the intensity of an inhomogeneous Poisson process. The key tool for this is a concentration inequality for some precise functional of the Poisson process. The oracle inequalities allows us to prove the adaptive properties of the estimator.

Speaker: Stravos Garoufalidis, School of Mathematics, Georgia Tech

Title: A probabilistic view of the Jones polynomial

Abstract: A planar projection of a knot is a directed planar graph, such that each vertex has two incoming edges and two outgoing edges. The vertices are decorated with a plus or a minus sign to indicate the crossing. All this is intuitively obvious. The talk will discuss two invariants of knots, namely the Alexander polynomial (about 75 years old) and the Jones polynomial (about 25 years old) as invariants of random walks on the planar projection of a knot. The main tools are combinatorial formulas for determinants, permanents, and elementary physics tricks. The talk involves no analysis, almost no probability and very little physics.

Speaker: Liang Peng, School of Mathematics, Georgia Tech

Title: Empirical likelihood methods with heavy tails

Abstract: Heavy tailed distributions have recently appeared in numerous applications: teletraffic data modeling, community size estimation, value-at-risk in finance, etc. For the estimation of the tail index, one of the well-known estimator is the so called Hill estimator. One obvious way to construct a confidence interval for this index is via the normal approximation of the Hill estimator. In the first part of this talk we present an empirical likelihood based confidence interval for the tail index. In the second part we propose an empirical likelihood based confidence interval for the mean when the underlying distribution has heavy tails, which includes the case of infinite variance.

Applied Mathematics | CDSNS Seminar | CDSNS/ACELab | Colloquium | Combinatorics | Geometry | Geometry/Topology/Algebra | Nonlinear Science | Research Horizons | Stochastics | Topology |