Seminars 2011

whose coefficients are time-independent, and is easily parallelizable. Applications include solving integro-differential equations backward parabolic problems, and option pricing. The numerical schemes proposed and analyzed in these papers are based fundamentally on the line of thoughts from the earlier work of D. Sheen, I. Sloan and V. Thomee (2000, 2003).
However, the method has been doubtful to be applicable to any nonlinear or linear problems whose coefficients are time-dependent. The reason is that the Laplace transform of two time-dependent functions leads to a convolution of the Laplace transformed functions in the dual variable. In this paper, we propose a method of Laplace transformation to linear parabolic problems with time-dependent coefficients, which is as efficient as the method for parabolic problems time-independent coefficients. Several numerical results are provided, which support the efficiency of the proposed scheme. This is a joint work with Hyoseop Lee (Bell Labs Seoul) and Jinwoo Lee (Kwangwoon University).

Extracting a few tuples from the database to support multi-criteria decision making is an important functionality for database systems. This is important in many application domains where the end-users are more interested in the most important query answers in the potentially massive answer space. In this setting, we think of the database as a large set of multi-dimensional points stored in a disk and want to help the users to quickly find the most desired point. For example, the user might want to find a hotel with the best price-rating combination out of hundreds of hotels or a car that has a high miles per gallon and a high horse power. In this talk, I will present several results aiming towards an efficient way to support multi-criteria decision making. 

1) Skyline Computation: The skyline query is one classic approach to assist the users. Its computation has been extensively studied earlier. In this talk, I will present an algorithm that is significantly faster than previous algorithms both theoretically and experimentally. The insight is to solve the problem on the streaming model which helps emphasize the benefits of sequential access over random disk access. The result can be extended to the distributed setting and is complemented with a lower bound.

2) Beyond Skyline: Skyline and its sibling, Top-k, are known to have some drawbacks and there have been several approaches to deal with these drawbacks recently. I will present a few new directions towards this goal. One important direction is the notion of minimum regret ratio which is used to identify the output quality in terms of ``user satisfaction''. Using a simple geometric analysis, it can be shown that this seemingly too ambitious goal of satisfying every user is, in fact, possible. I will also present approaches that help making the process more efficient by interacting with users called interactive regret minimization and by assuming some distributions called threshold-based preference distributions.

This talk will be self-contained (no database background will be assumed). This talk is based on joint works with Atish Das Sarma, Ashwin Lall, Kazuhisa Makino, Jun Xu, and Richard J. Lipton.

This paper considers a general model specification between a parametric nonlinear and a nonparametric nonlinear time series model through involving a stochastically nonstationary time series. A new test is proposed and the resulting asymptotic theory is established. The test statistic is constructed based on a natural distance function between a nonparametric estimate and a smoothed parametric counterpart. The asymptotic distribution of the test statistic under the parametric specification is proportional to a local-time random variable with a known distribution. By comparison, existing results for the stationary time series case show that a standardized version of such a test statistic can converge to a normal distribution. The proposed models and testing procedures have applications in various disciplines, including climatology, economics and finance.

In this talk we will consider optimisation problems on regular graphs. The main focus will be on Independent Set but we will also mention some other problems such as variants of the Maximum Matching Problem and various types of domination problems. In particular we will consider the development of a randomised greedy algorithm for finding a large independent set of vertices of a regular graph. By analysing algorithms using the so-called "Differential Equation Method" we are able to assess the average-case performance of the algorithms.

Representation theory is a part of mathematics that enables us to study abstract mathematical objects (such as groups, rings, Lie algebras etc.) by representing their elements as linear transformations of vector spaces. A representation makes an abstract object more concrete because matrices are more familiar objects. The aim of the talk is to present some basic ideas and open problems in representation theory of finite groups. The emphasis will be on the representations of symmetric groups. The talk will be introductory, only basic knowledge of undergraduate level algebra will be assumed.

Motivated by the first differencing method for linear panel data models, we propose a class of iterative local polynomial estimators for nonparametric dynamic panel data models with or without exogeous regressors. The estimators utilize the additive structure of the first-differenced model, the fact that the two additive components have the same functional form, and the unknown function of interest is implicitly defined as a solution of a Fredholm integral equation of the second kind. We establish the uniform consistency and asymptotic normality of the estimators. We also propose a consistent test for the correct specification of linearity in typical dynamic panel data models based on the L₂distance of our nonparametric estimates and the parametric estimates under the linear restriction. We derive the asymptotic distributions of the test statistic under the null hypothesis and a sequence of Pitman local alternatives, and prove its consistency against global alternatives. Simulations suggest that the proposed estimators and tests perform well in finite samples. We apply our new methods to study the relation between economic growth, initial economic condition and capital accumulation and find the nonlinear relation between economic growth and initial economic condition.
Key words: Additive models, Dynamic panel data models, Fredholm integral equation, Iterative estimator, Linearity, Local polynomial regression, Specification test.

In a great many regression problems the explanatory variable, $X$, represents the value taken by a treatment, for example a dosage, and the conditional mean of the response, $Y$, is anticipated to be a monotone function of $X$. Indeed, if this regression mean is not monotone (in the appropriate direction) then the medical or commercial value of the treatment is likely to be significantly curtailed, at least for values of $X$ that lie beyond the point at which monotonicity fails. Addressing these problems requires a method for testing the hypothesis that the regression mean is monotone, and, if the conclusion of the test is positive, a technique for estimating the mean response subject to the constraint that it is monotone. Methodology for solving these problems already exists, but it ignores the potential for errors in measuring $X$. In this talk we outline an approach that accommodates those errors, using statistical tilting.

Optimal design theory and ideas are increasingly applied to many research areas, including education, biomedical sciences, chemical engineering and bioengineering, health services and food science. In this talk, I present an overview of the optimal design methodology and recent advances in the field. The statistical foundation is briefly reviewed and discussed in the context of practical problems in the biomedical sciences.

To promote optimal design ideas, I present a website that allows practitioners to generate a variety of optimal designs easily and freely. After selecting a suitable model from a list of statistical models on the site and an optimality criterion, the practitioner inputs design parameters for his or her problem. The site returns the optimal design and the efficiency of any selected design. I will give demonstrations using problems in the biomedical sciences and hope that the site will facilitate practitioners implement more informed designs that provide improved statistical inference at minimal cost.

KEY WORDS: approximate design, convex analysis, design efficiency, equivalence theorem, multiple-objective optimal design, particle swarm optimization method.

Large-scale computer experiments are becoming increasingly important in science. A multi-step procedure is introduced to statisticians for modeling such experiments, which builds an accurate interpolator in multiple steps. In practice, the procedure shows substantial improvements in overall accuracy but its theoretical properties are not well established. We introduce the terms nominal and numeric error and decompose the overall error of an interpolator into nominal and numeric portions. Bounds on the numeric and nominal error are developed to show theoretically that substantial gains in overall accuracy can be attained with the multi-step approach.

The density conjecture of Katz and Sarnak suggests that the distribution of zeros near 1/2 of a family of L-functions is the same as that of eigenvalues near 1 of a corresponding classical compact group. This has been confirmed for various families. In this talk, I will discuss my work on the n-level density of low-lying zeros of some families of L-functions.

MAS Executive Classroom 1, MAS-03-06

Suppose we have independent $R^d$ random vectors $X_1,X_2,...$, dxd matrices $A_1,A_2,...$, and $R^d$ vectors $x_1,x_2,...$ such that the normalized partial sums:

       $A_n (X_1+X_2+...+X_n) + x_n$     (*)

have limiting distribution $\mu$. What can one say about $\mu$ when the summands in (*) are uniformly infinitesimal ? To answer this probabilistic question we need to introduce completely new algebraic tools such as the Urbanik decomposability semigroups $D (\mu)$. 
This lecture will be based on the monograph "Operator-limit distributions in probability theory" by Z. J.J urek and J. David Mason published by Wiley in 1993.

In Jurek 1985 and 1988 the random integral representation conjecture was stated. It claims that (some) limit laws can be written as probability distributions of random integrals of the form
$\int_{(a,b]}h(t)dY_{\nu}(r(t))$, for some deterministic functions h, r and a Levy process $Y_{\nu}(t), t \ge 0$. Here we review situations where such a claim holds true. Each theorem is followed by a remark which gives references to other related papers, results as well as some
historical comments. Moreover, some open questions are stated.
Key words and phrases: Class L distributions or selfdecomposable distributions; infinite divisibility; Levy-Khintchine formula; class U distributions or s-selfdecomposable distributions; Euclidean space; Levy process; random integral; Banach space.

We consider 2-colorings in k-regular k-uniform hypergraphs. We relate such colorings to results in digraphs (such as even cycles in regular digraphs and disjoint cycles in digraphs). We furthermore relate these results to areas such as domination in graphs.

Asymptotic stability and boundedness have been two of most popular topics in the study of stochastic functional differential equations (SFDEs). In general, the existing results on asymptotic stability and boundedness of SFDEs require (i) the coefficients of the SFDEs obey the local Lipschitz condition and the linear growth condition; (ii) the diffusion operator of the SFDEs acting on a $C^{2,1}$-function be bounded by a polynomial with the same order as the $C^{2,1}$-function.

However, there are many SFDEs which do not obey the linear growth condition. Moreover, for such highly nonlinear SFDEs, the diffusion operator acting on a $C^{2,1}$-function is generally bounded by a polynomial with a higher order than the $C^{2,1}$-function. Hence the existing criteria on stability and boundedness for SFDEs are not applicable and we see the necessity to develop new criteria. Our main aim of this paper is to establish new criteria where the linear growth condition is no longer needed while the up-bound for the diffusion operator may take a much more general form.

By late 60's particle theorists had given up on Quantum Field Theory as the language of fundamental particles. An incredible suggestion was made by Nambu, Nielsen and Susskind that all strongly interacting particles may be the states of a relativistic quantum string. But very soon severe problems arose as consistency required space-time dimensions to be 26(10) and also the spectrum of strings contained particles clearly not associated with strong interactions.

Theorists began to favour Quantum Chromodynamics (QCD) as the correct Quantum Field Theory to describe hadron physics. But here too severe problems arose as quarks could not be liberated from the hadrons. The \emph{Dual Superconductor}picture was proposed by t'Hooft and Mandelstam for the QCD vacuum as a resolution. Such a picture naturally predicted \emph{Flux Tubes} connecting quarks. Evidence for such flux tubes has been unambiguously provided by Lattice Gauge Simulations. A few years ago, an algorithmic breakthrough by Luscher and Weisz permitted very accurate studies of the flux tube which showed that sub-dominant term in ground state energy was characterstic of \emph{Bosonic Strings}. Pushan Majumdar and myself carried out massive simulations using Luscher-Weisz algorithm and showed that even the next higher order term is exactly what is expected of a free bosonic string. We subsequently showed this result analytically. Implications of these results for every system with one-dimensional defects will be briefly discussed.

In order to measure risks, financial analysts take account of variability and dependence for the returns on assets held in their portfolio. Many financial models for dependent risks are based on assumption of multivariate normality. However, observed financial data are usually not normally distributed and tend to have marginal distributions with heavier tails.

Multivariate copulas (distributions on unit hypercube with uniform margins) have recently become very popular in this context as they allow specification of a joint distribution with given marginal distributions that is not restricted to any parametric form. To specify a multivariate copula model, a set of parameters of this copula has to be provided. It is in general not obvious which parameter values specify consistent model and which dependence structures can be obtained with given copula (Joe 1997). 

In (Cooke 1997) a graphical model called regular vine was introduced. Vines allow specification of a joint distribution on n variables with given margins by specifying n choose 2 bivariate copulas and conditional copulas. This is a very rich dependence model which was shown to perform better than competitors in modeling financial data (Aas et al 2006). Distributions specified by vines with copulas are easily sampled (Kurowicka and Cooke 2006).  

In this talk we introduce the vine copula construction and explain how it compares with few other existing dependence models. 

Semidefinite Programming (SDP) may be seen as a generalization of Linear Programming (LP). In particular, one may extend interior point algorithms for LP to SDP, but it has proven much more difficult to exploit structure in the SDP data during computation.

We will give an overview of existing techniques to exploit this structure in the data. The basic idea is that we may assume that some optimal solution of the SDP lies in the matrix *-algebra that contains the data matrices. Thus one may sometimes greatly reduce the size of SDP instances with suitable algebraic symmetry.

To illustrate this technique, we will consider several applications, including: 

Through restoration of the light source information in small animals in vivo, optical molecular imaging, such as fluorescence molecular tomography (FMT) and bioluminescence tomography (BLT), can depict biological and physiological changes observed using molecular probes. A priori information plays an indispensable role in tomographic reconstruction. We consider both Gaussian and Poisson MAP restoration model together with sparsity a priori information.  In Gaussian noise case, the proposed L^1 minimization is validated with simulated data, Monte Carlo-based synthetic data and a mouse-shaped phantom.  Testing with different noise levels and single/multiple source settings at different depths demonstrates the improved performance of this algorithm. In Poisson noise case, we show numerically that although several L^1 minimization algorithms are efficient for compressive sensing reconstruction, they fail for BLT reconstruction. Instead, we propose a novel greedy algorithm for L^0 regularization to reconstruct sparse solutions for BLT problem. Numerical experiments on simulated data both by forward system matrix and Monte-Carlo method show the accuracy and efficiency of the proposed method.

Starting from Feynman's informal path integral ideas in quantum mechanics, we show how to make sense of them using well defined stochastic diffusion processes. Then we use a subclass of these processes to stochastically deform the notion of classical (two dimensional) integrable systems. Some open problems will be mentioned in relation with the expected conjugacy between the time and energy variables in this dynamical context.

We  introduce   the   proposition induced functions    and  define  the  truth degrees  of propositions , by means of infinite product of evenly distributed probability spaces and integrated semantics,  respectively  for  discrete and continuous  cases. Next, a graded approximate reasoning theory is established. Finally,  the  consistency degrees of finite theories is also proposed. 

The wide availability of data observed over time and space, in particular through inexpensive geographical information systems, has stimulated many studies in a variety of disciplines such as economics, environmental science, political science, demography and geography. In these studies, specific statistical methods have been developed to deal with the spatial structure reflected in the distribution of the dependent variable; see e.g. Ripley (1981), Anselin (1988), Cressie (1993), Guyon (1995), Stein (1999), Diggle (2003), Arbia (2006) and LeSage & Pace (2009) for systematic reviews in spatial statistics and spatial econometrics.

In this talk I will first review some developments in exploring the nonlinearity in spatial and spatio-temporal data, in particular the challenges we have faced and the work that we have recently done. For example, nonparametric methods have been very popular in the last couple of decades in time series econometrics, but no such development has taken place for spatial models. A rather obvious reason for this is the curse of dimensionality. We proposed an adaptive varying-coefficient spatio-temporal model to avoid this problem for data observed irregularly over space and regularly in time. It is capable of catching possible nonlinearity (both in space and in time) and nonstationarity (in space) by allowing the autoregressive coefficients to vary with both spatial location and an unknown index variable. An application of the methodology to a climatological data set in the North Sea illustrates that our adaptive coefficient model outperforms all other naive and linear forecasts, with the smallest overall mean predictive error. An outlook with nonlinear spatial econometric modelling is also provided.

In 1984, Bannai and Ito conjectured that for given valency $k$, at least three, there are finitely many distance-regular graphs with valency $k$. In this talk I will discuss a proof for this conjecture and some generalizations.