Numerical Analysis and Computation Commons^™

Gauss-Seidel Estimation Of Generalized Linear Mixed Models With Application To Poisson Modeling Of Spatially Varying Disease Rates, Subharup Guha, Louise Ryan

Harvard University Biostatistics Working Paper Series

Generalized linear mixed models (GLMMs) provide an elegant framework for the analysis of correlated data. Due to the non-closed form of the likelihood, GLMMs are often fit by computational procedures like penalized quasi-likelihood (PQL). Special cases of these models are generalized linear models (GLMs), which are often fit using algorithms like iterative weighted least squares (IWLS). High computational costs and memory space constraints often make it difficult to apply these iterative procedures to data sets with very large number of cases.

This paper proposes a computationally efficient strategy based on the Gauss-Seidel algorithm that iteratively fits sub-models of the GLMM …

Computational Techniques For Spatial Logistic Regression With Large Datasets, Christopher J. Paciorek, Louise Ryan

Harvard University Biostatistics Working Paper Series

In epidemiological work, outcomes are frequently non-normal, sample sizes may be large, and effects are often small. To relate health outcomes to geographic risk factors, fast and powerful methods for fitting spatial models, particularly for non-normal data, are required. We focus on binary outcomes, with the risk surface a smooth function of space. We compare penalized likelihood models, including the penalized quasi-likelihood (PQL) approach, and Bayesian models based on fit, speed, and ease of implementation.

A Bayesian model using a spectral basis representation of the spatial surface provides the best tradeoff of sensitivity and specificity in simulations, detecting real spatial …

Remarks On Risk-Sensitive Control Problems, José Luis Menaldi, Maurice Robin

Mathematics Faculty Research Publications

The main purpose of this paper is to investigate the asymptotic behavior of the discounted risk-sensitive control problem for periodic diffusion processes when the discount factor α goes to zero. If u_α(θ, x) denotes the optimal cost function, being the risk factor, then it is shown that lim_α→0αu_α(θ, x) = ξ(θ) where ξ(θ) is the average on ]0, θ[ of the optimal cost of the (usual) in nite horizon risk-sensitive control problem.

The Interquartile Range: Theory And Estimation., Dewey Lonzo Whaley

Electronic Theses and Dissertations

The interquartile range (IQR) is used to describe the spread of a distribution. In an introductory statistics course, the IQR might be introduced as simply the “range within which the middle half of the data points lie.” In other words, it is the distance between the two quartiles, IQR = Q₃ - Q₁. We will compute the population IQR, the expected value, and the variance of the sample IQR for various continuous distributions. In addition, a bootstrap confidence interval for the population IQR will be evaluated.

Time-Dependent Thermal Imaging Of Circular Inclusions, Donald L. Brouwn, Mark Hubenthal

Mathematical Sciences Technical Reports (MSTR)

This paper considers the inverse problem of locating one or more circular inclusions in a two-dimensional domain using thermal boundary data, specifically, the input heat flux and measured boundary temperature. The forward problem is governed by the heat equation. We show how the position and size of such defects can be recovered using the boundary data and various approximations of the solution to the forward problem. We also consider the stability of the algorithm involved to recover the defects.

Compression Of Laser Radiation In Plasmas Using Electromagnetic Cascading, Serguei Y. Kalmykov, Gennady Shvets

Serge Youri Kalmykov

Compressing high-power laser beams in plasmas via generation of a coherent cascade of electromagnetic sidebands is described. The technique requires two copropagating beams detuned by a near-resonant frequency, \Omega < \omega_{p}. The ponderomotive force of the laser beat wave drives an electron plasma wave which modifies the refractive index of plasma so as to produce a periodic phase modulation of the laser field with the beat period t_b = 2\pi/\Omega. A train of chirped laser beat notes (each of duration t_b) is thus created. The group velocity dispersion of radiation in plasma can then compress each beat note to a few-laser-cycle duration. As a result, a train of sharp electromagnetic spikes separated in time by t_b is formed. Depending on the plasma and laser parameters, chirping and compression can be implemented either concurrently in the same plasma or sequentially in different plasmas.

Survival Model And Estimation For Lung Cancer Patients., Xingchen Yuan

Electronic Theses and Dissertations

Lung cancer is the most frequent fatal cancer in the United States. Following the notion in actuarial math analysis, we assume an exponential form for the baseline hazard function and combine Cox proportional hazard regression for the survival study of a group of lung cancer patients. The covariates in the hazard function are estimated by maximum likelihood estimation following the proportional hazards regression analysis. Although the proportional hazards model does not give an explicit baseline hazard function, the baseline hazard function can be estimated by fitting the data with a non-linear least square technique. The survival model is then examined …

Strongly Coupled Large-Angle Stimulated Raman Scattering Of Short Laser Pulse In Plasma-Filled Capillary, Serguei Y. Kalmykov, Patrick Mora

Serge Youri Kalmykov

Strongly coupled large-angle stimulated Raman scattering sLA SRSd of a short intense laser pulse develops in a plane plasma-filled capillary differently than in a plasma with open boundaries. Coupling the laser pulse to a capillary seeds the LA SRS in the forward direction (scattering angle smaller than \pi / 2 ) and can thus produce a high instability level in the vicinity of the entrance plane. In addition, oblique mirror reflections off capillary walls partly suppress the lateral convection of scattered radiation and increase the growth rate of the SRS under arbitrary (not too small) angle. Hence, the saturated convective …

Lower Bounds For Simplicial Covers And Triangulations Of Cubes, Adam Bliss '03, Francis E. Su

All HMC Faculty Publications and Research

We show that the size of a minimal simplicial cover of a polytope P is a lower bound for the size of a minimal triangulation of P, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and we improve lower bounds for covers and triangulations in dimensions 4 through at least 12 (and possibly more dimensions as well). Important ingredients are an analysis of the number of exterior faces that a simplex in the cube can have of a specified dimension and volume, and a characterization of corner simplices in terms of their …

Laser Wakefield Acceleration By Petawatt Ultrashort Laser Pulses, Leonid M. Gorbunov, Serguei Y. Kalmykov, Patrick Mora

Serge Youri Kalmykov

An ultrashort (about 30 fs) petawatt laser pulse focused with a wide focal spot (about 100 mm) in a rarefied plasma (n_0 ~ 10^{17} cm^{−3}) excites a nonlinear plasma wakefield which can accelerate injected electrons up to GeV energies without any pulse channeling. Under these conditions, propagation of the laser pulse with an overcritical power for relativistic self-focusing is almost the same as in vacuum. The nonlinear quasiplane plasma wave, whose amplitude and phase velocity vary along the laser path, effectively traps and accelerates injected electrons with a wide range of initial energies. Electrons accelerated over two Rayleigh lengths (about …

Reconstruction Of An Unknown Boundary Portion From Cauchy Data In N- Dimensions, Kurt Bryan, Lester Caudill

Department of Math & Statistics Faculty Publications

We consider the inverse problem of determining the shape of some inaccessible portion of the boundary of a region in n dimensions from Cauchy data for the heat equation on an accessible portion of the boundary. The inverse problem is quite ill-posed, and nonlinear. We develop a Newton-like algorithm for solving the problem, with a simple and efficient means for computing the required derivatives, develop methods for regularizing the process, and provide computational examples.

Cluster Analysis Of Genomic Data With Applications In R, Katherine S. Pollard, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In this paper, we provide an overview of existing partitioning and hierarchical clustering algorithms in R. We discuss statistical issues and methods in choosing the number of clusters, the choice of clustering algorithm, and the choice of dissimilarity matrix. In particular, we illustrate how the bootstrap can be employed as a statistical method in cluster analysis to establish the reproducibility of the clusters and the overall variability of the followed procedure. We also show how to visualize a clustering result by plotting ordered dissimilarity matrices in R. We present a new R package, hopach, which implements the hybrid clustering method, …

Multiple Testing Procedures And Applications To Genomics, Merrill D. Birkner, Katherine S. Pollard, Mark J. Van Der Laan, Sandrine Dudoit

U.C. Berkeley Division of Biostatistics Working Paper Series

This chapter proposes widely applicable resampling-based single-step and stepwise multiple testing procedures (MTP) for controlling a broad class of Type I error rates, in testing problems involving general data generating distributions (with arbitrary dependence structures among variables), null hypotheses, and test statistics (Dudoit and van der Laan, 2005; Dudoit et al., 2004a,b; van der Laan et al., 2004a,b; Pollard and van der Laan, 2004; Pollard et al., 2005). Procedures are provided to control Type I error rates defined as tail probabilities for arbitrary functions of the numbers of Type I errors, V_n, and rejected hypotheses, R_n. These error rates include: …

Robust Inferences For Covariate Effects On Survival Time With Censored Linear Regression Models, Larry Leon, Tianxi Cai, L. J. Wei

Harvard University Biostatistics Working Paper Series

Various inference procedures for linear regression models with censored failure times have been studied extensively. Recent developments on efficient algorithms to implement these procedures enhance the practical usage of such models in survival analysis. In this article, we present robust inferences for certain covariate effects on the failure time in the presence of "nuisance" confounders under a semiparametric, partial linear regression setting. Specifically, the estimation procedures for the regression coefficients of interest are derived from a working linear model and are valid even when the function of the confounders in the model is not correctly specified. The new proposals are …

Penalty Approximation And Analytical Characterization Of The Problem Of Super-Replication Under Portfolio Constraints, Alain Bensoussan, Nizar Touzi, José Luis Menaldi

Mathematics Faculty Research Publications

In this paper, we consider the problem of super-replication under portfolio constraints in a Markov framework. More specifically, we assume that the portfolio is restricted to lie in a convex subset, and we show that the super-replication value is the smallest function which lies above the Black-Scholes price function and which is stable for the so-called face lifting operator. A natural approach to this problem is the penalty approximation, which not only provides a constructive smooth approximation, but also a way to proceed analytically.

A Posteriori Estimate For Tikhonov Regularization Parameter, S. Abbasbandy

Saeid Abbasbandy

This paper deals the numerical solution of integral equations of the first kind with using regularization method. There are many stopping rules based on discrepancy principle or discussed in [3]. Here a new stopping rule is described which uses SVD (Singular Value Decomposition) and condition number of matrices. Finally, we give a number of numerical examples showing that the method works well in practice.

A Method For Solving Fuzzy Linear Systems, S. Abbasbandy, M. Alavi

Saeid Abbasbandy

In this paper we present a method for solving fuzzy linear systems by two crisp linear systems. Also necessary and sufficient conditions for existence of solution are given. Some numerical examples illustrate the efficiency of the method.

A New Method For Solving Symmetric Fuzzy Linear Systems, S. Abbasbandy, M. Alavi

Saeid Abbasbandy

In this paper we represent a new method for solving a symmetric fuzzy linear system by two crisp linear systems. Also necessary and sufficient conditions for the solution existence are given.

Application Of Meshless Methods For Thermal Analysis, Darrell Pepper, Bozidar Sarler

Mechanical Engineering Faculty Research

Many numerical and analytical schemes exist for solving heat transfer problems. The meshless method is a particularly attractive method that is receiving attention in the engineering and scientific modeling communities. The meshless method is simple, accurate, and requires no polygonalisation. In this study, we focus on the application of meshless methods using radial basis functions (RBFs) — which are simple to implement — for thermal problems. Radial basis functions are the natural generalization of univariate polynomial splines to a multivariate setting that work for arbitrary geometry with high dimensions. RBF functions depend only on the distance from some center point. …

Transient Non-Linear Heat Conduction Solution By A Dual Reciprocity Boundary Element Method With An Effective Posteriori Error Estimator, Eduardo Divo, Alain J. Kassab

Publications

A Dual Reciprocity Boundary Element Method is formulated to solve non-linear heat conduction problems. The approach is based on using the Kirchhoff transform along with lagging of the effective non-linear thermal diffusivity. A posteriori error estimate is used to provide effective estimates of the temporal and spatial error. A numerical example is used to demonstrate the approach.