Physical Sciences and Mathematics Commons^™

A Collocation Method Via The Quasi-Affine Biorthogonal Systems For Solving Weakly Singular Type Of Volterra-Fredholm Integral Equations, Mutaz Mohammad, Carlo Cattani

All Works

© 2020 Faculty of Engineering, Alexandria University Tight framelet system is a recently developed tool in applied mathematics. Framelets, due to their nature, are widely used in the area of image manipulation, data compression, numerical analysis, engineering mathematical problems such as inverse problems, visco-elasticity or creep problems, and many more. In this manuscript we provide a numerical solution of important weakly singular type of Volterra - Fredholm integral equations WSVFIEs using the collocation type quasi-affine biorthogonal method. We present a new computational method based on special B-spline tight framelets and use it to introduce our numerical scheme. The method provides …

A Numerical Solution Of Fredholm Integral Equations Of The Second Kind Based On Tight Framelets Generated By The Oblique Extension Principle, Mutaz Mohammad

All Works

© 2019 by the authors. In this paper, we present a new computational method for solving linear Fredholm integral equations of the second kind, which is based on the use of B-spline quasi-affine tight framelet systems generated by the unitary and oblique extension principles. We convert the integral equation to a system of linear equations. We provide an example of the construction of quasi-affine tight framelet systems. We also give some numerical evidence to illustrate our method. The numerical results confirm that the method is efficient, very effective and accurate.

An Investigation Of Nurbs-Based Deformable Image Registration, Travis J. Jacobson

Theses and Dissertations

Deformable image registration (DIR) is an essential tool in medical image processing. It provides a means to combine image datasets, allowing for intra-subject, inter-subject, multi-modality, and multi-instance analysis, as well as motion detection and compensation. One of the most popular DIR algorithms models the displacement vector field (DVF) as B-splines, a sum of piecewise polynomials with coefficients that enable local shape control. B-splines have many advantageous properties in the context of DIR, but they often struggle to adequately model steep local gradients and discontinuities. This dissertation addresses that limitation by proposing the replacement of conventional B-splines with a generalized formulation …

Functional Generalized Linear Models With Images As Predictors, Philip T. Reiss, R. Todd Ogden

Philip T. Reiss

Functional principal component regression (FPCR) is a promising new method for regressing scalar outcomes on functional predictors. In this paper we present a theoretical justification for the use of principal components in functional regression. FPCR is then extended in two directions: from linear to the generalized linear modeling, and from univariate signal predictors to high-resolution image predictors. We show how to implement the method efficiently by adapting generalized additive model technology to the functional regression context. A technique is proposed for estimating simultaneous confidence bands for the coefficient function; in the neuroimaging setting, this yields a novel means to identify …

Smoothing Parameter Selection For A Class Of Semiparametric Linear Models, Philip T. Reiss, R. Todd Ogden

Philip T. Reiss

Spline-based approaches to nonparametric and semiparametric regression, as well as to regression of scalar outcomes on functional predictors, entail choosing a parameter controlling the extent to which roughness of the fitted function is penalized. In this paper we demonstrate that the equations determining two popular methods for smoothing parameter selection, generalized cross-validation and restricted maximum likelihood, share a similar form that allows us to prove several results common to both, and to derive a condition under which they yield identical values. These ideas are illustrated by application of functional principal component regression, a method for regressing scalars on functions, to …

Functional Principal Component Regression And Functional Partial Least Squares, Philip T. Reiss, R. Todd Ogden

Philip T. Reiss

Regression of a scalar response on signal predictors, such as near-infrared (NIR) spectra of chemical samples, presents a major challenge when, as is typically the case, the dimension of the signals far exceeds their number. Most solutions to this problem reduce the dimension of the predictors either by regressing on components--e.g. principal component regression (PCR) and partial least squares (PLS)--or by smoothing methods which restrict the coefficient function to the span of a spline basis. This paper introduces functional versions of PCR and PLS, which combine both of the above dimension reduction approaches. Two versions of functional PCR are developed, …

A Spline-Based Lack-Of-Fit Test For Independent Variable Effect, Chin-Shang Li, Wanzhu Tu

Journal of Modern Applied Statistical Methods

In regression analysis of count data, independent variables are often modeled by their linear effects under the assumption of log-linearity. In reality, the validity of such an assumption is rarely tested, and its use is at times unjustifiable. A lack-of-fit test is proposed for the adequacy of a postulated functional form of an independent variable within the framework of semiparametric Poisson regression models based on penalized splines. It offers added flexibility in accommodating the potentially non-loglinear effect of the independent variable. A likelihood ratio test is constructed for the adequacy of the postulated parametric form, for example log-linearity, of the …

A Nonstationary Negative Binomial Time Series With Time-Dependent Covariates: Enterococcus Counts In Boston Harbor, E. Andres Houseman, Brent Coull, James P. Shine

Harvard University Biostatistics Working Paper Series

Boston Harbor has had a history of poor water quality, including contamination by enteric pathogens. We conduct a statistical analysis of data collected by the Massachusetts Water Resources Authority (MWRA) between 1996 and 2002 to evaluate the effects of court-mandated improvements in sewage treatment. Motivated by the ineffectiveness of standard Poisson mixture models and their zero-inflated counterparts, we propose a new negative binomial model for time series of Enterococcus counts in Boston Harbor, where nonstationarity and autocorrelation are modeled using a nonparametric smooth function of time in the predictor. Without further restrictions, this function is not identifiable in the presence …

Algebraic Grid Generation Using Tensor Product B-Splines, Bonita Valerie Saunders

Mathematics & Statistics Theses & Dissertations

In general, finite difference methods are more successful if the accompanying grid has lines which are smooth and nearly orthogonal. This thesis discusses the development of an algorithm which produces such a grid when given the boundary description.

Topological considerations in structuring the grid generation mapping are discussed. In particular, this thesis examines the concept of the degree of a mapping and how it can be used to determine what requirements are necessary if a mapping is to produce a suitable grid.

The grid generation algorithm uses a mapping composed of bicubic B-splines. Boundary coefficients are chosen so that the …