The reference interval is the most widely used medical decision‐making, constituting a central tool in determining whether an individual is healthy or not. When the results of several continuous ...diagnostic tests are available for the same patient, their clinical interpretation is more reliable if a multivariate reference region (MVR) is available rather than multiple univariate reference intervals. MVRs, defined as regions containing 95% of the results of healthy subjects, extend the concept of the reference interval to the multivariate setting. However, they are rarely used in clinical practice owing to difficulties associated with their interpretability and the restrictions inherent to the assumption of a Gaussian distribution. Further statistical research is thus needed to make MVRs more applicable and easier for physicians to interpret. Since the joint distribution of diagnostic test results may well change with patient characteristics independent of disease status, MVRs adjusted for covariates are desirable. The present work introduces a novel formulation for MVRs based on multivariate conditional transformation models (MCTMs). Additionally, we take into account the estimation uncertainty of such MVRs by means of tolerance regions. These conditional MVRs imply no parametric restriction on the response, and potentially nonlinear continuous covariate effects can be estimated. MCTMs allow the estimation of the effects of covariates on the joint distribution of multivariate response variables and on these variables' marginal distributions, via the use of most likely transformation estimation. Our contributions proved reliable when tested with simulated data and for a real data application with two glycemic markers.
A fast and accurate algorithm to compute the bidiagonal decomposition of rectangular totally positive h-Bernstein-Vandermonde matrices is presented. The error analysis of the algorithm and the ...perturbation theory for the bidiagonal decomposition of totally positive h-Bernstein-Vandermonde matrices are addressed. The computation of this bidiagonal decomposition is used as the first step for the accurate and efficient computation of the singular values of rectangular totally positive h-Bernstein-Vandermonde matrices and for solving least squares problems whose coefficient matrices are such matrices.
In this paper, a new numerical technique is introduced to find the solution of the system of Volterra integral equations based on symmetric Bernstein polynomials. The use of Bernstein polynomials to ...find the numerical solutions of differential and integral equations increased due to its fast convergence. Here, the numerical solution of the system of Volterra integral equations on any finite interval m,n is obtained by replacing the unknown functions with the generalized Bernstein basis functions. The proposed technique converts the given system of equations into the system of algebraic equations which can be solved by using any standard rule. Further, Hyers–Ulam stability criteria are used to check the stability of the given technique. The comparison between exact and numerical solution for the distinct nodes is demonstrated to show its fast convergence.
•New bivariate h-Bernstein basis on a triangular domain with a shape parameter h.•Interesting algebraic and geometric properties of the h-Bernstein basis.•h-Bernstein parametric surfaces and h-de ...Casteljau algorithm.•h-blossoming and Marsden identity.
In this paper, we introduce and study new h-Bernstein basis functions over a triangular domain. In particular, after defining the h-Bernstein polynomial functions of degree n, we prove their algebraic and geometric properties, such as partition of unity and degree elevation and we show that they form a basis for the space of polynomials of total degree less than or equal to n on a triangle. Then, we propose the h-de Casteljau algorithm and we prove the Marsden identity.
A reduced difference polynomialf(u,v)=(p(u)−p(v))/(u−v) may be associated with any given univariate polynomial p(t), t ∈ 0, 1 such that the locus f(u,v)=0 identifies the pairs of distinct values u ...and v satisfying p(u)=p(v). The Bernstein coefficients of f(u, v) on 0, 1 2 can be determined from those of p(t) through a simple algorithm, and can be restricted to any subdomain of 0, 1 2 by standard subdivision methods. By constructing the reduced difference polynomials f(u, v) and g(u, v) associated with the coordinate components of a polynomial curve r(t)=(x(t),y(t)), a quadtree decomposition of 0, 1 2 guided by the variation-diminishing property yields a numerically stable scheme for isolating real solutions of the system f(u,v)=g(u,v)=0, which identify self-intersections of the curve r(t). Through the Kantorovich theorem for guaranteed convergence of Newton–Raphson iterations to a unique solution, the self-intersections can be efficiently computed to machine precision. By generalizing the reduced difference polynomial to encompass products of univariate polynomials, the method can be readily extended to compute the self-intersections of rational curves, and of the rational offsets to Pythagorean–hodograph curves.
Degree reduction of composite Bézier curves Gospodarczyk, Przemysław; Lewanowicz, Stanisław; Woźny, Paweł
Applied mathematics and computation,
01/2017, Volume:
293
Journal Article
Peer reviewed
This paper deals with the problem of multi-degree reduction of a composite Bézier curve with the parametric continuity constraints at the endpoints of the segments. We present a novel method which is ...based on the idea of using constrained dual Bernstein polynomials to compute the control points of the reduced composite curve. In contrast to other methods, ours minimizes the L2-error for the whole composite curve instead of minimizing the L2-errors for each segment separately. As a result, an additional optimization is possible. Examples show that the new method gives much better results than multiple application of the degree reduction of a single Bézier curve.
The (q,h)-Bernstein–Bézier curves are generalizations of both the h-Bernstein–Bézier curves and the q-Bernstein–Bézier curves. We investigate two essential features of (q,h)-Bernstein bases and ...(q,h)-Bézier curves: the variation diminishing property and the degree elevation algorithm. We show that the (q,h)-Bernstein bases for a non-empty interval a,b satisfy Descartes' law of signs on a,b when q>−1, q≠0, and h≤min{(1−q)a,(1−q)b}. We conclude that the corresponding (q,h)-Bézier curves are variation diminishing. We also derive a degree elevation formula for (q,h)-Bernstein bases and (q,h)-Bézier curves over arbitrary intervals a,b. We show that these degree elevation formulas depend only on the parameter q and are independent of both the parameter h and the interval a,b. We investigate the convergence of the control polygons generated by repeated degree elevation. We show that unlike classical Bézier curves, the control polygons generated by repeated degree elevation for (q,h)-Bézier curves with 0<q<1 do not converge to the original (q,h)-Bézier curve, but rather to a piecewise linear curve with vertices that depend only on q and the monomial coefficients of the q-Bézier curve having the same control points as the original (q,h)-Bézier curve. A similar result holds when q>1. Here the control polygons generated by repeated degree elevation converge to a piecewise linear curve that depends only on q and the monomial coefficients of the 1/q-Bézier curve with the control points of the original (q,h)-Bézier curve in reverse order.
Explicit formulae for the Bézier coefficients of the constrained dual Bernstein basis polynomials are derived in terms of the Hahn orthogonal polynomials. Using difference properties of the latter ...polynomials, efficient recursive scheme is obtained to compute these coefficients. Applications of this result to some problems of CAGD is discussed.
Fuzzy transforms (or F-transforms for short) are an approximation technique recently introduced. The main application is referred to image and data compression. There are really few works devoted to ...the use of F-transform for solving ordinary differential equations. In the present paper, an F-transform-based Picard-like numerical scheme is proposed in order to solve a class of delay differential equations. For linear cases, the proposed approach leads to a non-recursive approximate solution by means of operational matrices and vectors of known quantities. Numerical results show the good performance of the proposed method against known solutions.
This work is a contribution in the approximation theory for studying and analyzing piecewise polynomial functions (splines), which uses the blossoming approach. Some existing results in the ...literature are reformulated, such as the smoothness conditions between polynomials of a spline, by using the affinity property of the blossom. Some definitions of sub-splines are proposed which can be very useful in the study and the construction of splines such as macro-elements or quasi-interpolants. As an application of the proposed results, a quartic spline quasi-interpolant with optimal approximation order is defined without using any mask for smoothness or B-spline basis. Numerical results are presented and compared with other methods given in the literature.
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