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1
Oruç, Halil. "Generalized Bernstein polynomials and total positivity." Thesis, University of St Andrews, 1999. http://hdl.handle.net/10023/11183.
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This thesis deals mainly with geometric properties of generalized Bernstein polynomials which replace the single Bernstein polynomial by a one-parameter family of polynomials. It also provides a triangular decomposition and 1-banded factorization of the Vandermonde matrix. We first establish the generalized Bernstein polynomials for monomials, which leads to a definition of Stirling polynomials of the second kind. These are q-analogues of Stirling numbers of the second kind. Some of the properties of the Stirling numbers are generalized to their q-analogues. We show that the generalized Bernstein polynomials are monotonic in degree n when the function ƒ is convex ... Shape preserving properties of the generalized Bernstein polynomials are studied by making use of the concept of total positivity. It is proved that monotonic and convex functions produce monotonic and convex generalized Bernstein polynomials. It is also shown that the generalized Bernstein polynomials are monotonic in the parameter q for the class of convex functions. Finally, we look into the degree elevation and degree reduction processes on the generalized Bernstein polynomials.
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Liang, Jie Ling. "Approximation by Bernstein polynomials at the point of discontinuity." Honors in the Major Thesis, University of Central Florida, 2011. http://digital.library.ucf.edu/cdm/ref/collection/ETH/id/460.
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Chlodovsky showed that if x₀ is a point of discontinuity of the first kind of the function f, then the Bernstein polynomials Bsubscript n](f, x₀) converge to the average of the one-sided limits on the right and on the left of the function f at the point x₀. In 2009, Telyakovskii in (5) extended the asymptotic formulas for the deviations of the Bernstein polynomials from the differentiable functions at the first-kind discontinuity points of the highest derivatives of even order and demonstrated the same result fails for the odd order case. Then in 2010, Tonkov in (6) found the right formulation and proved the result that was missing in the odd-order case. It turned out that the limit in the odd order case is related to the jump of the highest derivative. The proofs in these two cases look similar but have many subtle differences, so it is desirable to find out if there is a unifying principle for treating both cases. In this thesis, we obtain a unified formulation and proof for the asymptotic results of both Telyakovskii and Tonkov and discuss extension of these results in the case where the highest derivative of the function is only assumed to be bounded at the point under study.B.S.
Bachelors
Sciences
Mathematics
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Yang, Ning. "Structured matrix methods for computations on Bernstein basis polynomials." Thesis, University of Sheffield, 2013. http://etheses.whiterose.ac.uk/3311/.
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This thesis considers structure preserving matrix methods for computations on Bernstein polynomials whose coefficients are corrupted by noise. The ill-posed operations of greatest common divisor computations and polynomial division are considered, and it is shown that structure preserving matrix methods yield excellent results. With respect to greatest common divisor computations, the most difficult part is the computation of its degree, and several methods for its determination are presented. These are based on the Sylvester resultant matrix, and it is shown that a new form of the Sylvester resultant matrix in the modified Bernstein basis yields the best results. The B´ezout resultant matrix in the modified Bernstein basis is also considered, and it is shown that the results from it are inferior to those from the Sylvester resultant matrix in the modified Bernstein basis.
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Herath, Dushanthi N. "Nonparametric Estimation of Receiver Operating Characteristic Surfaces Via Bernstein Polynomials." Thesis, University of North Texas, 2012. https://digital.library.unt.edu/ark:/67531/metadc177212/.
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Receiver operating characteristic (ROC) analysis is one of the most widely used methods in evaluating the accuracy of a classification method. It is used in many areas of decision making such as radiology, cardiology, machine learning as well as many other areas of medical sciences. The dissertation proposes a novel nonparametric estimation method of the ROC surface for the three-class classification problem via Bernstein polynomials. The proposed ROC surface estimator is shown to be uniformly consistent for estimating the true ROC surface. In addition, it is shown that the map from which the proposed estimator is constructed is Hadamard differentiable. The proposed ROC surface estimator is also demonstrated to lead to the explicit expression for the estimated volume under the ROC surface . Moreover, the exact mean squared error of the volume estimator is derived and some related results for the mean integrated squared error are also obtained. To assess the performance and accuracy of the proposed ROC and volume estimators, Monte-Carlo simulations are conducted. Finally, the method is applied to the analysis of two real data sets.
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Piazzon, Federico. "Bernstein Markov Properties and Applications." Doctoral thesis, Università degli studi di Padova, 2016. http://hdl.handle.net/11577/3424517.
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The Bernstein Markov Property for a compact set E and a positive finite mea-
sure μ supported on E is a strong comparability assumption between L μ 2 and uni-
form norms on E of polynomials (or other nested families of functions) as their
degree tends to infinity.
Admissible meshes are sequences of sampling sets A k ⊂ E whose cardinality
is growing sub-exponentially with respect to k and for which there exists a positive
finite constant C such that max E |p| ≤ C max A k |p| for any polynomial of degree at
most k.
These two mathematical objects have several applications and motivations from
Approximation Theory and Pluripotential Theory, the study of plurisubharmonic
functions in several complex variables.
The properties of Bernstein Markov measures and admissible meshes for a
given compact set E are very similar, indeed they may be seen as the continuous
and the discrete approach to the same problem.
This work is concerned on providing sufficient conditions for some different
instances of the Bernstein Markov property and explicitly constructing admissible
meshes.
As first problem, we study sufficient conditions for a version of the Bernstein
Markov property for rational functions on the complex plane and its relation with
the polynomial Bernstein Markov property.
In Chapter 5, we consider the case of a compact subset E of an algebraic pure
m-dimensional subset A of C n and we prove a sufficient condition for the Bernstein
Markov property for the traces of polynomials on E.
To this aim, we provide two new results in Pluripotential Theory regarding the
convergence and the comparability of the relative capacities, the relative and global
extremal functions and the Chebyshev constants on a (possibly non-smooth) pure
m-dimensional algebraic variety in C n , which are of independent interest.
In the last part of the dissertation, we provide some construction procedures
for admissible meshes on some classes of real compact sets.
Finally, we present some algorithms, based on admissible meshes, for the
numerical approximation of the most relevant objects in Pluripotential Theory,
namely, the transfinite diameter, the Siciak Zaharjuta extremal function and the
pluripotential equilibrium measure.La proprietà di Bernstein Markov per un compatto E ed una misura positiva finita μ avente supporto in E è un’ assunzione di comparabilità asintotica tra le norme uniformi ed L μ 2 dei polinomi di grado al più k (o altre famiglie innestate di funzioni) al tendere all’ infinito di k. Le Admissible Meshes sono sequenze di sottoinsiemi finiti A k del compatto E la cui cardinalità cresce in modo subesponenziale rispetto a k e per i quali esiste una costante positiva C tale che max E |p| ≤ C max A k |p| per ogni polinomi di grado al più k. Questi due oggetti matematici hanno molte appliicazioni e motivazioni prove- nienti dalla Teoria dell’ Approssimazione e dalla Teoria del Pluripotenziale, lo stu- dio delle funzioni plurisubarmoniche in più variabili complesse. Le proprietà delle misure di Bernstein Markov e delle admissible meshes per un dato compatto E sono molto simili, infatti le due definizioni possono essere viste come gli approcci rispettivamente continuo e discreto dello stesso problema. Questo lavoro si concentra nel fornire condizioni sufficienti per la proprietà di Bernstein Markov in diverse situazioni e nella costruzione esplicita di admissible meshes. Come primo problema vengono studiate condizioni sufficienti per una versione della proprietà di Bernstein Markov per successioni di funzioni razionali nel piano complesso in relazione alla stessa proprietà per i polinomi. Nel Capitolo 5 viene considerato il caso di un compatto E sottoinsieme di una varietà algebrica A ⊂ C n di dimensione pura m < n ed irriducibile e quindi provata una condizione sufficiente per la proprietà di Bernstein Markov per le tracce dei polinomi su E. A questo scopo vengono provati due risultati nuovi in Teoria del Pluripoten- ziale riguardanti la convergenza e la comparabilità della capacità relativa (di Monge Ampère), delle funzioni plurisubarmoniche estremali globali e relative e delle co- stanti di Chebyshev per sottoinsiemi E j di un dato compatto E della varietà alge- brica A, anche nel caso A sia singolare. Tali risultati sono di interesse indipendente. Nell’ultima parte della tesi vengono provate ed illustrate alcune procedure per la costruzione di admissible meshes per alcune classi di compatti reali. In ultimo vengono presentati alcuni nuovi algoritmi, basati sulle admissible meshes, per l’ approssimazione numerica delle più rilevanti grandezze in Teoria del Pluripotenziale: il diametro transfinito, la funzione estremale di Siciak-Zaharjuta e la misura di equilibrio pluripotenziale.
6
Bourne, Martin. "Structure-preserving matrix methods for computations on univariate and bivariate Bernstein polynomials." Thesis, University of Sheffield, 2017. http://etheses.whiterose.ac.uk/20860/.
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Curve and surface intersection finding is a fundamental problem in computer-aided geometric design (CAGD). This practical problem motivates the undertaken study into methods for computing the square-free factorisation of univariate and bivariate polynomials in Bernstein form. It will be shown how these two problems are intrinsically linked and how finding univariate polynomial roots and bivariate polynomial factors is equivalent to finding curve and surface intersection points. The multiplicities of a polynomial’s factors are maintained through the use of a square free factorisation algorithm and this is analogous to the maintenance of smooth intersections between curves and surfaces, an important property in curve and surface design. Several aspects of the univariate and bivariate polynomial factorisation problem will be considered. This thesis examines the structure of the greatest common divisor (GCD) problem within the context of the square-free factorisation problem. It is shown that an accurate approximation of the GCD can be computed from inexact polynomials even in the presence of significant levels of noise. Polynomial GCD computations are ill-posed, in that noise in the coefficients of two polynomials which have a common factor typically causes the polynomials to become coprime. Therefore, a method for determining the approximate greatest common divisor (AGCD) is developed, where the AGCD is defined to have the same degree as the GCD and its coefficients are sufficiently close to those of the exact GCD. The algorithms proposed assume no prior knowledge of the level of noise added to the exact polynomials, differentiating this method from others which require derived threshold values in the GCD computation. The methods of polynomial factorisation devised in this thesis utilise the Sylvester matrix and a sequence of subresultant matrices for the GCD finding component. The classical definition of the Sylvester matrix is extended to compute the GCD of two and three bivariate polynomials defined in Bernstein form, and a new method of GCD computation is devised specifically for bivariate polynomials in Bernstein form which have been defined over a rectangular domain. These extensions are necessary for the computation of the factorisation of bivariate polynomials defined in the Bernstein form.
7
Kebede, Sebsibew. "On Bernstein-Sato ideals and Decomposition of D-modules over Hyperplane Arrangements." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-129493.
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Harden, Lisa A. Govil N. K. "On the growth of polynomials and entire functions of exponential type." Auburn, Ala., 2004. http://repo.lib.auburn.edu/EtdRoot/2004/FALL/Mathematics/Thesis/hardeli_58_Thesis.pdf.
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Hamadneh, Tareq [Verfasser]. "Bounding Polynomials and Rational Functions in the Tensorial and Simplicial Bernstein Forms / Tareq Hamadneh." Konstanz : Bibliothek der Universität Konstanz, 2018. http://d-nb.info/1151075027/34.
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Stahlke, Colin. "Bernstein-Polynom und Tjurinazahl von [mu]-konstant-Deformationen der Singularitäten xa̲ + yb̲." Bonn : [s.n.], 1998. http://catalog.hathitrust.org/api/volumes/oclc/41464676.html.
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Fossaluza, Victor. "Estimação de distribuições discretas via cópulas de Bernstein." Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/45/45133/tde-30042012-160407/.
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As relações de dependência entre variáveis aleatórias é um dos assuntos mais discutidos em probabilidade e estatística e a forma mais abrangente de estudar essas relações é por meio da distribuição conjunta. Nos últimos anos vem crescendo a utilização de cópulas para representar a estrutura de dependência entre variáveis aleatórias em uma distribuição multivariada. Contudo, ainda existe pouca literatura sobre cópulas quando as distribuições marginais são discretas. No presente trabalho será apresentada uma proposta não-paramétrica de estimação da distribuição conjunta bivariada de variáveis aleatórias discretas utilizando cópulas e polinômios de Bernstein.The relations of dependence between random variables is one of the most discussed topics in probability and statistics and the best way to study these relationships is through the joint distribution. In the last years has increased the use of copulas to represent the dependence structure among random variables in a multivariate distribution. However, there is still little literature on copulas when the marginal distributions are discrete. In this work we present a non-parametric approach for the estimation of the bivariate joint distribution of discrete random variables using copulas and Bernstein polynomials.
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Almeida, Evert Elvis Batista de. "Curvas de Bézier." Universidade Federal da Paraíba, 2015. http://tede.biblioteca.ufpb.br:8080/handle/tede/8049.
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In this work we will make an introduction an important application mathematics called Bézier curves. The history of this curve originated in industry automobile French , and found many applications in various elds of science. Revisit some concepts such as parametric functions, polynomials Bernstein and interpolation for de nition the curves Bézier. We will discuss the algorithm Casteljau which facilitates the construction of the curve and determine derivative. Throughout the text we will implement some examples with Geogebra software and LATEX in addition to discuss relevant issues that arouse public interest.
Neste trabalho fazemos uma introdução às Curvas de Bézier, importante item da aplicação matemática que originou-se na indústria automobilística francesa e que têm aplicações em várias áreas cientí cas. Diversos conceitos básicos são revisitados tais como curvas de nidas parametricamente, polinômios de Bernstein e polinômios de interpolação. Ao longo do texto, é abordado o algoritmo de Casteljau para construção de curva e suas derivadas. São implementados exemplos de construção usando o GeoGebra e LATEX.
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Jemai, Asma. "Estimation fonctionnelle non paramétrique au voisinage du bord." Thesis, Poitiers, 2018. http://www.theses.fr/2018POIT2257/document.
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L’objectif de cette thèse est de construire des estimateurs non-paramétriques d’une fonction de distribution, d’une densité de probabilité et d’une fonction de régression en utilisant les méthodes d’approximation stochastiques afin de corriger l’effet du bord créé par les estimateurs à noyaux continus classiques. Dans le premier chapitre, on donne quelques propriétés asymptotiques des estimateurs continus à noyaux. Puis, on présente l’algorithme stochastique de Robbins-Monro qui permet d’introduire les estimateurs récursifs. Enfin, on rappelle les méthodes utilisées par Vitale, Leblanc et Kakizawa pour définir des estimateurs d’une fonction de distribution et d’une densité de probabilité en se basant sur les polynômes de Bernstein.Dans le deuxième chapitre, on a introduit un estimateur récursif d’une fonction de distribution en se basant sur l’approche de Vitale. On a étudié les propriétés de cet estimateur : biais, variance, erreur quadratique intégré (MISE) et on a établi sa convergence ponctuelle faible. On a comparé la performance de notre estimateur avec celle de Vitale et on a montré qu’avec le bon choix du pas et de l’ordre qui lui correspond notre estimateur domine en terme de MISE. On a confirmé ces résultatsthéoriques à l’aide des simulations. Pour la recherche pratique de l’ordre optimal, on a utilisé la méthode de validation croisée. Enfin, on a confirmé les meilleures qualités de notre estimateur à l’aide des données réelles. Dans le troisième chapitre, on a estimé une densité de probabilité d’une manière récursive en utilisant toujours les polynômes de Bernstein. On a donné les caractéristiques de cet estimateur et on les a comparées avec celles de l’estimateur de Vitale, de Leblanc et l’estimateur donné par Kakizawa en utilisant la méthode multiplicative de correction du biais. On a appliqué notre estimateur sur des données réelles. Dans le quatrième chapitre, on a introduit un estimateur récursif et non récursif d’une fonction de régression en utilisant les polynômes de Bernstein. On a donné les caractéristiques de cet estimateur et on les a comparées avec celles de l’estimateur à noyau classique. Ensuite, on a utilisé notre estimateur pour interpréter des données réellesThe aim of this thesis is to construct nonparametric estimators of distribution, density and regression functions using stochastic approximation methods in order to correct the edge effect created by kernels estimators. In the first chapter, we givesome asymptotic properties of kernel estimators. Then, we introduce the Robbins-Monro stochastic algorithm which creates the recursive estimators. Finally, we recall the methods used by Vitale, Leblanc and Kakizawa to define estimators of distribution and density functions based on Bernstein polynomials. In the second chapter, we introduced a recursive estimator of a distribution function based on Vitale’s approach. We studied the properties of this estimator : bias, variance, mean integratedsquared error (MISE) and we established a weak pointwise convergence. We compared the performance of our estimator with that of Vitale and we showed that, with the right choice of the stepsize and its corresponding order, our estimator dominatesin terms of MISE. These theoretical results were confirmed using simulations. We used the cross-validation method to search the optimal order. Finally, we applied our estimator to interpret real dataset. In the third chapter, we introduced a recursive estimator of a density function using Bernstein polynomials. We established the characteristics of this estimator and we compared them with those of the estimators of Vitale, Leblanc and Kakizawa. To highlight our proposed estimator, we used real dataset. In the fourth chapter, we introduced a recursive and non-recursive estimator of a regression function using Bernstein polynomials. We studied the characteristics of this estimator. Then, we compared our proposed estimator with the classical kernel estimator using real dataset
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Spencer, Melvin R. "Polynomial Real Root Finding in Bernstein Form." BYU ScholarsArchive, 1994. https://scholarsarchive.byu.edu/etd/4246.
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This dissertation addresses the problem of approximating, in floating-point arithmetic, all real roots (simple, clustered, and multiple) over the unit interval of polynomials in Bernstein form with real coefficients.
15
Blanco, Fernández Guillem. "Bernstein-Sato polynomial of plane curves and Yano's conjecture." Doctoral thesis, Universitat Politècnica de Catalunya, 2020. http://hdl.handle.net/10803/669107.
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The main aim of this thesis is the study of the Bernstein-Sato polynomial of plane curve singularities. In this context, we prove a conjecture posed by Yano about the generic b-exponents of a plane irreducible curve.
In a part of the thesis, we study the Bernstein-Sato polynomial through the analytic continuation of the complex zeta function of a singularity. We obtain several results on the vanishing and non-vanishing of the residues of the complex zeta function. Using these results we obtain a proof of Yano's conjecture under the hypothesis that the eigenvalues of the monodromy are pair-wise different. In another part of the thesis, we study the periods of integrals in the Milnor fiber and their asymptotic expansion. These periods of integrals can be related to the b-exponents and can be constructed in terms of resolution of singularities. Using these techniques, we can present a proof for the general case of Yano's conjecture.
In addition to the Bernstein-Sato polynomial, we also study the minimal Tjurina number of a plane irreducible curve and we answer in the positive a question raised by Dimca and Greuel on the quotient between the Milnor and Tjurina numbers. More precisely, we prove a formula for the minimal Tjurina number of a plane irreducible curve in terms of the multiplicities of the strict transform along its minimal resolution. From this formula, we obtain the positive answer to Dimca and Greuel question.
This thesis also contains computational results for the theory of singularities on smooth complex surfaces. First, we describe an algorithm to compute log-resolutions of ideals on a smooth complex surface. Secondly, we provide an algorithm to compute generators for complete ideals on a smooth complex surface. These algorithms have several applications, for instance, in the computation of the multiplier ideals associated to an ideal on a smooth complex surface.El principal objectiu d'aquesta tesi és l'estudi del polinomi de Bernstein-Sato de singularitats de corbes planes. En aquest context, es demostra una conjectura proposada per Yano el 1982 sobre els \( b \)-exponents genèrics d'una corba plana irreductible. En una part d'aquesta tesi, s'estudia el polinomi de Bernstein-Sato utilitzant la continuació analítica de la funció zeta complexa d'una singularitat. S'obtenen diversos resultat sobre l'anul·lació i no anul·lació del residu de la funció zeta complexa d'una corba plana. Utilitzant aquests resultats, s'obté una demostració de la conjectura de Yano sota la hipòtesi de que els valors propis de la monodromia siguin diferents dos a dos. En un altre part de la tesi, s'estudien els períodes d'integrals en la fibra de Milnor i la seva expansió asimptòtica. Aquesta expansió asimptòtica dels períodes pot ser relacionada amb els b-exponents i pot ser construïda en termes de la resolució de singularitats. Utilitzant aquestes tècniques, es presenta una prova del cas general de la conjectura de Yano. A més a més del polinomi de Bernstein-Sato, també s'estudia el nombre de Tjurina mínim d'una corba plana irreductible i responem positivament a una pregunta formulada per Dimca i Greuel sobre el quocient entre els nombres de Milnor i Tjurina. Concretament, es demostra una fórmula pel nombre de Tjurina mínim en un classe d'equisingularitat de corbes planes irreductibles en termes de la seqüència de multiplicitats de la transformada estricta al llarg de la resolució minimal. A partir d'aquesta fórmula, s'obté la resposta positiva a la pregunta de Dimca i Greuel. Aquesta tesi també conté resultats computacionals per la teoria de singularitats en superfícies complexes llises. Primer, es descriu un algorisme que calcula la log-resolució d'ideals en un superfície complexa llisa. En segon lloc, es dona un algorisme per calcular generadors per ideals complets en una superfície complexa llisa. Aquests algorismes tenen diverses aplicacions, com per exemple, en el càlcul d'ideals multiplicadors associats a un ideal en una superfície complexa llisa.
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Tencaliec, Patricia. "Developments in statistics applied to hydrometeorology : imputation of streamflow data and semiparametric precipitation modeling." Thesis, Université Grenoble Alpes (ComUE), 2017. http://www.theses.fr/2017GREAM006/document.
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Les précipitations et les débits des cours d'eau constituent les deux variables hydrométéorologiques les plus importantes pour l'analyse des bassins versants. Ils fournissent des informations fondamentales pour la gestion intégrée des ressources en eau, telles que l’approvisionnement en eau potable, l'hydroélectricité, les prévisions d'inondations ou de sécheresses ou les systèmes d'irrigation.Dans cette thèse de doctorat sont abordés deux problèmes distincts. Le premier prend sa source dans l’étude des débits des cours d’eau. Dans le but de bien caractériser le comportement global d'un bassin versant, de longues séries temporelles de débit couvrant plusieurs dizaines d'années sont nécessaires. Cependant les données manquantes constatées dans les séries représentent une perte d'information et de fiabilité, et peuvent entraîner une interprétation erronée des caractéristiques statistiques des données. La méthode que nous proposons pour aborder le problème de l'imputation des débits se base sur des modèles de régression dynamique (DRM), plus spécifiquement, une régression linéaire multiple couplée à une modélisation des résidus de type ARIMA. Contrairement aux études antérieures portant sur l'inclusion de variables explicatives multiples ou la modélisation des résidus à partir d'une régression linéaire simple, l'utilisation des DRMs permet de prendre en compte les deux aspects. Nous appliquons cette méthode pour reconstruire les données journalières de débit à huit stations situées dans le bassin versant de la Durance (France), sur une période de 107 ans. En appliquant la méthode proposée, nous parvenons à reconstituer les débits sans utiliser d'autres variables explicatives. Nous comparons les résultats de notre modèle avec ceux obtenus à partir d'un modèle complexe basé sur les analogues et la modélisation hydrologique et d'une approche basée sur le plus proche voisin. Dans la majorité des cas, les DRMs montrent une meilleure performance lors de la reconstitution de périodes de données manquantes de tailles différentes, dans certains cas pouvant allant jusqu'à 20 ans.Le deuxième problème que nous considérons dans cette thèse concerne la modélisation statistique des quantités de précipitations. La recherche dans ce domaine est actuellement très active car la distribution des précipitations exhibe une queue supérieure lourde et, au début de cette thèse, il n'existait aucune méthode satisfaisante permettant de modéliser toute la gamme des précipitations. Récemment, une nouvelle classe de distribution paramétrique, appelée distribution généralisée de Pareto étendue (EGPD), a été développée dans ce but. Cette distribution exhibe une meilleure performance, mais elle manque de flexibilité pour modéliser la partie centrale de la distribution. Dans le but d’améliorer la flexibilité, nous développons, deux nouveaux modèles reposant sur des méthodes semiparamétriques.Le premier estimateur développé transforme d'abord les données avec la distribution cumulative EGPD puis estime la densité des données transformées en appliquant un estimateur nonparamétrique par noyau. Nous comparons les résultats de la méthode proposée avec ceux obtenus en appliquant la distribution EGPD paramétrique sur plusieurs simulations, ainsi que sur deux séries de précipitations au sud-est de la France. Les résultats montrent que la méthode proposée se comporte mieux que l'EGPD, l’erreur absolue moyenne intégrée (MIAE) de la densité étant dans tous les cas presque deux fois inférieure.Le deuxième modèle considère une distribution EGPD semiparamétrique basée sur les polynômes de Bernstein. Plus précisément, nous utilisons un mélange creuse de densités béta. De même, nous comparons nos résultats avec ceux obtenus par la distribution EGPD paramétrique sur des jeux de données simulés et réels. Comme précédemment, le MIAE de la densité est considérablement réduit, cet effet étant encore plus évident à mesure que la taille de l'échantillon augmentePrecipitation and streamflow are the two most important meteorological and hydrological variables when analyzing river watersheds. They provide fundamental insights for water resources management, design, or planning, such as urban water supplies, hydropower, forecast of flood or droughts events, or irrigation systems for agriculture.In this PhD thesis we approach two different problems. The first one originates from the study of observed streamflow data. In order to properly characterize the overall behavior of a watershed, long datasets spanning tens of years are needed. However, the quality of the measurement dataset decreases the further we go back in time, and blocks of data of different lengths are missing from the dataset. These missing intervals represent a loss of information and can cause erroneous summary data interpretation or unreliable scientific analysis.The method that we propose for approaching the problem of streamflow imputation is based on dynamic regression models (DRMs), more specifically, a multiple linear regression with ARIMA residual modeling. Unlike previous studies that address either the inclusion of multiple explanatory variables or the modeling of the residuals from a simple linear regression, the use of DRMs allows to take into account both aspects. We apply this method for reconstructing the data of eight stations situated in the Durance watershed in the south-east of France, each containing daily streamflow measurements over a period of 107 years. By applying the proposed method, we manage to reconstruct the data without making use of additional variables, like other models require. We compare the results of our model with the ones obtained from a complex approach based on analogs coupled to a hydrological model and a nearest-neighbor approach, respectively. In the majority of cases, DRMs show an increased performance when reconstructing missing values blocks of various lengths, in some of the cases ranging up to 20 years.The second problem that we approach in this PhD thesis addresses the statistical modeling of precipitation amounts. The research area regarding this topic is currently very active as the distribution of precipitation is a heavy-tailed one, and at the moment, there is no general method for modeling the entire range of data with high performance. Recently, in order to propose a method that models the full-range precipitation amounts, a new class of distribution called extended generalized Pareto distribution (EGPD) was introduced, specifically with focus on the EGPD models based on parametric families. These models provide an improved performance when compared to previously proposed distributions, however, they lack flexibility in modeling the bulk of the distribution. We want to improve, through, this aspect by proposing in the second part of the thesis, two new models relying on semiparametric methods.The first method that we develop is the transformed kernel estimator based on the EGPD transformation. That is, we propose an estimator obtained by, first, transforming the data with the EGPD cdf, and then, estimating the density of the transformed data by applying a nonparametric kernel density estimator. We compare the results of the proposed method with the ones obtained by applying EGPD on several simulated scenarios, as well as on two precipitation datasets from south-east of France. The results show that the proposed method behaves better than parametric EGPD, the MIAE of the density being in all the cases almost twice as small.A second approach consists of a new model from the general EGPD class, i.e., we consider a semiparametric EGPD based on Bernstein polynomials, more specifically, we use a sparse mixture of beta densities. Once again, we compare our results with the ones obtained by EGPD on both simulated and real datasets. As before, the MIAE of the density is considerably reduced, this effect being even more obvious as the sample size increases
17
Leroy, Richard. "Certificats de positivité et minimisation polynomiale dans la base de Bernstein multivariée." Phd thesis, Université Rennes 1, 2008. http://tel.archives-ouvertes.fr/tel-00349444.
Full textAbstract:
L'étude des polynômes réels en plusieurs variables est un problème classique en géométrie algébrique réelle et en calcul formel. Plusieurs questions sont naturelles : positivité éventuelle, calcul du minimum...Nous nous proposons, dans cette thèse, d'étudier ces questions dans le cas particulier où l'étude est menée sur un simplexe de $\R^k$.
L'outil essentiel dans notre travail est la base de Bernstein, plus adaptée à la situation que la traditionnelle base des monômes. Elle jouit notamment de propriétés de positivité et d'encadrement essentielles à notre étude.
Elle permet tout d'abord d'obtenir un algorithme décidant si un polynôme $f$ est positif sur un simplexe $V$, et le cas échéant, fournissant une écriture de $f$ rendant triviale cette positivité : on parle de certificat de positivité.
En outre, elle est à l'origine d'un algorithme de minimisation polynomiale sur un simplexe. Ces deux algorithmes sont certifiés, et l'étude de leur complexité est menée dans cette thèse. Ils ont également fait l'objet d'implémentation sur ordinateur.
18
Sadik, Mohamed. "Inégalités de Markov-Bernstein en L2 : les outils mathématiques d'encadrement de la constante de Markov-Bernstein." Phd thesis, INSA de Rouen, 2010. http://tel.archives-ouvertes.fr/tel-00557914.
Full textAbstract:
Les travaux de recherche de cette thèse concernent l'encadrement de la constante de Markov Bernstein pour la norme L2 associée aux mesures de Jacobi et Gegenbauer généralisée. Ce travail est composé de deux parties : dans la première partie, nous avons développé une généralisation de l'algorithme qd pour les matrices symétriques définies positives à largeur de bande $\ell$ et nous avons construit l'algorithme qd pour les matrices de Jacobi par blocs. Ensuite, nous l'avons généralisé aux cas des matrices par bloc à largeur de bande $\ell$. Ces algorithmes nous permettent de trouver un majorant de la constante. Enfin, nous avons développé le déterminant caractéristique d'une matrice symétrique définie positive pentadiagonale, ce qui nous permet d'obtenir un minorant de la constante en utilisant la méthode de Newton. La deuxième partie est consacrée à l'application de tous les outils développés à l'encadrement de la constante de Markov Bernstein pour la norme L2 associée à la mesure de Gegenbauer généralisée.
19
Abbas, Lamia. "Inégalités de Landau-Kolmogorov dans des espaces de Sobolev." Phd thesis, INSA de Rouen, 2012. http://tel.archives-ouvertes.fr/tel-00776349.
Full textAbstract:
Ce travail est dédié à l'étude des inégalités de type Landau-Kolmogorov en normes L2. Les mesures utilisées sont celles d'Hermite, de Laguerre-Sonin et de Jacobi. Ces inégalités sont obtenues en utilisant une méthode variationnelle. Elles font intervenir la norme d'un polynômes p et celles de ces dérivées. Dans un premier temps, on s'intéresse aux inégalités en une variable réelle qui font intervenir un nombre quelconque de normes. Les constantes correspondantes sont prises dans le domaine où une certaine forme bilinéaire est définie positive. Ensuite, on généralise ces résultats aux polynômes à plusieurs variables réelles en utilisant le produit tensoriel dans L2 et en faisant intervenir au plus les dérivées partielles secondes. Pour les mesures d'Hermite et de Laguerre-Sonin, ces inégalités sont étendues à toutes les fonctions d'un espace de Sobolev. Pour la mesure de Jacobi on donne des inégalités uniquement pour les polynômes d'un degré fixé par rapport à chaque variable.
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Tomek, Peter. "Approximation of Terrain Data Utilizing Splines." Master's thesis, Vysoké učení technické v Brně. Fakulta informačních technologií, 2012. http://www.nusl.cz/ntk/nusl-236488.
Full textAbstract:
Pro optimalizaci letových trajektorií ve velmi malé nadmorské výšce, terenní vlastnosti musí být zahrnuty velice přesne. Proto rychlá a efektivní evaluace terenních dat je velice důležitá vzhledem nato, že čas potrebný pro optimalizaci musí být co nejkratší. Navyše, na optimalizaci letové trajektorie se využívájí metody založené na výpočtu gradientu. Proto musí být aproximační funkce terenních dat spojitá do určitého stupne derivace. Velice nádejná metoda na aproximaci terenních dat je aplikace víceroměrných simplex polynomů. Cílem této práce je implementovat funkci, která vyhodnotí dané terenní data na určitých bodech spolu s gradientem pomocí vícerozměrných splajnů. Program by měl vyčíslit více bodů najednou a měl by pracovat v $n$-dimensionálním prostoru.
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BIGNALET, CAZALET REMI. "Riguardo le trasformazione determinantale." Doctoral thesis, Università degli studi di Genova, 2021. http://hdl.handle.net/11567/1062495.
Full textAbstract:
Studiamo trasformazioni di Cremona determinantale, cioè trasformazioni birazionale il cui ideale di base è l'ideale dei minori massimali di una matrice Phi, via la risoluzioni dei sistemi di polinomi definiti da Phi. Usando geometria convessa, questo approccio porta in particolare a descrivere i gradi proiettivi di alcuni trasformazioni di Cremona determinantale raccolte.We study determinantal Cremona maps, i.e. birational maps whose base ideal is the maximal minors ideal of a given matrix Phi, via the resolution of the polynomials systems defined by Phi. Using convex geometry, this approach leads in particular to describe the projective degrees of some glued determinantal maps.
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HSU, YUAN-MING, and 許元銘. "Set Valued Bernstein Polynomials." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/15013077537806784455.
Full textAbstract:
碩士中原大學
應用數學研究所
97
In the theory of approximation,it is well-known that a continuous single-valued function defined on a compact interval [a, b] can be approached uniformly by a sequence of polynomials.In this paper, we show a similar result for set-valued case. In section2,we give some concepts of convergence for sequences of sets in a metric space.In section3,we give the concetps of continuity for set-valued functions of a metric space into another metric space.In section4,we prove that a continuous set-valued function on a compact interval can be approached uniformly by a sequence of set-valued polynomials under the Hausdorff distance.
23
Hsiao, Ai-ling, and 蕭愛齡. "Binary regression with Bernstein polynomials." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/7jxnyg.
Full textAbstract:
碩士國立中央大學
數學研究所
96
Data analysis of binary response variables are often conducted by logistic regression model. Logistic regression model assumes that the conditional probability function of success is a monotonic function. In order to eliminate this sometimes unnecessary monotone restriction, we propose to use Bernstein polynomials to model the conditional probability of success. As a Bayesian approach, we put a prior on the space of Bernstein polynomials having values in [0,1] through their coe cients. The sample from the posterior distribution for inference purpose is obtained by MCMC methods. We conduct simulation studies to examine the e ects of sample size and priors, to indicate that the numerical performance of this method is generally good and to show that our model performs better than the logistic regression model when the regression function is not monotone.
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Xian, Shanshan. "Kernel smoothing based on Bernstein polynomials." 2005. http://hdl.handle.net/1993/20231.
Full text
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Wang, Tso-Kang, and 王佐剛. "Bayesian Regression with Isotonic Random Bernstein Polynomials." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/78302206956818146078.
Full textAbstract:
碩士中原大學
應用數學研究所
97
The problem of regression which alike linear regression is very important since remotest time. This paper is about using Bayesian method to estimate increasing regression curve.Because of continuous functions be approached by Bernstein polynomails which can be found the coefficient to show that it is increasing and approach all of the insreasing continuous functions.On the other hand, it's easy to have the prior and is helpful in calculate the posterior by using Bernstein polynomials.So the model using Bernstein polynomials to describe the graph of increasing curve is more complicate than using linear regression,without saying, it's also more difficult to estimate the M.L.E. Above all,we decide to use Bayesian method and calculate the posterior by using Markov Chain Monte Carlo(M.C.M.C.) method. We have introduction on model algorithm and the theorem of reduction completely in this paper.And using the package software Matlab writing the program to get the estimator can be very nice. The theorem is in 4. It's really complicate by using M.L.E. mothod.We may use this to be our research title in the future.Compare with our estimation,the paper also can be expended to estimation of 2-dimention surface regression,all of them can be direction of research in our future.
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Wei, Tzung-Wei, and 魏宗緯. "Bayesian Regression with Sigmodial Random Bernstein Polynomials." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/28387366353574645492.
Full textAbstract:
碩士中原大學
數學研究所
96
This paper is about the regression curve with sigmodial using the Bayesian method. Use random Bernstein polynomials as the prior, and calculate the posterior by using Markov Chain Monte Carlo (M.C.M.C.) method. Because we discovered that Bernstein polynomials may describe the geometric graph easily, therefore we also controled the coefficient of Bernstein polynomials. Then obtained the regression curve with sigmodial which we want to study. We had also proven that all continuous functions with sigmodial can be approached by contrained Bernstein polynomials. Therefore we can use it to be the model of regression curve which is going to be estimated. According to analyze the data which we got, we can confirm that graph is sigmodial. Hence we can use the method of this paper to estimate it. The method of Bayesian estimation of this paper is using Markov Chain Monte Carlo (M.C.M.C.) method to calculate the posterior. Our algorithm is easy to be written into the program (using the package software Matlab), whence it is easy to calculate. In fact the estimation which we obtain is quite good, present in the last of this paper. Because there is still no one estimating it by M.L.E., we can not compare with it. But it can be a good subject for the research in the future.
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Huang, Wan-Wen, and 黃琬雯. "Bayesian Regression with Concave Random Bernstein Polynomials." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/54330048256377789489.
Full textAbstract:
碩士中原大學
應用數學研究所
96
We want to investigate the relations between independent variable and dependent variable, especially to make a study of regression curve in statistics. So this paper mainly provides to estimate method of regression curve. If the regression curve is unimodal concave down, it use like relations between crops harvest and fertilizer in economics. When the fertilizers are too few or too many, it can create the crops harvest reduction. Therefore the fertilizer amount used can form relations the unimodal to crops harvest (HILDRETH, 1954). This paper aims at the regression curve to make for concave down each kind of graph have the system of the reorganization and the research,that includes concave down and increasing、unimodal and concave down, makes the estimate with the Bayesian method, and also writes down their integrity to develop the algorithm. It develops the algorithm to divide into independent Metropolis Algorithm and Metropolis-Hastings-Green Algorithm, and makes the simulation to compare its difference. The MCMC method is to applied estimate calculates that is pretty good and presents in final part of the paper.
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KANITA. "BERNSTEIN OPERATOR AND ITS MODIFICATIONS." Thesis, 2021. http://dspace.dtu.ac.in:8080/jspui/handle/repository/20487.
Full textAbstract:
S.N. Bernstein was the mathematician who proved the Weierstrass Theorem
by defining Bernstein Polynomials and Operators. This paper illustrates the
various forms of the Bernstein Operator and the different modifications done
by other mathematicians in order to study and prove more theorems in
Approximation Theory.
In this paper, we will be dealing with the classical Bernstein Operator,
Bernstein-Kantorovich Operator, q-Bernstein Operator and Bernstein Durrmeyer Operator.
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Kuo, Yu-Cheng, and 郭育成. "Estimation for Survival Hazard Rate using Bernstein Polynomials." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/19459024223066157927.
Full textAbstract:
碩士淡江大學
數學學系碩士班
96
In this thesis, we study the maximum likelihood estimator for a survival hazard rate with right censored data, in which the hazard rate is specified by the Bernstein polynomial. Our estimation procedure can provide a smooth estimator of the survival hazard rate. We develop an efficient Newton-Raphson based algorithm for the computation of the maximum likelihood estimate. The success of this method is demonstrated in simulation studies and in the analysis of Leukemia remission-time data. In addition, the comparison with Nelson-Aalen method is presented and the selection of the degree for Bernstein polynomial is discussed.
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Klurman, Oleksiy. "On constrained Markov-Nikolskii and Bernstein type inequalities." 2011. http://hdl.handle.net/1993/4820.
Full textAbstract:
This thesis is devoted to polynomial inequalities with constraints. We present a history of the development of this subject together with recent progress.
In the first part, we solve an analog of classical Markov's problem for monotone polynomials. More precisely, if ∆n denotes the set of all monotone polynomials on [-1,1] of degree n, then for Pn ϵ ∆n and x ϵ [-1,1] the following sharp inequalities hold:
│P’n(x)│≤ 2 max(Sk(-x),Sk(-x))║Pn║,
for n = 2k + 2, k ≥ 0, and
│P'n(x)│ ≤ 2 max (Fk(x), Hk(x))║Pn║,
for n = 2k + 1, k ≥ 0, where
Sk(x) := (1+x)∑_(l=0 )^k▒(J_l (0,1)(x^2)) ;
S_k (x) &:=(1+x)\sum\limits_{l=0}^{k} (J^{(0,1)}_l (x))^2;\\
H_k (x) &:=(1-x^2)\sum\limits_{l=0}^{k-1} (J_l ^{(1,1)} (x))^2;\\
F_k(x) &:=\sum\limits_{l=0}^{k} (J_l ^{(0,0)} (x))^2,
\end{align*}
and $J_l^{(\alpha,\beta)}(x),$ $l\ge 1$ are the Jacobi polynomials.
Let ∆n(1) be the set of all monotone nonnegative polynomials on $[-1,1]$ of degree $n.$ In the second part, we investigate the asymptotic behavior of the constants $$M_{q,p}^{(1)}(n,1):=\sup_{P_n\in\triangle^{(1)}_n}\frac{\|P'_n\|_{L_q
[-1,1]}}{\|P_n\|_{L_p [-1,1]}},$$ in constrained Markov-Nikolskii type inequalities.
Our conjecture is that
\[M^{(1)}_{q,p} (n,1)\asymp
\left\{
\begin{array}{ll}
n^{2+2/p-2/q} , & \mbox{\rm if } 1>1/q-1/p ,\\
\log{n} , & \mbox{\rm if } 1=1/q-1/p, \\
1 , & \mbox{\rm if } 1< 1/q - 1/p .
\end{array}
\right.
\]
We prove this conjecture for all values of p,q > 0, except for the case 0 < q < 1, 1/2 ≤ 1/q- 1/p ≤ 1, p ≠ 1
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Boeringer, Daniel Wilharm. "Multi-objective particle swarm optimization of a modified Bernstein polynomial for curved phased array synthesis using Bézier curves, surfaces, and volumes." 2004. http://etda.libraries.psu.edu/theses/approved/WorldWideIndex/ETD-583/index.html.
Full text
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Chen, Jeng-wen, and 陳正文. "Numerical Solutions of Hypersingular Integral Equations Using Bernstein Polynomials." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/75942563564986610612.
Full textAbstract:
碩士大同大學
應用數學學系(所)
94
The Bernstein polynomials are applied to solve the numerical solutions of singular integral equations. We also solve a hypersingular integral equation which arises in the study of the scattering acoustic wave by transforming the Bernstein polynomials into Chebyshev polynomials. Then using the collocation methods, we obtain the numerical solutions of the integral equations. To compute more accurate numerical solutions, some useful formulas in singular integrals are derived. Four numerical examples are given to illustrate our numerical techniques.
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Chen, Jen-Wen, and 陳正文. "Numerical Solutions of Hypersingular Integral Equations Using Bernstein Polynomials." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/97273477122461024953.
Full textAbstract:
碩士大同大學
應用數學研究所
93
The Bernstein polynomials are applied to solve the numerical solutions of singular integral equations. We also solve a hypersingular integral equation which arises in the study of the scattering acoustic wave by transforming the Bernstein polynomials into Chebyshev polynomials. Then using the collocation methods, we obtain the numerical solutions of the integral equations. To compute more accurate numerical solutions, some useful formulas in singular integrals are derived. Four numerical examples are given to illustrate our numerical techniques.
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Hong, Li-Syuan, and 洪立軒. "Estimation and Prediction for Mortality Rate using Bernstein Polynomials." Thesis, 2018. http://ndltd.ncl.edu.tw/handle/636t9b.
Full textAbstract:
碩士中原大學
應用數學研究所
106
In this study, the two-dimensional Bernstein polynomials are used to describe and predict mortality rates for the some particular years and age groups that we are interested in. We mainly use the shape-restricted two-dimensional Bernstein polynomials to build a mathematical model and use the simulated annealing algorithm to find the maximum likelihood estimates of the model parameters, in order to estimate and predict the mortality rate for the some particular years and age groups that we are interested in. We use the analysis of the simulated datasets and real data examples to examine the effectiveness of our method by comparing the existing methods, the P-spline smoothing method and the Lee-Carter model. The Swedish mortality data from the Human Mortality Database (http://www.mortality.org/) are illustrated in the analysis of the real data examples.
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Huang, Yung-Ching, and 黃永青. "Maximum Likelihood Estimator in Regression Analysis with Unimodal Random Bernstein Polynomials." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/04022652491255014032.
Full textAbstract:
碩士中原大學
應用數學研究所
98
We want to investigate the relations between independent variable and dependent variable, especially to make a study of unimodal regression curve in statistics. So this paper mainly provides to estimate method of regression curve by MLE. If the regression curve is unimodal concave down, it use like relations between crops harvest and fertilizer in economics. When the fertilizers are too few or too many, it can create the crops harvest reduction. Therefore the fertilizer amount used can form relations the unimodal to crops harvest (HILDRETH, 1954). This paper aims at the regression curve to make for unimodal each kind of graph have the system of the reorganization and the research, seen in (Chang et al 2007),that includes unimodal and concave down, makes the estimate with MLE, and also writes down their integrity to develop the algorithm. named independent Metropolis Algorithm . The MCMC method is to applied estimate calculates that is pretty good and presents in final part of the paper.
36
Chuang, Sheng-Tu, and 莊昇都. "A Maximum Likelihood Estimator of Bernstein Polynomials with the Monotonic Regression." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/75325306615231655201.
Full textAbstract:
碩士國立高雄第一科技大學
電腦與通訊工程所
99
When talking about the problem of regression, it plays an important role in many scientific areas. This paper is focused on the maximum likelihood estimator of monotone regression model departure from Wang’s in 2008, used a Bayesian method to estimate parameters.Then comparing with the classical regression model. We use Bernstein polynomial to illustrate our model whose monotonicity can be depended on the coefficients. However it is difficult to compute maximum likelihood estimator because of the unfixed degree of polynomials and the curve shape of the regression models.Therefore Chang et al. proposed Bayesian method and algorithms to approximate the parameters of interest in 2008. We will discuss the similar method with them. Here we useMarkov ChainMonte Carlo algorithmto find maximum likelihood estimator for convenience. In this paper, we will introduce related theory of the models and estimated step of the parameters.We use Matlab in simulations and computations and obtain a good estimated result in comparison with classical model.
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Yang, Ya-Wen, and 楊雅雯. "Maximum Likelihood Estimator of Survival Analysis for Current Status Data Using Bernstein Polynomials." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/53815717369818224940.
Full textAbstract:
碩士中原大學
應用數學研究所
97
Survival analysis of current status data is studied using Bernstein polynomials Estimation Cumulate hazard functions Markov chain Monte Carlo (M.C.M.C.) methods estimation Maximum Likelihood Estimator (M.L.E.). These Bernstein polynomials easily take into consideration geometric information like concave or initial guess on the cumulative hazard functions , select only smooth functions , can have large enough support , and can be easily specified and generated. We use these M.C.M.C. methods Estimation MLE are quite satisfactory. Survival analysis is a important research in the statistics. It is applied to calculate the survival rate and the mean survival time in the medicine , also estimate important information of insurance , so we want to study it. To study survival analysis using Bayes method like Sinha & Dey (1997, 1998), as well as Ibrahim et al. (2001). Estimating cumulate hazard functions with the step functions like McKeague & right; Tighiouart (2000, 2002). To estimate cumulate hazard functions of current status data using Bernstein polynomials and find M.L.E. with M.C.M.C. methods in this paper. This paper be organized as follows : Chapter 2 introduce the relations between polynomial coefficients and graphic structures , we discuss some problems about counter statements. Chapter 3 derive model and the likelihood function. Chapter 4 introduce algorithm : Metropolis-Hastings Green method. Chapter 5 is simulation study , we will compare Bernstein polynomials M.L.E. with Step M.L.E.. Chapter 6 is the conclusion and suggestion.
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Liu, Mao-Ting, and 劉懋婷. "Maximum Likelihood Estimator of Proportional Odds Modelwith Right Censored Data Using Bernstein Polynomials." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/40438995626699915385.
Full textAbstract:
碩士中原大學
應用數學研究所
101
The thesis focused on adding covariates to the right censored data with Proportional Odds Model, refer to Pettiet(1982) and Bennett(1983a,b) for more detail. For survival analysis, the semi-parametric regression has been widely used to calculate the correlation of covariate Z and the failure time T, such as the proportional hazards model and proportional odds model. The proportional odds model was also taken by Wu (2012) to calculate the nonparametric estimators per Bernstein polynomials,Bayesian Methodology and Makov Chain Monte Carlo (M.C.M.C.). The model taken byWu is used for this thesis as well to get the nonparametric estimator by maximum likelihood estimation (M.L.E.) and M.C.M.C.. We proved the model workable with excellent results by simulations.
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Chuang, Ya-Wen, and 莊雅雯. "Maximum Likelihood Estimator of Proportional Odds Model withCurrent Status Data Using Bernstein Polynomials." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/27326391726701645672.
Full textAbstract:
碩士中原大學
應用數學研究所
101
In this paper, we use the status quo data proportional odds model (see Bennett (1983a, b)) to carry out research in the Ke Xingjie’s Thesis (2011), he used the maximum likelihood estimation method and the use of proportional odds ratio model, research data likelihood ratio test statistic. In Ni Yu Cheng’s (2012) paper is to come and go with the Bayesian approach to estimate, but only estimated the proportional odds model, and propose a Bernstein polynomial used in Bayesian survival analysis, this thesis the model assumptions and Ni Yu Cheng (2012) paper is the same, but the estimated parameters, we are using the maximum likelihood estimation method. The nonparametric parameters of the model part, as with Bernstein polynomials in the covariate Z discrete value of 0 or 1, with the Markov chain Monte Carlo method to find the parameters of maximum likelihood estimation (MLE). And simulation, we have a good performance.
40
Chak, Pok Man. "Approximation and consistent estimation of shape-restricted functions and their derivatives." 2001. http://wwwlib.umi.com/cr/yorku/fullcit?pNQ67896.
Full textAbstract:
Thesis (Ph. D.)--York University, 2001. Graduate Programme in Economics.Typescript. Includes bibliographical references (leaves 116-121). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNQ67896.
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Pan, Chun-hao, and 潘君豪. "Maximum likelihood estimation for a shape-restricted regression model by sieve of Bernstein polynomials." Thesis, 2012. http://ndltd.ncl.edu.tw/handle/23297762672995240571.
Full textAbstract:
博士國立中央大學
數學研究所
100
We consider maximum likelihood estimation (MLE) of a regression function using sieves defined by Bernstein polynomials, in terms of their order and coefficients. In case, that we know the regression function satisfies certain shape-restriction like monotonicity or convexity, we can impose corresponding restriction through the coefficients of the Bernstein polynomials in the sieves so that the estimate also satisfies the desired shape-restriction. For sieve MLE of this type, we establish its consistency when the regression function is continuous and its rate of convergence when its derivative satisfies Lipschitz condition. Under the same condition, we also show that the integral of the estimate converges weakly to that of the regression function at rate of root n. Simulation studies are presented to evaluate its numerical performance. In addition to excellent confidence interval estimates of area under the regression function, sieve MLE performs better than the Bayesian method based on Bernstein polynomials and density-regression method, reported in Chang et al. (2007).
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Микора, М. Н., and M. N. Mikora. "Неравенство Бернштейна для тригонометрических полиномов для пары пространств L0 и L2 : магистерская диссертация." Master's thesis, 2015. http://hdl.handle.net/10995/35766.
Full textAbstract:
We study the best constant C(n) in the Bernstein inequality between the L0-norm of the first derivative of a trigonometric polynomial and the L2-norm of the polynomial itself on the set of trigonometric polynomials of a given degree n ≥1 with real coefficients. We prove that on the subset of polynomials from Tn such that all zeros of the derivative of a polynomial are real, the Bernstein inequality holds with the constant n/√2. In the general case, we obtain the close two-sided estimates: n/√2≤C(n)≤n.Изучается наилучшая константа C(n) в неравенстве Бернштейна между L0-нормой первой производной тригонометрического полинома и L2-нормой самого полинома на множестве Tn тригонометрических полиномов заданного порядка n ≥1 с вещественными коэффициентами. Показано, что на подмножестве полиномов из Tn, все нули производной которых вещественные, неравенство Бернштейна имеет место с константой n/√2. В общем случае для константы C(n) получены близкие двусторонние оценки n/√2≤C(n)≤n.
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Hachani, Mohamed Amine. "Certain problems concerning polynomials and transcendental entire functions of exponential type." Thèse, 2014. http://hdl.handle.net/1866/11087.
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Soit P(z):=\sum_{\nu=0}^na_\nu z^{\nu}$ un polynôme de degré n et M:=\sup_{|z|=1}|P(z)|.$ Sans aucne restriction suplémentaire, on sait que $|P'(z)|\leq Mn$ pour $|z|\leq 1$ (inégalité de Bernstein). Si nous supposons maintenant que les zéros du polynôme $P$ sont à l'extérieur du cercle $|z|=k,$ quelle amélioration peut-on apporter à l'inégalité de Bernstein? Il est déjà connu [{\bf \ref{Mal1}}] que dans le cas où $k\geq 1$ on a $$(*) \qquad |P'(z)|\leq \frac{n}{1+k}M \qquad (|z|\leq 1),$$ qu'en est-il pour le cas où $k < 1$? Quelle est l'inégalité analogue à $(*)$ pour une fonction entière de type exponentiel $\tau ?$
D'autre part, si on suppose que $P$ a tous ses zéros dans $|z|\geq k \, \, (k\geq 1),$ quelle est l'estimation de $|P'(z)|$ sur le cercle unité, en terme des quatre premiers termes de son développement en série entière autour de l'origine. Cette thèse constitue une contribution à la théorie analytique des polynômes à la lumière de ces questions.Let P(z):=\sum_{\nu=0}^na_\nu z^{\nu}$ a polynomial of degree n and M:=\sup_{|z|=1}|P(z)|$. Without any additional restriction, we know that $|P '(z) | \leq Mn$ for $| z | \leq 1$ (Bernstein's inequality). Now if we assume that the zeros of the polynomial $P$ are outside the circle $| z | = k$, which improvement could be made to the Bernstein inequality? It is already known [{\bf \ref{Mal1}}] that in the case where $k \geq 1$, one has$$ (*) \qquad | P '(z) | \leq \frac{n}{1 + k} M \qquad (| z | \leq 1),$$ what would it be in the case where $k < 1$? What is the analogous inequality for an entire function of exponential type $\tau$? On the other hand, if we assume that $P$ has all its zeros in $| z | \geq k \, \, (k \geq 1),$ which is the estimate of $| P '(z) |$ on the unit circle, in terms of the first four terms of its Maclaurin series expansion. This thesis comprises a contribution to the analytic theory of polynomials in the light of these problems.
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Wang, Shu-Fen, and 王淑芬. "Bernstein Polynomial In Statistic Application." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/00072034931003056871.
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碩士中原大學
應用數學研究所
96
This paper mainly searches that provides the model of the function graph, describes with Bernstein polynomial, and promotes from a dimension to the n dimension. The structure of this paper as follow. The first section introduces Bernstein polynomial correlation background. The second section explains the shape of function graph and Bernstein polynomial coefficient relations. Proposition 1 provides sufficiency of Bernstein polynomial geometry character. Proposition 2 describes the continuous function geometry character by Bernstein polynomial viewpoint. The third section uses the probability method showed the high dimension Bernstein polynomial approximation theo- rem. The fourth section explains the function uniform convergence theory with the analysis method. The fifth section explains the shape of function graph and the high dimension Bernstein polynomial coefficient relations. Discusses the coefficient take two variables as the example in certain condition limit minor function graphs change situations, may use to describe in the curved surface geometry shape, applies in curved surface regression analysisestimate. The sixth section is a discussion. After this paper regarding how to describe the continuous function graph uses Bernstein polynomial, roughly has the system method, also has the quite good description method to the curved surface graph, has the greatest help to the curved surface graph estimate.
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Леонтьева, А. О., and A. O. Leont’eva. "Оценки норм линейных операторов на множестве тригонометрических полиномов в пространстве L0 : магистерская диссертация." Master's thesis, 2015. http://hdl.handle.net/10995/35765.
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We study a Bernstein inequality for a fractional derivative of order α ≥ 0 of a trigonometric polynomial in the space L0. In the case of zero order derivative, we obtain two-sided estimates for a sharp constant in this inequality, which show its behavior with respect to n. For positive and sufficiently small α, we obtain an upper estimate for a constant in the Bernstein inequality in L0. In the second part of the dissertation, we obtain estimates for norms in the space L0 of operators that set several higher or lower coefficients of a trigonometric polynomial to be zero.Изучается неравенство Бернштейна для дробной производной порядка α ≥ 0 тригонометрических полиномов в пространстве L0. В случае производной нулевого порядка получены двусторонние оценки точной константы в этом неравенстве, дающие порядок ее поведения по n. Для положительных, достаточно малых значений α получена оценка сверху константы в неравенстве Бернштейна в L0. Во второй части работы получены оценки норм в пространстве L0 операторов, которые зануляют несколько старших или младших коэффициентов тригонометрического полинома.
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Cheng, Li-Hsueh, and 鄭麗雪. "Partial Unimodal Bayesian Regression Using Bernstein Polynomial." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/35386600473955400866.
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碩士中原大學
應用數學研究所
98
The estimated regression function is an important statistical problem, and partially linear models have many applications. Such as Hardle et al (2000) example, Engle, Granger, Rice and Weiss (1986) were among the first to consider the partially linear model. They analyzed the relationship between temperature and electricity usage. We rst mention several examples from the existing literature. Most of the examples are concerned with practical problems involving partially linear models. They used data based on the monthly electricity sales yi for four cities, the monthly price of electricity x1, income x2, and average daily temperature t. They modeled the electricity demand y as the sum of a smooth function g of monthly temperature t, and a linear function of x1 and x2, as well as with 11 monthly dummy variables x3, . . . , x13. In this paper, mainly used in agriculture, the first application for the amount of fertilizer and crops harvest relationship. Suppose the amount of fertilizer as the x-axis and the crops harvest as the y-axis, the graph is unimodal, if there are two or more kind of the fertilizer and crops harvest relationship. The second application is the crops harvest and two or more kind of the fertilizer and the relationship between seasons. We use partially linear regression model for analysis and comparison, the unimodal model using Bernstein polynomials to describe, using Bayesian methods to estimate, algorithm with Markov Chain Monte Carlo method to calculate.
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Liao, Wei-Ting, and 廖偉廷. "Bayesian partial monotone regression by using Bernstein polynomial." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/78716532250812719470.
Full textAbstract:
碩士中原大學
應用數學研究所
98
The topic of linear regression is very important so far away. Especially when we are doing the linear research on the statistic by the relationship between the dependent variable and the independent variable that can help us to evaluate the past and to predict the forecast. Besides, we can also calculate the function of linear regression to get rid of error and to analysis all of the possibility, then make the best choice by estimator. The main in the thesis, is to study the curve line of regression under the restrict condition which is called partial linear regression. The curve line can be used in the functions which are having the property with monotone increasing or monotone decreasing. We can adopt Bernstein polynomial to apply our regression functions because of the reason that the property of Bernstein polynomial can be limited all of continuous functions. All we have to do is finding the coefficient of Bernstein polynomial that can conform to our conditions of partial linear regression and infer to posterior easier. But it is harder to estimate it by using M.L.E.; these functions are more complicating than others general linear regression functions. Therefore, we provide a method to estimate the partial linear regression. The method is using Markov Chain Monte Carlo (M.C.M.C.) to calculate the posterior of regression data after using Bayesian inference to estimate the function. We have already written down all of detail by IMA algorithm steps in the “Section 4” in the thesis. Above all, we not only introduce the model complete but also explain the application of theorem. Besides, we simulate the program by software R of statistic with different numbers of data, then calculate the estimated result to compare with real data for our conclusion.
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Lin, Hui-Hsin, and 林慧欣. "Maximum Likelihood Estimator of Partial Monotone Regression Using Bernstein Polynomial." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/99252463316938339255.
Full textAbstract:
碩士中原大學
應用數學研究所
99
Estimating the linear regression helps us to study the relationship between the dependent variable and the independent variable, and we can also predict the forecast which make us to find the best method. The main point in the thesis is to probe the partial monotone linear regression, and joins the covariance. That is, we may change the factor of its curve and moreover this curve has the monotonous nature. We use Bernstein polynomial to describe our graph. Because these functions are more complex than other general linear regression functions, so we estimate it by using M.L.E. We also provide a method to estimate the partial monotone linear regression. We simulated the sample with different numbers of data, and used independent calculating method to calculate the conclusion with software- R. We not only completely introduce model establishment and analogue the result, but also compare with real data for our conclusion.
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Peng, Hsin-Ju, and 彭欣茹. "Maximum Likelihood Estimator of Partial Unimodal Regression Using Bernstein Polynomial." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/84757550942849249261.
Full textAbstract:
碩士中原大學
應用數學研究所
99
”The estimator of regression curve” is an important issue in the field of statistics, and there are many applications in ”partial linear model”, like examples given by Hardle (2000). Engle, Granger, Rice and Weiss(1986) are the people who first took ”partial linear model” into considerarion. They analyzed the relationship between temparature and the use of electricity. In the master thesis/dissertaion of Cheng (2010), if regression curve is unimodal, we can use Bernstein Polynomial to describe its shape and estimate it by Bayesian method. The main contribution of this thesis is to use a different statistic method-”Maximun Likelihood Estimator” to estimate ”parameter value” under the same construction, as well as making a comparison. The method of calculation is to apply ”M.C.M.C.” in search of M.L.E., using the program ”R.”
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Hsu, Chia-Te, and 徐嘉德. "Bayesian regression for two or more variables using multivariate Bernstein polynomial." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/45230986736057199782.
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碩士中原大學
應用數學研究所
97
In this paper, Bayes regression for two or more variables is proposed using priors on multivariate Bernstein polynomials, since multivariate Bernstein polynomials can be used to approximate to an arbitrary continuous function of several variables [ see Altomare and Campiti (1994) ]. From our Literature survey, this approach has not been considered before; so far what has been done was any for functions which are monotone and generated sampling from the posterior distribution using Markov chain Monte Carlo methods. These priors easily take into consideration geometric information like convexity, increasing in x for fixed y and increasing in y for fixed x , increasing in x for fixed y and convex in y for fixed x , convex in x for fixed y and convex in y for fixed x , as well select only smooth function, can have large enough support, and can be easily specified and generated. Simulation studies to evaluate the performance of these Bayes methods.