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Metin Demiralp - One of the best experts on this subject based on the ideXlab platform.
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separate node ascending derivatives expansion snade on complex plane
ICNPAA World Congress 2018, 2018Co-Authors: Derya Bodur, Metin DemiralpAbstract:\begin{abstract} In this study we focus on the method called \lq\lq Separate % Node Ascending Derivatives Expansion \rq\rq ( SNADE ) % \cite {MD1,NABMD1,NABEG1,EGNAB1}which is % obtained as a result of the studies recently carried out in % Group for Science and Methods of Computing ( G4S \&MC ) under % the leadership of Metin Demiralp . SNADE is considered as a new % type power series involving denumerable infinitely many nodes, % like Taylor Series Expansion. However, each position is accom % panied by a different derivative value in this method, different % ly from Taylor Series Expansion. SNADE is based on the use of % derivative integration formula for a Univariate Function in dif % ferent nodal values. In this constructed structure, when integ % ral operators defined as % %--------------------------------(1)----------------------------------% \begin {eqnarray} \label{eq:1} \mathcal {I} _m ( x_1 , \cdots,x_m )g(x) \equiv \int_ { x_1 }^x d \xi_1 % \int_ { x_2 }^{ \xi_1 }d \xi_2 % \cdots\int_ { x_m }^{ \xi_ {m-1}} d \xi_m g( \xi_m ), % \nonumber \\ % m=1,2, \cdots, \quad \mathcal {I} _0g (x) \equiv g(x) \nonumber \end {eqnarray} %--------------------------------(1)----------------------------------% are used, it is possible to write the following formula for SNADE . % %--------------------------------(2)----------------------------------% \begin {eqnarray} \label{eq:2} f(x)= \sum_ {i=0}^ \infty f^{(i)}( x_ {i+1}) \mathcal {I} _i ( x_1 , % \cdots,x_i ) 1_f \nonumber % \end {eqnarray} %--------------------------------(2)----------------------------------% Heretofore, relations related to constructions which are called as % SNADE polynomials were obtained and the theoretical structure of % the method was composed on these findings \cite { DBMD1 , % DBMD2 }. By using these constructs, convergence of the method was % investigated and findings were supported by numerical % implementations. In this presentation we will focus on the transition from real- val % uedness to complex- valuedness and relations related with this meth % od will be reconstructed by taking domains in complex plane into % account. Convergence issues will also be considered on a complex % plane. \end{abstract}
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certain implementative applications of separate node ascending derivatives expansion snade
International Conference on Mathematical Problems in Engineering Aerospace and Sciences, 2017Co-Authors: Derya Bodur, Metin DemiralpAbstract:In this work we have focused on a very recently developed method called as Separate Node Ascending Derivatives Expansion (SNADE). SNADE can be considered as an infinite interpolation like Taylor Series Expansion. A Taylor Series is an infinite sum representation whose terms are calculated from the values of the Functions derivatives at a single point. This newly proposed method involves denumerable infinitely many nodes in contrast to Taylor Series Expansion. SNADE is based on derivative integration formula for a Univariate Function. Integral of derivative identity is not only required to be used for the target Function but repetitiously for its all derivatives. It may not be required to be used in the same interval. In addition to all these, each derivative value becomes evaluated at a different independent variable value. This work is designed to emphasize on the methods interpolatory nature. For this purpose certain implementation results are given and compared with well-known interpolation methods.
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Separate Node Ascending Derivatives Expansion (SNADE) as a Univariate Function Representation
2016Co-Authors: Derya Bodur, Metin DemiralpAbstract:This work focuses on the novel approach which is named as “Seperate Node Ascending Derivatives (SNADE)”. SNADE is a very recently developed Univariate Function representation. This method has a similar structure to the Taylor series expansion. Two specific cases related recurrent nodes with reference to certain rules are handled in this paper.
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arrowheading enhanced multivariance products representation for a kernel aemprk in a bivariate taylor series expansion
INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING 2015 (ICCMSE 2015), 2015Co-Authors: Ayla Okan, Metin DemiralpAbstract:Our recent efforts to increase the capabilities of Enhanced Multivariate Products Representation (EMPR) has become fruitful and resulted in the birth of several new representation methods like Tridiagonal Matrix EMPR, Tridiagonal Kernel EMPR. We have also observed that the EMPR of a sum of outer products globally results in a three factor product representation whose kernel matrix is in an arrowheaded form. We have also proven that the same thing happens to be existing for the EMPR of bivariate Functions composed of a sum of binary Univariate Function products. This work specifically focuses on the case of a bivariate Taylor expansion’s EMPR.
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two very specific cases for separate node ascending derivatives expansion snade
INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING 2015 (ICCMSE 2015), 2015Co-Authors: Derya Bodur, Metin DemiralpAbstract:This paper presents the application of very recently developed Univariate Function representation on two very specific cases: (1) the case where all nodes are same, (2) the case where all odd indexed nodes are same while the even indexed nodes are all same but equal to a different value. The purpose of (1) is to show the equivalence of SNADE to well-known Taylor expansion whereas (2) is presented to reveal the difference of SNADE as being a novel approach, from Taylor expansion. The relation between SNADE and known multinode Taylor expansions based on some identities could have also been given here. However we do not intend to mention them here.
Fabrizio Durante - One of the best experts on this subject based on the ideXlab platform.
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Marshall–Olkin type copulas generated by a global shock
Journal of Computational and Applied Mathematics, 2016Co-Authors: Fabrizio Durante, Stéphane Girard, Gildas MazoAbstract:A way to transform a given copula by means of a Univariate Function is presented. The resulting copula can be interpreted as the result of a global shock affecting all the components of a system modeled by the original copula. The properties of this copula transformation from the perspective of semi–group action are presented, together with some investigations on the impact on the tail behavior. Finally, the whole methodology is applied to model risk assessment.
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Invariant dependence structures and Archimedean copulas
Statistics & Probability Letters, 2011Co-Authors: Fabrizio Durante, Piotr Jaworski, Radko MesiarAbstract:We consider a family of copulas that are invariant under Univariate truncation. Such a family has some distinguishing properties: it is generated by means of a Univariate Function; it can capture non-exchangeable dependence structures; it can be easily simulated. Moreover, such a class presents strong probabilistic similarities with the class of Archimedean copulas from a theoretical and practical point of view.
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on a family of multivariate copulas for aggregation processes
Information Sciences, 2007Co-Authors: Fabrizio Durante, Jose Juan Quesadamolina, Manuel IbedafloresAbstract:We introduce a family of multivariate copulas - a special type of n-ary aggregation operations - depending on a Univariate Function. This family is used in the construction of a special aggregation operation that satisfies a Lipschitz condition. Several examples are provided and some statistical properties are studied.
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A new family of symmetric bivariate copulas
Comptes Rendus Mathematique, 2007Co-Authors: Fabrizio DuranteAbstract:Abstract A new class of copulas, depending on an Univariate Function, is introduced and its properties (dependence, ordering, symmetry) are studied. To cite this article: F. Durante, C. R. Acad. Sci. Paris, Ser. I 344 (2007).
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EUSFLAT Conf. (1) - A Method for Constructing Multivariate Copulas.
2007Co-Authors: Fabrizio Durante, José Juan Quesada-molina, Manuel Úbeda-floresAbstract:We provide a method for constructing a class of multivariate copulas depending on a Univariate Function. We study some properties of this class and present several examples. The same circle of ideas is used in a similar construction of quasi–copulas.
Martin J Wainwright - One of the best experts on this subject based on the ideXlab platform.
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minimax optimal rates for sparse additive models over kernel classes via convex programming
Journal of Machine Learning Research, 2012Co-Authors: Garvesh Raskutti, Martin J WainwrightAbstract:Sparse additive models are families of d-variate Functions with the additive decomposition f* = Σj∈S fj*, where S is an unknown subset of cardinality s < d. In this paper, we consider the case where each Univariate component Function fj* lies in a reproducing kernel Hilbert space (RKHS), and analyze a method for estimating the unknown Function f* based on kernels combined with l1-type convex regularization. Working within a high-dimensional framework that allows both the dimension d and sparsity s to increase with n, we derive convergence rates in the L2(P) and L2(Pn) norms over the class Fd,s,H of sparse additive models with each Univariate Function fj* in the unit ball of a Univariate RKHS with bounded kernel Function. We complement our upper bounds by deriving minimax lower bounds on the L2(P) error, thereby showing the optimality of our method. Thus, we obtain optimal minimax rates for many interesting classes of sparse additive models, including polynomials, splines, and Sobolev classes. We also show that if, in contrast to our Univariate conditions, the d-variate Function class is assumed to be globally bounded, then much faster estimation rates are possible for any sparsity s = Ω(√n), showing that global boundedness is a significant restriction in the high-dimensional setting.
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minimax optimal rates for sparse additive models over kernel classes via convex programming
arXiv: Statistics Theory, 2010Co-Authors: Garvesh Raskutti, Martin J WainwrightAbstract:Sparse additive models are families of $d$-variate Functions that have the additive decomposition $f^* = \sum_{j \in S} f^*_j$, where $S$ is an unknown subset of cardinality $s \ll d$. In this paper, we consider the case where each Univariate component Function $f^*_j$ lies in a reproducing kernel Hilbert space (RKHS), and analyze a method for estimating the unknown Function $f^*$ based on kernels combined with $\ell_1$-type convex regularization. Working within a high-dimensional framework that allows both the dimension $d$ and sparsity $s$ to increase with $n$, we derive convergence rates (upper bounds) in the $L^2(\mathbb{P})$ and $L^2(\mathbb{P}_n)$ norms over the class $\MyBigClass$ of sparse additive models with each Univariate Function $f^*_j$ in the unit ball of a Univariate RKHS with bounded kernel Function. We complement our upper bounds by deriving minimax lower bounds on the $L^2(\mathbb{P})$ error, thereby showing the optimality of our method. Thus, we obtain optimal minimax rates for many interesting classes of sparse additive models, including polynomials, splines, and Sobolev classes. We also show that if, in contrast to our Univariate conditions, the multivariate Function class is assumed to be globally bounded, then much faster estimation rates are possible for any sparsity $s = \Omega(\sqrt{n})$, showing that global boundedness is a significant restriction in the high-dimensional setting.
Garvesh Raskutti - One of the best experts on this subject based on the ideXlab platform.
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minimax optimal rates for sparse additive models over kernel classes via convex programming
Journal of Machine Learning Research, 2012Co-Authors: Garvesh Raskutti, Martin J WainwrightAbstract:Sparse additive models are families of d-variate Functions with the additive decomposition f* = Σj∈S fj*, where S is an unknown subset of cardinality s < d. In this paper, we consider the case where each Univariate component Function fj* lies in a reproducing kernel Hilbert space (RKHS), and analyze a method for estimating the unknown Function f* based on kernels combined with l1-type convex regularization. Working within a high-dimensional framework that allows both the dimension d and sparsity s to increase with n, we derive convergence rates in the L2(P) and L2(Pn) norms over the class Fd,s,H of sparse additive models with each Univariate Function fj* in the unit ball of a Univariate RKHS with bounded kernel Function. We complement our upper bounds by deriving minimax lower bounds on the L2(P) error, thereby showing the optimality of our method. Thus, we obtain optimal minimax rates for many interesting classes of sparse additive models, including polynomials, splines, and Sobolev classes. We also show that if, in contrast to our Univariate conditions, the d-variate Function class is assumed to be globally bounded, then much faster estimation rates are possible for any sparsity s = Ω(√n), showing that global boundedness is a significant restriction in the high-dimensional setting.
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minimax optimal rates for sparse additive models over kernel classes via convex programming
arXiv: Statistics Theory, 2010Co-Authors: Garvesh Raskutti, Martin J WainwrightAbstract:Sparse additive models are families of $d$-variate Functions that have the additive decomposition $f^* = \sum_{j \in S} f^*_j$, where $S$ is an unknown subset of cardinality $s \ll d$. In this paper, we consider the case where each Univariate component Function $f^*_j$ lies in a reproducing kernel Hilbert space (RKHS), and analyze a method for estimating the unknown Function $f^*$ based on kernels combined with $\ell_1$-type convex regularization. Working within a high-dimensional framework that allows both the dimension $d$ and sparsity $s$ to increase with $n$, we derive convergence rates (upper bounds) in the $L^2(\mathbb{P})$ and $L^2(\mathbb{P}_n)$ norms over the class $\MyBigClass$ of sparse additive models with each Univariate Function $f^*_j$ in the unit ball of a Univariate RKHS with bounded kernel Function. We complement our upper bounds by deriving minimax lower bounds on the $L^2(\mathbb{P})$ error, thereby showing the optimality of our method. Thus, we obtain optimal minimax rates for many interesting classes of sparse additive models, including polynomials, splines, and Sobolev classes. We also show that if, in contrast to our Univariate conditions, the multivariate Function class is assumed to be globally bounded, then much faster estimation rates are possible for any sparsity $s = \Omega(\sqrt{n})$, showing that global boundedness is a significant restriction in the high-dimensional setting.
Sun Yuqinag - One of the best experts on this subject based on the ideXlab platform.
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a new algorithm for the fitting of a sine curve with therr sample points
Computer Simulation, 2006Co-Authors: Sun YuqinagAbstract:Fitting a sine curve with three sample points is a basic problem encountered in signal measuring and dynamic control fields.Traditionally this kind of problem is solved with minimum squarer error method,B-sample curves based fitting methods and genetic algorithms.The problem is a combined optimal problem in which three variables are involved.A new algorithm is presented which reduces the problem into a minimum problem of an Univariate Function according to the corresponding relationship between a sine cure and an ellipse on a cylinder surface.The new algorithm has a good geological interpretation.It is more reliable than traditional algorithms.
