Stochastic processes with sample paths in reproducing kernel Hilbert spaces

Lukić, Milan N.; Beder, Jay H.

doi:10.1090/s0002-9947-01-02852-5

Cited by 127 publications

(83 citation statements)

References 9 publications

(3 reference statements)

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“…This isomorphism preserves the inner product; i.e., it is a congruence. This congruence is referred to as Loève's isometry (Lukić and Beder 2001). It maps Z(s) onto K(s, t) ψ(Z(s))(t) = K(s, t), s, t ∈ I.…”

Section: Stochastic Processes and Rkhs'smentioning

confidence: 99%

“…If the RKHS is infinite dimensional, the trajectories of the random process do not belong to H K with probability one (Kailath 1971;Lukić and Beder 2001;Berlinet and Thomas-Agnan 2004). In such case, X, m K cannot represent an inner product in H K .…”

Section: Non-singular Homoscedastic Classificationmentioning

confidence: 99%

“…Since H BM is an infinite-dimensional RKHS, the sample trajectories X do not belong to that space with probability one (Lukić and Beder 2001). In the case of Brownian motion it is clear that the sample trajectories, which are continuous but non-differentiable, are not in H BM .…”

Section: Brownian Processes With Different Meansmentioning

confidence: 99%

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Optimal classification of Gaussian processes in homo- and heteroscedastic settings

et al. 2020

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A procedure to derive optimal discrimination rules is formulated for binary functional classification problems in which the instances available for induction are characterized by random trajectories sampled from different Gaussian processes, depending on the class label. Specifically, these optimal rules are derived as the asymptotic form of the quadratic discriminant for the discretely monitored trajectories in the limit that the set of monitoring points becomes dense in the interval on which the processes are defined. The main goal of this work is to provide a detailed analysis of such optimal rules in the dense monitoring limit, with a particular focus on elucidating the mechanisms by which near perfect classification arises. In the general case, the quadratic discriminant includes terms that are singular in this limit. If such singularities do not cancel out, one obtains near perfect classification, which means that the error approaches zero asymptotically, for infinite sample sizes. This singular limit is a consequence of the orthogonality of the probability measures associated to the stochas-

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Section: Stochastic Processes and Rkhs'smentioning

confidence: 99%

Section: Non-singular Homoscedastic Classificationmentioning

confidence: 99%

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Optimal classification of Gaussian processes in homo- and heteroscedastic settings

et al. 2020

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“…As Γ is a symmetric and non-negative-definite function, we can consider the vector-valued RKHS associated with the kernel Γ, HΓ (Aronszajn, 1950). Assuming that Γ is continuous, it is known that HΓ⊂HK (Lukić and Beder, 2001), and then vj∈HK.…”

Section: Bases Of Functions In Hkmentioning

confidence: 99%

Generalized linear models for geometrical current predictors: An application to predict garment fit

Centella

Gual‐Arnau

Ibáñez

et al. 2019

Statistical Modelling

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The aim of this paper is to model an ordinal response variable in terms of vector-valued functional data included on a vector-valued RKHS. In particular, we focus on the vector-valued RKHS obtained when a geometrical object (body) is characterized by a current and on the ordinal regression model. A common way to solve this problem in functional data analysis is to express the data in the orthonormal basis given by decomposition of the covariance operator. But our data present very important differences with respect to the usual functional data setting. On the one hand, they are vector-valued functions, and on the other, they are functions in an RKHS with a previously defined norm. We propose to use three different bases: the orthonormal basis given by the kernel that defines the RKHS, a basis obtained from decomposition of the integral operator defined using the covariance function, and a third basis that combines the previous two. The three approaches are compared and applied to an interesting problem: building a model to predict the fit of children's garment sizes, based on a 3D database of the Spanish child population.

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“…As in the proof of [4, Lemma 2.2], in order to "match the spaces", any other kernel that "dominates" K (in the sense of [8]) could play the role of the integral-type kernel *…”

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confidence: 99%

Kernel-Based Collocation Methods Versus Galerkin Finite Element Methods for Approximating Elliptic Stochastic Partial Differential Equations

Fasshauer

2012

Meshfree Methods for Partial Differential Equations VI

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Summary. We compare a kernel-based collocation method (meshfree approximation method) with a Galerkin finite element method for solving elliptic stochastic partial differential equations driven by Gaussian noise. The kernel-based collocation solution is a linear combination of reproducing kernels obtained from related differential and boundary operators centered at chosen collocation points. Its random coefficients are obtained by solving a system of linear equations with multiple random right-hand sides. The finite element solution is given as a tensor product of triangular finite elements and Lagrange polynomials defined on a finite-dimensional probability space. Its coefficients are obtained by solving several deterministic finite element problems. For the kernel-based collocation method, we directly simulate the (infinite-dimensional) Gaussian noise at the collocation points. For the finite element method, however, we need to truncate the Gaussian noise into finite-dimensional noises. According to our numerical experiments, the finite element method has the same convergence rate as the kernel-based collocation method provided the Gaussian noise is truncated using a suitable number terms.

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Stochastic processes with sample paths in reproducing kernel Hilbert spaces

Cited by 127 publications

References 9 publications

Optimal classification of Gaussian processes in homo- and heteroscedastic settings

Optimal classification of Gaussian processes in homo- and heteroscedastic settings

Generalized linear models for geometrical current predictors: An application to predict garment fit

Kernel-Based Collocation Methods Versus Galerkin Finite Element Methods for Approximating Elliptic Stochastic Partial Differential Equations

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