Strictly Positive Definite Research Articles

Strictly positive definite functions by integral transforms

We study the continuity and strict positive definiteness of positive definite functions on quasi-metric spaces given by integral transforms. We apply some of our findings to positive definite functions on the Euclidean space Rm which are given by cosine transforms (m=1) and Fourier–Bessel transforms (m>1). We also apply the results to positive definite functions on a general quasi-metric space realized as extensions of certain real Laplace transforms defined by conditionally negative definite functions on the quasi-metric space itself.

Positive definite functions on products of metric spaces by integral transforms

An influential theorem proved by T. Gneiting in the beginning of the century provides a large class of continuous positive definite functions on the product Rd×R commonly used in theory and applications. It turns out that the positive definite functions given by this theorem fit into what is frequently called the scale mixture approach. In other words, they are generated by certain integral transforms defined by products of parameterized positive definite functions on Rd and R. In this paper, we consider positive definite functions on a product of metric spaces which are given by general integral transforms. We provide conditions under which the positive definite functions are either continuous or strictly positive definite. In the case in which one of the metric spaces is Rd, we offer constructions of continuous strictly positive definite functions defined by certain hypergeometric functions and conditionally negative definite functions. They complement and generalize the original Gneiting's contribution. Additionally, we present necessary and sufficient conditions for the strict positive definiteness of the aforementioned generalizations.

Strictly positive definite kernels on the 2-sphere: from radial symmetry to eigenvalue block structure

Abstract The paper introduces a new characterization of strict positive definiteness for kernels on the 2-sphere without assuming the kernel to be radially (isotropic) or axially symmetric. The results use the series expansion of the kernel in spherical harmonics. Then additional sufficient conditions are proven for kernels with a block structure of expansion coefficients. These generalize the result derived by Chen et al. (2003, A necessary and sufficient condition for strictly positive definite functions on spheres. Proc. Amer. Math. Soc., 131, 2733–2740) for radial kernels to nonradial kernels.

Positive definite functions on products of metric spaces via generalized Stieltjes functions

For quasi-metric spaces ( X , ρ ) (X,\rho ) and ( Y , σ ) (Y,\sigma ) and a positive real number λ \lambda , we propose a model for generating positive definite functions G r : { ρ ( x , x ′ ) : x , x ′ ∈ X } × { σ ( y , y ′ ) : y , y ′ ∈ Y } ↦ R G_r: \{\rho (x,x’):x,x’\in X\} \times \{\sigma (y,y’):y,y’ \in Y\} \mapsto \mathbb {R} having the form G r ( t , u ) = 1 h ( u ) r f ( g ( t ) h ( u ) ) , \begin{equation*} G_r(t,u)=\frac {1}{h(u)^r} f\left (\frac {g(t)}{h(u)}\right ), \end{equation*} where r ≥ λ r\geq \lambda , f f belongs to a convex cone S λ b \mathcal {S}_\lambda ^b of bounded completely monotone functions, g g is a nonnegative valued conditionally negative definite function on ( X , ρ ) (X,\rho ) , and h h is a positive valued conditionally negative definite function on ( Y , σ ) (Y,\sigma ) . In the case where ( X , ρ ) (X,\rho ) and ( Y , σ ) (Y,\sigma ) are metric spaces, we determine necessary and sufficient conditions for the strict positive definiteness of the model. The cone S λ b \mathcal {S}_\lambda ^b possesses well-established stability properties that allow alternative formulations of the model leading to many classes of positive definite and strictly positive definite functions on X × Y X\times Y . If X = R d X=\mathbb {R}^d , Y = R Y=\mathbb {R} , ρ \rho is the Euclidean distance on X X , σ 1 / 2 \sigma ^{1/2} is the Euclidean distance on Y Y , g ( t ) = t 2 g(t)=t^2 , t ≥ 0 t\geq 0 , h h is a positive valued function with a completely monotone derivative, and λ = d / 2 \lambda =d/2 , then { G r : r ≥ λ } \{G_r:r\geq \lambda \} is a subset of the Gneiting’s class of covariance space-time functions on X × Y X\times Y frequently dealt with in the literature.

Strict positive definiteness under axial symmetry on the sphere

Axial symmetry for covariance functions defined over spheres has been a very popular assumption for climate, atmospheric, and environmental modeling. For Gaussian random fields defined over spheres embedded in a three-dimensional Euclidean space, maximum likelihood estimation techiques as well kriging interpolation rely on the inverse of the covariance matrix. For any collection of points where data are observed, the covariance matrix is determined through the realizations of the covariance function associated with the underlying Gaussian random field. If the covariance function is not strictly positive definite, then the associated covariance matrix might be singular. We provide conditions for strict positive definiteness of any axially symmetric covariance function. Furthermore, we find conditions for reducibility of an axially symmetric covariance function into a geodesically isotropic covariance. Finally, we provide conditions that legitimate Fourier inversion in the series expansion associated with an axially symmetric covariance function.

Pólya-type criteria for conditional strict positive definiteness of functions on spheres

Identifying (conditionally) strictly positive definite functions is of great importance as they allow the unique solution of certain interpolation problems. We introduce new sufficient (and some necessary) conditions for functions to be conditionally strictly positive definite on all spheres Sd−1, d>2, only employing monotonicity properties. For strictly positive definite and conditionally negative definite functions on all spheres we give a characterisation in terms of monotonicity properties. Further, we prove that multiply monotone functions are positive definite on spheres up to a certain dimension.

Characterization of Strict Positive Definiteness on products of complex spheres

In this paper we consider Positive Definite functions on products $\Omega_{2q}\times\Omega_{2p}$ of complex spheres, and we obtain a condition, in terms of the coefficients in their disc polynomial expansions, which is necessary and sufficient for the function to be Strictly Positive Definite. The result includes also the more delicate cases in which $p$ and/or $q$ can be $1$ or $\infty$. The condition we obtain states that a suitable set in $\mathbb{Z}^2$, containing the indexes of the strictly positive coefficients in the expansion, must intersect every product of arithmetic progressions.

Strictly positive definite multivariate covariance functions on spheres

We study the strict positive definiteness of matrix-valued covariance functions associated to multivariate random fields defined over d-dimensional spheres of the (d+1)-dimensional Euclidean space. Characterization of strict positive definiteness is crucial to both estimation and cokriging prediction in classical geostatistical routines. We provide characterization theorems for high dimensional spheres as well as for the Hilbert sphere. We offer a necessary condition for positive definiteness on the circle. Finally, we discuss a parametric example which might turn to be useful for geostatistical applications.

Strict positive definiteness in geostatistics

Geostatistical modeling is often based on the use of covariance functions, i.e., positive definite functions. However, when interpolation problems have to be solved, it is advisable to consider the subset of strictly positive definite functions. Indeed, it will be argued that ensuring strict positive definiteness for a covariance function is convenient from a theoretical and practical point of view. In this paper, an extensive analysis on strictly positive definite covariance functions has been given. The closure of the set of strictly positive definite functions with respect to the sum and the product of covariance functions defined on the same Euclidean dimensional space or on factor spaces, as well as on partially overlapped lower dimensional spaces, has been analyzed. These results are particularly useful (a) to extend strict positive definiteness in higher dimensional spaces starting from covariance functions which are only defined on lower dimensional spaces and/or are only strictly positive definite in lower dimensional spaces, (b) to construct strictly positive definite covariance functions in space–time as well as (c) to obtain new asymmetric and strictly positive definite covariance functions.

Strict positive definiteness on products of compact two-point homogeneous spaces

ABSTRACTWe present an explicit characterization for the real, continuous, isotropic and strictly positive definite kernels on a product of compact two-point homogeneous spaces, covering almost all possible choices for the spaces. The result complements similar characterizations previously obtained for products of high-dimensional spheres.

Strictly Positive Definite Kernels on a Product of Spheres II

We present, among other things, a necessary and sufficient condition for the strict positive definiteness of an isotropic and positive definite kernel on the cartesian product of a circle and a higher dimensional sphere. The result complements similar results previously obtained for strict positive definiteness on a product of circles [Positivity, to appear, arXiv:1505.01169] and on a product of high dimensional spheres [J. Math. Anal. Appl. 435 (2016), 286-301, arXiv:1505.03695].

Strict positive definiteness of multivariate covariance functions on compact two-point homogeneous spaces

The authors provide a characterization of the continuous and isotropic multivariate covariance functions associated to a Gaussian random field with index set varying over a compact two-point homogeneous space. Sufficient conditions for the strict positive definiteness based on this characterization are presented. Under the assumption that the space is not a sphere, a necessary and sufficient condition is given for the continuous and isotropic multivariate covariance function to be strictly positive definite. Under the same assumption, an alternative necessary and sufficient condition is also provided for the strict positive definiteness of a continuous and isotropic bivariate covariance function based on the main diagonal entries in the matrix representation for the covariance function.

Strictly positive definite kernels on a product of spheres

For the real, continuous, isotropic and positive definite kernels on a product of spheres, one may consider not only its usual strict positive definiteness but also strict positive definiteness restrict to the points of the product that have distinct components. In this paper, we provide a characterization for strict positive definiteness in these two cases, settling all the cases but those in which one of the spheres is a circle.

Strict Positive Definiteness of a Product of Covariance Functions

Positive definiteness represents an admissibility condition for a function to be a covariance. Nevertheless, the more restricted condition of strict positive definiteness has received attention in literature, especially in spatial statistics, since it ensures that the kriging system has a unique solution. Most known covariance functions are isotropic but there are applications where isotropy is not appropriate, e.g., space-time covariance functions. One way to construct non-isotropic covariance functions is to use a product or a product-sum. In this article, it is given a necessary as well as a sufficient condition for a product of two covariance functions to be strictly positive definite. This result is extended to the well-known product-sum covariance model.

On strict positive definiteness of product and product–sum covariance models

Although positive definiteness is a sufficient condition for a function to be a covariance, the stronger strict positive definiteness is important for many applications, especially in spatial statistics, since it ensures that the kriging equations have a unique solution. In particular, spatial–temporal prediction has received a lot of attention, hence strictly positive definite spatial–temporal covariance models (or equivalently strictly conditionally negative definite variogram models) are needed. In this paper the necessary and sufficient condition for the product and the product–sum space–time covariance models to be strictly positive definite (or the variogram function to be strictly conditionally negative definite) is given. In addition it is shown that an example appeared in the recent literature which purports to show that product–sum covariance functions may be only semi-definite is itself invalid. Strict positive definiteness of the sum of products model is also discussed.

Strictly positive definite kernels on subsets of the complex plane

Let ( z, w) ∈ ℂ × ℂ ( zw) be a positive definite kernel and B a subset of ℂ. In this paper, we seek conditions in order that the restriction ( z, w) ∈ B × B( zw) be strictly positive definite. Since this problem has been solved recently in the cases in which B is either ℂ or the unit circle in ℂ, our purpose here is twofold: to present some results we obtained when attempting to solve the problem for the above and other choices of B and to acquaint the audience with some other questions that remain. For two different classes of subsets, we completely characterize the strict positive definiteness of the kernel. We include a complete discussion of the case in which B is the unit circle of ℂ, making a comparison with the classical problem of strict positive definiteness on the real circle.

Perturbed kernel approximation on homogeneous manifolds

Current methods for interpolation and approximation within a native space rely heavily on the strict positive-definiteness of the underlying kernels. If the domains of approximation are the unit spheres in euclidean spaces, then zonal kernels (kernels that are invariant under the orthogonal group action) are strongly favored. In the implementation of these methods to handle real world problems, however, some or all of the symmetries and positive-definiteness may be lost in digitalization due to small random errors that occur unpredictably during various stages of the execution. Perturbation analysis is therefore needed to address the stability problem encountered. In this paper we study two kinds of perturbations of positive-definite kernels: small random perturbations and perturbations by Dunkl's intertwining operators [C. Dunkl, Y. Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications, vol. 81, Cambridge University Press, Cambridge, 2001]. We show that with some reasonable assumptions, a small random perturbation of a strictly positive-definite kernel can still provide vehicles for interpolation and enjoy the same error estimates. We examine the actions of the Dunkl intertwining operators on zonal (strictly) positive-definite kernels on spheres. We show that the resulted kernels are (strictly) positive-definite on spheres of lower dimensions.

A necessary and sufficient condition for strictly positive definite functions on spheres

We give a necessary and sufficient condition for the strict positive-definiteness of real and continuous functions on spheres of dimension greater than one.

Strict positive definiteness on spheres via disk polynomials

We characterize complex strictly positive definite functions on spheres in two cases, the unit sphere of ℂq, q ≥ 3, and the unit sphere of the complex ℓ2. The results depend upon the Fourier‐like expansion of the functions in terms of disk polynomials and, among other things, they enlarge the classes of strictly positive definite functions on real spheres studied in many recent papers.

A complex approach to strict positive definiteness on spheres

We study strictly positive definite functions on the unit sphere of either Cq , q≥2 or its complex extension l 2. We present several characterizations of strict positive deflniteness and use them to actually identify strictly positive definite functions. The role of strict positive deflniteness on real spheres in this new context is examined.