A function class of strictly positive definite and logarithmically completely monotonic functions related to the modified Bessel functions

We investigate a connection between spatial statistics and statistical physics to obtain new covariance functions with direct physical interpretation for spatial random fields. These covariance functions are based on the exponential Boltzmann-Gibbs representation and use an energy functional to represent interactions between the values of the random field at different points in space. This formulation results in closed-form generalized covariance functions, which display infinite variance in Euclidean spaces of dimension larger than one. We propose regularization schemes in real and reciprocal (spectral) space that lead to well-behaved covariance structures. The real-space regularization parameter allows a continuous interpolation between the Boltzmann-Gibbs covariance and the exponential covariance. We also propose discretized approximations on regular grids, and we show that they represent reparametrized versions of the well-known Besag and Leroux lattice models. We then discuss parameter estimation and spatial prediction for the regularized Boltzmann-Gibbs covariance model in two dimensions. We recommend using the pairwise difference likelihood that combines satisfactory estimation performance and good scalability with many observation points. The predictive performance of the regularized covariance function is assessed by means of cross-validation statistics. Irregularlyspaced samples from the Walker Lake dataset are used, and spatial prediction is conducted by means of ordinary kriging. The regularized Boltzmann-Gibbs covariance yields improved predictive performance compared to the exponential covariance model.

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“…The negative exponential is known to be positive definite on R for any ∈ N * . Following [14,Proposition 4], the function…”

Section: Theoretical Resultsmentioning

confidence: 99%

Linking physics and spatial statistics: A new family of Boltzmann-Gibbs random fields

Allard

Hristopulos

Opitz

2021

Electron. J. Statist.

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“…(1 − t 2 ) α−1/2 cos(xt) dt, see e.g. [19,34]. This representation admits the natural extension to the complex plane…”

Section: Example 4 Consider the Real-variable Bessel Function Of The First Kind J α (X) For α > −mentioning

confidence: 99%

Positive-definiteness and integral representations for special functions

Buescu¹,

Paixão²

2020

Positivity

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It is known that a holomorphic positive definite function f defined on a horizontal strip of the complex plane may be characterized as the Fourier-Laplace transform of a unique exponentially finite measure on R. In this paper we apply this complex integral representation to specific families of special functions, including the , ζ and Bessel functions, and construct explicitly the corresponding measures, thus providing new insight into the nature of complex positive and co-positive definite functions. In the case of the ζ function this process leads to a new proof of an integral representation on the critical strip.

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“…There have been plenty of literature dedicating to the study and applications of completely monotonic functions, logarithmically completely monotonic functions, and completely monotonic degrees. For better motivation of this paper, we would like to recommend three papers [5][6][7] in 2018, three papers [8][9][10] in 2019, three papers [11][12][13] in 2020, three papers [14][15][16] in 2021, three papers [17][18][19] in 2022, three papers [20][21][22] in 2023, and the newlypublished expository, survey, and review article [23] in 2024.…”

Section: Introductionmentioning

confidence: 99%

Monotonicity and complete monotonicity of two functions constituted via three derivatives of a function involving trigamma function

Han

Yin

2020

Preprint

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In the paper, by convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein's theorem for completely monotonic functions, and other analytic techniques, the author (1) presents the decreasing monotonicity of a ratio constituted via three derivatives of a function involving trigamma function; (2) discovers necessary and sufficient conditions for a function constituted via three derivatives of a function involving trigamma function to be completely monotonic. These results conform previous guesses posed by the author.

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A function class of strictly positive definite and logarithmically completely monotonic functions related to the modified Bessel functions

Cited by 10 publications

References 5 publications

Linking physics and spatial statistics: A new family of Boltzmann-Gibbs random fields

Linking physics and spatial statistics: A new family of Boltzmann-Gibbs random fields

Positive-definiteness and integral representations for special functions

Monotonicity and complete monotonicity of two functions constituted via three derivatives of a function involving trigamma function

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