arXiv subjects

Tamio Koyama

Publications and source records attributed to Tamio Koyama.

10 recordsLinked to original sources

arXiv2020

Holonomic gradient method for the probability content of a simplex region with a multivariate normal distribution

We use the holonomic gradient method to evaluate the probability content of a simplex region under a multivariate normal distribution. This probability equals to the integral of the probability density function of the multivariate Gaussian distribution on the simplex region. For this purpose, we generalize the inclusion--exclusion identity which was given for polyhedra, to the faces of a polyhedron. This extended inclusion--exclusion identity enables us to calculate the derivatives of the function associated with the probability content of a polyhedron in general position. We show that these derivatives can be written as integrals of the faces of the polyhedron.

math.ST↗

arXiv2019

The Annihilating Ideal of the Fisher Integral

In this paper, we discuss a system of differential equations for the Fisher integral on the special orthogonal group. Especially, we explicitly give a set of linear differential operators which generates the annihilating ideal of the Fisher integral, and we prove that the annihilating ideal is a maximal left ideal of the ring of differential operators with polynomial coefficients. Our proof is given by a discussion concerned with an annihilating ideal of a Schwartz distribution associated with the Haar measure on the special orthogonal group. We also give differential operators annihilating the Fisher integral for the diagonal matrix by a new approach.

math.CA↗

arXiv2019

Holonomic Gradient Method for Two Way Contingency Tables

The holonomic gradient method gives an algorithm to efficiently and accurately evaluate normalizing constants and their derivatives. We apply the holonomic gradient method in the case of the conditional Poisson or multinomial distribution on two way contingency tables. We utilize the modular method in computer algebra for an efficient and exact evaluation, and we discuss on complexities of these algorithms and their implementation. We also discuss on a theoretical aspect of the distribution from the viewpoint of the conditional maximum likelihood estimation.

math.CA↗

arXiv2018

An integral formula for the powered sum of the independent, identically and normally distributed random variables

The distribution of the sum of r-th power of standard normal random variables is a generalization of the chi-squared distribution. In this paper, we represent the probability density function of the random variable by an one-dimensional absolutely convergent integral with the characteristic function. Our integral formula is expected to be applied for evaluation of the density function. Our integral formula is based on the inversion formula, and we utilize a summation method. We also discuss on our formula in the view point of hyperfunctions.

math.CA↗

arXiv2015

Holonomic gradient method for distribution function of a weighted sum of noncentral chi-square random variables

We apply the holonomic gradient method to compute the distribution function of a weighted sum of independent noncentral chi-square random variables. It is the distribution function of the squared length of a multivariate normal random vector. We treat this distribution as an integral of the normalizing constant of the Fisher-Bingham distribution on the unit sphere and make use of the partial differential equations for the Fisher-Bingham distribution.

math.ST↗

arXiv2015

Holonomic modules associated with multivariate normal probabilities of polyhedra

The probability content of a convex polyhedron with a multivariate normal distribution can be regarded as a real analytic function. We give a system of linear partial differential equations with polynomial coefficients for the function and show that the system induces a holonomic module. The rank of the holonomic module is equal to the number of nonempty faces of the convex polyhedron, and we provide an explicit Pfaffian equation (an integrable connection) that is associated with the holonomic module. These are generalizations of results for the Schläfli function that were given by Aomoto.

math.CA↗

arXiv2013

Holonomic Gradient Descent for the Fisher-Bingham Distribution on the $d$-dimensional Sphere

We propose an accelerated version of the holonomic gradient descent and apply it to calculating the maximum likelihood estimate (MLE) of the Fisher-Bingham distribution on a $d$-dimensional sphere. We derive a Pfaffian system (an integrable connection) and a series expansion associated with the normalizing constant with an error estimation. These enable us to solve some MLE problems up to dimension $d=7$ with a specified accuracy.

math.ST↗

arXiv2013

The Holonomic Rank of the Fisher-Bingham System of Differential Equations

The Fisher-Bingham system is a system of linear partial differential equations satisfied by the Fisher-Bingham integral for the $n$-dimensional sphere $S^n$. The system is given in [Nakayama et al. (2011), Theorem 2] and it is shown that it is a holonomic system [Koyama]. We show that the holonomic rank of the system is equal to $2n+2$.

math.CA↗

arXiv2013

Calculation of orthant probabilities by the holonomic gradient method

We apply the holonomic gradient method (HGM) introduced by [9] to the calculation of orthant probabilities of multivariate normal distribution. The holonomic gradient method applied to orthant probabilities is found to be a variant of Plackett's recurrence relation ([14]). However an implementation of the method yields recurrence relations more suitable for numerical computation than Plackett's recurrence relation. We derive some theoretical results on the holonomic system for the orthant probabilities. These results show that multivariate normal orthant probabilities possess some remarkable properties from the viewpoint of holonomic systems. Finally we show that numerical performance of our method is comparable or superior compared to existing methods.

math.NA↗

arXiv2012

A Holonomic Ideal Annihilating the Fisher-Bingham Integral

We calculate the integration ideal of annihilating differential operators of the non-normalized Fisher-Bingham distribution and show that the ideal agrees with the set of operators for the Fisher-Bingham integral given in "Holonomic Gradient Descent and its Application to the Fisher-Bingham Integral". They conjectured that the set generates a holonomic ideal and we prove their conjecture.

math.CA↗