📂Probability Distribution

Reasons Why the Modified Bessel Function of the First Kind Appears in Directional Statistics

Buildup

Modified Bessel Functions

$$J_{\nu}(x) = \sum \limits_{n=0}^{\infty} \frac{(-1)^{n} }{\Gamma (n+1) \Gamma (n+\nu+1)} \left(\frac{x}{2} \right)^{2n+\nu}$$

The $I_{\nu}$ defined as follows for the Bessel function of the first kind $J_{\nu}$ is called the modified Bessel function of the first kind¹.

$$\begin{align*} I_{\nu} (z) :=& i^{-\nu} J_{\nu} \left( iz \right) \\ =& \left( {{ z } \over { 2 }} \right)^{\nu} \sum_{k=0}^{\infty} {{ {{ z } \over { 2 }}^{2k} } \over { k! \Gamma \left( \nu + k + 1 \right) }} \\ =& {{ \left( {{ z } \over { 2 }} \right)^{\nu} } \over { \sqrt{\pi} \Gamma \left( \nu + {{ 1 } \over { 2 }} \right) }} \int_{-1}^{1} e^{zt} \left( 1 - t^{2} \right)^{\nu - {{ 1 } \over { 2 }}} dt \end{align*}$$

Directional Statistics

On the other hand, Directional Statistics is a field that studies probability distributions and statistical inferences in manifolds, rather than in the usual Euclidean space. For example, it deals with data placed on spheres like the Earth, represented by Spheres, and on tori reflected by $2 \pi$ modulus, like Torus, and can be applied to spatial statistics (on spheres) such as the distance between points, as well as to angles between molecules (on tori), showing a bright future for this subdivision. However, the distributions that appear here all have strange probability density functions as follows. $$f_{p} \left( \mathbf{x} ; \mu , \kappa \right) := \left( {{ \kappa } \over { 2 }} \right)^{p/2 - 1} {{ 1 } \over { \Gamma \left( p/2 \right) I_{p/2 - 1} \left( \kappa \right) }} \exp \left( \kappa \mu^{T} \mathbf{x} \right) \qquad , \mathbf{x} \in S^{p-1}$$ The complex factor in front is a constant that normalizes $\int_{S^{p-1}} f d \mathbf{x} = 1$, including the modified Bessel function of the first kind $I_{\nu}$. This complex function is used for a simple reason.

Solutions

Derivation of the Bessel Function of the First Kind: For $\nu \in \mathbb{R}$, the differential equation of the following form is called the $\nu$ order Bessel equation. $$\begin{align*} && x^{2} y^{\prime \prime} +xy^{\prime}+(x^{2}-\nu^{2})y &= 0 \\ \text{or} && y^{\prime \prime}+\frac{1}{x} y^{\prime} + \left( 1-\frac{\nu^{2}}{x^{2}} \right)y &= 0 \end{align*}$$ Bessel’s equation is a differential equation that appears when solving the wave equation in spherical coordinates. The coefficient is not constant and depends on the independent variable $x$. Solutions can be found using the Frobenius Method, and the series solution looks as follows. $$\begin{align*} J_{\nu}(x) &= \sum \limits_{n=0}^{\infty} \frac{(-1)^{n} }{\Gamma (n+1) \Gamma (n+\nu+1)} \left(\frac{x}{2} \right)^{2n+\nu} \\ J_{-\nu}(x) & =\sum \limits_{n=0}^{\infty}\frac{(-1)^{n}}{\Gamma (n+1)\Gamma (n-\nu+1)} \left( \frac{x}{2} \right)^{2n-\nu} \end{align*}$$

Bessel functions and their derivation involve complex expressions, like Bessel equation being one of the differential equations and its solutions. However, in the provided quote about directional statistics, only one sentence is of importance.

“Bessel’s equation is a differential equation that appears when solving the wave equation in spherical coordinates.”

In mathematical physics, there might be discussions about wave equations and such, but what we really need is only $\int_{S^{p-1}} f d \mathbf{x} = 1$. The issue is, unlike in ordinary Euclidean space, the probability density function values on the sphere do not experience the phenomenon of ‘getting farther from the center and closer to $0$,’ making integration itself not an easy task. Imagine the shape covered by the probability density function of the normal distribution wrapping around a circle $S^{1}$.

Looking at when $\tau = 1$ in the above figure, the infinitely long tail of the normal distribution is adding infinitely thin layers while circling around $S^{1}$ infinitely, not being $0$. ² This repeats with $2 \pi$, twice the pi, as its period, and this is exactly why the Bessel function can be used, as it creates a ‘wave on the sphere’ like shape.