# Directional statistics - Entropy of wrapped normal (Jacobi theta) distribution

• Rap
In summary, the conversation discusses the wrapped distribution around the unit circle, which is periodic with a period of 2π. The (differential) entropy for this distribution is calculated using an integral with a Jacobi theta function and a given interval of length 2π. The speaker is having trouble finding the entropy for a specific wrapped normal distribution and is seeking guidance on how to calculate it, potentially using numerical integration or an asymptotic expansion.
Rap
If p(x) is a probability distribution on the real number line, the "wrapped" distribution around the unit circle is:
$$p_w(\theta)=\sum_{n=-\infty}^\infty p(\theta+2\pi n)$$
which is periodic with period 2π. The (differential) entropy is:
$$H=-\int_\Gamma p_w(\theta)\ln[p_w(\theta)]\,d\theta$$
where $\Gamma$ is any interval of length 2π. I am having trouble finding the entropy for the wrapped normal distribution. For the wrapped normal distribution with zero mean,
$$p_w(\theta)=\frac{\vartheta _3\left(\frac{\theta }{2},e^{-\frac{\sigma ^2}{2}}\right)}{2 \pi }$$
which is in Mathematica notation, $\vartheta _3(\cdot)$ is the Jacobi theta function:
$$\vartheta _3(u,q)=1+2\sum_{n=1}^\infty q^{n^2}\cos(2nu)$$
Is anyone familiar enough with Jacobi theta functions to give some guidance in calculating the entropy? Mathematica is not giving much help.

Last edited:
You'd be lucky if a closed form expression exists. Perhaps try numerical integration or an asymptotic expansion?

## 1. What is directional statistics?

Directional statistics is a branch of statistics that deals with data that is directional or angular in nature. This includes data such as wind direction, compass headings, and angles of rotation. Unlike traditional statistics that assume data is linear, directional statistics takes into account the circular nature of this type of data.

## 2. What is entropy of a wrapped normal distribution?

The entropy of a wrapped normal distribution is a measure of the uncertainty or randomness of the data. It is a way to quantify the amount of information contained in a set of directional data. Entropy is calculated using the probability density function of the wrapped normal distribution.

## 3. What is the Jacobi theta function?

The Jacobi theta function is a special function used in mathematics and physics. It is defined as a sum of an infinite series and is often used in the study of elliptic functions and modular forms. In directional statistics, the Jacobi theta function is used to calculate the probability density function of the wrapped normal distribution.

## 4. How is the wrapped normal distribution different from the normal distribution?

The wrapped normal distribution is different from the normal distribution in that it takes into account the circular nature of data. Unlike the normal distribution, which assumes data is linear, the wrapped normal distribution can be used to model data on a circular scale, such as compass headings or wind directions.

## 5. What are some applications of directional statistics?

Directional statistics has many applications in various fields, including meteorology, geology, biology, and robotics. It can be used to model wind direction and speed, animal migration patterns, and orientation of geological structures. In robotics, directional statistics is used to determine the orientation and movement of robots in space.

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