Wrapped Normal Distribution - Definition

Definition

The probability density function of the wrapped normal distribution is

f_{WN}(\theta;\mu,\sigma)=\frac{1}{\sigma \sqrt{2\pi}} \sum^{\infty}_{k=-\infty} \exp \left

where μ and σ are the mean and standard deviation of the unwrapped distribution, respectively. Expressing the above density function in terms of the characteristic function of the normal distribution yields:

f_{WN}(\theta;\mu,\sigma)=\frac{1}{2\pi}\sum_{n=-\infty}^\infty e^{-\sigma^2n^2/2+in(\theta-\mu)} =\frac{1}{2\pi}\vartheta\left(\frac{\theta-\mu}{2\pi},\frac{i\sigma^2}{2\pi}\right) ,

where is the Jacobi theta function, given by

\vartheta(\theta,\tau)=\sum_{n=-\infty}^\infty (w^2)^n q^{n^2} \text{ where } w \equiv e^{i\pi \theta} and

The wrapped normal distribution may also be expressed in terms of the Jacobi triple product:

where and

Read more about this topic:  Wrapped Normal Distribution

Famous quotes containing the word definition:

    “One definition of man is “an intelligence served by organs.””
    —Ralph Waldo Emerson (1803–1882)

    “No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
    —Florence Nightingale (1820–1910)

    “Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.”
    —The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)