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)

    “The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
    —Samuel Taylor Coleridge (1772–1834)

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)