2.2.2. Wrapped Distributions

The easiest to visualize circular distribution, or rather group of distributions, that is discussed is the set of wrapped distributions. The wrapped distributions take a distribution on the real line and wrap it onto the unit circle according to

$$p\_w(\theta) = \sum\_{j=-\infty}^{\infty} p(\theta + 2\pi j)\,\mu$$

where the support of *p*(·) is an interval of R, and the domain of *pw* is an interval on R with length 2*π*. For example, wrapping a normal distribution takes the pdf

$$p\_{\rm tr}(\mathbf{x};\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left\{-\frac{(\mathbf{x}-\mu)^2}{2\sigma^2}\right\},$$

where the domain of *x* is R, and *μ* and *σ* are the mean and standard deviation, respectively, and wraps it, resulting in the wrapped pdf (in this case wrapped normal)

$$p\_{wn}(\theta; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \sum\_{j=-\infty}^{\infty} \exp\left\{ -\frac{(\theta - \mu + 2\pi j)^2}{2\sigma^2} \right\},$$

where the domain of *θ* is an interval on R with length 2*π*. Zero-mean normal distributions with varying values of *σ* are wrapped, with the results shown in Figure 1b.

Recall the weight function of the Rogers-Szeg˝o polynomials in Equation (4). As the log function is monotonically increasing, the term log(1/*q*) increases monotonically as *q* decreases. Observing the extremes of *q*: as *q* approaches 1, log(1/*q*) approaches 0, and as *q* approaches 0, log(1/*q*) approaches ∞. Letting log(1/*q*) = *σ*<sup>2</sup> and *μ* = 0, this becomes a zero-mean wrapped normal distribution.

It is clear from Figure 1 that both distributions described previously have strong similarities to the unwrapped normal distribution. Figure 1 also shows the difference in the standard deviation parameter. Whereas the wrapped normal distribution directly uses the standard deviation of the unwrapped distribution, the von Mises distribution is with respect to concentration parameter that is inversely related to the dispersion of the random variable. This makes the wrapped normal distribution slightly more intuitive when comparing with an unwrapped normal distribution.

## 2.2.3. Directional Statistics

The estimation of stochastic variables generally relies on calculating the statistics of that variable. Most notable of these statistics are the mean and variance, the first two central moments. For pdfs (*p*(*x*)) on the real line that are continuously integrable, the central moments are given as

$$\begin{array}{ll} \mu\_1 &= \int\_{\mathbb{S}} \mathbf{x} p(\mathbf{x}) d\mathbf{x} \\\\ \mu\_k &= \int\_{\mathbb{S}} (\mathbf{x} - \mu\_1)^k p(\mathbf{x}) d\mathbf{x} \quad k \ge 2, \end{array} \tag{6}$$

where S is the support of *p*(*x*). Although less utilized in general, raw moments are commonly used in directional statistics and are given as

$$
\rho\_k = \int\_{\mathbb{S}} \mathbf{x}^k p(\mathbf{x}) d\mathbf{x},
$$

or

$$\rho\_k = \int\_{\mathbb{S}} \theta^k p(\theta) d\theta,$$

where the slight distinction is that the integration variable is within the variable *ϑ* = *<sup>e</sup>iθ*. In addition, mean direction and circular variance are not the first and second central moments [24]. Instead, both are calculated from the first moment's angle (*θ*1) and length (*R*1):

$$\theta\_1 = \arg(\rho\_1) = \tan^{-1} \frac{\text{imag}(\rho\_1)}{\text{real}(\rho\_1)} \tag{7a}$$

$$\mathcal{R}\_1 = \left\| \rho\_1 \right\|\,\prime\,\tag{7b}$$

where · is the *l*2-norm. From Mardia [24], the length can be used to calculate the circular variance *V*1 and circular standard deviation *σ*1 according to

$$V\_1 = 1 - R\_1 \tag{8a}$$

$$
\sigma\_1 = \sqrt{-2\ln(\mathcal{R}\_1)}\,. \tag{8b}
$$

Effectively, as the length of the moment decreases, the concentration of the pdf about the mean direction decreases and the unwrapped standard deviation (USTD) increases. Note that while the subscript in Equations (7) and (8) is 1, there are corresponding mean directions and lengths associated with all moments; however, these are rarely used in applications.
