Directional statistics

Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Rⁿ), axes (lines through the origin in Rⁿ) or rotations in Rⁿ. More generally, directional statistics deals with observations on compact Riemannian manifolds.

The overall shape of a protein can be parameterized as a sequence of points on the unit sphere. Shown are two views of the spherical histogram of such points for a large collection of protein structures. The statistical treatment of such data is in the realm of directional statistics.^[1]

The fact that 0 degrees and 360 degrees are identical angles, so that for example 180 degrees is not a sensible mean of 2 degrees and 358 degrees, provides one illustration that special statistical methods are required for the analysis of some types of data (in this case, angular data). Other examples of data that may be regarded as directional include statistics involving temporal periods (e.g. time of day, week, month, year, etc.), compass directions, dihedral angles in molecules, orientations, rotations and so on.

Circular and higher-dimensional distributions

Any probability density function $p(x)$ on the line can be "wrapped" around the circumference of a circle of unit radius.^[2] That is, the pdf of the wrapped variable

\theta =x_{w}=x\mod 2\pi \ \ \in (-\pi ,\pi ]

p_{w}(\theta )=\sum _{k=-\infty }^{\infty }{p(\theta +2\pi k)}.

This concept can be extended to the multivariate context by an extension of the simple sum to a number of $F$ sums that cover all dimensions in the feature space:

p_{w}({\vec {\theta }})=\sum _{k_{1}=-\infty }^{\infty }\cdots \sum _{k_{F}=-\infty }^{\infty }{p({\vec {\theta }}+2\pi k_{1}\mathbf {e} _{1}+\dots +2\pi k_{F}\mathbf {e} _{F})}

where $\mathbf {e} _{k}=(0,\dots ,0,1,0,\dots ,0)^{\mathsf {T}}$ is the $k$ th Euclidean basis vector.

The following sections show some relevant circular distributions.

von Mises circular distribution

The von Mises distribution is a circular distribution which, like any other circular distribution, may be thought of as a wrapping of a certain linear probability distribution around the circle. The underlying linear probability distribution for the von Mises distribution is mathematically intractable; however, for statistical purposes, there is no need to deal with the underlying linear distribution. The usefulness of the von Mises distribution is twofold: it is the most mathematically tractable of all circular distributions, allowing simpler statistical analysis, and it is a close approximation to the wrapped normal distribution, which, analogously to the linear normal distribution, is important because it is the limiting case for the sum of a large number of small angular deviations. In fact, the von Mises distribution is often known as the "circular normal" distribution because of its ease of use and its close relationship to the wrapped normal distribution (Fisher, 1993).

The pdf of the von Mises distribution is:

f(\theta ;\mu ,\kappa )={\frac {e^{\kappa \cos(\theta -\mu )}}{2\pi I_{0}(\kappa )}}

where

I_{0}

is the modified Bessel function of order 0.

Circular uniform distribution

The probability density function (pdf) of the circular uniform distribution is given by

U(\theta )=1/(2\pi ).\,

Wrapped normal distribution

The pdf of the wrapped normal distribution (WN) is:

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

where μ and σ are the mean and standard deviation of the unwrapped distribution, respectively and

\vartheta (\theta ,\tau )

is the Jacobi theta function:

\vartheta (\theta ,\tau )=\sum _{n=-\infty }^{\infty }(w^{2})^{n}q^{n^{2}}

where

w\equiv e^{i\pi \theta }

and

q\equiv e^{i\pi \tau }.

Wrapped Cauchy distribution

The pdf of the wrapped Cauchy distribution (WC) is:

WC(\theta ;\theta _{0},\gamma )=\sum _{n=-\infty }^{\infty }{\frac {\gamma }{\pi (\gamma ^{2}+(\theta +2\pi n-\theta _{0})^{2})}}={\frac {1}{2\pi }}\,\,{\frac {\sinh \gamma }{\cosh \gamma -\cos(\theta -\theta _{0})}}

where

\gamma

is the scale factor and

\theta _{0}

is the peak position.

Wrapped Lévy distribution

The pdf of the Wrapped Lévy distribution (WL) is:

f_{WL}(\theta ;\mu ,c)=\sum _{n=-\infty }^{\infty }{\sqrt {\frac {c}{2\pi }}}\,{\frac {e^{-c/2(\theta +2\pi n-\mu )}}{(\theta +2\pi n-\mu )^{3/2}}}

where the value of the summand is taken to be zero when

\theta +2\pi n-\mu \leq 0

c

is the scale factor and

\mu

is the location parameter.

Distributions on higher-dimensional manifolds

Three points sets sampled from different Kent distributions on the sphere.

There also exist distributions on the two-dimensional sphere (such as the Kent distribution^[3]), the N-dimensional sphere (the von Mises-Fisher distribution^[4]) or the torus (the bivariate von Mises distribution^[5]).

The von Mises–Fisher distribution is a distribution on the Stiefel manifold, and can be used to construct probability distributions over rotation matrices.^[6]

The Bingham distribution is a distribution over axes in N dimensions, or equivalently, over points on the (N − 1)-dimensional sphere with the antipodes identified.^[7] For example, if N = 2, the axes are undirected lines through the origin in the plane. In this case, each axis cuts the unit circle in the plane (which is the one-dimensional sphere) at two points that are each other's antipodes. For N = 4, the Bingham distribution is a distribution over the space of unit quaternions. Since a unit quaternion corresponds to a rotation matrix, the Bingham distribution for N = 4 can be used to construct probability distributions over the space of rotations, just like the Matrix-von Mises–Fisher distribution.

These distributions are for example used in geology,^[8] crystallography^[9] and bioinformatics.^[1] ^[10] ^[11]

Moments

The raw vector (or trigonometric) moments of a circular distribution are defined as

m_{n}=E(z^{n})=\int _{\Gamma }P(\theta )z^{n}d\theta \,

where $\Gamma$ is any interval of length $2\pi$ , $P(\theta )$ is the PDF of the circular distribution, and $z=e^{i\theta }$ . Since the integral $P(\theta )$ is unity, and the integration interval is finite, it follows that the moments of any circular distribution are always finite and well defined.

Sample moments are analogously defined:

{\overline {m}}_{n}={\frac {1}{N}}\sum _{i=1}^{N}z_{i}^{n}.

The population resultant vector, length, and mean angle are defined in analogy with the corresponding sample parameters.

\rho =m_{1}\,

R=|m_{1}|\,

\theta _{\mu }=\mathrm {Arg} (m_{1}).\,

In addition, the lengths of the higher moments are defined as:

R_{n}=|m_{n}|\,

while the angular parts of the higher moments are just $(n\theta _{\mu })\mod 2\pi$ . The lengths of the higher moments will all lie between 0 and 1.

Measures of location and spread

Main article: Mean of circular quantities

Various measures of location and spread may be defined for both the population and a sample drawn from that population.^[12] The most common measure of location is the circular mean. The population circular mean is simply the first moment of the distribution while the sample mean is the first moment of the sample. The sample mean will serve as an unbiased estimator of the population mean.

When data is concentrated, the median and mode may be defined by analogy to the linear case, but for more dispersed or multi-modal data, these concepts are not useful.

The most common measures of circular spread are:

The circular variance. For the sample the circular variance is defined as:

{\overline {\mathrm {Var} (z)}}=1-{\overline {R}}\,

and for the population

\mathrm {Var} (z)=1-R\,

Both will have values between 0 and 1.

The circular standard deviation

S(z)={\sqrt {\ln(1/R^{2})}}={\sqrt {-2\ln(R)}}\,

{\overline {S}}(z)={\sqrt {\ln(1/{\overline {R}}^{2})}}={\sqrt {-2\ln({\overline {R}})}}\,

with values between 0 and infinity. This definition of the standard deviation (rather than the square root of the variance) is useful because for a wrapped normal distribution, it is an estimator of the standard deviation of the underlying normal distribution. It will therefore allow the circular distribution to be standardized as in the linear case, for small values of the standard deviation. This also applies to the von Mises distribution which closely approximates the wrapped normal distribution. Note that for small

S(z)

, we have

S(z)^{2}=2\mathrm {Var} (z)

The circular dispersion

\delta ={\frac {1-R_{2}}{2R^{2}}}

{\overline {\delta }}={\frac {1-{{\overline {R}}_{2}}}{2{\overline {R}}^{2}}}

with values between 0 and infinity. This measure of spread is found useful in the statistical analysis of variance.

Distribution of the mean

Given a set of N measurements $z_{n}=e^{i\theta _{n}}$ the mean value of z is defined as:

{\overline {z}}={\frac {1}{N}}\sum _{n=1}^{N}z_{n}

which may be expressed as

{\overline {z}}={\overline {C}}+i{\overline {S}}

where

{\overline {C}}={\frac {1}{N}}\sum _{n=1}^{N}\cos(\theta _{n}){\text{ and }}{\overline {S}}={\frac {1}{N}}\sum _{n=1}^{N}\sin(\theta _{n})

or, alternatively as:

{\overline {z}}={\overline {R}}e^{i{\overline {\theta }}}

where

{\overline {R}}={\sqrt {{\overline {C}}^{2}+{\overline {S}}^{2}}}\,\,\,\mathrm {and} \,\,\,\,{\overline {\theta }}=\mathrm {ArcTan} ({\overline {S}},{\overline {C}}).

The distribution of the mean ( ${\overline {\theta }}$ ) for a circular pdf P(θ) will be given by:

P({\overline {C}},{\overline {S}})\,d{\overline {C}}\,d{\overline {S}}=P({\overline {R}},{\overline {\theta }})\,d{\overline {R}}\,d{\overline {\theta }}=\int _{\Gamma }...\int _{\Gamma }\prod _{n=1}^{N}\left[P(\theta _{n})\,d\theta _{n}\right]

where $\Gamma$ is over any interval of length $2\pi$ and the integral is subject to the constraint that ${\overline {S}}$ and ${\overline {C}}$ are constant, or, alternatively, that ${\overline {R}}$ and ${\overline {\theta }}$ are constant.

The calculation of the distribution of the mean for most circular distributions is not analytically possible, and in order to carry out an analysis of variance, numerical or mathematical approximations are needed.^[13]

The central limit theorem may be applied to the distribution of the sample means. (main article: Central limit theorem for directional statistics). It can be shown^[14] that the distribution of $[{\overline {C}},{\overline {S}}]$ approaches a bivariate normal distribution in the limit of large sample size.

Goodness of fit and significance testing

For cyclic data - (e.g., is it uniformly distributed) :

Rayleigh test for a unimodal cluster
Kuiper's test for possibly multimodal data.

References

1 2 "Hamelryck, T., Kent, J., Krogh, A. (2006) Sampling realistic protein conformations using local structural bias. PLoS Comput. Biol., 2(9): e131". Public Library of Science (PLoS). Retrieved 2008-02-01.
↑ Bahlmann, C., (2006), Directional features in online handwriting recognition, Pattern Recognition, 39
↑ Kent, J (1982) The Fisher–Bingham distribution on the sphere. J Royal Stat Soc, 44, 71–80.
↑ Fisher, RA (1953) Dispersion on a sphere. Proc. Roy. Soc. London Ser. A., 217, 295–305
↑ Mardia, KM. Taylor, CC., Subramaniam, GK. (2007) Protein Bioinformatics and Mixtures of Bivariate von Mises Distributions for Angular Data. Biometrics, 63, 505–512
↑ Downs, (1972) Orientational statistics. Biometrica, 59, 665–676
↑ Bingham, C. (1974) An Antipodally Symmetric Distribution on the Sphere. Ann. Stat., 2, 1201-1225.
↑ Peel, D., Whiten, WJ., McLachlan, GJ. (2001) Fitting mixtures of Kent distributions to aid in joint set identification. J. Am. Stat. Ass., 96, 56–63
↑ Krieger Lassen, N. C., Juul Jensen, D. & Conradsen, K. (1994) On the statistical analysis of orientation data. Acta Crystallogr., A50, 741–748.
↑ Kent, J.T., Hamelryck, T. (2005). Using the Fisher–Bingham distribution in stochastic models for protein structure. In S. Barber, P.D. Baxter, K.V.Mardia, & R.E. Walls (Eds.), Quantitative Biology, Shape Analysis, and Wavelets, pp. 57–60. Leeds, Leeds University Press
↑ "Boomsma, W., Mardia, KV., Taylor, CC., Ferkinghoff-Borg, J., Krogh, A., Hamelryck, T. (2008) A generative, probabilistic model of local protein structure. Proc. Natl. Acad. Sci. USA, 105(26), 8932-8937". Retrieved 2008-06-26.
↑ Fisher, NI., Statistical Analysis of Circular Data, Cambridge University Press, 1993. ISBN 0-521-35018-2
↑ Jammalamadaka, S. Rao; Sengupta, A. (2001). Topics in Circular Statistics. World Scientific Publishing Company. ISBN 978-981-02-3778-3. Retrieved 2010-03-03.
↑ Jammalamadaka, S. Rao; SenGupta, A. (2001). Topics in circular statistics. New Jersey: World Scientific. ISBN 981-02-3778-2. Retrieved 2011-05-15.

Books on directional statistics

Batschelet, E. Circular statistics in biology, Academic Press, London, 1981. ISBN 0-12-081050-6.
Fisher, NI., Statistical Analysis of Circular Data, Cambridge University Press, 1993. ISBN 0-521-35018-2
Fisher, NI., Lewis, T., Embleton, BJJ. Statistical Analysis of Spherical Data, Cambridge University Press, 1993. ISBN 0-521-45699-1
Mardia, KV. and Jupp P., Directional Statistics (2nd edition), John Wiley and Sons Ltd., 2000. ISBN 0-471-95333-4

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Directional statistics

Circular and higher-dimensional distributions

von Mises circular distribution

Circular uniform distribution

Wrapped normal distribution

Wrapped Cauchy distribution

Wrapped Lévy distribution

Distributions on higher-dimensional manifolds

Moments

Measures of location and spread

Distribution of the mean

Goodness of fit and significance testing

See also

References

Books on directional statistics