kent distribution

Author

kishore

ABSTRACT:

Circular data is data that is measured on a circle in degrees or radians. It is fundamentally different from linear data due to its periodic nature (0° = 360°). Circular data arise in various ways. The two main ways correspond to the two principal circular measuring instruments, the compas and the clock.

Typical observations measured by the compass include wind directions and directions of migrating birds. Data of a similar type arise from measurements by spirit,level or protractor. Typical observations measured by the clock include the arrival times (on a 24-hour clock) of patients at a casualty unit in a hospital. Data of a similar type arise as times of year (or times of month) of appropriate events.

Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions,axes (lines through the origin in Rn) or rotations in Rn. More generally, directional statistics deals with observations on compact Riemannian manifolds including the Stiefel manifold. One of the directional distribution is Kent Distribution, which is discussed in this article.

INTRODUCTION:

The Kent distribution, also known as the 5-parameter Fisher-Bingham distribution (named after John T. Kent, Ronald Fisher, and Christopher Bingham), is a probability distribution in ℜ3, the real three dimensional coordinate space, of a two-dimensional unit sphere.

The Bingham, and more generally the Fisher–Bingham distribution, are constructed by constraining multivariate normal distributions to lie on the surface of a sphere of unit radius. They are used in modeling spherical data which usually represent directions but in some cases they can also be used in shape analysis.

FORMULA:

The Kent distribution’s probability density function, f(x), is given by the equation:

Here,

x is a three dimensional unit value.

c(κ, β) is a normalizing constant, and is given by the equation

In the above equation, Iv(κ) represents what is called the modified Bessel function.

Generalization to higher dimensions:

The Kent distribution can be easily generalized to spheres in higher dimensions. If x is a point on the unit sphere S^{{p-1}} in{R}^{p} , then the density function of the p-dimensional Kent distribution is proportional to:

where sum _{j=2}^{p}\beta _{j}=0} and 0<=2|beta _{j}|<k and the vectors {gamma_{j}| j=1… p}are orthonormal. However, the normalization constant becomes very difficult to work with for p>3.

APPLICATION:

1)Gaussian tracking with Kent-distributed direction-of-arrival measurements

Target tracking and localisation based on noisy direction-ofarrival (DOA) measurements is important in many applications such as advanced vehicular systems, mobile communication systems and sonar. This problem is usually posed in a Bayesian framework in which the objective is to calculatethe posterior density, which refers to the density of the current state given current and past measurements, as it contains all information of interest.

In non-linear/non-Gaussian systems, the posterior does not have a closed-form expression and must be approximated. For these systems, it is appealing to develop computationally efficient Gaussian filters, which provide a Gaussian approximation to the posterior with a suitable performance. Examples of these

Gaussian filters are the extended Kalman filter (EKF) and sigma-point Kalman filters, e.g. the unscented Kalman filter (UKF).

2)Modelling of directional data using Kent distributions

The modelling of data on a spherical surface requires the consideration of directional probability distributions. To model asymmetrically distributed data on a three-dimensional sphere, Kent distributions are often used. The moment estimates of the parameters are typically used in modelling tasks involving Kent distributions. However, these lack a rigorous statistical treatment.

PROBLEMS AND SOLUTION:

1)Modelling of directional data using Kent distributions

CONCLUSION:

Kent distribution is a Directional statistics where it has multiple applications such as modelling of data, gaussian tracking, and also in deep learning and AI. It is used on circular data effectively .

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