Tsallis distribution
Encyclopedia
In q-analog
theory and statistical mechanics
, a Tsallis distribution is a probability distribution
derived from the maximization of the Tsallis entropy
under appropriate constraints. There are several different families of Tsallis distributions, yet different sources may reference an individual family as "the Tsallis distribution". The q-Gaussian is a generalization of the Gaussian in the same way that Tsallis entropy
is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. Similarly, if the domain of the variable is constrained to be positive in the maximum entropy
procedure, the q-exponential distribution is derived.
The Tsallis distributions have been applied to problems in the fields of statistical mechanics
, geology
, anatomy
, astronomy
, economics
, finance
, and machine learning
. The distributions are often used for their heavy tails.
Q-analog
Roughly speaking, in mathematics, specifically in the areas of combinatorics and special functions, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1...
theory and statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, a Tsallis distribution is a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
derived from the maximization of the Tsallis entropy
Tsallis entropy
In physics, the Tsallis entropy is a generalization of the standard Boltzmann-Gibbs entropy. In the scientific literature, the physical relevance of the Tsallis entropy is highly debated...
under appropriate constraints. There are several different families of Tsallis distributions, yet different sources may reference an individual family as "the Tsallis distribution". The q-Gaussian is a generalization of the Gaussian in the same way that Tsallis entropy
Tsallis entropy
In physics, the Tsallis entropy is a generalization of the standard Boltzmann-Gibbs entropy. In the scientific literature, the physical relevance of the Tsallis entropy is highly debated...
is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. Similarly, if the domain of the variable is constrained to be positive in the maximum entropy
Maximum entropy probability distribution
In statistics and information theory, a maximum entropy probability distribution is a probability distribution whose entropy is at least as great as that of all other members of a specified class of distributions....
procedure, the q-exponential distribution is derived.
The Tsallis distributions have been applied to problems in the fields of statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, geology
Geology
Geology is the science comprising the study of solid Earth, the rocks of which it is composed, and the processes by which it evolves. Geology gives insight into the history of the Earth, as it provides the primary evidence for plate tectonics, the evolutionary history of life, and past climates...
, anatomy
Anatomy
Anatomy is a branch of biology and medicine that is the consideration of the structure of living things. It is a general term that includes human anatomy, animal anatomy , and plant anatomy...
, astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, finance
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...
, and machine learning
Machine learning
Machine learning, a branch of artificial intelligence, is a scientific discipline concerned with the design and development of algorithms that allow computers to evolve behaviors based on empirical data, such as from sensor data or databases...
. The distributions are often used for their heavy tails.
Procedure
In a similar procedure to how the normal distribution can be derived using the standard Boltzmann–Gibbs entropy or Shannon entropy, the q-Gaussian can be derived from a maximization of the Tsallis entropy subject to the appropriate constraints.Further reading
- Juniper, J. (2007) "The Tsallis Distribution and Generalised Entropy: Prospects for Future Research into Decision-Making under Uncertainty", Centre of Full Employment and Equity, The University of Newcastle, Australia