Quantum dilogarithm

My name is Jestine (34 years old) and my hobbies are Origami and Microscopy.

Here is my web site; http://Www.hostgator1centcoupon.info/ (support.file1.com) In information theory, dual total correlation (Han 1978) or excess entropy (Olbrich 2008) is one of the two known non-negative generalizations of mutual information. While total correlation is bounded by the sum entropies of the n elements, the dual total correlation is bounded by the joint-entropy of the n elements. Although well behaved, dual total correlation has received much less attention than the total correlation. A measure known as "TSE-complexity" defines a continuum between the total correlation and dual total correlation (Ay 2001).

Definition

For a set of n random variables $\{X_{1},\ldots ,X_{n}\}$, the dual total correlation $D(X_{1},\ldots ,X_{n})$ is given by

$D(X_{1},\ldots ,X_{n})=H\left(X_{1},\ldots ,X_{n}\right)-\sum _{i=1}^{n}H\left(X_{i}|X_{1},\ldots ,X_{i-1},X_{i+1},\ldots ,X_{n}\right),$

where $H(X_{1},\ldots ,X_{n})$ is the joint entropy of the variable set $\{X_{1},\ldots ,X_{n}\}$ and $H(X_{i}|...)$ is the conditional entropy of variable $X_{i}$, given the rest.

Normalized

The dual total correlation normalized between [0,1] is simply the dual total correlation divided by its maximum value $H(X_{1},\ldots ,X_{n})$,

$ND(X_{1},\ldots ,X_{n})={\frac {D(X_{1},\ldots ,X_{n})}{H(X_{1},\ldots ,X_{n})}}.$

Bounds

Dual total correlation is non-negative and bounded above by the joint entropy $H(X_{1},\ldots ,X_{n})$.

$0\leq D(X_{1},\ldots ,X_{n})\leq H(X_{1},\ldots ,X_{n}).$

Secondly, Dual total correlation has a close relationship with total correlation, $C(X_{1},\ldots ,X_{n})$. In particular,

${\frac {C(X_{1},\ldots ,X_{n})}{n-1}}\leq D(X_{1},\ldots ,X_{n})\leq (n-1)\;C(X_{1},\ldots ,X_{n}).$

History

Han (1978) originally defined the dual total correlation as,

${\begin{aligned}&{}\qquad D(X_{1},\ldots ,X_{n})\\[10pt]&\equiv \left[\sum _{i=1}^{n}H(X_{1},\ldots ,X_{i-1},X_{i+1},\ldots ,X_{n})\right]-(n-1)\;H(X_{1},\ldots ,X_{n})\;.\end{aligned}}$

However Abdallah and Plumbley (2010) showed its equivalence to the easier-to-understand form of the joint entropy minus the sum of conditional entropies via the following:

${\begin{aligned}&{}\qquad D(X_{1},\ldots ,X_{n})\\[10pt]&\equiv \left[\sum _{i=1}^{n}H(X_{1},\ldots ,X_{i-1},X_{i+1},\ldots ,X_{n})\right]-(n-1)\;H(X_{1},\ldots ,X_{n})\\&=\left[\sum _{i=1}^{n}H(X_{1},\ldots ,X_{i-1},X_{i+1},\ldots ,X_{n})\right]+(1-n)\;H(X_{1},\ldots ,X_{n})\\&=H(X_{1},\ldots ,X_{n})+\left[\sum _{i=1}^{n}H(X_{1},\ldots ,X_{i-1},X_{i+1},\ldots ,X_{n})-H(X_{1},\ldots ,X_{n})\right]\\&=H\left(X_{1},\ldots ,X_{n}\right)-\sum _{i=1}^{n}H\left(X_{i}|X_{1},\ldots ,X_{i-1},X_{i+1},\ldots ,X_{n}\right)\;.\end{aligned}}$

See also

• Mutual information
• Total correlation

References

• Han T. S. (1978). Nonnegative entropy measures of multivariate symmetric correlations, Information and Control 36, 133–156.
• Fujishige Satoru (1978). Polymatroidal Dependence Structure of a Set of Random Variables, Information and Control 39, 55–72. 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park..
• Olbrich, E. and Bertschinger, N. and Ay, N. and Jost, J. (2008). How should complexity scale with system size?, The European Physical Journal B - Condensed Matter and Complex Systems. 21 year-old Glazier James Grippo from Edam, enjoys hang gliding, industrial property developers in singapore developers in singapore and camping. Finds the entire world an motivating place we have spent 4 months at Alejandro de Humboldt National Park..
• Abdallah S. A. and Plumbley, M. D. (2010). A measure of statistical complexity based on predictive information, ArXiv e-prints. Template:Arxiv.

• Nihat Ay, E. Olbrich, N. Bertschinger (2001). A unifying framework for complexity measures of finite systems. European Conference on Complex Systems. pdf.