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 $\{X_1,\dots ,X_n\}$, the dual total correlation $D(X_1,\dots ,X_n)$ is given by

$D(X_1,\dots ,X_n)=H(X_1,\dots ,X_n)-\sum _{i=1}^{n}H\left(X_i\right|X_1,\dots ,X_{i-1},X_{i+1},\dots ,X_n),$

where $H(X_1,\dots ,X_n)$ is the joint entropy of the variable set $\{X_1,\dots ,X_n\}$ 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 $H(X_1,\dots ,X_n)$,

$ND(X_1,\dots ,X_n)=\frac{D(X_1,\dots ,X_n)}{H(X_1,\dots ,X_n)}.$

Bounds

Dual total correlation is non-negative and bounded above by the joint entropy $H(X_1,\dots ,X_n)$.

$0\le D(X_1,\dots ,X_n)\le H(X_1,\dots ,X_n).$

Secondly, Dual total correlation has a close relationship with total correlation, $C(X_1,\dots ,X_n)$. In particular,

$\frac{C(X_1,\dots ,X_n)}{n-1}\le D(X_1,\dots ,X_n)\le (n-1)C(X_1,\dots ,X_n).$

History

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

$\begin{array}{rl}& D(X_1,\dots ,X_n)\\ & \equiv [\sum _{i=1}^{n}H(X_1,\dots ,X_{i-1},X_{i+1},\dots ,X_n)]-(n-1)H(X_1,\dots ,X_n).\end{array}$

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:

$\begin{array}{rl}& D(X_1,\dots ,X_n)\\ & \equiv [\sum _{i=1}^{n}H(X_1,\dots ,X_{i-1},X_{i+1},\dots ,X_n)]-(n-1)H(X_1,\dots ,X_n)\\ & =[\sum _{i=1}^{n}H(X_1,\dots ,X_{i-1},X_{i+1},\dots ,X_n)]+(1-n)H(X_1,\dots ,X_n)\\ & =H(X_1,\dots ,X_n)+[\sum _{i=1}^{n}H(X_1,\dots ,X_{i-1},X_{i+1},\dots ,X_n)-H(X_1,\dots ,X_n)]\\ & =H(X_1,\dots ,X_n)-\sum _{i=1}^{n}H\left(X_i\right|X_1,\dots ,X_{i-1},X_{i+1},\dots ,X_n).\end{array}$

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.