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| | {{context|date=February 2012}} |
| | 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 variable]]s <math>\{X_1,\ldots,X_n\}</math>, the dual total correlation <math>D(X_1,\ldots,X_n)</math> is given by |
| | |
| | :<math> 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) ,</math> |
| | |
| | where <math>H(X_{1},\ldots,X_{n})</math> is the [[joint entropy]] of the variable set <math>\{X_{1},\ldots,X_{n}\}</math> and <math>H(X_{i}| ... )</math> is the [[conditional entropy]] of variable <math>X_{i}</math>, given the rest. |
| | |
| | ==Normalized== |
| | The dual total correlation normalized between [0,1] is simply the dual total correlation divided by its maximum value <math>H(X_{1}, \ldots, X_{n})</math>, |
| | |
| | :<math>ND(X_1,\ldots,X_n) = \frac{D(X_1,\ldots,X_n)}{H(X_1,\ldots,X_n)} .</math> |
| | |
| | ==Bounds== |
| | Dual total correlation is non-negative and bounded above by the joint entropy <math>H(X_1, \ldots, X_n)</math>. |
| | |
| | :<math> 0 \leq D(X_1, \ldots, X_n) \leq H(X_1, \ldots, X_n) .</math> |
| | |
| | Secondly, Dual total correlation has a close relationship with total correlation, <math>C(X_1, \ldots, X_n)</math>. In particular, |
| | |
| | :<math> \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) .</math> |
| | |
| | ==History== |
| | Han (1978) originally defined the dual total correlation as, |
| | : <math> |
| | \begin{align} |
| | & {} \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{align} |
| | </math> |
| | 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: |
| | |
| | : <math> |
| | \begin{align} |
| | & {} \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{align} |
| | </math> |
| | |
| | ==See also== |
| | *[[Mutual information]] |
| | *[[Total correlation]] |
| | |
| | {{no footnotes|date=June 2011}} |
| | |
| | ==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. {{doi|10.1016/S0019-9958(78)91063-X}}. |
| | * 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''. {{doi|10.1140/epjb/e2008-00134-9}}. |
| | * Abdallah S. A. and Plumbley, M. D. (2010). A measure of statistical complexity based on predictive information, ''ArXiv e-prints''. {{arxiv|1012.1890v1}}. |
| | |
| | * Nihat Ay, E. Olbrich, N. Bertschinger (2001). A unifying framework for complexity measures of finite systems. European Conference on Complex Systems. [http://www.cabdyn.ox.ac.uk/complexity_PDFs/ECCS06/Conference_Proceedings/PDF/p202.pdf pdf]. |
| | |
| | [[Category:Information theory]] |
| | [[Category:Probability theory]] |
| | [[Category:Statistical dependence]] |
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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 
, the dual total correlation 
is given by
where 
is the joint entropy of the variable set and 
is the conditional entropy of variable 
, given the rest.
Normalized
The dual total correlation normalized between [0,1] is simply the dual total correlation divided by its maximum value ,
Bounds
Dual total correlation is non-negative and bounded above by the joint entropy .
Secondly, Dual total correlation has a close relationship with total correlation, 
. In particular,
History
Han (1978) originally defined the dual total correlation as,
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c150d73b5cc049c2b6d5f400a91dd5ef8cc4b9ac)
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:
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33514cd7e6d9c48841a022cb018a7dd494ad937d)
See also
Template:No footnotes
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.