# Permutation Complexity and Coupling Measures in Hidden Markov Models

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## Abstract

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## 1. Introduction

## 2. The Duality between Words and Permutations

- (i)
- Given a permutation, $\pi \in {\varphi}_{n,L}\left({A}_{n}^{L}\right)\subseteq {\mathcal{S}}_{L}$, we decompose the sequence, $\pi (1)\cdots \pi (L)$, of length L into maximal ascending subsequences. A subsequence, ${i}_{j}\cdots {i}_{j+k}$, of a sequence, ${i}_{1}\cdots {i}_{L}$, of length L is called a maximal ascending subsequence if it is ascending, namely, ${i}_{j}\le {i}_{j+1}\le \cdots \le {i}_{j+k}$, and neither ${i}_{j-1}{i}_{j}\cdots {i}_{j+k}$ nor ${i}_{j}{i}_{j+1}\cdots {i}_{j+k+1}$ is ascending;
- (ii)
- If $\pi (1)\cdots \pi ({i}_{1}),\phantom{\rule{4pt}{0ex}}\pi ({i}_{1}+1)\cdots \pi ({i}_{2}),\cdots ,\pi ({i}_{k-1}+1)\cdots \pi (L)$ is a decomposition of $\pi (1)\cdots \pi (L)$ into maximal ascending subsequences, then a word, ${x}_{1:L}\in {A}_{n}^{L}$, is defined by:$$\begin{array}{c}\hfill {x}_{\pi (1)}=\cdots ={x}_{\pi ({i}_{1})}=1,{x}_{\pi ({i}_{1}+1)}=\cdots ={x}_{\pi ({i}_{2})}=2,\cdots ,{x}_{\pi ({i}_{k-1})+1}=\cdots ={x}_{\pi (L)}=k.\end{array}$$

**Theorem 1**

- (i)
- For every $\pi \in {\mathcal{S}}_{L}$,$$|{\varphi}_{n,L}^{-1}(\pi )|=\left(\genfrac{}{}{0pt}{}{L+n-\text{Desc}(\pi )-1}{L}\right),$$
- (ii)
- Let us put:$$\begin{array}{ccc}\hfill {B}_{n,L}& :=& \{{x}_{1:L}\in {A}_{n}^{L}|{\varphi}_{n,L}^{-1}(\pi )=\{{x}_{1:L}\}\phantom{\rule{4.pt}{0ex}}\mathit{\text{for}}\phantom{\rule{4.pt}{0ex}}\mathit{\text{some}}\phantom{\rule{4.pt}{0ex}}\pi \in {\mathcal{S}}_{L}\},\hfill \\ \hfill {C}_{n,L}& :=& \{\pi \in {\mathcal{S}}_{L}||{\varphi}_{n,L}^{-1}(\pi )|=1\}.\hfill \end{array}$$$$\begin{array}{ccc}\hfill {B}_{n,L}& =& \{{x}_{1:L}\in {A}_{n}^{L}|1\le \forall i\le n-1,\phantom{\rule{4pt}{0ex}}1\le \exists j<k\le L\phantom{\rule{4.pt}{0ex}}\text{s.}\phantom{\rule{4.pt}{0ex}}\text{t.}\phantom{\rule{4.pt}{0ex}}{x}_{j}=i+1,{x}_{k}=i\},\hfill \\ \hfill {C}_{n,L}& =& \{\pi \in {\mathcal{S}}_{L}|\text{Desc}(\pi )=n-1\}.\hfill \end{array}$$

**Lemma 2**

## 3. A Result on Finite-State Finite-Alphabet Hidden Markov Models

- (i)
- ${T}_{s{s}^{\prime}}^{(a)}\ge 0$ for any $s,{s}^{\prime}\in \Sigma $ and $a\in A$;
- (ii)
- ${\sum}_{{s}^{\prime},a}{T}_{s{s}^{\prime}}^{(a)}=1$ for any $s\in \Sigma $;
- (iii)
- and $\mu ({s}^{\prime})={\sum}_{s,a}\mu (s){T}_{s{s}^{\prime}}^{(a)}$ for any ${s}^{\prime}\in \Sigma $.

**Lemma 3**

## 4. Permutation Complexity and Coupling Measures

#### 4.1. Fundamental Lemma

**Lemma 4**

#### 4.2. Excess Entropy

**Proposition 5**

#### 4.3. Transfer Entropy and Momentary Information Transfer

**Proposition 6**

**Proposition 7**

#### 4.4. Directed Information

**Proposition 8**

- (i)
- $${I}_{\infty}^{*}(\mathbf{Y}\to \mathbf{X})=\underset{L\to \infty}{lim}{I}^{*}({X}_{L};{Y}_{1:L}|{X}_{1:L-1}).$$$${I}_{\infty}(\mathbf{Y}\to \mathbf{X})=\underset{L\to \infty}{lim}I({X}_{L};{Y}_{1:L}|{X}_{1:L-1});$$
- (ii)
- $$\begin{array}{c}\hfill {I}_{\infty}(D\mathbf{Y}\to \mathbf{X})={I}_{\infty}^{*}(D\mathbf{Y}\to \mathbf{X})=\underset{L\to \infty}{lim}{I}^{*}({X}_{L};{Y}_{1:L-1}|{X}_{1:L-1}).\end{array}$$$$\begin{array}{c}\hfill {I}_{\infty}(D\mathbf{Y}\to \mathbf{X}):=\underset{L\to \infty}{lim}\frac{1}{L}I(D{Y}_{1:L}\to {X}_{1:L})\end{array}$$$$\begin{array}{c}\hfill I(D{Y}_{1:L}\to {X}_{1:L}):=\sum _{i=1}^{L}I({X}_{i};{Y}_{1:i-1}|{X}_{1:i-1}).\end{array}$$
- (iii)
- $$\begin{array}{c}\hfill {I}_{\infty}(\mathbf{Y}\to \mathbf{X}||D\mathbf{Y})={I}_{\infty}^{*}(\mathbf{Y}\to \mathbf{X}||D\mathbf{Y})=\underset{L\to \infty}{lim}{I}^{*}({X}_{L};{Y}_{L}|{X}_{1:L-1},{Y}_{1:L-1}),\end{array}$$$$\begin{array}{c}\hfill {I}_{\infty}(\mathbf{Y}\to \mathbf{X}||D\mathbf{Y}):=\underset{L\to \infty}{lim}\frac{1}{L}I({Y}_{1:L}\to {X}_{1:L}||D{Y}_{1:L})\end{array}$$$$\begin{array}{ccc}\hfill I({Y}_{1:L}\to {X}_{1:L}||D{Y}_{1:L})& =& H({X}_{1:L}||D{Y}_{1:L})-H({X}_{1:L}||{Y}_{1:L},D{Y}_{1:L})\hfill \\ & =& \sum _{i=1}^{L}I({X}_{i};{Y}_{1:i}|{X}_{1:i-1},{Y}_{1:i-1})\hfill \\ & =& \sum _{i=1}^{L}I({X}_{i};{Y}_{i}|{X}_{1:i-1},{Y}_{1:i-1}).\hfill \end{array}$$$$\begin{array}{c}\hfill {I}_{\infty}(\mathbf{Y}\to \mathbf{X}||D\mathbf{Y})=\underset{L\to \infty}{lim}I({X}_{L};{Y}_{L}|{X}_{1:L-1},{Y}_{1:L-1}).\end{array}$$
- (iv)
- $$\begin{array}{c}\hfill {I}_{\infty}^{*}(\mathbf{Y}\to \mathbf{X})={I}_{\infty}^{*}(D\mathbf{Y}\to \mathbf{X})+{I}_{\infty}^{*}(\mathbf{Y}\to \mathbf{X}||D\mathbf{Y}).\end{array}$$$$\begin{array}{c}\hfill {I}_{\infty}(\mathbf{Y}\to \mathbf{X})={I}_{\infty}(D\mathbf{Y}\to \mathbf{X})+{I}_{\infty}(\mathbf{Y}\to \mathbf{X}||D\mathbf{Y});\end{array}$$
- (v)
- $$\begin{array}{c}\hfill {I}_{\infty}^{*}(\mathbf{Y}\to \mathbf{X})+{I}_{\infty}^{*}(D\mathbf{X}\to \mathbf{Y})={I}_{\infty}^{*}(\mathbf{X};\mathbf{Y}).\end{array}$$$$\begin{array}{c}\hfill {I}_{\infty}(\mathbf{Y}\to \mathbf{X})+{I}_{\infty}(D\mathbf{X}\to \mathbf{Y})={I}_{\infty}(\mathbf{X};\mathbf{Y}),\end{array}$$$$\begin{array}{c}\hfill {I}_{\infty}(\mathbf{X};\mathbf{Y}):=\underset{L\to \infty}{lim}\frac{1}{L}I({X}_{1:L};{Y}_{1:L})\end{array}$$

- (i’)
- $${I}_{\infty}^{*}(\mathbf{Y}\to \mathbf{X}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})=\underset{L\to \infty}{lim}{I}^{*}({X}_{L};{Y}_{1:L}|{X}_{1:L-1},{Z}_{1:L}^{1},\cdots ,{Z}_{1:L}^{k}).$$$${I}_{\infty}(\mathbf{Y}\to \mathbf{X}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})=\underset{L\to \infty}{lim}I({X}_{L};{Y}_{1:L}|{X}_{1:L-1},{Z}_{1:L}^{1},\cdots ,{Z}_{1:L}^{k});$$
- (ii’)
- $$\begin{array}{ccc}\hfill {I}_{\infty}(D\mathbf{Y}\to \mathbf{X}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})& =& {I}_{\infty}^{*}(D\mathbf{Y}\to \mathbf{X}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})\hfill \\ & =& \underset{L\to \infty}{lim}{I}^{*}({X}_{L};{Y}_{1:L-1}|{X}_{1:L-1},{Z}_{1:L}^{1},\cdots ,{Z}_{1:L}^{k}).\hfill \end{array}$$$$\begin{array}{c}\hfill {I}_{\infty}(D\mathbf{Y}\to \mathbf{X}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})=\underset{L\to \infty}{lim}I({X}_{L};{Y}_{1:L-1}|{X}_{1:L-1},{Z}_{1:L}^{1},\cdots ,{Z}_{1:L}^{k});\end{array}$$
- (iii’)
- $$\begin{array}{ccc}\hfill {I}_{\infty}(\mathbf{Y}\to \mathbf{X}||D\mathbf{Y},{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})& =& {I}_{\infty}^{*}(\mathbf{Y}\to \mathbf{X}||D\mathbf{Y},{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})\hfill \\ & =& \underset{L\to \infty}{lim}{I}^{*}({X}_{L};{Y}_{L}|{X}_{1:L-1},{Y}_{1:L-1},{Z}_{1:L}^{1},\cdots ,{Z}_{1:L}^{k}),\hfill \end{array}$$$$\begin{array}{c}\hfill {I}_{\infty}(\mathbf{Y}\to \mathbf{X}||D\mathbf{Y},{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})=\underset{L\to \infty}{lim}I({X}_{L};{Y}_{L}|{X}_{1:L-1},{Y}_{1:L-1},{Z}_{1:L}^{1},\cdots ,{Z}_{1:L}^{k}).\end{array}$$
- (iv’)
- $$\begin{array}{c}\hfill {I}_{\infty}^{*}(\mathbf{Y}\to \mathbf{X}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})={I}_{\infty}^{*}(D\mathbf{Y}\to \mathbf{X}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})+{I}_{\infty}^{*}(\mathbf{Y}\to \mathbf{X}||D\mathbf{Y},{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k}).\end{array}$$$$\begin{array}{c}\hfill {I}_{\infty}(\mathbf{Y}\to \mathbf{X}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})={I}_{\infty}(D\mathbf{Y}\to \mathbf{X}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})+{I}_{\infty}(\mathbf{Y}\to \mathbf{X}||D\mathbf{Y},{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k});\end{array}$$
- (v’)
- $$\begin{array}{c}\hfill {I}_{\infty}^{*}(\mathbf{Y}\to \mathbf{X}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})+{I}_{\infty}^{*}(D\mathbf{X}\to \mathbf{Y}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})={I}_{\infty}^{*}(\mathbf{X};\mathbf{Y}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k}).\end{array}$$$$\begin{array}{c}\hfill {I}_{\infty}(\mathbf{Y}\to \mathbf{X}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})+{I}_{\infty}(D\mathbf{X}\to \mathbf{Y}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})={I}_{\infty}(\mathbf{X};\mathbf{Y}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k}),\end{array}$$$$\begin{array}{ccc}\hfill {I}_{\infty}(\mathbf{X};\mathbf{Y}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})& :=& \underset{L\to \infty}{lim}\frac{1}{L}(H({X}_{1:L}||{Z}_{1:L}^{1},\cdots ,{Z}_{1:L}^{k})\hfill \\ & & +H({Y}_{1:L}||{Z}_{1:L}^{1},\cdots ,{Z}_{1:L}^{k})-H({X}_{1:L},{Y}_{1:L}||{Z}_{1:L}^{1},\cdots ,{Z}_{1:L}^{k}))\hfill \end{array}$$$$\begin{array}{c}\hfill {I}_{\infty}(\mathbf{X};\mathbf{Y}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})={I}_{\infty}^{*}(\mathbf{X};\mathbf{Y}||{\mathbf{Z}}^{1},\cdots ,{\mathbf{Z}}^{k})\end{array}$$

## 5. Discussion

- (i)
- for any $i,j\in V$, $(i,j)\notin {E}_{d}$, if and only if ${I}_{\infty}(D{\mathbf{X}}^{i}\to {\mathbf{X}}^{j}||\mathcal{X}\setminus \{{\mathbf{X}}^{i},{\mathbf{X}}^{j}\})=0$;
- (ii)
- for any $i,j\in V$, $(i,j)\notin {E}_{u}$, if and only if ${I}_{\infty}({\mathbf{X}}^{i}\to {\mathbf{X}}^{j}||D{\mathbf{X}}^{i},\mathcal{X}\setminus \{{\mathbf{X}}^{i},{\mathbf{X}}^{j}\})=0$.

- (i’)
- for any $i,j\in V$, $(i,j)\notin {E}_{d}$, if and only if ${I}_{\infty}^{*}(D{\mathbf{X}}^{i}\to {\mathbf{X}}^{j}||\mathcal{X}\setminus \{{\mathbf{X}}^{i},{\mathbf{X}}^{j}\})=0$;
- (ii’)
- for any $i,j\in V$, $(i,j)\notin {E}_{u}$, if and only if ${I}_{\infty}^{*}({\mathbf{X}}^{i}\to {\mathbf{X}}^{j}||D{\mathbf{X}}^{i},\mathcal{X}\setminus \{{\mathbf{X}}^{i},{\mathbf{X}}^{j}\})=0$.

## Acknowledgments

## Conflicts of Interest

## References

- Bandt, C.; Pompe, B. Permutation entropy: A natural complexity measure for time series. Phys. Rev. Lett.
**2002**, 88, e174102. [Google Scholar] [CrossRef] - Amigó, J.M. Permutation Complexity in Dynamical Systems; Springer-Verlag: Berlin/ Heidelberg, Germany, 2010. [Google Scholar]
- Bahraminasab, A.; Ghasemi, F.; Stefanovska, A.; McClintock, P.V.E.; Kantz, H. Direction of coupling from phases of interacting oscillators: A permutation information approach. Phys. Rev. Lett.
**2008**, 100, e084101. [Google Scholar] [CrossRef] - Cao, Y.H.; Tung, W.W.; Gao, J.B.; Protopopescu, V.A.; Hively, L.M. Detecting dynamical changes in time series using the permutation entropy. Phys. Rev. E
**2004**, 70, e046217. [Google Scholar] [CrossRef] - Kugiumtzis, D. Partial transfer entropy on rank vectors. Eur. Phys. J. Special Topics
**2013**, 222, 401–420. [Google Scholar] [CrossRef] - Nakajima, K.; Haruna, T. Symbolic local information transfer. Eur. Phys. J. Special Topics
**2013**, 222, 421–439. [Google Scholar] [CrossRef] - Rosso, O.A.; Larrondo, H.A.; Martin, M.T.; Plastino, A.; Fuentes, M.A. Distinguishing noise from chaos. Phys. Rev. Lett.
**2007**, 99, e154102. [Google Scholar] [CrossRef] - Amigó, J.M.; Keller, K. Permutation entropy: One concept, two approaches. Eur. Phys. J. Special Topics
**2013**, 222, 263–273. [Google Scholar] [CrossRef] - Bandt, C.; Keller, G.; Pompe, B. Entropy of interval maps via permutations. Nonlinearity
**2002**, 15, 1595–1602. [Google Scholar] [CrossRef] - Keller, K.; Sinn, M. A standardized approach to the Kolmogorov-Sinai entropy. Nonlinearity
**2009**, 22, 2417–2422. [Google Scholar] [CrossRef] - Keller, K.; Sinn, M. Kolmogorov-Sinai entropy from the ordinal viewpoint. Phys. D
**2010**, 239, 997–1000. [Google Scholar] [CrossRef] - Keller, K. Permutations and the Kolmogorov-Sinai entropy. Discr. Cont. Dyn. Syst.
**2012**, 32, 891–900. [Google Scholar] [CrossRef] - Keller, K.; Unakafov, A.M.; Unakafova, V.A. On the relation of KS entropy and permutation entropy. Phys. D
**2012**, 241, 1477–1481. [Google Scholar] [CrossRef] - Unakafova, V.A.; Unakafov, A.M.; Keller, K. An approach to comparing Kolmogorov-Sinai and permutation entropy. Eur. Phys. J. Special Topics
**2013**, 222, 353–361. [Google Scholar] [CrossRef] - Amigó, J.M.; Kennel, M.B.; Kocarev, L. The permutation entropy rate equals the metric entropy rate for ergodic information sources and ergodic dynamical systems. Phys. D
**2005**, 210, 77–95. [Google Scholar] [CrossRef] - Amigó, J.M. The equality of Kolmogorov-Sinai entropy and metric permutation entropy generalized. Phys. D
**2012**, 241, 789–793. [Google Scholar] [CrossRef] - Haruna, T.; Nakajima, K. Permutation complexity via duality between values and orderings. Phys. D
**2011**, 240, 1370–1377. [Google Scholar] [CrossRef] - Haruna, T.; Nakajima, K. Permutation excess entropy and mutual information between the past and future. Int. J. Comput. Ant. Sys.
**2012**, in press. [Google Scholar] - Haruna, T.; Nakajima, K. Symbolic transfer entropy rate is equal to transfer entropy rate for bivariate finite-alphabet stationary ergodic Markov processes. Eur. Phys. J. B
**2013**, 86, e230. [Google Scholar] [CrossRef] - Haruna, T.; Nakajima, K. Permutation approach to finite-alphabet stationary stochastic processes based on the duality between values and orderings. Eur. Phys. J. Special Topics
**2013**, 222, 383–399. [Google Scholar] [CrossRef] - Crutchfield, J.P.; Feldman, D.P. Regularities unseen, randomness observed: Levels of entropy convergence. Chaos
**2003**, 15, 25–54. [Google Scholar] [CrossRef] - Schreiber, T. Measuring information transfer. Phys. Rev. Lett.
**2000**, 85, 461–464. [Google Scholar] [CrossRef] [PubMed] - Kaiser, A.; Schreiber, T. Information transfer in continuous processes. Phys. D
**2002**, 166, 43–62. [Google Scholar] [CrossRef] - Pompe, B.; Runge, J. Momentary information transfer as a coupling measure of time series. Phys. Rev. E
**2011**, 83, e051122. [Google Scholar] [CrossRef] - Marko, H. The bidirectional communication theory—A generalization of information theory. IEEE Trans. Commun.
**1973**, 21, 1345–1351. [Google Scholar] [CrossRef] - Massey, J.L. Causality, Feedback and Directed Information. In Proceedings of International Symposium on Information Theory and Its Applications, Waikiki, HI, USA, 27–30 November 1990.
- Anderson, B.D.O. The realization problem for hidden Markov models. Math. Control Signals Syst.
**1999**, 12, 80–120. [Google Scholar] [CrossRef] - Walters, P. An Introduction to Ergodic Theory; Springer-Verlag: New York, NY, USA, 1982. [Google Scholar]
- Seneta, E. Non-Negative Matrices and Markov Chains; Springer: New York, NY, USA, 1981. [Google Scholar]
- Horn, R.A.; Johnson, C.R. Matrix Analysis; Cambridge University Press: Cambridge, UK, 1985. [Google Scholar]
- Cover, T.M.; Thomas, J.A. Elements of Information Theory, 2nd ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2006. [Google Scholar]
- Arnold, D.V. Information-theoretic analysis of phase transitions. Complex Syst.
**1996**, 10, 143–155. [Google Scholar] - Bialek, W.; Nemenman, I.; Tishby, N. Predictability, complexity, and learning. Neural Comput.
**2001**, 13, 2409–2463. [Google Scholar] [CrossRef] [PubMed] - Feldman, D.P.; McTague, C.S.; Crutchfield, J.P. The organization of intrinsic computation: Complexity-entropy diagrams and the diversity of natural information processing. Chaos
**2008**, 18, e043106. [Google Scholar] [CrossRef] [PubMed] - Grassberger, P. Toward a quantitative theory of self-generated complexity. Int. J. Theor. Phys.
**1986**, 25, 907–938. [Google Scholar] [CrossRef] - Li, W. On the relationship between complexity and entropy for Markov chains and regular languages. Complex Syst.
**1991**, 5, 381–399. [Google Scholar] - Shaw, R. The Dripping Faucet as a Model Chaotic System; Aerial Press: Santa Cruz, CA, USA, 1984. [Google Scholar]
- Löhr, W. Models of Discrete Time Stochastic Processes and Associated Complexity Measures. Ph.D. Thesis, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, 2010. [Google Scholar]
- Frenzel, S.; Pompe, B. Partial mutual information for coupling analysis of multivariate time series. Phys. Rev. Lett.
**2007**, 99, e204101. [Google Scholar] [CrossRef] - Amblard, P.O.; Michel, O.J.J. On directed information theory and Granger causality graphs. J. Comput. Neurosci.
**2011**, 30, 7–16. [Google Scholar] [CrossRef] [PubMed] - Ash, R. Information Theory; Wiley Interscience: New York, NY, USA, 1965. [Google Scholar]
- Staniek, M.; Lehnertz, K. Symbolic transfer entropy. Phys. Rev. Lett.
**2008**, 100, e158101. [Google Scholar] [CrossRef] - Kugiumtzis, D. Transfer entropy on rank vectors. J. Nonlin. Sys. Appl.
**2012**, 3, 73–81. [Google Scholar] - Kramer, G. Directed Information for Channels with Feedback. Ph.D. Thesis, Swiss Federal Institute of Technology, Zurich, Switzerland, 1998. [Google Scholar]
- Amblard, P.O.; Michel, O.J.J. Relating Granger causality to directed information theory for networks of stochastic processes. 2011; arXiv:0911.2873v4. [Google Scholar]
- Dahlaus, R.; Eichler, M. Causality and graphical models in time series analysis. In Highly Structured Stochastic Systems; Green, P., Hjort, N., Richardson, S., Eds.; Oxford University Press: New York, NY, USA, 2003; pp. 115–137. [Google Scholar]
- European Physical Journal Special Topics on Recent Progress in Symbolic Dynamics and Permutation Complexity. Eur. Phys. J.
**2013**, 222, 241–598.

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Haruna, T.; Nakajima, K. Permutation Complexity and Coupling Measures in Hidden Markov Models. *Entropy* **2013**, *15*, 3910-3930.
https://doi.org/10.3390/e15093910

**AMA Style**

Haruna T, Nakajima K. Permutation Complexity and Coupling Measures in Hidden Markov Models. *Entropy*. 2013; 15(9):3910-3930.
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**Chicago/Turabian Style**

Haruna, Taichi, and Kohei Nakajima. 2013. "Permutation Complexity and Coupling Measures in Hidden Markov Models" *Entropy* 15, no. 9: 3910-3930.
https://doi.org/10.3390/e15093910