# 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

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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.
https://doi.org/10.3390/e15093910

**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