# Entropy and Recurrence Measures of a Financial Dynamic System by an Interacting Voter System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Price Process Modeling by a Finite-Range Voter System

^{d}might have one of two possible opinions on a political issue (in favor or against), at independent exponential times; an individual reassesses his view by choosing a neighbor at random with certain probabilities and then adopting his position. Let η

_{s}(x) be the position of voter x at time s (s ≥ 0); the political position of the voters can be denoted by η

_{s}={ η

_{s}(x) : x ∈ ℤ

^{d}}, which is a mapping function from ℤ

^{d}to {0, 1}, i.e., η

_{s}(x) = 1 if x ∈ η

_{s}and η

_{s}(x) = 0 if x ∉ η

_{s}. Then, the dynamics of the voter model can be formulated as follows: (i) if x ∈ η

_{s}, then x becomes vacant at a rate equal to the number of vacant neighbors; (ii) if x ∉ η

_{s}, then x becomes occupied at a rate equal to λ times the number of occupied neighbors, where λ(> 1) is an intensity called the “carcinogenic advantage”. When λ = 1, the model is called the basic voter model, and it is called the biased voter model when λ > 1. Let ${\eta}_{s}^{A}$ denote the state at time s with the initial state set ${\eta}_{0}^{A}=A$ and ${\eta}_{s}^{\{0\}}\left(x\right)$ be the state of x ∈ ℤ

^{d}at time s for ${\eta}_{0}^{\{0\}}=\{0\}$. More generally, we consider the initial distribution as v

_{θ}, the product measure with density θ (each site is independently occupied with probability θ) and let ${\eta}_{s}^{\theta}$ denote the voter model with initial distribution v

_{θ}. More formally, the stochastic dynamics of voter model η

_{s}is a Markov process on a configuration space ${\{0,1\}}^{{\mathbb{Z}}^{d}}$, whose generator has the form:

^{x}(z) = η(z) if z ≠ x, η

^{x}(z) = 1 − η(x) if z = x, for x, z ∈ ℤ

^{d}. c(x, η) is the transition rate function for the process, which is given by the following (see [30–32]). For any x ∈ ℤ

^{d}, the state of x ∈ ℤ

^{d}flips according to the transition rates:

^{d}and ${\sum}_{y\u03f5{\mathbb{Z}}^{d}}p\left(x,y\right)}=1$ for all x ∈ ℤ

^{d}. Here, we suppose that the transition probability p(x, y) is translation invariant and symmetric and such that the Markov chain with those transition probabilities is irreducible [15,43]. If a site x ∈ ℤ

^{d}is occupied by a one (resp. zero), then at rate one (resp. λ), it picks a site y ∈ ℤ

^{d}with probability p(x, y) and adopts the state of the individual at y. For the biased voter model (λ > 1), there exists a “critical value” for the process, which is defined as ${\mathrm{\lambda}}_{\mathrm{c}}=\text{inf}\{\mathrm{\lambda}:P(|{\eta}_{s}^{\{0\}}|\phantom{\rule{0.2em}{0ex}}>0$, for all s ≥ 0) > 0}, where $|{\eta}_{s}^{\{0\}}|$ is the cardinality of ${\eta}_{s}^{\{0\}}$. Assume λ > λ

_{c}, then there is a convex set C, so that on ${\mathrm{\Omega}}_{\infty}=\{{\eta}_{s}^{\{0\}}\ne \varnothing ,\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}\text{all}\phantom{\rule{0.2em}{0ex}}s\}$, we have for any ϵ > 0 and for all s sufficiently large, $(1-\u03f5)sC\phantom{\rule{0.2em}{0ex}}\cap \phantom{\rule{0.2em}{0ex}}{\mathbb{Z}}^{d}\subset {\eta}_{s}^{\{0\}}\subset (1+\u03f5)sC\cap {\mathbb{Z}}^{d}$. If λ < λ

_{c}, for some positive ρ(λ), we have $P({\eta}_{s}^{\{0\}}\ne \varnothing )\le {e}^{-\rho s}$. The above results imply that, on a d-dimensional lattice, the process becomes vacant exponentially for λ < λ

_{c}; the process survives with the positive probability for λ > λ

_{c}.

^{ℤ}[31,32], since the graphical representation is very useful to illustrate and simulate the model. We start by constructing the process η

_{s}from a collection of Poisson processes in the case λ ≥ 1. For each pair x, y ∈ ℤ with |x − y| ≤ R (R is the finite-range), let $\{{T}_{n}^{(x,y)}:n\ge 1\}$ and $\{{U}_{n}^{(x,y)}:n\ge 1\}$ be independent Poisson processes with rate one and λ − 1, respectively. At times ${T}_{n}^{(x,y)}$, we draw an arrow from y to x and put a δ at x. At times ${U}_{n}^{(x,y)}$, we just draw an arrow from y to x. Then, the process is obtained from the graphical representation as follows: At time ${T}_{n}^{(x,y)}$, the state of site x imitates the state of site y, i.e., becomes occupied by a one (resp. zero) if site y is occupied by a one (resp. zero). At time ${U}_{n}^{(x,y)}$, the site x becomes occupied by a one if y is occupied by a one, and the state of site x is not affected if y is occupied by a zero. A figure illustration of the construction of the graphical representation for a one-dimensional-biased voter model with neighbor range R = 3 is presented in Figure 1. We imagine fluid entering the bottom and flowing up the structure. The δ’s are like dams, and the arrows are like pipes, which allow the fluid to flow in the indicated direction.

_{t}(s), where s ∈ [0, l]. Suppose that the stock market consists of 2M + 1 (M is large enough) traders, who are located in a line {−M,⋯, −1, 0, 1,⋯, M} ⊂ ℤ (similarly for a d-dimensional lattice ℤ

^{d}). At the very beginning of each trading day, we select a certain proportion of traders (with the initial distribution v

_{θ}) randomly in the system and consider them as those who receive some market news. We define a random variable ζ

_{t}with the values +1, −1, 0 to represent that these investors hold a buying opinion, selling opinion or neutral opinion with probabilities p

_{+1}, p

_{−1}or 1 − (p

_{+1}+ p

_{−1}), respectively. Then, these investors send a bullish, bearish or neutral signal to their finite-range neighbors. According to the d-dimensional voter process system, investors can affect each other or the news can be spread, which is supposed as the main factor of price fluctuations for the market. The aggregate excess demand for the asset at time t is defined by:

_{0}is the stock price at Time 0. The corresponding stock logarithmic return and absolute return from t − 1 to t are defined by:

_{c}and gradually change the value to make the simulated returns be as close as possible to the real market indices. For each setting, 20 repetitions are performed, and the averaged time series is the simulation data for analysis. Meanwhile, we select the daily prices of two important actual market stock indexes in comparison with the simulation ones, the Hang Seng Index (HSI) from 31 December 1990 to 18 March 2013, with 5530 data points, and the Shanghai Stock Exchange (SSE) Composite Index from the period 3 March 1993 to 22 February 2013, with 5142 data points. The normalized plots of these prices and the logarithmic return for simulation data with parameter setting {λ = 1.7, R = 3} are shown in Figure 2.

## 3. Comparison Empirical Analysis of Complexity Behaviors

#### 3.1. Composite Multiscale Entropy Analysis

- For an one-dimensional time series x = {x
_{1}, x_{2}, ⋯, x_{N}}, consecutive coarse-grained time series are constructed by averaging a successively increasing number of points within non-overlapping windows. Unlike the MSE algorithm in which each of the coarse-grained time series {y^{(}^{τ}^{)}} is computed as ${y}_{j}^{(\tau )}=\frac{1}{\tau}{\displaystyle \sum _{i=(j-1)\tau +1}^{j\tau}{x}_{i}}$, the k-th coarse-grained time series in the CMSE method for a scale factor τ, ${\mathrm{y}}_{k}^{(\tau )}=\{{y}_{k,1}^{(\tau )},{y}_{k,2}^{(\tau )},\cdots ,{y}_{k,p}^{(\tau )}\}$ is defined as:$${y}_{k,j}^{(\tau )}=\frac{1}{\tau}{\displaystyle \sum _{i=(j-1)\tau +k}^{j\tau +k-1}{x}_{i},\phantom{\rule{1em}{0ex}}1\le j\le N/\tau ,\phantom{\rule{0.2em}{0ex}}1\le k}\le \tau .$$Note that for τ = 1, the coarse-grained time series is simply the original time series. Figure 3 shows a schematic illustration of the coarse-graining procedure for both MSE (a) and CMSE (b) with τ = 2 and τ = 3, respectively, from which a clear difference between these two methods can be seen. - The entropy measure, the sample entropy (SampEn), is calculated for each coarse-grained time series and then plotted as a function of the scale factor. SampEn quantifies the regularity or predictability of a time series, which is defined as the negative logarithm of the conditional probability that a point that repeats itself within a tolerance of ϵ in an m-dimensional phase space will repeat itself in an m + 1-dimensional phase space:$$\text{SampEn}=-\mathrm{log}[C(m+1,\u03f5)/C(m,\u03f5)]$$$$\text{CMSE}(\mathrm{x},\tau ,m,\u03f5)=\frac{1}{\tau}{\displaystyle \sum _{k=1}^{\tau}\text{SampEn}}({\mathrm{y}}_{k}^{(\tau )},m,\u03f5)$$

^{q}, for simulation data and actual data from Scale 1 to Scale 30 (τ = 1 to 30), where q is taken as 0.25, 0.5, 0.75, 1, 1.5 and 2, respectively. It is known that the absolute return is a proxy of the volatility of time series, and |r|

^{q}exhibit obvious different volatility behaviors for different q [19]. The entropy value of each coarse-grained time series is calculated with phase space embedding dimension m = 6 for minimizing the fraction of false neighbors [38] and ϵ = 0.15σ, where σ denotes the standard deviation of the original time series. From the figure, we find that values of the composite multiscale entropy of return series monotonically decrease as the scale factor increases for either actual indexes or simulation data from the financial price model. For the absolute return with different power exponents |r(t)|

^{q}, it is observable that the entropy values decrease gradually and, finally, remain almost constant as the scale factor becomes larger, which indicate that each of these time series, unlike the return series, contains correlations and complex structures across multiple time scales. The behavior for each of these volatility time series is similar to that of 1/f series, which has the correlated fluctuations (but the degrees of correlations are different). Meanwhile, it is observed that the entropy values for all given scales become smaller with the power exponents q becoming larger, suggesting decreasing complex structures. The simulative data for the financial price model show similar fluctuation behaviors to the actual SSE and HSI data.

^{q}series for both the financial price model and actual market indexes. The results can be found in Table 1. It is seen that the Hurst exponents of return series for both the simulation data and the actual ones are around 0.5, indicating weak correlations. The Hurst exponents for |r(t)|

^{q}are all much larger than 0.5, which means that the time series are long-range auto-correlated. However, for each time series, the Hurst exponents do not show the obvious monotonic relationship with power exponent q. Therefore, we cannot intuitively conclude that the stronger correlations of time series correspond to higher complex structures or entropy values.

#### 3.2. Recurrence Plot and Recurrence Quantification Analysis

_{t}= {x

_{t}, x

_{t}

_{−Δ}

_{t}, x

_{t}

_{−2Δ}

_{t}, ⋯, x

_{t}

_{−(}

_{m}

_{−1)Δ}

_{t}}, where m denotes the embedding dimension and Δt is the time delay. Then, the Euclidean distance matrix

**R**using the independence time-delayed coordinates is calculated:

_{min}is the minimal length of a diagonal line that is defined by l

_{min}= 2. DET provides an indication of determinism and predictability in the system; thus the larger the value of DET, the more predictable the system with diagonal lines in RP. Another measure on the diagonal line is the Shannon information entropy (L

_{ENT}) defined for diagonal line collections:

_{ENT}refers to the increase of complexity of the time series. Moreover, the mean length of the diagonal lines ${L}_{mean}={\displaystyle {\sum}_{l={l}_{\mathrm{min}}}^{{N}_{R}}{l}_{p}(l)}$ is a parameter indicating the system stability. Instead of considering diagonal lines, we measure vertical recurrence lines. In analogy to determinism, the laminarity (LAM) is defined for vertical line patterns:

_{min}= 2. The larger the laminarity parameter, the more stable the behavior of the system. Finally, the average vertical line length (TT) is given as $\mathrm{TT}={\displaystyle {\sum}_{v={v}_{\mathrm{min}}}^{{N}_{R}}vp(v)}$, and it estimates the mean time that the system remains at a specific state.

_{mean}values decrease when the h changes from 0.02 to 0.04, except the DET value for HSI, which becomes 0.7793 for h = 0.04 from 0.7685 for h = 0.02. Since the higher DET and L

_{mean}correspond to a more predictable and stable system, thus the results imply the reduction of determinism of these return series in the case of h = 0.04. It is also noticed that the L

_{mean}for HIS under h = 0.02 is much smaller than those for other series, indicating the weakest predictability among them. Furthermore, it is observable that all of the values of L

_{ENT}have increased for h = 0.04, and its value for the return series with R = 1 has a relatively considerable rise, which indicate the increasing of the complexity of these time series. In terms of measures of vertical recurrence lines, the values of LAM and TT enlarge when h becomes 0.04, which means a rise in the fraction of recurrence points forming vertical lines, and it can also obviously be seen in Figure 7.

## 4. Conclusions

^{q}for different values of q are also observable. Finally, the recurrence plots and recurrence quantification analysis are utilized to further explore the complexity of the return series. For recurrence thresholds h = 0.02 and h = 0.04, the RQA measures present apparently different values, indicating different complex determinism behaviors. Based on the statistical research, we can observe that there is some evidence of similar complex behaviors of the returns of the financial model derived from the finite-range voter system to the real stock markets; this shows that the proposed model can grasp some nature of the real stock market in certain respects and is reasonable for real stock price modeling.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 2.**(

**a**) Normalized price series for simulation data and actual data; (

**b**) return and absolute return series for simulation data with parameter {λ = 1.7, R = 3}.

**Figure 3.**(

**a**) Schematic illustrations of the MSE and (

**b**) composite multiscale entropy (CMSE) procedures, respectively.

**Figure 4.**Composite multiscale entropy of return series r and absolute return form |r|

^{q}with q = 0.25, 0.5, 0.75, 1, 1.5, 2 for simulation data and actual indexes, respectively.

**Figure 5.**(

**a**) CMSE of return series and the corresponding shuffled time series; (

**b**) CMSE of absolute returns and the corresponding shuffled time series.

**Table 1.**Hurst exponent of |r|

^{q}for simulation data and actual data. HSI, Hang Seng Index; SSE, Shanghai Stock Exchange.

data | return | q = 0.25 | q = 0.5 | q = 0.75 | q = 1 | q = 1.5 | q = 2 |
---|---|---|---|---|---|---|---|

{λ = 1.3, R = 1} | 0.52026 | 0.63220 | 0.74669 | 0.81593 | 0.85684 | 0.88665 | 0.87644 |

{λ = 1.5, R = 2} | 0.51066 | 0.65271 | 0.68973 | 0.76012 | 0.80596 | 0.84566 | 0.84251 |

{λ = 1.7, R = 3} | 0.44587 | 0.63266 | 0.74921 | 0.82767 | 0.87982 | 0.92975 | 0.93350 |

HSI | 0.52570 | 0.85341 | 0.88780 | 0.89769 | 0.89280 | 0.85217 | 0.79179 |

SSE | 0.52271 | 0.72742 | 0.79722 | 0.81659 | 0.81297 | 0.76844 | 0.70621 |

Data | RR (h = 0.02, 0.04) | DET (h = 0.02, 0.04) | L_{mean}(h = 0.02, 0.04) | |||
---|---|---|---|---|---|---|

R = 1 | 0.0005 | 0.0027 | 0.9044 | 0.6649 | 46.7234 | 3.6071 |

R = 2 | 0.0007 | 0.0046 | 0.8373 | 0.6817 | 15.6688 | 3.2662 |

R = 3 | 0.0005 | 0.0018 | 0.9336 | 0.6779 | 51.0233 | 4.5073 |

HSI | 0.0010 | 0.0163 | 0.7685 | 0.7793 | 7.2597 | 3.5210 |

SSE | 0.0006 | 0.0021 | 0.8503 | 0.5994 | 20.4435 | 3.9045 |

Data | L_{ENT}(h = 0.02, 0.04) | LAM (h = 0.02, 0.04) | TT (h = 0.02, 0.04) | |||
---|---|---|---|---|---|---|

R = 1 | 0.3921 | 1.1084 | 0.0049 | 0.0671 | 2.0000 | 2.3217 |

R = 2 | 0.7699 | 1.2140 | 0.0095 | 0.0989 | 2.3333 | 2.4383 |

R = 3 | 0.7629 | 1.1562 | 0.0000 | 0.0449 | 0.0000 | 2.6015 |

HSI | 1.1184 | 1.6163 | 0.0072 | 0.2814 | 2.0000 | 2.8932 |

SSE | 0.4499 | 0.8834 | 0.1154 | 0.1334 | 2.3116 | 2.6532 |

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Niu, H.-L.; Wang, J. Entropy and Recurrence Measures of a Financial Dynamic System by an Interacting Voter System. *Entropy* **2015**, *17*, 2590-2605.
https://doi.org/10.3390/e17052590

**AMA Style**

Niu H-L, Wang J. Entropy and Recurrence Measures of a Financial Dynamic System by an Interacting Voter System. *Entropy*. 2015; 17(5):2590-2605.
https://doi.org/10.3390/e17052590

**Chicago/Turabian Style**

Niu, Hong-Li, and Jun Wang. 2015. "Entropy and Recurrence Measures of a Financial Dynamic System by an Interacting Voter System" *Entropy* 17, no. 5: 2590-2605.
https://doi.org/10.3390/e17052590