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Markov-binary visibility graph: A new method for analyzing complex systems
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Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins 5 6
Markov-binary visibility graph: A new method for analyzing complex systems
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Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran
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Sodeif Ahadpour ⇑, Yaser Sadra, Zahra ArastehFard
a r t i c l e Q3
i n f o
Article history: Received 2 April 2012 Received in revised form 4 April 2013 Accepted 3 March 2014 Available online xxxx
a b s t r a c t In order to stu te dynamics of complex systems has been used information recorded in the form of time series. These time series can be investigated from a complex network perspective. Using two-state Markov chain and the binary visibility graph, we investigate these time series. Moreover, several topological aspects of the constructed graph, such as degree distribution, clustering coefficient, and mean visibility length are studied. Our results show that the Markov-binary visibility algorithm stands as a simple method to discriminate statistically dependent and independent systems. Some remarkable examples confirm the reliability of Markov-binary visibility graph for time series analysis. Ó 2014 Published by Elsevier Inc.
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Keywords: Binary visibility graph Markov chain Time series Complex network
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1. Introduction
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Describing the dynamics of complex systems using time series is a fundamental problem in a wide variety of fields that there are different measures to analyze the complex, such as Lyapunov exponent, entropies, and correlation dimension [33]. Each of the above can only deal with a view to analyze the dynamics but complex networks can deal with different aspects to analyze the dynamics [15,33]. In the past few years, numerous transformations to analysis of the dynamics of complex systems based on complex networks have been proposed [15,33,20,17,7,9,14,24,25,31]. Complex network theory has fascinated much attention in the study of social, informational, technological and biological systems, resulting in a deeper understanding of complex systems [33,16,2,5,10,11,21]. One of these transformations is binary visibility algorithm. The binary visibility algorithm is applied in various fields such as stock market indices [22,29,34], human heartbeat dynamics [8,23], energy dissipation rates in three-dimensional fully developed turbulence [19], and foreign exchange rates [32]. The binary visibility algorithm transforms a time series into a binary visibility graph [1]. We are used geometrical visibility of top of the bars in the binary sequence bar diagram to produce binary visibility graphs of time series. On the other hand, there is a linear interpolation between data which have mutual visibility. This algorithm depends on the statistical persistence of the time series [30]. Therefore, this dependence is less, if visibility be low. In this way, we introduce the application of complex networks in testing the randomness of binary sequences. Another method uses of fluctuations of the time series’s line diagram for time series analysis. On the other words, we transform fluctuations of the time series’s line diagram into two-state Markov chain and then using binary visibility graph, we are mapped the two-state Markov chain into complex networks. This algorithm is defined as a Markov-binary visibility algorithm (MBVA). Whereas this algorithm uses the two-state Markov chains for transform the time series into the complex networks and in a two-state Markov chain, the next state only depends on the current state and not on the sequence of events that preceded it (memoryless), thus, this algorithm is less dependent on the statistical persistence of the time series. Consequently, this algorithm be obtained more precise results. The rest of
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⇑ Corresponding author. Tel.: +98 9379940398; fax: +98 9151041295. E-mail address: [email protected] (S. Ahadpour). http://dx.doi.org/10.1016/j.ins.2014.03.007 0020-0255/Ó 2014 Published by Elsevier Inc.
Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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this paper is organized as follows: In Section 2 we introduce the MBVG algorithm. In Section 3 we derive results for statistical properties of the MBVG such as degree distribution, clustering coefficient, visibility length and mean visibility. In Section 4, numerical results of the model are reported with an analysis of the human heartbeat dynamics. The conclusions are given in Section 5.
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2. Structure of Markov-binary visibility graph
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A Markov chain is a discrete-time process for which the future behavior, given the past and the present, only depends on the present and not on the past. Also, a Markov chain is characterized by a set of states S and the transition probabilities P i!j between the states. Using this property, we show how Markov-binary visibility algorithm (MBVA) maps the time series into a complex network (see Fig. 1). Here, we briefly describe the Markov-binary visibility algorithm: First, consider fyi gi¼1;...;N be a time series of N data. Next, to draw line diagram corresponding to time series. Then, we get slope ðmÞ between any two consecutive data in the time series’s line diagram (see Fig. 1). Now, the set of states S of Markov chain is defined as follows: For the stochastic systems,
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S¼
1 m>0 0
m60
ð1Þ
and for the deterministic systems,
S¼
0
mP0
1 m<0
ð2Þ
Fig. 1. An example of the Markov binary sequence depicted in the upper part. In the lower part we represent the graph generated through the binary visibility algorithm. The algorithm assigns each bit of the binary sequence to a node in the Markov-binary visibility graph (MBVG). Two nodes i and j in the MBVG are connected if one can draw a visibility line in the binary sequence joining the neighboring xi and xj that does not intersect any intermediate bits height.
Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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Then, to use of states S of two-state Markov chain obtained fluctuations of the time series, we construct two-state Markov chain of states 0 and 1. Thus, we have Markov-binary sequence of N bits (MBS ¼ fxi gi¼1;...;N ). With regard to the above contents, the Markov-binary visibility algorithm assigns each fluctuation of the time series to a bit of the Markov-binary sequence and then assigns each bit of the Markov-binary sequence to a node in the MBVG. Two nodes i and j in the MBVG are connected if one can draw a visibility line in the binary sequence joining the neighboring xi and xj that does not intersect any intermediate bits height. xi ðxj Þ can only be 0 or 1. Therefore, i and j are two connected nodes if the successive geometrical criterion is satisfied within the binary sequence:
xi þ xj > xn that xn ¼ 0 for all n such that i < n < j:
ð3Þ
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The MBVG associated with a time series is always connected and undirected. In what follows, we will show the geometrical simplicity of the Markov-binary version of the visibility algorithm and that it grants an analytically easier solubility; this new method can attest to perfect distinguish between statistically independent and dependent systems. Also, the MBVG is able to analysis of the fractal systems that will be explored in detail in further work. This point should also be noted that a MBVG, like a visibility graph is invariant under affine transformation of the series data since the visibility criterion is invariant under re-scaling of both horizontal and vertical axes, and under horizontal and vertical transformation [28]. MBVG is a planar graph which is connective, undirected and a combination of two linear and maximal planar subgraphs (see Fig. 1). The constituting subgraphs are as follows: Maximal Planar Graph (MPGn ) is a connective simple planar graph. But it is not a simple planar graph on adding an extra edge e, i.e., e R E ðMPGÞ, where n and E ðMPGÞ are the number of nodes and edges of the graph, respectively. Linear Planar Graph (LPGn ) is a connective planar graph without loops and multiple edges. The degree of all of the nodes is 2 except for the first and the last nodes which have a degree of 1. As will be discussed later, MBVA can be employed as a new stringent criterion for time series analysis. In order to show the significance of MBVG and its ability, analysis of its statistical properties is necessary.
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3. Statistical (Topological) properties of the MBVG
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A MBVG inherits the statistical (topological) properties of the associated Markov binary sequence. Analyzing MBVG from statistical point of view, we have studied its visibility degree and length probability distributions, clustering coefficient, and mean visibility length. These properties will be proven to have the useful capability for analysis of time series.
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3.1. Probability distribution for the visibility degree P(k)
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As we mentioned before, a Markov chain is characterized by a set of states S ¼ f0; 1g and the transition probabilities Pi!j between the states. Here, Pi!j is the probability that the Markov-binary sequence is at the next time point in state j, given that it is at the present time point at state i. The matrix P MBS with elements P i!j is called the transition probability matrix of the Markov-binary sequence and as be defined follows [18]:
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PMBS ¼
P1!1
P1!0
P0!1
P0!0
¼
ð1 pÞ p q
ð1 qÞ
:
ð4Þ
Let us consider a Markov-binary sequence of time series such that MBS 2 f0; 1g. Considering the bits of MBS ¼ fxi gi¼1;...;N , we take an arbitrary bit according to which, we are going to define the degree distribution. This bit is the one to which, all the other bits are connected by ‘‘visibility line’’. Therefore, we call that the ‘‘observer’’ bit here after with the label x0 . In order to obtain the degree distribution PðkÞ [6] of the associated graph, we are going to estimate the probability if x0 can observe k other bits. k is the degree of the MBVG. For any configuration of the degree k, there always exist two bounding bits both with the value 1. This implies that the minimum possible visibility degree is k ¼ 2. We call the bits between the bounding bits and the observer as the inner bits: some can be observed by the observer called the ‘‘visible bits’’ and some cannot be observed called the ‘‘nonvisible bits’’. It is worthwhile to mention that although both the visible and nonvisible bits are counted in calculating the degree distribution probability, it is only the visible bits who determine the degree of the visibility graph. In line with what is mentioned above, for a given k, there are exactly k 1 different possible configurations fF i gi¼0;...;k2 , where the index i determines the number of visible bits on the left-hand side of x0 (see Fig. 2). According to the nature of the configurations, the case where k ¼ 4 and x0 ¼ 0 has a different behavior, since the observer should always remain between two visible bits. Also, for k P 5, the observer can only be the bit with value 1. It is due to the fact that the observer x0 ¼ 0 may not observe more than 4 bits. In order to make the construction of the F i more clear, we have calculated as an example, a set of possible configurations for k ¼ 4 with the results denoted in Fig. 2. In x0 ¼ 1; F 0 is the configuration where, none of the k 2 ¼ 2 visible bits are located on the left of x0 , hence the left bounding bit is labeled as x1 and the right bounding bit is labeled as x3 . For x0 ¼ 0; F 0 is the configuration where one of the k 2 ¼ 2 visible bits is located on the left of x0 , so the left bounding bit is x2 and the right bounding bit accounting for n nonvisible bits in the way, is labeled as xnþ1 . In x0 ¼ 1, F 1 is the configuration for which, x0 is located between two visible bits. For x0 ¼ 0; F 1 is the configuration for which, n1 nonvisible bits are located on the left of x0 and n2 nonvisible bits are located on the right. Finally, in x0 ¼ 1; F 2 is the configuration for which, both visible bits are located Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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Fig. 2. The configurations for k = 2, 3, 4, 5. We can see that the sign of the subindex in xi indicates if the bit are located whether at the left-hand side of x0 (sign minus) or at right-hand side. Consequently, the bounding’s bit subindex directly indicates the amount of bit located in that side. We call the bits between the bounding bits and the observer as the inner bits: some are visible and others are nonvisible bits.
Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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on the left of the observer. For x0 ¼ 0; F 2 is the configuration where one of the k 2 ¼ 2 visible bits is located on the right of x0 , and therefore the right bounding bit is labeled as x2 and the left bounding bit is labeled as xðnþ1Þ . Notice that n nonvisible bits can be located on the right of the observer bit (see Fig. 2). Consequently, F i corresponds to the configuration for which i visible bits are placed on the left of x0 , and k 2 i visible bits are placed on the right. Each of these possible configurations have an associated probability g F i P F i ðx0 Þ that will result in PðkÞ such that
PðkÞ ¼
k2 X g F i PF i ðx0 Þ
where g F i is the probability density function for each of these possible configurations. For k P 2, the total probability that x0 observes is 1:
X PðkÞ ¼ 1
148
kP2
To consider in Eq. (4), q ¼ p1 and p ¼ ð1 p1 Þ, thus, we have,
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ð5Þ
i¼0
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PMBS ¼
p1
ð1 p1 Þ
p1
ð1 p1 Þ
ð6Þ
where p1 and ð1 p1 Þ are probability of bits of 1 and probability of bits of 0 in the Markov-binary sequence, respectively. Note that the definition of the Pi!j implies that the row sums of P MBS are equal to 1 [18]. We propose the relation of PF i ðx0 Þ as following:
PF i ðx0 Þ ¼
m Y ðPMBS Þh h¼1
that is the joint probability distribution of the bits of MBS ¼ fxh gh¼1;...;m corresponds to the configuration F i . Therefore, We show that the general relation for PðkÞ as following: k2 Y m X PðkÞ ¼ gðkÞ ðPMBS Þh i¼0
! ð7Þ
h¼1
where gðkÞ is the probability density function for degree of k. Starting with k ¼ 2, i.e. the probability that the observer bit has two and only two visible bits: For x0 ¼ 0:
0
1
PF 0 ðx0 ¼ 0Þ ¼ P @1 |{z} 0 1A ¼ Pð1Þ P 1!0 P0!1 ¼ p21 ð1 p1 Þ x0
For x0 ¼ 1:
0
1
PF 0 ðx0 ¼ 1Þ ¼ P@1 |{z} 1 1A ¼ Pð1Þ P1!1 P1!1 ¼ p31 x0
Then,
Pðk ¼ 2Þ ¼ gð2Þ p21
ð8Þ
The second case is k ¼ 3, i.e., for the observer which can see three and only three visible bits. In this way, we encounter two different configurations: F 0 and F 1 . As can be seen from Fig. 2, for x0 ¼ 0, an arbitrary number of nonvisible bits (n for F 0 and n0 for F 1 ) are located between the observer and the bounding bit. It is crucial to note that the nonvisible bits must be taken into account in the probability calculation. For F 0 , the nonvisible bits are bj ¼ 0 (j = 1, . . . , n) and for F 1 , they are dj ¼ 0 (j = 1, . . . , n0 ). Hence,
0
1 b1 ;...;bn n zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ B C PF 0 ðx0 ¼ 0Þ ¼ P @1 |{z} 0 0 00 . . . 00 1A ¼ Pð1Þ P1!0 P0!0 P0!0 . . . P0!0 P0!1 x0
0
1 d1 ;...;dn n zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ B zfflfflfflfflffl}|fflfflfflfflffl{ C PF 1 ðx0 ¼ 0Þ ¼ P@1 00 . . . 00 0 |{z} 0 1A ¼ Pð1Þ P1!0 P 0!0 . . . P0!0 P0!0 P0!1 ; x0
Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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Considering all the configurations for x0 ¼ 0 with and without nonvisible bits (F 0 without nonvisible bits, F 0 with a single nonvisible bit, F 0 with two nonvisible bits, and so on, and the same for F 1 ):
" PF 0 ðx0 ¼ 0Þ ¼
p21 ð1
2
p1 Þ 1 þ
1 X
# ð1 p1 Þ
n
¼ p1 ð1 p1 Þ2
n¼1
Where the first term in the square bracket in Eq. (4) corresponds to the contribution of a configuration with no nonvisible bits and the second is a sum over the contributions of n nonvisible bits. For x0 ¼ 1, there are no nonvisible bits therefore:
0
1
PF 0 ðx0 ¼ 1Þ ¼ P@1 |{z} 1 01A ¼ Pð1Þ P1!1 P1!0 P0!1 ¼ p31 ð1 p1 Þ x0
Similar results can be reached for P F 1 , then, one gets
Pðk ¼ 3Þ ¼ gð3Þ 2p1 ð1 p1 Þðp21 p1 þ 1Þ
ð9Þ
The third step is to calculate the probability for k ¼ 4, i.e., for the observer which has four and only four visible bits. There are three different configurations: F 0 ; F 1 ; F 2 . This case is different from the others in that, for x0 ¼ 0, there always exists a visible bit between the observer and the boundary. This stems from the nature of observer and the geometrical criterion for k ¼ 4. Accordingly, for F 0 [F 2 ], all the nonvisible bits bj ¼ 0 (j = 2, . . . , n)[bi ¼ 0 (i = 2, . . . , n)] can be on the right[left] of the observer. With respect to the symmetry in the visibility geometrical criterion of BVG, it will be sufficient to calculate the probability for one configuration:
0
1 b1 ;...;bn n zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ B C PF 0 ðx0 ¼ 0Þ ¼ P@10 |{z} 0 0 00 . . . 00 1A ¼ Pð1Þ P1!0 P0!0 P 0!0 P0!0 . . . P0!0 P0!1 ; x0
From which, we can get:
PF 0 ðx0 ¼ 0Þ ¼
p21 ð1
" # 1 X n p1 Þ ð1 p1 Þ ¼ p1 ð1 p1 Þ4 3
n¼1
However, for F 1 ; n1 nonvisible bits bi ¼ 0 (i = 2, . . . , n1) are to the left of the observer and n2 nonvisible bits bj ¼ 0 (j = 2, . . . , n2) are to the right. Hence,
0
1 b1 ;...;bn2 zfflffl ffl }|fflffl ffl { zfflfflfflfflffl}|fflfflfflfflffl{ B C PF 1 ðx0 ¼ 0Þ ¼ P@1 0 . . . 0 0 |{z} 0 0 00 . . . 00 1A b1 ;...;bn1
x0
n1
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n2
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ ¼ Pð1Þ P1!0 P0!0 . . . P 0!0 P0!0 P0!0 P0!0 . . . P0!0 P0!1 ; From which, we can get:
" PF 1 ðx0 ¼ 0Þ ¼ p21 ð1 p1 Þ3 1 þ
# 1 1 1X 1X ð1 p1 Þn1 þ ð1 p1 Þn2 ¼ p1 ð1 p1 Þ3 2 n ¼1 2 n ¼1 1
2
The first term corresponds to the state without nonvisible bits while the next two terms represent the configurations with nonvisible bits. Regarding the symmetry in the BVG visibility geometrical criterion for x0 ¼ 1, we may write P F 0 ¼ PF 1 ¼ PF 2 . Therefore, it suffices to calculate P F 0 ðx0 ¼ 1Þ:
0
1
PF 0 ðx0 ¼ 1Þ ¼ P@1 |{z} 1 001A ¼ Pð1Þ P1!1 P1!0 P0!0 P0!1 ¼ p31 ð1 p1 Þ2 x0
and then, it is straightforward to obtain:
Pðk ¼ 4Þ ¼ gð4Þ 3p1 ð1 p1 Þ2
5 2 5 p1 p1 þ 1 3 3
ð10Þ
The next case at issue is k P 5. Inasmuch as the zero observer (x0 ¼ 0) may not see more than four bits in BVG, we are not going to witness x0 ¼ 0 when k P 5. Consequently, in all the configurations for k P 5, we are only left with the observer x0 ¼ 1. When k ¼ 5, we can only have three inner bits so, there will be four configurations (see Fig. 2).
0
1
PF 0 ðx0 ¼ 1Þ ¼ P@1 |{z} 1 0001A ¼ Pð1Þ P 1!1 P1!0 P0!0 P0!0 P0!1 ¼ p31 ð1 p1 Þ3 : x0
Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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Ultimately, one gets
Pðk ¼ 5Þ ¼ gð5Þ 4 p31 ð1 p1 Þ3
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ð11Þ
Eventually, the probability distribution for the visibility degree is as follows:
PðkÞ ¼ ðk 1ÞgðkÞ p1 ð1 p1 Þk2 Fðp1 Þ
248
250
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ð12Þ
where
8 p1 > > > 2 < p p1 þ 1 Fðp1 Þ ¼ 5 1 2 5 > > 3 p1 3 p1 þ 1 > : 2 p1
k¼2 k¼3 k¼4 kP5
As we know, if the component of system be statistically independent, it’s probability density function is equal to 1. Since the MBVG inherit this property, then, for the statistically independent systems, gðkÞ for all degrees is equal to 1 (gðkÞ ¼ 1). For the statistically dependent systems, the results show that the probability density function gðkÞ is equal to 1 (gðkÞ ¼ 1) unless value of degree k is more than or equal to 5, i.e. k P 5, in which case the gðkÞ can be defined as follows: 2 ðk2ÞÞd
gðk P 5Þ ¼ a eðr
ð13Þ
where a and r2 are constant value and variance of the Markov-binary sequence corresponding to the statistically dependent systems, respectively. The results show that the value of d is proportional with correlation exponent s in correlation stochastic systems and correlation dimension D in chaotic systems. We check the reliability of the above equations using the following examples: For the statistically independent systems: Linear piecewise map, that is a piecewise linear map with stochastic specificity which defined as following [4]:
( xn xnþ1 ¼
p xn p 1p
0 6 xn < p p 6 xn 1
where p 2 ½0; 1. (Here, p ¼ 0:30; p ¼ 0:51; p ¼ 0:70) For the statistically dependent systems: 1. Chaotic logistic map (r ’ 4): it is a nonlinear chaotic map and the correlation dimension D is 1.016 [26]. 2. Henon map: it is a two-dimension nonlinear chaotic map and the correlation dimension D is 1:220 (In this case a ¼ 1:4 and b ¼ 0:3) [26]. 3. Lozi map: it is a piecewise linear variant of the Henon map and the correlation dimension D is 1:384 (In this case a ¼ 1:7 and b ¼ 0:5) [26]. 4. Ornstein–Uhlenbeck (O–U) process (Red noise): it is stationary, Gaussian, and Markovian [27]. It is a Fokker–Planck equation (FPE) with the damping parameter (correlated exponent) j ¼ 1s (s is the ‘‘memory’’ of the system) that generate short-range correlated series [12] (Here, s ¼ 0:5; s ¼ 1:0, s ¼ 1:5). 5. Long-Range stochastic (L-R) process: it is a random process with a flat power spectral density that the power-law correlation function is proportional with power of time, i.e. RðtÞ t s [12]. (Here, s ¼ 0:5; s ¼ 1:0; s ¼ 1:5).
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Numerical results show that the degree distribution PðkÞ of the MBVG corresponding to time series of above examples can be predicted by Eq. (12) (see Figs. 3 and 4). According to the figures, whatever the data of time series are more independent, the Diagrams are more linear.
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3.2. Probability distribution of the clustering coefficient P(C)
286
If the node (i) in a MBVG has ki edges, it is connective with ki other nodes. If the first neighbors of this node (i) is a part of a ki edges between the neighbors. The ratio of the number of edges ðEi Þ existing between ki nodes to all cluster, there are 2 ki the edges gives the clustering coefficient for that node [2,3]. In other words, it is the number of the nodes that observe 2 (i) nodes and have mutual visibility (the triangles) and is normalized by the possible set of triangles. Thus:
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Ci ¼
2Ei ki ðki 1Þ
ð14Þ
In MBVG, the first neighbors of the node i are the ones seen by that node, i.e. the visibility degree (ki ) of that node. In order to analyze behavior of systems using clustering coefficient of MBVG, we focus on degrees are more than and equal to 5, i.e. Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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Fig. 3. Experimental degree distribution of the Markov-binary visibility graph corresponding to time series (105 data) of the statistically independent and dependent systems. Solid line corresponding to Eq. (12). In this equation, for chaotic systems a ¼ 4, for Ornstein–Uhlenbeck (O–U) processes a ¼ 3:5 s (s is the ‘‘memory’’ of the system), and for Long-Range stochastic (L-R) processes a ¼ 2:5.
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k P 5. First we calculate it for low degrees and then generalize them for all MBVGs. For a clear analysis, we propose Table 1 corresponding to x0 ¼ 1 observer bits. Table 1 is in accord with two intermediate and side states of the observer bit. According to Table 1, it is not hard to detect a precise order for the number of configurations and edges as well as for the 2 clustering coefficient and the probability distribution of the configuration (bðkÞ). The bðkÞ is equal to k1 in the side states and k3 is equal to k1 in the intermediate states. Also, the number of edges in these states are equal to 2k 5 for side states and 4k10 4k12 2k 6 for intermediate states. Consequently, according to the Eq. (14), we have, C S ¼ kðk1Þ for side states and C M ¼ kðk1Þ for intermediate states. With regard to the above contents, the visibility degree of nodes ðkÞ in the MBVG can be defined as following:
k¼ 305
8 < F ðC S ÞþC S
Side states
: F ðC M ÞþCM 2C M
Intermediate states
2C S
ð15Þ 1
306 307 308 309 310 311 312 313
3.2.1. Side states The states that observer bit located at side a bounding bit (see Table 1). With regard to Eqs. (14) and (15), we have,
314 316 317
318
320
1
where F ðC S Þ and F ðC M Þ are equal to 4 ðC 2S 32C S þ 16Þ2 and 4 ðC 2M 40C M þ 16Þ2 , respectively. With regard to the obtained results, we are going to calculate the clustering coefficient probability P(C) in different conditions. We must carry out the calculations for two situations: intermediate and side states. In general, when k P 5, in order to compute the clustering coefficient probability P(C), it would be sufficient to multiply the visibility degree probability distribution P(k) by the one for the configuration bðkÞ. As mentioned earlier, we are going to consider first side and second the intermediate states as follows:
PS ðk P 5Þ ¼ bS ðkÞ Pðk P 5Þ ¼ 2gðkÞ p31 ð1 p1 Þk2 ¼ P
F ðC S Þ þ C S : 2C S
Assuming the following conditions:
8 < kþ ¼ F þ ðCS ÞþCS 2C S
kþ P 5
: k ¼ F ðCS ÞþCS 2C S
k P 5
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Fig. 4. Experimental probability distribution of the clustering coefficient of the Markov-binary visibility graph corresponding to time series (105 data) of the statistically independent and dependent systems. Solid line corresponding to Eq. (16) and point line corresponding to Eq. (17). In this equations, for chaotic systems a ¼ 4 and also, d ¼ 1D for logistic map and d ¼ 2D for henon and lozi maps (D is correlation dimension).
Table 1 Probable configurations for one-observer in intermediate and side states.
E k=5 E k=6 E k=7 .. .
Side states
Intermediate states
5 110001
4 101001
7 1100001 9 11000001 .. .
6 1010001 8 10100001 .. .
Side states 4 100101
6 1001001 8 10010001
8 10001001
5 100011 6 1000101 8 10000101
7 1000011 9 10000011 .. .
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and kþ ; k 2 Z. we can get the subsequent relation for clustering coefficient probability:
PðC S Þ ¼ 2g
F þ ðC S Þ3C S F þ ðC S Þ þ C S p31 ð1 p1 Þ 2CS 2C S
ð16Þ
325
where 0 < C A 0:5.
326
3.2.2. Intermediate states The states that observer bit located at between the inner bits (see Table 1). With regard to Eqs. (14) and (15), we have,
327
328 330
PM ðk P 5Þ ¼ bM ðkÞ Pðk P 5Þ ¼ ðk 3ÞgðkÞ p31 ð1 p1 Þk2 ¼ P
F ðC M Þ þ C M : 2C M
Fig. 5. Experimental probability distribution of the clustering coefficient of the Markov-binary visibility graph corresponding to time series (105 data) of the statistically dependent systems. The Ornstein–Uhlenbeck (O–U) processes are in left column and the Long-Range stochastic (L-R) processes are in right column. Solid line corresponding to Eq. (16) and point line corresponding to Eq. (17). In this equations, for Ornstein–Uhlenbeck (O–U) processes a ¼ 3:5 s; d ¼ 1:5 for s ¼ 0:5; d ¼ 2:0 for s ¼ 1:0 and d ¼ 2:5 for s ¼ 1:5; for Long-Range stochastic (L-R) processes, a ¼ 2:5 and also, d ¼ 2.
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Fig. 6. Experimental probability distribution for the MPGn (visibility length) of the Markov-binary visibility graph corresponding to time series (105 data) of the statistically independent and dependent systems. Solid line corresponding to Eq. (18). In this equation, a ¼ 0:5 for logistic map and a ¼ 1:0 for henon and lozi maps; for Ornstein–Uhlenbeck (O–U) processes, a ¼ 3:5 s and for Long-Range stochastic (L-R) processes, a ¼ 2:5 .
Fig. 7. Experimental mean subgraphs for the MPGn (mean of the visibility length) of the Markov-binary visibility graph corresponding to time series (N ¼ 27 ; . . . ; 216 data) of the statistically independent systems. The lines corresponding to Eq. (19). In this equation, a ¼ 0:18; 1:10; 5:50 for the statistically independent systems (piecewise map) p ¼ 0:30; 0:51; 0:70, respectively.
Fig. 8. Three RR-interval time series of three groups from subjects: Normally, Congestive Heart Failure (CHF) and Atrial Fibrillation (AF) subjects.
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Fig. 9. Experimental degree distribution of the Markov-binary visibility graph corresponding to RR-interval time series (105 data) of three groups from subjects: Normally, Congestive Heart Failure (CHF) and Atrial Fibrillation (AF) subjects (black squares) and the statistically independent and dependent systems (black circles). Solid line corresponding to Eq. (12). In this equation, a ¼ 4 for RR-interval time series of CHF subjects, also, a ¼ 2:5 for RR-interval time series of AF subjects. 331
332
334 335
Assuming the following conditions:
8 < kþ ¼ F þ ðCM ÞþC M 2C M
kþ P 5
: k ¼ F ðCM ÞþC M 2C M
k P 5
and kþ ; k 2 Z. Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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If these conditions are true,
337
PðC M Þ ¼ 339 340 341 342 343
13
Fþ ðC M Þ X FðC M Þ¼F ðC M
FðC M Þ3C M FðC M Þ 5C M FðC M Þ þ C M g p31 ð1 p1 Þ 2CM 2C M 2C M Þ
ð17Þ
where 0 < C M 0:4 and if any of the above mentioned conditions (kþ ; k 2 Z and kþ ; k P 5) are violated, the probability will Q4 be zero (see Fig. 5). Numerical results show that the probability distribution of clustering coefficient PðCÞ of the MBVG corresponding to time series of above examples (the examples of pervious Section (3.1)) can be predicted by Eqs. (16) and (17) (see Figs. 6 and 7).
Fig. 10. Experimental probability distribution of the clustering coefficient of the Markov-binary visibility graph corresponding to RR-interval time series (105 data) of three groups from subjects: Normally, Congestive Heart Failure (CHF) and Atrial Fibrillation (AF) subjects (black circles). Solid line corresponding to Eq. (16) and point line corresponding to Eq. (17). In this equation, a ¼ 4 and d ¼ 1:016 for RR-interval time series of CHF subjects, also, a ¼ 2:5 and d ¼ 2 for RR-interval time series of AF subjects.
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3.3. Probability distribution of the visibility length and the mean of the visibility length
345
The visibility length is defined for the ‘‘one’’ observer. It is the number of intermediate zero bits between two consecutive one-observers in a way that the two one-observers are visible to each other. In graph terminology, it corresponds to the order of the MPG graph which is a subgraph of MBVG. Similar to the Eq. (7) and the above contents, we have,
346 347
348
Fig. 11. Experimental probability distribution for the MPGn (visibility length) of the Markov-binary visibility graph corresponding to RR-interval time series (105 data) of three groups from subjects: Normally, Congestive Heart Failure (CHF) and Atrial Fibrillation (AF) subjects (black squares) and the statistically independent and dependent systems (black circles). Solid line corresponding to Eq. (18). In this equation, a ¼ 0:5 for normally subjects and CHF subjects, also a ¼ 2:5 for RR-interval time series of AF subjects.
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Fig. 12. Experimental mean subgraphs for the MPGn (mean of the visibility length) of the Markov-binary visibility graph corresponding to RR-interval time series (N ¼ 27 ; . . . ; 216 data) of normally subjects (black circles). The lines corresponding to Eq. (19). In this equation, a ¼ 0:5 for normally subjects.
P1!1 ðnÞ ¼ gðnÞ
350 351 352
353
h¼1
where gðnÞ is the probability density function for visibility of n and ðPMBS Þh is the joint probability distribution of the bits of MBS ¼ fxh gh¼1;...;nþ1 . Consequently, it be defined as follows:
0 1 n n nþ1 zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Y ðPMBS Þh ¼ P@00 . . . 00A ¼ P0!0 P0!0
355 356
357 359
nþ1 Y ðPMBS Þh
h¼1
then,
P1
1 ðnÞ
¼ gðnÞ ð1 p1 Þn :
360
The value of gðnÞ for the systems is different and can be classified as following:
361
Statistically independent systems: The gðnÞ is constant value. For n ¼ 1; gðnÞ ¼ 1 and for n P 2 is as follows:
362
364 365
366
8 > < a > 1 p1 > 0:5 gðnÞ ¼ a ¼ 1 p1 ¼ 0:5 > : a < 1 p1 < 0:5 Statistically dependent systems: For correlated stochastic systems,
( gðnÞ ¼
369
372 373 374
1
n ¼ 1; 2
a eðn r2 Þ
368 370
ð18Þ
d
nP3
and also, for chaotic systems,
gðnÞ ¼
1
ae
ðn
r2 Þ
d
n¼1 nP2
where a and r are constant value and variance of the Markov-binary sequence, respectively. The value of d is proportional with correlation exponent s in correlation stochastic systems and correlation dimension D in chaotic systems. 2
375 376 377 378 379
380
382
The probability distribution of the visibility length equals the probability distributions for the MPG. Numerical results confirm that the probability distribution for the MPGs of the MBVG corresponding to time series of above examples (the examples of pervious Section (3.1)) can be predicted by Eq. (18) (see Figs. 8 and 9). According to the above results, the mean visibility length for uncorrelated stochastic systems can be defined as follows:
LðNÞ
N X nP1 n¼1
1 ðnÞ
¼ gðnÞ
N X nð1 p1 Þn : n¼1
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In which, N is the number of bits in an Markov-binary sequence. This can be reduced to:
383
384
LðNÞ ¼ gðnÞ
386
ðp1 1Þ ð1 p1 ÞN ð1 þ Np1 Þ 1 p21
:
ð19Þ
389
where the gðnÞ is constant value. This equation gives the mean graph order of MPGn that equal the mean visibility length for uncorrelated stochastic systems. Numerical results confirm that the mean graph order of MPGn of the MBVG corresponding to time series of uncorrelated stochastic systems can be predicted by Eq. (19) (see Fig. 10).
390
4. Network analysis of the human heartbeat dynamics
387 388
391 392 393 394 395 396 397 398 399 400 401 402 403 404 405
Recently, the human heartbeat (RR-interval) dynamics has attracted much attention from researchers [33,16,11]. As we know, the RR-interval is the time elapsing between two consecutive R waves in the electrocardiogram. The RR-interval time series are different in the normal and patient subjects (see Fig. 11). Here, we investigate the RR-interval time series of three groups from subjects: Normally, Congestive Heart Failure (CHF) and Atrial Fibrillation (AF) subjects. Each RR-interval time series is about 24 h long (roughly 105 intervals). All of the time series were derived from continuous ambulatory (Holter) electrocardiograms (ECGs) that are available on the PhysioNet web site (http://www.physionet.org/challenge/chaos). Our analysis are on the fluctuations of time in the RR-interval time series using the MBVA. With regarded to above section, the results show that the degree distribution PðkÞ of the MBVG corresponding to RR-interval time series of normally, Q5 CHF and AF subjects can be predicted by Eq. (12) (see Figs. 12 and 13). To better analyze the RR-interval time series, we obtain other properties of the MBVG corresponding to RR-interval time series. results indicate that the probability distribution of clustering coefficient PðCÞ of the MBVG corresponding to RR-interval time series of normally, CHF and AF subjects can be predicted by Eqs. (16) and (17) (see Fig. 14). Also, the probability distribution of the visibility length PðnÞ of the MBVG corresponding to RR-interval time series of normally, CHF and AF subjects and the mean of the visibility length LðNÞ of the MBVG corresponding to RR-interval time series of normally subjects can be predicted by Eqs. (18) and (19) (see Figs. 15 and 16). The results are in agreement with previous results [13].
406
5. Conclusion
407
412
In conclusion, we introduced the MBVG which can be used to analyze the complex systems. There have been many studies dedicated to Markov binary sequences analysis however, using MBVG, we have been able to analyze and test complex systems from the complex networks perspective. Our results on topological (statistical) properties of the graph confirm this statement. An important result that needs to be highlighted in this paper is that the MBVG can be used easily by any computer and because of its binary status it can be used in broad situations. In addition, this graph has the capability of analyzing normally and statistically independent stochastic, chaotic and statistically dependent stochastic systems.
413
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Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins 5 6
Markov-binary visibility graph: A new method for analyzing complex systems
3 4 7
Q1
8
Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran
9 10 1 4 2 2 13 14 15 16 17
Sodeif Ahadpour ⇑, Yaser Sadra, Zahra ArastehFard
a r t i c l e Q3
i n f o
Article history: Received 2 April 2012 Received in revised form 4 April 2013 Accepted 3 March 2014 Available online xxxx
a b s t r a c t In order to stu te dynamics of complex systems has been used information recorded in the form of time series. These time series can be investigated from a complex network perspective. Using two-state Markov chain and the binary visibility graph, we investigate these time series. Moreover, several topological aspects of the constructed graph, such as degree distribution, clustering coefficient, and mean visibility length are studied. Our results show that the Markov-binary visibility algorithm stands as a simple method to discriminate statistically dependent and independent systems. Some remarkable examples confirm the reliability of Markov-binary visibility graph for time series analysis. Ó 2014 Published by Elsevier Inc.
18 19 20 21 22 23
Keywords: Binary visibility graph Markov chain Time series Complex network
37 36
1. Introduction
38
Describing the dynamics of complex systems using time series is a fundamental problem in a wide variety of fields that there are different measures to analyze the complex, such as Lyapunov exponent, entropies, and correlation dimension [33]. Each of the above can only deal with a view to analyze the dynamics but complex networks can deal with different aspects to analyze the dynamics [15,33]. In the past few years, numerous transformations to analysis of the dynamics of complex systems based on complex networks have been proposed [15,33,20,17,7,9,14,24,25,31]. Complex network theory has fascinated much attention in the study of social, informational, technological and biological systems, resulting in a deeper understanding of complex systems [33,16,2,5,10,11,21]. One of these transformations is binary visibility algorithm. The binary visibility algorithm is applied in various fields such as stock market indices [22,29,34], human heartbeat dynamics [8,23], energy dissipation rates in three-dimensional fully developed turbulence [19], and foreign exchange rates [32]. The binary visibility algorithm transforms a time series into a binary visibility graph [1]. We are used geometrical visibility of top of the bars in the binary sequence bar diagram to produce binary visibility graphs of time series. On the other hand, there is a linear interpolation between data which have mutual visibility. This algorithm depends on the statistical persistence of the time series [30]. Therefore, this dependence is less, if visibility be low. In this way, we introduce the application of complex networks in testing the randomness of binary sequences. Another method uses of fluctuations of the time series’s line diagram for time series analysis. On the other words, we transform fluctuations of the time series’s line diagram into two-state Markov chain and then using binary visibility graph, we are mapped the two-state Markov chain into complex networks. This algorithm is defined as a Markov-binary visibility algorithm (MBVA). Whereas this algorithm uses the two-state Markov chains for transform the time series into the complex networks and in a two-state Markov chain, the next state only depends on the current state and not on the sequence of events that preceded it (memoryless), thus, this algorithm is less dependent on the statistical persistence of the time series. Consequently, this algorithm be obtained more precise results. The rest of
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⇑ Corresponding author. Tel.: +98 9379940398; fax: +98 9151041295. E-mail address: [email protected] (S. Ahadpour). http://dx.doi.org/10.1016/j.ins.2014.03.007 0020-0255/Ó 2014 Published by Elsevier Inc.
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this paper is organized as follows: In Section 2 we introduce the MBVG algorithm. In Section 3 we derive results for statistical properties of the MBVG such as degree distribution, clustering coefficient, visibility length and mean visibility. In Section 4, numerical results of the model are reported with an analysis of the human heartbeat dynamics. The conclusions are given in Section 5.
62
2. Structure of Markov-binary visibility graph
63
A Markov chain is a discrete-time process for which the future behavior, given the past and the present, only depends on the present and not on the past. Also, a Markov chain is characterized by a set of states S and the transition probabilities P i!j between the states. Using this property, we show how Markov-binary visibility algorithm (MBVA) maps the time series into a complex network (see Fig. 1). Here, we briefly describe the Markov-binary visibility algorithm: First, consider fyi gi¼1;...;N be a time series of N data. Next, to draw line diagram corresponding to time series. Then, we get slope ðmÞ between any two consecutive data in the time series’s line diagram (see Fig. 1). Now, the set of states S of Markov chain is defined as follows: For the stochastic systems,
58 59 60
64 65 66 67 68 69 70
71 73
74
75 77
S¼
1 m>0 0
m60
ð1Þ
and for the deterministic systems,
S¼
0
mP0
1 m<0
ð2Þ
Fig. 1. An example of the Markov binary sequence depicted in the upper part. In the lower part we represent the graph generated through the binary visibility algorithm. The algorithm assigns each bit of the binary sequence to a node in the Markov-binary visibility graph (MBVG). Two nodes i and j in the MBVG are connected if one can draw a visibility line in the binary sequence joining the neighboring xi and xj that does not intersect any intermediate bits height.
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Then, to use of states S of two-state Markov chain obtained fluctuations of the time series, we construct two-state Markov chain of states 0 and 1. Thus, we have Markov-binary sequence of N bits (MBS ¼ fxi gi¼1;...;N ). With regard to the above contents, the Markov-binary visibility algorithm assigns each fluctuation of the time series to a bit of the Markov-binary sequence and then assigns each bit of the Markov-binary sequence to a node in the MBVG. Two nodes i and j in the MBVG are connected if one can draw a visibility line in the binary sequence joining the neighboring xi and xj that does not intersect any intermediate bits height. xi ðxj Þ can only be 0 or 1. Therefore, i and j are two connected nodes if the successive geometrical criterion is satisfied within the binary sequence:
xi þ xj > xn that xn ¼ 0 for all n such that i < n < j:
ð3Þ
101
The MBVG associated with a time series is always connected and undirected. In what follows, we will show the geometrical simplicity of the Markov-binary version of the visibility algorithm and that it grants an analytically easier solubility; this new method can attest to perfect distinguish between statistically independent and dependent systems. Also, the MBVG is able to analysis of the fractal systems that will be explored in detail in further work. This point should also be noted that a MBVG, like a visibility graph is invariant under affine transformation of the series data since the visibility criterion is invariant under re-scaling of both horizontal and vertical axes, and under horizontal and vertical transformation [28]. MBVG is a planar graph which is connective, undirected and a combination of two linear and maximal planar subgraphs (see Fig. 1). The constituting subgraphs are as follows: Maximal Planar Graph (MPGn ) is a connective simple planar graph. But it is not a simple planar graph on adding an extra edge e, i.e., e R E ðMPGÞ, where n and E ðMPGÞ are the number of nodes and edges of the graph, respectively. Linear Planar Graph (LPGn ) is a connective planar graph without loops and multiple edges. The degree of all of the nodes is 2 except for the first and the last nodes which have a degree of 1. As will be discussed later, MBVA can be employed as a new stringent criterion for time series analysis. In order to show the significance of MBVG and its ability, analysis of its statistical properties is necessary.
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3. Statistical (Topological) properties of the MBVG
103 105
A MBVG inherits the statistical (topological) properties of the associated Markov binary sequence. Analyzing MBVG from statistical point of view, we have studied its visibility degree and length probability distributions, clustering coefficient, and mean visibility length. These properties will be proven to have the useful capability for analysis of time series.
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3.1. Probability distribution for the visibility degree P(k)
107
As we mentioned before, a Markov chain is characterized by a set of states S ¼ f0; 1g and the transition probabilities Pi!j between the states. Here, Pi!j is the probability that the Markov-binary sequence is at the next time point in state j, given that it is at the present time point at state i. The matrix P MBS with elements P i!j is called the transition probability matrix of the Markov-binary sequence and as be defined follows [18]:
88 89 90 91 92 93 94 95 96 97 98 99 100
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108 109 110
111 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134
PMBS ¼
P1!1
P1!0
P0!1
P0!0
¼
ð1 pÞ p q
ð1 qÞ
:
ð4Þ
Let us consider a Markov-binary sequence of time series such that MBS 2 f0; 1g. Considering the bits of MBS ¼ fxi gi¼1;...;N , we take an arbitrary bit according to which, we are going to define the degree distribution. This bit is the one to which, all the other bits are connected by ‘‘visibility line’’. Therefore, we call that the ‘‘observer’’ bit here after with the label x0 . In order to obtain the degree distribution PðkÞ [6] of the associated graph, we are going to estimate the probability if x0 can observe k other bits. k is the degree of the MBVG. For any configuration of the degree k, there always exist two bounding bits both with the value 1. This implies that the minimum possible visibility degree is k ¼ 2. We call the bits between the bounding bits and the observer as the inner bits: some can be observed by the observer called the ‘‘visible bits’’ and some cannot be observed called the ‘‘nonvisible bits’’. It is worthwhile to mention that although both the visible and nonvisible bits are counted in calculating the degree distribution probability, it is only the visible bits who determine the degree of the visibility graph. In line with what is mentioned above, for a given k, there are exactly k 1 different possible configurations fF i gi¼0;...;k2 , where the index i determines the number of visible bits on the left-hand side of x0 (see Fig. 2). According to the nature of the configurations, the case where k ¼ 4 and x0 ¼ 0 has a different behavior, since the observer should always remain between two visible bits. Also, for k P 5, the observer can only be the bit with value 1. It is due to the fact that the observer x0 ¼ 0 may not observe more than 4 bits. In order to make the construction of the F i more clear, we have calculated as an example, a set of possible configurations for k ¼ 4 with the results denoted in Fig. 2. In x0 ¼ 1; F 0 is the configuration where, none of the k 2 ¼ 2 visible bits are located on the left of x0 , hence the left bounding bit is labeled as x1 and the right bounding bit is labeled as x3 . For x0 ¼ 0; F 0 is the configuration where one of the k 2 ¼ 2 visible bits is located on the left of x0 , so the left bounding bit is x2 and the right bounding bit accounting for n nonvisible bits in the way, is labeled as xnþ1 . In x0 ¼ 1, F 1 is the configuration for which, x0 is located between two visible bits. For x0 ¼ 0; F 1 is the configuration for which, n1 nonvisible bits are located on the left of x0 and n2 nonvisible bits are located on the right. Finally, in x0 ¼ 1; F 2 is the configuration for which, both visible bits are located Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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Fig. 2. The configurations for k = 2, 3, 4, 5. We can see that the sign of the subindex in xi indicates if the bit are located whether at the left-hand side of x0 (sign minus) or at right-hand side. Consequently, the bounding’s bit subindex directly indicates the amount of bit located in that side. We call the bits between the bounding bits and the observer as the inner bits: some are visible and others are nonvisible bits.
Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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on the left of the observer. For x0 ¼ 0; F 2 is the configuration where one of the k 2 ¼ 2 visible bits is located on the right of x0 , and therefore the right bounding bit is labeled as x2 and the left bounding bit is labeled as xðnþ1Þ . Notice that n nonvisible bits can be located on the right of the observer bit (see Fig. 2). Consequently, F i corresponds to the configuration for which i visible bits are placed on the left of x0 , and k 2 i visible bits are placed on the right. Each of these possible configurations have an associated probability g F i P F i ðx0 Þ that will result in PðkÞ such that
PðkÞ ¼
k2 X g F i PF i ðx0 Þ
where g F i is the probability density function for each of these possible configurations. For k P 2, the total probability that x0 observes is 1:
X PðkÞ ¼ 1
148
kP2
To consider in Eq. (4), q ¼ p1 and p ¼ ð1 p1 Þ, thus, we have,
150 152 153 154 155
156
158 159 160
161
163 164 165 166
167
169 170
171
173 174
175 177
178 179 180 181 182
183
185 186
188
ð5Þ
i¼0
146
149
5
PMBS ¼
p1
ð1 p1 Þ
p1
ð1 p1 Þ
ð6Þ
where p1 and ð1 p1 Þ are probability of bits of 1 and probability of bits of 0 in the Markov-binary sequence, respectively. Note that the definition of the Pi!j implies that the row sums of P MBS are equal to 1 [18]. We propose the relation of PF i ðx0 Þ as following:
PF i ðx0 Þ ¼
m Y ðPMBS Þh h¼1
that is the joint probability distribution of the bits of MBS ¼ fxh gh¼1;...;m corresponds to the configuration F i . Therefore, We show that the general relation for PðkÞ as following: k2 Y m X PðkÞ ¼ gðkÞ ðPMBS Þh i¼0
! ð7Þ
h¼1
where gðkÞ is the probability density function for degree of k. Starting with k ¼ 2, i.e. the probability that the observer bit has two and only two visible bits: For x0 ¼ 0:
0
1
PF 0 ðx0 ¼ 0Þ ¼ P @1 |{z} 0 1A ¼ Pð1Þ P 1!0 P0!1 ¼ p21 ð1 p1 Þ x0
For x0 ¼ 1:
0
1
PF 0 ðx0 ¼ 1Þ ¼ P@1 |{z} 1 1A ¼ Pð1Þ P1!1 P1!1 ¼ p31 x0
Then,
Pðk ¼ 2Þ ¼ gð2Þ p21
ð8Þ
The second case is k ¼ 3, i.e., for the observer which can see three and only three visible bits. In this way, we encounter two different configurations: F 0 and F 1 . As can be seen from Fig. 2, for x0 ¼ 0, an arbitrary number of nonvisible bits (n for F 0 and n0 for F 1 ) are located between the observer and the bounding bit. It is crucial to note that the nonvisible bits must be taken into account in the probability calculation. For F 0 , the nonvisible bits are bj ¼ 0 (j = 1, . . . , n) and for F 1 , they are dj ¼ 0 (j = 1, . . . , n0 ). Hence,
0
1 b1 ;...;bn n zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ B C PF 0 ðx0 ¼ 0Þ ¼ P @1 |{z} 0 0 00 . . . 00 1A ¼ Pð1Þ P1!0 P0!0 P0!0 . . . P0!0 P0!1 x0
0
1 d1 ;...;dn n zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ B zfflfflfflfflffl}|fflfflfflfflffl{ C PF 1 ðx0 ¼ 0Þ ¼ P@1 00 . . . 00 0 |{z} 0 1A ¼ Pð1Þ P1!0 P 0!0 . . . P0!0 P0!0 P0!1 ; x0
Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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Considering all the configurations for x0 ¼ 0 with and without nonvisible bits (F 0 without nonvisible bits, F 0 with a single nonvisible bit, F 0 with two nonvisible bits, and so on, and the same for F 1 ):
" PF 0 ðx0 ¼ 0Þ ¼
p21 ð1
2
p1 Þ 1 þ
1 X
# ð1 p1 Þ
n
¼ p1 ð1 p1 Þ2
n¼1
Where the first term in the square bracket in Eq. (4) corresponds to the contribution of a configuration with no nonvisible bits and the second is a sum over the contributions of n nonvisible bits. For x0 ¼ 1, there are no nonvisible bits therefore:
0
1
PF 0 ðx0 ¼ 1Þ ¼ P@1 |{z} 1 01A ¼ Pð1Þ P1!1 P1!0 P0!1 ¼ p31 ð1 p1 Þ x0
Similar results can be reached for P F 1 , then, one gets
Pðk ¼ 3Þ ¼ gð3Þ 2p1 ð1 p1 Þðp21 p1 þ 1Þ
ð9Þ
The third step is to calculate the probability for k ¼ 4, i.e., for the observer which has four and only four visible bits. There are three different configurations: F 0 ; F 1 ; F 2 . This case is different from the others in that, for x0 ¼ 0, there always exists a visible bit between the observer and the boundary. This stems from the nature of observer and the geometrical criterion for k ¼ 4. Accordingly, for F 0 [F 2 ], all the nonvisible bits bj ¼ 0 (j = 2, . . . , n)[bi ¼ 0 (i = 2, . . . , n)] can be on the right[left] of the observer. With respect to the symmetry in the visibility geometrical criterion of BVG, it will be sufficient to calculate the probability for one configuration:
0
1 b1 ;...;bn n zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ B C PF 0 ðx0 ¼ 0Þ ¼ P@10 |{z} 0 0 00 . . . 00 1A ¼ Pð1Þ P1!0 P0!0 P 0!0 P0!0 . . . P0!0 P0!1 ; x0
From which, we can get:
PF 0 ðx0 ¼ 0Þ ¼
p21 ð1
" # 1 X n p1 Þ ð1 p1 Þ ¼ p1 ð1 p1 Þ4 3
n¼1
However, for F 1 ; n1 nonvisible bits bi ¼ 0 (i = 2, . . . , n1) are to the left of the observer and n2 nonvisible bits bj ¼ 0 (j = 2, . . . , n2) are to the right. Hence,
0
1 b1 ;...;bn2 zfflffl ffl }|fflffl ffl { zfflfflfflfflffl}|fflfflfflfflffl{ B C PF 1 ðx0 ¼ 0Þ ¼ P@1 0 . . . 0 0 |{z} 0 0 00 . . . 00 1A b1 ;...;bn1
x0
n1
220 221
222
224 225 226 227
228
230 231
232 234
235 236 237
238
240
n2
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ ¼ Pð1Þ P1!0 P0!0 . . . P 0!0 P0!0 P0!0 P0!0 . . . P0!0 P0!1 ; From which, we can get:
" PF 1 ðx0 ¼ 0Þ ¼ p21 ð1 p1 Þ3 1 þ
# 1 1 1X 1X ð1 p1 Þn1 þ ð1 p1 Þn2 ¼ p1 ð1 p1 Þ3 2 n ¼1 2 n ¼1 1
2
The first term corresponds to the state without nonvisible bits while the next two terms represent the configurations with nonvisible bits. Regarding the symmetry in the BVG visibility geometrical criterion for x0 ¼ 1, we may write P F 0 ¼ PF 1 ¼ PF 2 . Therefore, it suffices to calculate P F 0 ðx0 ¼ 1Þ:
0
1
PF 0 ðx0 ¼ 1Þ ¼ P@1 |{z} 1 001A ¼ Pð1Þ P1!1 P1!0 P0!0 P0!1 ¼ p31 ð1 p1 Þ2 x0
and then, it is straightforward to obtain:
Pðk ¼ 4Þ ¼ gð4Þ 3p1 ð1 p1 Þ2
5 2 5 p1 p1 þ 1 3 3
ð10Þ
The next case at issue is k P 5. Inasmuch as the zero observer (x0 ¼ 0) may not see more than four bits in BVG, we are not going to witness x0 ¼ 0 when k P 5. Consequently, in all the configurations for k P 5, we are only left with the observer x0 ¼ 1. When k ¼ 5, we can only have three inner bits so, there will be four configurations (see Fig. 2).
0
1
PF 0 ðx0 ¼ 1Þ ¼ P@1 |{z} 1 0001A ¼ Pð1Þ P 1!1 P1!0 P0!0 P0!0 P0!1 ¼ p31 ð1 p1 Þ3 : x0
Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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Ultimately, one gets
Pðk ¼ 5Þ ¼ gð5Þ 4 p31 ð1 p1 Þ3
244 245
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249
252 253 254 255 256
257 259 260 261 262 263 264 265
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268 269 270 271 272 273 274 275 276 277 278 279 280
ð11Þ
Eventually, the probability distribution for the visibility degree is as follows:
PðkÞ ¼ ðk 1ÞgðkÞ p1 ð1 p1 Þk2 Fðp1 Þ
248
250
7
ð12Þ
where
8 p1 > > > 2 < p p1 þ 1 Fðp1 Þ ¼ 5 1 2 5 > > 3 p1 3 p1 þ 1 > : 2 p1
k¼2 k¼3 k¼4 kP5
As we know, if the component of system be statistically independent, it’s probability density function is equal to 1. Since the MBVG inherit this property, then, for the statistically independent systems, gðkÞ for all degrees is equal to 1 (gðkÞ ¼ 1). For the statistically dependent systems, the results show that the probability density function gðkÞ is equal to 1 (gðkÞ ¼ 1) unless value of degree k is more than or equal to 5, i.e. k P 5, in which case the gðkÞ can be defined as follows: 2 ðk2ÞÞd
gðk P 5Þ ¼ a eðr
ð13Þ
where a and r2 are constant value and variance of the Markov-binary sequence corresponding to the statistically dependent systems, respectively. The results show that the value of d is proportional with correlation exponent s in correlation stochastic systems and correlation dimension D in chaotic systems. We check the reliability of the above equations using the following examples: For the statistically independent systems: Linear piecewise map, that is a piecewise linear map with stochastic specificity which defined as following [4]:
( xn xnþ1 ¼
p xn p 1p
0 6 xn < p p 6 xn 1
where p 2 ½0; 1. (Here, p ¼ 0:30; p ¼ 0:51; p ¼ 0:70) For the statistically dependent systems: 1. Chaotic logistic map (r ’ 4): it is a nonlinear chaotic map and the correlation dimension D is 1.016 [26]. 2. Henon map: it is a two-dimension nonlinear chaotic map and the correlation dimension D is 1:220 (In this case a ¼ 1:4 and b ¼ 0:3) [26]. 3. Lozi map: it is a piecewise linear variant of the Henon map and the correlation dimension D is 1:384 (In this case a ¼ 1:7 and b ¼ 0:5) [26]. 4. Ornstein–Uhlenbeck (O–U) process (Red noise): it is stationary, Gaussian, and Markovian [27]. It is a Fokker–Planck equation (FPE) with the damping parameter (correlated exponent) j ¼ 1s (s is the ‘‘memory’’ of the system) that generate short-range correlated series [12] (Here, s ¼ 0:5; s ¼ 1:0, s ¼ 1:5). 5. Long-Range stochastic (L-R) process: it is a random process with a flat power spectral density that the power-law correlation function is proportional with power of time, i.e. RðtÞ t s [12]. (Here, s ¼ 0:5; s ¼ 1:0; s ¼ 1:5).
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Numerical results show that the degree distribution PðkÞ of the MBVG corresponding to time series of above examples can be predicted by Eq. (12) (see Figs. 3 and 4). According to the figures, whatever the data of time series are more independent, the Diagrams are more linear.
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3.2. Probability distribution of the clustering coefficient P(C)
286
If the node (i) in a MBVG has ki edges, it is connective with ki other nodes. If the first neighbors of this node (i) is a part of a ki edges between the neighbors. The ratio of the number of edges ðEi Þ existing between ki nodes to all cluster, there are 2 ki the edges gives the clustering coefficient for that node [2,3]. In other words, it is the number of the nodes that observe 2 (i) nodes and have mutual visibility (the triangles) and is normalized by the possible set of triangles. Thus:
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Ci ¼
2Ei ki ðki 1Þ
ð14Þ
In MBVG, the first neighbors of the node i are the ones seen by that node, i.e. the visibility degree (ki ) of that node. In order to analyze behavior of systems using clustering coefficient of MBVG, we focus on degrees are more than and equal to 5, i.e. Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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Fig. 3. Experimental degree distribution of the Markov-binary visibility graph corresponding to time series (105 data) of the statistically independent and dependent systems. Solid line corresponding to Eq. (12). In this equation, for chaotic systems a ¼ 4, for Ornstein–Uhlenbeck (O–U) processes a ¼ 3:5 s (s is the ‘‘memory’’ of the system), and for Long-Range stochastic (L-R) processes a ¼ 2:5.
295 296 297 298 299 300 301 302
303
k P 5. First we calculate it for low degrees and then generalize them for all MBVGs. For a clear analysis, we propose Table 1 corresponding to x0 ¼ 1 observer bits. Table 1 is in accord with two intermediate and side states of the observer bit. According to Table 1, it is not hard to detect a precise order for the number of configurations and edges as well as for the 2 clustering coefficient and the probability distribution of the configuration (bðkÞ). The bðkÞ is equal to k1 in the side states and k3 is equal to k1 in the intermediate states. Also, the number of edges in these states are equal to 2k 5 for side states and 4k10 4k12 2k 6 for intermediate states. Consequently, according to the Eq. (14), we have, C S ¼ kðk1Þ for side states and C M ¼ kðk1Þ for intermediate states. With regard to the above contents, the visibility degree of nodes ðkÞ in the MBVG can be defined as following:
k¼ 305
8 < F ðC S ÞþC S
Side states
: F ðC M ÞþCM 2C M
Intermediate states
2C S
ð15Þ 1
306 307 308 309 310 311 312 313
3.2.1. Side states The states that observer bit located at side a bounding bit (see Table 1). With regard to Eqs. (14) and (15), we have,
314 316 317
318
320
1
where F ðC S Þ and F ðC M Þ are equal to 4 ðC 2S 32C S þ 16Þ2 and 4 ðC 2M 40C M þ 16Þ2 , respectively. With regard to the obtained results, we are going to calculate the clustering coefficient probability P(C) in different conditions. We must carry out the calculations for two situations: intermediate and side states. In general, when k P 5, in order to compute the clustering coefficient probability P(C), it would be sufficient to multiply the visibility degree probability distribution P(k) by the one for the configuration bðkÞ. As mentioned earlier, we are going to consider first side and second the intermediate states as follows:
PS ðk P 5Þ ¼ bS ðkÞ Pðk P 5Þ ¼ 2gðkÞ p31 ð1 p1 Þk2 ¼ P
F ðC S Þ þ C S : 2C S
Assuming the following conditions:
8 < kþ ¼ F þ ðCS ÞþCS 2C S
kþ P 5
: k ¼ F ðCS ÞþCS 2C S
k P 5
Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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Fig. 4. Experimental probability distribution of the clustering coefficient of the Markov-binary visibility graph corresponding to time series (105 data) of the statistically independent and dependent systems. Solid line corresponding to Eq. (16) and point line corresponding to Eq. (17). In this equations, for chaotic systems a ¼ 4 and also, d ¼ 1D for logistic map and d ¼ 2D for henon and lozi maps (D is correlation dimension).
Table 1 Probable configurations for one-observer in intermediate and side states.
E k=5 E k=6 E k=7 .. .
Side states
Intermediate states
5 110001
4 101001
7 1100001 9 11000001 .. .
6 1010001 8 10100001 .. .
Side states 4 100101
6 1001001 8 10010001
8 10001001
5 100011 6 1000101 8 10000101
7 1000011 9 10000011 .. .
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and kþ ; k 2 Z. we can get the subsequent relation for clustering coefficient probability:
PðC S Þ ¼ 2g
F þ ðC S Þ3C S F þ ðC S Þ þ C S p31 ð1 p1 Þ 2CS 2C S
ð16Þ
325
where 0 < C A 0:5.
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3.2.2. Intermediate states The states that observer bit located at between the inner bits (see Table 1). With regard to Eqs. (14) and (15), we have,
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PM ðk P 5Þ ¼ bM ðkÞ Pðk P 5Þ ¼ ðk 3ÞgðkÞ p31 ð1 p1 Þk2 ¼ P
F ðC M Þ þ C M : 2C M
Fig. 5. Experimental probability distribution of the clustering coefficient of the Markov-binary visibility graph corresponding to time series (105 data) of the statistically dependent systems. The Ornstein–Uhlenbeck (O–U) processes are in left column and the Long-Range stochastic (L-R) processes are in right column. Solid line corresponding to Eq. (16) and point line corresponding to Eq. (17). In this equations, for Ornstein–Uhlenbeck (O–U) processes a ¼ 3:5 s; d ¼ 1:5 for s ¼ 0:5; d ¼ 2:0 for s ¼ 1:0 and d ¼ 2:5 for s ¼ 1:5; for Long-Range stochastic (L-R) processes, a ¼ 2:5 and also, d ¼ 2.
Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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11
Fig. 6. Experimental probability distribution for the MPGn (visibility length) of the Markov-binary visibility graph corresponding to time series (105 data) of the statistically independent and dependent systems. Solid line corresponding to Eq. (18). In this equation, a ¼ 0:5 for logistic map and a ¼ 1:0 for henon and lozi maps; for Ornstein–Uhlenbeck (O–U) processes, a ¼ 3:5 s and for Long-Range stochastic (L-R) processes, a ¼ 2:5 .
Fig. 7. Experimental mean subgraphs for the MPGn (mean of the visibility length) of the Markov-binary visibility graph corresponding to time series (N ¼ 27 ; . . . ; 216 data) of the statistically independent systems. The lines corresponding to Eq. (19). In this equation, a ¼ 0:18; 1:10; 5:50 for the statistically independent systems (piecewise map) p ¼ 0:30; 0:51; 0:70, respectively.
Fig. 8. Three RR-interval time series of three groups from subjects: Normally, Congestive Heart Failure (CHF) and Atrial Fibrillation (AF) subjects.
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Fig. 9. Experimental degree distribution of the Markov-binary visibility graph corresponding to RR-interval time series (105 data) of three groups from subjects: Normally, Congestive Heart Failure (CHF) and Atrial Fibrillation (AF) subjects (black squares) and the statistically independent and dependent systems (black circles). Solid line corresponding to Eq. (12). In this equation, a ¼ 4 for RR-interval time series of CHF subjects, also, a ¼ 2:5 for RR-interval time series of AF subjects. 331
332
334 335
Assuming the following conditions:
8 < kþ ¼ F þ ðCM ÞþC M 2C M
kþ P 5
: k ¼ F ðCM ÞþC M 2C M
k P 5
and kþ ; k 2 Z. Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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If these conditions are true,
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PðC M Þ ¼ 339 340 341 342 343
13
Fþ ðC M Þ X FðC M Þ¼F ðC M
FðC M Þ3C M FðC M Þ 5C M FðC M Þ þ C M g p31 ð1 p1 Þ 2CM 2C M 2C M Þ
ð17Þ
where 0 < C M 0:4 and if any of the above mentioned conditions (kþ ; k 2 Z and kþ ; k P 5) are violated, the probability will Q4 be zero (see Fig. 5). Numerical results show that the probability distribution of clustering coefficient PðCÞ of the MBVG corresponding to time series of above examples (the examples of pervious Section (3.1)) can be predicted by Eqs. (16) and (17) (see Figs. 6 and 7).
Fig. 10. Experimental probability distribution of the clustering coefficient of the Markov-binary visibility graph corresponding to RR-interval time series (105 data) of three groups from subjects: Normally, Congestive Heart Failure (CHF) and Atrial Fibrillation (AF) subjects (black circles). Solid line corresponding to Eq. (16) and point line corresponding to Eq. (17). In this equation, a ¼ 4 and d ¼ 1:016 for RR-interval time series of CHF subjects, also, a ¼ 2:5 and d ¼ 2 for RR-interval time series of AF subjects.
Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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3.3. Probability distribution of the visibility length and the mean of the visibility length
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The visibility length is defined for the ‘‘one’’ observer. It is the number of intermediate zero bits between two consecutive one-observers in a way that the two one-observers are visible to each other. In graph terminology, it corresponds to the order of the MPG graph which is a subgraph of MBVG. Similar to the Eq. (7) and the above contents, we have,
346 347
348
Fig. 11. Experimental probability distribution for the MPGn (visibility length) of the Markov-binary visibility graph corresponding to RR-interval time series (105 data) of three groups from subjects: Normally, Congestive Heart Failure (CHF) and Atrial Fibrillation (AF) subjects (black squares) and the statistically independent and dependent systems (black circles). Solid line corresponding to Eq. (18). In this equation, a ¼ 0:5 for normally subjects and CHF subjects, also a ¼ 2:5 for RR-interval time series of AF subjects.
Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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Fig. 12. Experimental mean subgraphs for the MPGn (mean of the visibility length) of the Markov-binary visibility graph corresponding to RR-interval time series (N ¼ 27 ; . . . ; 216 data) of normally subjects (black circles). The lines corresponding to Eq. (19). In this equation, a ¼ 0:5 for normally subjects.
P1!1 ðnÞ ¼ gðnÞ
350 351 352
353
h¼1
where gðnÞ is the probability density function for visibility of n and ðPMBS Þh is the joint probability distribution of the bits of MBS ¼ fxh gh¼1;...;nþ1 . Consequently, it be defined as follows:
0 1 n n nþ1 zfflfflfflfflffl}|fflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ Y ðPMBS Þh ¼ P@00 . . . 00A ¼ P0!0 P0!0
355 356
357 359
nþ1 Y ðPMBS Þh
h¼1
then,
P1
1 ðnÞ
¼ gðnÞ ð1 p1 Þn :
360
The value of gðnÞ for the systems is different and can be classified as following:
361
Statistically independent systems: The gðnÞ is constant value. For n ¼ 1; gðnÞ ¼ 1 and for n P 2 is as follows:
362
364 365
366
8 > < a > 1 p1 > 0:5 gðnÞ ¼ a ¼ 1 p1 ¼ 0:5 > : a < 1 p1 < 0:5 Statistically dependent systems: For correlated stochastic systems,
( gðnÞ ¼
369
372 373 374
1
n ¼ 1; 2
a eðn r2 Þ
368 370
ð18Þ
d
nP3
and also, for chaotic systems,
gðnÞ ¼
1
ae
ðn
r2 Þ
d
n¼1 nP2
where a and r are constant value and variance of the Markov-binary sequence, respectively. The value of d is proportional with correlation exponent s in correlation stochastic systems and correlation dimension D in chaotic systems. 2
375 376 377 378 379
380
382
The probability distribution of the visibility length equals the probability distributions for the MPG. Numerical results confirm that the probability distribution for the MPGs of the MBVG corresponding to time series of above examples (the examples of pervious Section (3.1)) can be predicted by Eq. (18) (see Figs. 8 and 9). According to the above results, the mean visibility length for uncorrelated stochastic systems can be defined as follows:
LðNÞ
N X nP1 n¼1
1 ðnÞ
¼ gðnÞ
N X nð1 p1 Þn : n¼1
Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007
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In which, N is the number of bits in an Markov-binary sequence. This can be reduced to:
383
384
LðNÞ ¼ gðnÞ
386
ðp1 1Þ ð1 p1 ÞN ð1 þ Np1 Þ 1 p21
:
ð19Þ
389
where the gðnÞ is constant value. This equation gives the mean graph order of MPGn that equal the mean visibility length for uncorrelated stochastic systems. Numerical results confirm that the mean graph order of MPGn of the MBVG corresponding to time series of uncorrelated stochastic systems can be predicted by Eq. (19) (see Fig. 10).
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4. Network analysis of the human heartbeat dynamics
387 388
391 392 393 394 395 396 397 398 399 400 401 402 403 404 405
Recently, the human heartbeat (RR-interval) dynamics has attracted much attention from researchers [33,16,11]. As we know, the RR-interval is the time elapsing between two consecutive R waves in the electrocardiogram. The RR-interval time series are different in the normal and patient subjects (see Fig. 11). Here, we investigate the RR-interval time series of three groups from subjects: Normally, Congestive Heart Failure (CHF) and Atrial Fibrillation (AF) subjects. Each RR-interval time series is about 24 h long (roughly 105 intervals). All of the time series were derived from continuous ambulatory (Holter) electrocardiograms (ECGs) that are available on the PhysioNet web site (http://www.physionet.org/challenge/chaos). Our analysis are on the fluctuations of time in the RR-interval time series using the MBVA. With regarded to above section, the results show that the degree distribution PðkÞ of the MBVG corresponding to RR-interval time series of normally, Q5 CHF and AF subjects can be predicted by Eq. (12) (see Figs. 12 and 13). To better analyze the RR-interval time series, we obtain other properties of the MBVG corresponding to RR-interval time series. results indicate that the probability distribution of clustering coefficient PðCÞ of the MBVG corresponding to RR-interval time series of normally, CHF and AF subjects can be predicted by Eqs. (16) and (17) (see Fig. 14). Also, the probability distribution of the visibility length PðnÞ of the MBVG corresponding to RR-interval time series of normally, CHF and AF subjects and the mean of the visibility length LðNÞ of the MBVG corresponding to RR-interval time series of normally subjects can be predicted by Eqs. (18) and (19) (see Figs. 15 and 16). The results are in agreement with previous results [13].
406
5. Conclusion
407
412
In conclusion, we introduced the MBVG which can be used to analyze the complex systems. There have been many studies dedicated to Markov binary sequences analysis however, using MBVG, we have been able to analyze and test complex systems from the complex networks perspective. Our results on topological (statistical) properties of the graph confirm this statement. An important result that needs to be highlighted in this paper is that the MBVG can be used easily by any computer and because of its binary status it can be used in broad situations. In addition, this graph has the capability of analyzing normally and statistically independent stochastic, chaotic and statistically dependent stochastic systems.
413
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Q1 Please cite this article in press as: S. Ahadpour et al., Markov-binary visibility graph: A new method for analyzing complex systems, Inform. Sci. (2014), http://dx.doi.org/10.1016/j.ins.2014.03.007