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Time-dependent limited penetrable visibility graph analysis of nonstationary time series
Accepted Manuscript Time-dependent limited penetrable visibility graph analysis of nonstationary time series Zhong-Ke Gao, Qing Cai, Yu-Xuan Yang, Wei-Dong Dang PII: DOI: Reference:
S0378-4371(17)30178-4 http://dx.doi.org/10.1016/j.physa.2017.02.038 PHYSA 18018
To appear in:
Physica A
Received date: 26 October 2016 Revised date: 27 December 2016 Please cite this article as: Z.-K. Gao, Q. Cai, Y.-X. Yang, W.-D. Dang, Time-dependent limited penetrable visibility graph analysis of nonstationary time series, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.02.038 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Research highlights to the manuscript entitled “Time-dependent limited penetrable visibility graph analysis of nonstationary time series” by Zhong-Ke Gao, Qing Cai, Yu-Xuan Yang, Wei-Dong Dang:
1. We develop a time-dependent limited penetrable visibility graph to analyze time series; 2. Our method enables to identify healthy, CHF and AF from RR interval time series; 3. Our method allows characterizing time-varying flow behaviors of gas-liquid flows; 4. Broader applicability of our method is demonstrated and articulated.
*Manuscript Click here to view linked References
Time-dependent limited penetrable visibility graph analysis of nonstationary time series Zhong-Ke Gao, Qing Cai, Yu-Xuan Yang, Wei-Dong Dang School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
Abstract Recent years have witnessed the development of visibility graph theory, which allows us to analyze a time series from the perspective of complex network. We in this paper develop a novel time-dependent limited penetrable visibility graph (TDLPVG). Two examples using nonstationary time series from RR intervals and gas-liquid flows are provided to demonstrate the effectiveness of our approach. The results of the first example suggest that our TDLPVG method allows characterizing the time-varying behaviors and classifying heart states of healthy, congestive heart failure and atrial fibrillation from RR interval time series. For the second example, we infer TDLPVGs from gas-liquid flow signals and interestingly find that the deviation of node degree of TDLPVGs enables to effectively uncover the time-varying dynamical flow behaviors of gas-liquid slug and bubble flow patterns. All these results render our TDLPVG method particularly powerful for characterizing the time-varying features underlying realistic complex systems from time series.
Keywords: Time series analysis; Limited penetrable visibility graph; Complex network; ECG
PACS: 05.45.Tp
1. Introduction Characterizing time-dependent complicated behavior from a nonstationary time series represents a challenging problem of significant importance. Methodologies from different research fields have been proposed to solve this challenge, e.g., fractal and multifractal analysis [1-2], recurrence analysis [3], detrended cross-correlation
Corresponding author.
E-mail address: [email protected] (ZK Gao)
analysis [4] and time-frequency representation [5]. During the last decade, a new multidisciplinary methodology using complex network has emerged for characterizing complex systems [6-15]. More importantly, complex network has been proved to be an effective analytical framework for probing dynamical behavior from time series [16-26]. In particular, visibility graph (VG) [18] allows us to construct a complex network from a time series in a fast manner, and it has been successfully implemented in different fields [27-35]. Recently, we developed a limited penetrable visibility graph (LPVG) [36-37] and found that LPVG not only inherits the merits of VG but also presents a good anti-noise ability, which render LPVG particularly useful for analyzing real signals polluted by unavoidable noise. Our LPVG method has been successfully adopted to analyze many real signals from different fields, e.g., EEG signals [37-39], experimental flow signals [40], electromechanical signals [41]. As a development of our previous works, we in this paper develop a novel time-dependent limited penetrable visibility graph (TDLPVG) to characterize the time-varying behavior from nonstationary time series. Two examples using nonstationary time series from RR intervals and gas-liquid flows are provided to demonstrate the effectiveness of our approach. Heart rate variability (HRV) characterizes the fluctuations in the intervals between heart beats, known as RR intervals with the irregular, nonstationary, and nonlinear features. Since RR intervals contain important information about the physiological state of the subject, the analysis of RR intervals has attracted a great deal of attention. Many methods have been developed to analyze HRV signals, e.g., generalized discriminant analysis [42], multifractality analysis[43], visibility graph [44-46], generalized sample entropy[47], multiscale entropy [48], permutation min-entropy [49], etc.. We in this paper focus on three different RR intervals from healthy, atrial fibrillation and congestive heart failure. Specifically, atrial fibrillation (AF) is one of the most common abnormal heart rhythm characterized by rapid and irregular RR intervals, which is usually associated with an increased risk of heart failure, dementia and stroke. Congestive heart failure (CHF) is a serious and complex disease in which the heart muscle has been damaged and it is rather difficult to efficiently pump enough oxygen-rich blood to and from the
body, resulting in potentially life-threatening congestion in the lungs and other tissues of the body. Identifying AF, CHF and healthy heart state from RR interval time series represents a challenge of significant importance and continuous interests. Our proposed TDLPVG allows distinguishing and classifying three RR interval time series from healthy, congestive heart failure (CHF), and atrial fibrillation (AF) subjects. Gas-liquid two-phase flow, as a typical complex system, is widely encountered in many industrial processes such as natural gas networks and nuclear reactor cooling. The fundamental difference between single phase flow and gas-liquid two-phase flow is the existence of flow pattern, which indicates the temporal-spatial distribution of gas phase with deformable shapes in a continuous water phase. The great interests in the study of flow patterns lie in the fact that different flow patterns exhibit distinct dynamical flow behaviors. How to uncover the time-varying dynamic behavior of different flow patterns from experimental measurements has constituted a challenge of significant importance. We carry out gas-liquid two-phase flow experiment to obtain the flow signals and then use TDLPVG method to identify and characterize time-varying dynamical flow behavior underlying gas-liquid slug flow pattern and bubble flow pattern. The results demonstrate that our method allows characterizing the intrinsic flow behaviors varying with the time for these two typical flow patterns, which enriches the understanding of the formation and evolution of gas-liquid flows. The above results suggest that our TDLPVG can serve as an effective method for characterizing time-varying behaviors underlying a given nonstationary time series.
2. Methodology The time-dependent limited penetrable visibility graph (TDLPVG) method can be implemented by the following steps: For a time series of length L,
x(i), i 1, 2,..., L , we first partition the time series into M non-overlap equal length windows in the order of time varying as follow:
y m ( j) x m 1 N j , m 1,2,..., M , j 1,2,..., N
(1)
where M represents the number of time window, N represents the length of window. Next we infer limited penetrable visibility graphs from the window time series
y
m
( j ), m 1, 2,..., M , j 1, 2,..., N , and then obtain a series of limited penetrable
visibility graphs as the window slides from the left to right (varying with the time). We in Fig. 1 show a schematic diagram for demonstrating how to infer limited penetrable visibility graph from a time series. For a continuous time series of length 10, we display them in the form of vertical bars in Fig.1(a). Considering this as a landscape, we link every bar (every point of the time series) with all those that can be seen from the top of the considered one (black lines), obtaining the associated visibility graph. In the visibility graph, two nodes (two vertical bars) can see each other and then a connection between these two nodes exists. That is, two nodes are connected only if a straight visibility line between them does not intersect any intermediate data height. More formally, we can establish the following visibility criteria: two arbitrary data values ( ta , Ea ) and ( tb , Eb ) will have visibility, and consequently will become two connected nodes of the associated graph, if any other data ( tc , Ec ) placed between them fulfills:
Ec Eb Ea Eb
tb t c tb t a
(2)
Our limited penetrable visibility graph is a development of visibility graph [18]. In particular, if we set the limited penetrable distance to be L, a connection between two nodes exists if the number of in-between nodes that block the visibility line is no more than L. As shown in Fig.1(b), the red lines are the new established connections when we infer the LPVG on the basis of VG with the limited penetrable distance being 1. Finally, based on the above procedure, we can obtain a series of TDLPVGs as the window slides from the left to right (varying with the time). The visibility graph leads to a natural graph-theoretical description of nonlinear systems with qualities in the spirit of symbolic dynamics. Representing each window time series through corresponding limited penetrable visibility graph, we can then
explore the dynamic behaviors in terms of network analysis, which is quantified via network statistical measures. In particular, we employ the global clustering coefficient (C), the average local clustering coefficient ( C ) [50], clustering coefficient entropy (EC) [26] and the deviation of node degree ( kstd ) to characterize the topological structure of inferred TDLPVGs. These network measures can be calculated as follows C
(3)
Ci
i , i
(4)
C
where
1 N
N
C i 1
(5)
i
N E PC ,i log( PC ,i ) C i 1 N PC ,i Ci Ci i 1
(6)
1/2 N 2 ( ki k ) k i 1 std N 1 N 1 k ki N i 1
(7)
denotes the number of closed triplets in a network,
denotes the number
of triplets (i.e., three nodes connected by two ties) in a network, number of closed triplets centered on node i,
i
i , denotes the
is the number of triplets centered on
node i, ki is the degree of a node i (connections of a node i). N is the node number of the derived TDLPVG.
3. TDLPVG analysis of RR interval time series We first use our TDLPVG method to analyze 15 RR interval time series from 5 healthy subjects, 5 congestive heart failure (CHF) subjects, and 5 atrial fibrillation (AF) subjects. The RR interval time series are derived from electrocardiograms
(ECG), available from the PhysioNet databases. Each time series is about 24 hours long, (roughly 100000 RR intervals). We first partition the RR interval time series into non-overlap windows with equal length of 2000. Then we infer the corresponding limited penetrable visibility graph from each time-dependent window time series and the node number of each derived TDLPVG is 2000. Totally, we obtain 225 TDLPVGs from healthy subjects, 224 TDLPVGs from the subjects suffering from congestive heart failure, and 289 TDLPVGs from the subjects suffering from atrial fibrillation. We calculate the global clustering coefficient, the average local clustering coefficient, clustering coefficient entropy and the deviation of node degree for all generated TDLPVGs. Fig.2 shows the boxplots of the four network features for all generated TDLPVGs. The one-way ANOVA analysis is employed to test the significance difference of TDLPVG features from three different RR interval time series groups. In addition, we compare in pairs by using the method of t-test, i.e., healthy vs. CHF, healthy vs. AF, and CHF vs. AF. The p-values of the TDLPVG features are presented in Table 1 to show the significance of the features in discriminating AF, CHF patients and healthy subjects. Note that, the p-value less than 0.05 indicates the existence of statistical significance. As shown in Table.1, all the p-values are much smaller than 0.05, indicating these TDLPVG features allow discriminating three type of RR interval time series groups. Furthermore, we combine the four TDLPVG features to generate four-dimensional feature vectors and employ leave-one-out cross-validation and random forest classifier to classify the three groups (healthy, CHF, AF). The leave-one-out cross-validation consists of removing one sample from the dataset, constructing the decision function on the basis only of the remaining dataset and then testing on the removed sample. In this fashion one tests all samples of the dataset and measures the fraction of errors over the total number of samples in the dataset. That is, for this classification, this process is repeated 738 times independently, with a different sample left out for testing every time. After 738 cross validations, we obtain the predicting labels for all samples and measure the fraction of correctly predicted samples over the total number of samples in the dataset. The classification accuracy of healthy, congestive heart failure (CHF), and atrial fibrillation (AF) subjects is 93.5%.
TDLPVG features of RR interval time series Groups
Analysis method
k std
C
C
EC
AF, CHF
10-86
10-15
10-133
10-173
t-test
AF, Healthy
10-32
10-18
10-51
10-86
t-test
CHF, Healthy
10-82
0.0241
10-33
10-85
t-test
AF, CHF, Healthy
10-147
10-17
10-134
10-224
One-way ANOVA
Table 1. P-values of the significance analysis of TDLPVG features
4. TDLPVG analysis of experimental gas-liquid flow signals We now apply our TDLPVG method to analyze gas-liquid flow signals, which are from our gas-water two-phase flow experiment carried out in a vertical upward 20 mm-inner-diameter plexiglass pipe. The experiential media are tap-water and air. These two immiscible media mix themselves and then flow together into a vertical testing pipe. The conductance sensor [51] is designed to capture the flow behavior and the measured flow signals are stored by data acquisition devices. The sampling rate is 4000 Hz. We use the high-speed camera to observe and define gas-water flow patterns. In this paper, we focus on the characterization of two typical gas-liquid flow patterns, i.e., slug flow and bubble flow. The experimental measurements are partitioned into non-overlap windows with equal length of 10000. We infer time-dependent limited penetrable visibility graphs from the non-overlap windows and the limited penetrable distance is 1. Then we employ the deviation of node degree to analyze the derived TDLPVGs from different flow conditions. The results are shown in Figs. 3-4. We can see that, the time-dependent distributions of the deviation of node degree for different flow patterns exhibit distinct features, which allows identifying gas-liquid slug and bubble flow patterns. In the gas-liquid slug flow, the gas slugs and small bubbles co-current flow in a water continuum, in which the flow behavior of gas slugs is dominant and exhibits the features of quasi-periodic oscillation. Consequently, the deviation of node degree of slug flows exhibit large values at different time windows, and the time-varying fluctuation behavior is the most drastic. The turbulent kinetic
energy is greatly enhanced with an increase in the water flow rate. As a result, the gas slugs are broken into numbers of small gas bubbles. That is, gas-liquid bubble flow occurs. In the bubble flow, the intermittent oscillation of gas slugs gradually disappears and the non-homogenous distribution of gas phase becomes weak; large numbers of small gas bubbles interact with each other and their movements exhibit stochastic features. Therefore, the fluctuation strength of the measured conductance signals is weakened. Correspondingly, the deviation of node degree of bubble flows decreases as the flow pattern evolves from slug flow to bubble flow and meanwhile the time-varying fluctuation behavior is also weakened. Furthermore, we notice that for the bubble flow pattern under a same water flow rate, with the increase of gas flow rate, the time-varying fluctuation behavior becomes intense, which attributes to the fact that many small gas bubbles coalesce with each other to form a relative big bubble resulting from the increased gas flow rate. Note that, such a formed big bubble is still much smaller than the gas slug, therefore the enhanced time-varying fluctuation is weaker than that of gas-liquid slug flow. In this regard, our proposed TDLPVG analysis enables to identify different flow states and further allows characterizing the time-varying flow behaviors underlying the evolution of gas-liquid slug and bubble flows.
5. Conclusions In summary, we have articulated a novel TDLPVG strategy (time-dependent limited penetrable visibility graph) for analyzing nonstationary time series. We apply the TDLPVG method to analyze two realistic data from RR intervals and gas-liquid flows, respectively. The results suggest that our TDLPVG method not only allows characterizing and classifying the time-varying features underlying three different RR interval time series, but also enables to faithfully reveal the time-varying flow behavior governing the transitions and evolutions of typical gas-liquid slug and bubble flow patterns. The effectiveness of our method is demonstrated and its applicability is articulated. Due to the generality of the TDLPVG method, we expect it to be useful for broader applications in science and engineering.
Acknowledgements This work was supported by National Natural Science Foundation of China under Grant No. 61473203 and the Natural Science Foundation of Tianjin, China under Grant No. 16JCYBJC18200.
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Figure legends: Fig. 1. (Color online) Example of (a) a time series and (b) its corresponding LPVG with the limited penetrable distance L being 1, where every node corresponds to time series data in the same order. The visibility lines between data points define the links connecting nodes in the graph.
Fig. 2. (Color online) Boxplots of the global clustering coefficient (a), the average local clustering coefficient (b), the clustering coefficient entropy (c) and the deviation of node degree (d) for healthy, CHF and AF subjects.
Fig. 3. (Color online) Time-varying distributions of TDLPVG measure for gas-liquid slug and bubble flow patterns under a fixed gas flow rate Usg = 0.055m/s.
Fig. 4. (Color online) Time-varying distributions of TDLPVG measure for gas-liquid slug and bubble flow patterns under a fixed gas flow rate Usg = 0.074m/s.
Fig. 1. (Color online) Example of (a) a time series and (b) its corresponding LPVG with the limited penetrable distance L being 1, where every node corresponds to time series data in the same order. The visibility lines between data points define the links connecting nodes in the graph.
(a)
(c)
(b)
(d)
Fig. 2. (Color online) Boxplots of the global clustering coefficient (a), the average local clustering coefficient (b), the clustering coefficient entropy (c) and the deviation of node degree (d) for healthy, CHF and AF subjects.
Usg=0.055m/s
220
Deviation of node degree
200 180
slug flow
Usw=0.037m/s
slug flow
Usw=0.074m/s
slug flow
Usw=0.147m/s
bubble flow Usw=0.737m/s bubble flow Usw=0.884m/s
160
bubble flow Usw=1.032m/s
140 120 100 80 60 40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time window Fig. 3. (Color online) Time-varying distributions of TDLPVG measure for gas-liquid slug and bubble flow patterns under a fixed gas flow rate Usg = 0.055m/s.
Usg=0.074m/s
200
Deviation of node degree
180
slug flow
Usw=0.037m/s
slug flow
Usw=0.074m/s
slug flow
Usw=0.147m/s
bubble flow Usw=0.737m/s bubble flow Usw=0.884m/s
160
bubble flow Usw=1.032m/s
140 120 100 80 60 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time window Fig. 4. (Color online) Time-varying distributions of TDLPVG measure for gas-liquid slug and bubble flow patterns under a fixed gas flow rate Usg = 0.074m/s.
S0378-4371(17)30178-4 http://dx.doi.org/10.1016/j.physa.2017.02.038 PHYSA 18018
To appear in:
Physica A
Received date: 26 October 2016 Revised date: 27 December 2016 Please cite this article as: Z.-K. Gao, Q. Cai, Y.-X. Yang, W.-D. Dang, Time-dependent limited penetrable visibility graph analysis of nonstationary time series, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.02.038 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
Research highlights to the manuscript entitled “Time-dependent limited penetrable visibility graph analysis of nonstationary time series” by Zhong-Ke Gao, Qing Cai, Yu-Xuan Yang, Wei-Dong Dang:
1. We develop a time-dependent limited penetrable visibility graph to analyze time series; 2. Our method enables to identify healthy, CHF and AF from RR interval time series; 3. Our method allows characterizing time-varying flow behaviors of gas-liquid flows; 4. Broader applicability of our method is demonstrated and articulated.
*Manuscript Click here to view linked References
Time-dependent limited penetrable visibility graph analysis of nonstationary time series Zhong-Ke Gao, Qing Cai, Yu-Xuan Yang, Wei-Dong Dang School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
Abstract Recent years have witnessed the development of visibility graph theory, which allows us to analyze a time series from the perspective of complex network. We in this paper develop a novel time-dependent limited penetrable visibility graph (TDLPVG). Two examples using nonstationary time series from RR intervals and gas-liquid flows are provided to demonstrate the effectiveness of our approach. The results of the first example suggest that our TDLPVG method allows characterizing the time-varying behaviors and classifying heart states of healthy, congestive heart failure and atrial fibrillation from RR interval time series. For the second example, we infer TDLPVGs from gas-liquid flow signals and interestingly find that the deviation of node degree of TDLPVGs enables to effectively uncover the time-varying dynamical flow behaviors of gas-liquid slug and bubble flow patterns. All these results render our TDLPVG method particularly powerful for characterizing the time-varying features underlying realistic complex systems from time series.
Keywords: Time series analysis; Limited penetrable visibility graph; Complex network; ECG
PACS: 05.45.Tp
1. Introduction Characterizing time-dependent complicated behavior from a nonstationary time series represents a challenging problem of significant importance. Methodologies from different research fields have been proposed to solve this challenge, e.g., fractal and multifractal analysis [1-2], recurrence analysis [3], detrended cross-correlation
Corresponding author.
E-mail address: [email protected] (ZK Gao)
analysis [4] and time-frequency representation [5]. During the last decade, a new multidisciplinary methodology using complex network has emerged for characterizing complex systems [6-15]. More importantly, complex network has been proved to be an effective analytical framework for probing dynamical behavior from time series [16-26]. In particular, visibility graph (VG) [18] allows us to construct a complex network from a time series in a fast manner, and it has been successfully implemented in different fields [27-35]. Recently, we developed a limited penetrable visibility graph (LPVG) [36-37] and found that LPVG not only inherits the merits of VG but also presents a good anti-noise ability, which render LPVG particularly useful for analyzing real signals polluted by unavoidable noise. Our LPVG method has been successfully adopted to analyze many real signals from different fields, e.g., EEG signals [37-39], experimental flow signals [40], electromechanical signals [41]. As a development of our previous works, we in this paper develop a novel time-dependent limited penetrable visibility graph (TDLPVG) to characterize the time-varying behavior from nonstationary time series. Two examples using nonstationary time series from RR intervals and gas-liquid flows are provided to demonstrate the effectiveness of our approach. Heart rate variability (HRV) characterizes the fluctuations in the intervals between heart beats, known as RR intervals with the irregular, nonstationary, and nonlinear features. Since RR intervals contain important information about the physiological state of the subject, the analysis of RR intervals has attracted a great deal of attention. Many methods have been developed to analyze HRV signals, e.g., generalized discriminant analysis [42], multifractality analysis[43], visibility graph [44-46], generalized sample entropy[47], multiscale entropy [48], permutation min-entropy [49], etc.. We in this paper focus on three different RR intervals from healthy, atrial fibrillation and congestive heart failure. Specifically, atrial fibrillation (AF) is one of the most common abnormal heart rhythm characterized by rapid and irregular RR intervals, which is usually associated with an increased risk of heart failure, dementia and stroke. Congestive heart failure (CHF) is a serious and complex disease in which the heart muscle has been damaged and it is rather difficult to efficiently pump enough oxygen-rich blood to and from the
body, resulting in potentially life-threatening congestion in the lungs and other tissues of the body. Identifying AF, CHF and healthy heart state from RR interval time series represents a challenge of significant importance and continuous interests. Our proposed TDLPVG allows distinguishing and classifying three RR interval time series from healthy, congestive heart failure (CHF), and atrial fibrillation (AF) subjects. Gas-liquid two-phase flow, as a typical complex system, is widely encountered in many industrial processes such as natural gas networks and nuclear reactor cooling. The fundamental difference between single phase flow and gas-liquid two-phase flow is the existence of flow pattern, which indicates the temporal-spatial distribution of gas phase with deformable shapes in a continuous water phase. The great interests in the study of flow patterns lie in the fact that different flow patterns exhibit distinct dynamical flow behaviors. How to uncover the time-varying dynamic behavior of different flow patterns from experimental measurements has constituted a challenge of significant importance. We carry out gas-liquid two-phase flow experiment to obtain the flow signals and then use TDLPVG method to identify and characterize time-varying dynamical flow behavior underlying gas-liquid slug flow pattern and bubble flow pattern. The results demonstrate that our method allows characterizing the intrinsic flow behaviors varying with the time for these two typical flow patterns, which enriches the understanding of the formation and evolution of gas-liquid flows. The above results suggest that our TDLPVG can serve as an effective method for characterizing time-varying behaviors underlying a given nonstationary time series.
2. Methodology The time-dependent limited penetrable visibility graph (TDLPVG) method can be implemented by the following steps: For a time series of length L,
x(i), i 1, 2,..., L , we first partition the time series into M non-overlap equal length windows in the order of time varying as follow:
y m ( j) x m 1 N j , m 1,2,..., M , j 1,2,..., N
(1)
where M represents the number of time window, N represents the length of window. Next we infer limited penetrable visibility graphs from the window time series
y
m
( j ), m 1, 2,..., M , j 1, 2,..., N , and then obtain a series of limited penetrable
visibility graphs as the window slides from the left to right (varying with the time). We in Fig. 1 show a schematic diagram for demonstrating how to infer limited penetrable visibility graph from a time series. For a continuous time series of length 10, we display them in the form of vertical bars in Fig.1(a). Considering this as a landscape, we link every bar (every point of the time series) with all those that can be seen from the top of the considered one (black lines), obtaining the associated visibility graph. In the visibility graph, two nodes (two vertical bars) can see each other and then a connection between these two nodes exists. That is, two nodes are connected only if a straight visibility line between them does not intersect any intermediate data height. More formally, we can establish the following visibility criteria: two arbitrary data values ( ta , Ea ) and ( tb , Eb ) will have visibility, and consequently will become two connected nodes of the associated graph, if any other data ( tc , Ec ) placed between them fulfills:
Ec Eb Ea Eb
tb t c tb t a
(2)
Our limited penetrable visibility graph is a development of visibility graph [18]. In particular, if we set the limited penetrable distance to be L, a connection between two nodes exists if the number of in-between nodes that block the visibility line is no more than L. As shown in Fig.1(b), the red lines are the new established connections when we infer the LPVG on the basis of VG with the limited penetrable distance being 1. Finally, based on the above procedure, we can obtain a series of TDLPVGs as the window slides from the left to right (varying with the time). The visibility graph leads to a natural graph-theoretical description of nonlinear systems with qualities in the spirit of symbolic dynamics. Representing each window time series through corresponding limited penetrable visibility graph, we can then
explore the dynamic behaviors in terms of network analysis, which is quantified via network statistical measures. In particular, we employ the global clustering coefficient (C), the average local clustering coefficient ( C ) [50], clustering coefficient entropy (EC) [26] and the deviation of node degree ( kstd ) to characterize the topological structure of inferred TDLPVGs. These network measures can be calculated as follows C
(3)
Ci
i , i
(4)
C
where
1 N
N
C i 1
(5)
i
N E PC ,i log( PC ,i ) C i 1 N PC ,i Ci Ci i 1
(6)
1/2 N 2 ( ki k ) k i 1 std N 1 N 1 k ki N i 1
(7)
denotes the number of closed triplets in a network,
denotes the number
of triplets (i.e., three nodes connected by two ties) in a network, number of closed triplets centered on node i,
i
i , denotes the
is the number of triplets centered on
node i, ki is the degree of a node i (connections of a node i). N is the node number of the derived TDLPVG.
3. TDLPVG analysis of RR interval time series We first use our TDLPVG method to analyze 15 RR interval time series from 5 healthy subjects, 5 congestive heart failure (CHF) subjects, and 5 atrial fibrillation (AF) subjects. The RR interval time series are derived from electrocardiograms
(ECG), available from the PhysioNet databases. Each time series is about 24 hours long, (roughly 100000 RR intervals). We first partition the RR interval time series into non-overlap windows with equal length of 2000. Then we infer the corresponding limited penetrable visibility graph from each time-dependent window time series and the node number of each derived TDLPVG is 2000. Totally, we obtain 225 TDLPVGs from healthy subjects, 224 TDLPVGs from the subjects suffering from congestive heart failure, and 289 TDLPVGs from the subjects suffering from atrial fibrillation. We calculate the global clustering coefficient, the average local clustering coefficient, clustering coefficient entropy and the deviation of node degree for all generated TDLPVGs. Fig.2 shows the boxplots of the four network features for all generated TDLPVGs. The one-way ANOVA analysis is employed to test the significance difference of TDLPVG features from three different RR interval time series groups. In addition, we compare in pairs by using the method of t-test, i.e., healthy vs. CHF, healthy vs. AF, and CHF vs. AF. The p-values of the TDLPVG features are presented in Table 1 to show the significance of the features in discriminating AF, CHF patients and healthy subjects. Note that, the p-value less than 0.05 indicates the existence of statistical significance. As shown in Table.1, all the p-values are much smaller than 0.05, indicating these TDLPVG features allow discriminating three type of RR interval time series groups. Furthermore, we combine the four TDLPVG features to generate four-dimensional feature vectors and employ leave-one-out cross-validation and random forest classifier to classify the three groups (healthy, CHF, AF). The leave-one-out cross-validation consists of removing one sample from the dataset, constructing the decision function on the basis only of the remaining dataset and then testing on the removed sample. In this fashion one tests all samples of the dataset and measures the fraction of errors over the total number of samples in the dataset. That is, for this classification, this process is repeated 738 times independently, with a different sample left out for testing every time. After 738 cross validations, we obtain the predicting labels for all samples and measure the fraction of correctly predicted samples over the total number of samples in the dataset. The classification accuracy of healthy, congestive heart failure (CHF), and atrial fibrillation (AF) subjects is 93.5%.
TDLPVG features of RR interval time series Groups
Analysis method
k std
C
C
EC
AF, CHF
10-86
10-15
10-133
10-173
t-test
AF, Healthy
10-32
10-18
10-51
10-86
t-test
CHF, Healthy
10-82
0.0241
10-33
10-85
t-test
AF, CHF, Healthy
10-147
10-17
10-134
10-224
One-way ANOVA
Table 1. P-values of the significance analysis of TDLPVG features
4. TDLPVG analysis of experimental gas-liquid flow signals We now apply our TDLPVG method to analyze gas-liquid flow signals, which are from our gas-water two-phase flow experiment carried out in a vertical upward 20 mm-inner-diameter plexiglass pipe. The experiential media are tap-water and air. These two immiscible media mix themselves and then flow together into a vertical testing pipe. The conductance sensor [51] is designed to capture the flow behavior and the measured flow signals are stored by data acquisition devices. The sampling rate is 4000 Hz. We use the high-speed camera to observe and define gas-water flow patterns. In this paper, we focus on the characterization of two typical gas-liquid flow patterns, i.e., slug flow and bubble flow. The experimental measurements are partitioned into non-overlap windows with equal length of 10000. We infer time-dependent limited penetrable visibility graphs from the non-overlap windows and the limited penetrable distance is 1. Then we employ the deviation of node degree to analyze the derived TDLPVGs from different flow conditions. The results are shown in Figs. 3-4. We can see that, the time-dependent distributions of the deviation of node degree for different flow patterns exhibit distinct features, which allows identifying gas-liquid slug and bubble flow patterns. In the gas-liquid slug flow, the gas slugs and small bubbles co-current flow in a water continuum, in which the flow behavior of gas slugs is dominant and exhibits the features of quasi-periodic oscillation. Consequently, the deviation of node degree of slug flows exhibit large values at different time windows, and the time-varying fluctuation behavior is the most drastic. The turbulent kinetic
energy is greatly enhanced with an increase in the water flow rate. As a result, the gas slugs are broken into numbers of small gas bubbles. That is, gas-liquid bubble flow occurs. In the bubble flow, the intermittent oscillation of gas slugs gradually disappears and the non-homogenous distribution of gas phase becomes weak; large numbers of small gas bubbles interact with each other and their movements exhibit stochastic features. Therefore, the fluctuation strength of the measured conductance signals is weakened. Correspondingly, the deviation of node degree of bubble flows decreases as the flow pattern evolves from slug flow to bubble flow and meanwhile the time-varying fluctuation behavior is also weakened. Furthermore, we notice that for the bubble flow pattern under a same water flow rate, with the increase of gas flow rate, the time-varying fluctuation behavior becomes intense, which attributes to the fact that many small gas bubbles coalesce with each other to form a relative big bubble resulting from the increased gas flow rate. Note that, such a formed big bubble is still much smaller than the gas slug, therefore the enhanced time-varying fluctuation is weaker than that of gas-liquid slug flow. In this regard, our proposed TDLPVG analysis enables to identify different flow states and further allows characterizing the time-varying flow behaviors underlying the evolution of gas-liquid slug and bubble flows.
5. Conclusions In summary, we have articulated a novel TDLPVG strategy (time-dependent limited penetrable visibility graph) for analyzing nonstationary time series. We apply the TDLPVG method to analyze two realistic data from RR intervals and gas-liquid flows, respectively. The results suggest that our TDLPVG method not only allows characterizing and classifying the time-varying features underlying three different RR interval time series, but also enables to faithfully reveal the time-varying flow behavior governing the transitions and evolutions of typical gas-liquid slug and bubble flow patterns. The effectiveness of our method is demonstrated and its applicability is articulated. Due to the generality of the TDLPVG method, we expect it to be useful for broader applications in science and engineering.
Acknowledgements This work was supported by National Natural Science Foundation of China under Grant No. 61473203 and the Natural Science Foundation of Tianjin, China under Grant No. 16JCYBJC18200.
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Figure legends: Fig. 1. (Color online) Example of (a) a time series and (b) its corresponding LPVG with the limited penetrable distance L being 1, where every node corresponds to time series data in the same order. The visibility lines between data points define the links connecting nodes in the graph.
Fig. 2. (Color online) Boxplots of the global clustering coefficient (a), the average local clustering coefficient (b), the clustering coefficient entropy (c) and the deviation of node degree (d) for healthy, CHF and AF subjects.
Fig. 3. (Color online) Time-varying distributions of TDLPVG measure for gas-liquid slug and bubble flow patterns under a fixed gas flow rate Usg = 0.055m/s.
Fig. 4. (Color online) Time-varying distributions of TDLPVG measure for gas-liquid slug and bubble flow patterns under a fixed gas flow rate Usg = 0.074m/s.
Fig. 1. (Color online) Example of (a) a time series and (b) its corresponding LPVG with the limited penetrable distance L being 1, where every node corresponds to time series data in the same order. The visibility lines between data points define the links connecting nodes in the graph.
(a)
(c)
(b)
(d)
Fig. 2. (Color online) Boxplots of the global clustering coefficient (a), the average local clustering coefficient (b), the clustering coefficient entropy (c) and the deviation of node degree (d) for healthy, CHF and AF subjects.
Usg=0.055m/s
220
Deviation of node degree
200 180
slug flow
Usw=0.037m/s
slug flow
Usw=0.074m/s
slug flow
Usw=0.147m/s
bubble flow Usw=0.737m/s bubble flow Usw=0.884m/s
160
bubble flow Usw=1.032m/s
140 120 100 80 60 40 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time window Fig. 3. (Color online) Time-varying distributions of TDLPVG measure for gas-liquid slug and bubble flow patterns under a fixed gas flow rate Usg = 0.055m/s.
Usg=0.074m/s
200
Deviation of node degree
180
slug flow
Usw=0.037m/s
slug flow
Usw=0.074m/s
slug flow
Usw=0.147m/s
bubble flow Usw=0.737m/s bubble flow Usw=0.884m/s
160
bubble flow Usw=1.032m/s
140 120 100 80 60 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Time window Fig. 4. (Color online) Time-varying distributions of TDLPVG measure for gas-liquid slug and bubble flow patterns under a fixed gas flow rate Usg = 0.074m/s.