Horizontal Visibility Graph Research Articles

Fault diagnosis of rolling bearings using weighted horizontal visibility graph and graph Fourier transform

Graph Fourier transform (GFT) has been proven to be an effective tool for impulse component extraction of rolling bearings, but its performance is closely related to the structure of underlying graph. Compared with the weighted path graph, the weighted horizontal visibility graph (WHVG) can reflect the dynamics characteristics of vibration signals better. When the fault bearing vibration signal is transformed into the WHVG, GFT can cluster most of the fault impulse component to the highest order range and has strong anti-interference ability. Based on WHVG and GFT, a novel fault diagnosis method for rolling bearings is proposed. In the proposed method, the graph spectrum coefficients in the highest order range are extracted to reconstruct the fault impulse component, and then the Hilbert envelope spectrum is used to diagnose the bearing fault. Simulation and experimental results demonstrate that the proposed fault diagnosis method for rolling bearings is noise tolerant and effective.

Improving methodology in heart rate variability analysis for the premature infants: Impact of the time length.

BackgroundHeart rate variability (HRV) has been emerging in neonatal medicine. It may help for the early diagnosis of pathology and estimation of autonomous maturation. There is a lack of standardization and automation in the selection of the sequences to analyze and some features have not been explored in this specific population. The main objective of this study was to analyze the impact of the time length of the sequences on the estimation of linear and non-linear HRV features, including horizontal visibility graphs (HVG).MethodsHRV features were repeatedly measured with linear and non-linear methods on 2-, 5-, 10-minute sequences selected from the longest 15-min sequence and recorded on a weekly basis in 39 infants less than 31 weeks at birth. The associations between HRV measurements were analyzed through principal component analysis and k-means clustering. The effects of the time lengths on HRV measurements and post-menstrual age (PMA) were analyzed by linear mixed effect model for repeated measures.ResultsThe domains of analysis were concordant for their descriptive parameters of short (rMSSD, SD1 and HF) and long-term (SD, SD2 and LF) variability. α1 was correlated with the LF/HF and SD2/SD1. DC and AC were correlated with short-term variability estimates and significantly increased with GA and PMA. Shortening the windows of analysis increased the random measurement error for all the features and increased the bias for all but short term features and HVGs.ConclusionThe linear and non-linear measurements of HRV are correlated each other. Shortening the windows of analysis increased the random error for all the features and increased the bias for all but short term features and HVGs. Short-term HRV can be an index for evaluating the maturation in whatever sequence length.

Nonuniversality of the horizontal visibility graph in inferring series periodicity

The filter horizontal visibility graph (fHVg) algorithm was recently proposed to detect the hidden periodicity of intrinsically periodic series under the pollution of noise. In this work, we evaluate the reliability of this algorithm by taking into account the effect of finite size and noise pollution, and something intriguing is found. The fHVg is first applied to logistic map with period 2 and 3, and numerical results suggest that the accuracy of fHVg is not affected by the length of tested series. It is effective in analyzing very short time series but sensitive to extrinsic noises. However, the fHVg has unexpected limitations that lead to spurious results. It lacks generality and shows inability when applied to logistic map with period 4 and to the monthly mean temperature dataset from real-world.

The effects generated by events on destination dynamics and topology

ABSTRACT This study investigates the effects generated by a mega-event (the World Expo 2015) on a destination (Milan), using network analytic methods. The horizontal visibility graph was used to transform the time series into a network. Two hypotheses are tested: first, the ability of the Milan Expo to generate a turning point; second, the ability of this event to increase the system’s stability. The findings are based on a longitudinal analysis (2004–2017) of three widely used metrics: occupancy, average daily rate, and revenue per available room. The empirical pieces of evidence confirm both hypotheses. The Milan Expo has generated a positive turning point increasing the system stability. In the conclusion section, some theoretical and practical advancements are proposed, along with the implications for future research, and the main study limitations.

Could network analysis of horizontal visibility graphs be faithfully used to infer long-term memory properties in real-world time series?

Mapping time series to complex networks has great potential to investigate their temporal structures. Some previous studies have been carried on time series generated from idealized stochastic models and found that there are inherent associations between long-term memory (LTM) of pure long-range correlated time series and topological parameters (TPs) of their networks, which is thought as a new prospective to extract information from time series. However, output time series from natural systems seldom take so idealized structures as those from idealized stochastic models, and there is usually some unconsidered information inherited in them, which makes these derived associations questionable. To check and generalize these conclusions acquired from idealized stochastic models, horizontal visibility graph (HVG) algorithm is employed to map time series to their horizontal visibility networks. Firstly synthetic time series with known mixed correlations (apart from LTM) have been analyzed and results indicate that topological parameters (TPs) of HVG networks are not solely dominated by the strength of LTM, other factors such as white noise, short-term correlation (STC) and nonlinear correlations are also playing crucial roles. Taking this fact into account, after some preprocessing treatments have been carried out, the LTM of daily mean air temperature series can indeed be inferred by means of the inherent associations between LTM of pure long-range correlated time series and TPs of their networks.

Construction and analysis of interdependent networks based on time series and horizontal visibility graphs

In this paper, the construction and statistical features of the directional complex networks with interdependencies (i.e. interdependent networks) are investigated, in which the intra-structure of each sub-network is not considered. Based on time series and horizontal visibility graph, two new visibility model methods are presented: many-to-many directional horizontal visibility algorithm and one-to-one directional horizontal visibility algorithm. Using R software, the interdependent structure of the directional interdependent network constructed by these two algorithms are analyzed and the statistical features are evaluated associated with two periodic time series, a periodic time series and a fractal time series, a periodic time series and a random series, separately. The results show that both algorithms ensure the simplicity of the interdependent structure of the networks and inherit the morphological characteristics of the time series.

Directed Connectivity in Large-scale Brain Networks for Precision Grip Force Control.

Precision grip requires accurate but dynamic control of the magnitude and direction of fingertip forces. It is still not well known whether directed information flow across activated cortical regions could change with different force outputs during precision grip. This study aimed to investigate the directed connectivity in large-scale brain networks for precision grip force control. Eight healthy volunteers participated in the experiment. Totally 32 channel electroencephalography (EEG) signals were recorded during a precision grip task that requires accurate force control following a dynamically changed visual target. The force target changed between 10% and 1% Maximal Voluntary Contraction (MVC). Horizontal visibility graph transfer entropy (HVG-TE) was used to measure the directed connectivity between pairs of EEG channels. In addition, the relative HVG-TE (rHVG-TE) was applied to evaluate the activation of each EEG channel. Results showed relatively higher rHVG-TE values in the posterior brain region but lower rHVG-TE values in the anterior brain region within the whole spectrum (4-70Hz), indicating a posterior-to-anterior information flow. This study revealed that the activation of posterior region is higher than the anterior region. There is a directed functional connectivity of cortex from posterior to anterior region for precision grip force control. This study shed light on how to quantify activation and large-scale connectivity of cortical regions, and reveal the in-depth central mechanisms for peripheral fine motor control.

Characterization of SSMVEP-based EEG signals using multiplex limited penetrable horizontal visibility graph.

The steady state motion visual evoked potential (SSMVEP)-based brain computer interface (BCI), which incorporates the motion perception capabilities of the human visual system to alleviate the negative effects caused by strong visual stimulation from steady-state VEP, has attracted a great deal of attention. In this paper, we design a SSMVEP-based experiment by Newton's ring paradigm. Then, we use the canonical correlation analysis and Support Vector Machines to classify SSMVEP signals for the SSMVEP-based electroencephalography (EEG) signal detection. We find that the classification accuracy of different subjects under fatigue state is much lower than that in the normal state. To probe into this, we develop a multiplex limited penetrable horizontal visibility graph method, which enables to infer a brain network from 62-channel EEG signals. Subsequently, we analyze the variation of the average weighted clustering coefficient and the weighted global efficiency corresponding to these two brain states and find that both network measures are lower under fatigue state. The results suggest that the associations and information transfer efficiency among different brain regions become weaker when the brain state changes from normal to fatigue, which provide new insights into the explanations for the reduced classification accuracy. The promising classification results and the findings render the proposed methods particularly useful for analyzing EEG recordings from SSMVEP-based BCI system.

Reversibility of Turbulent and Non-Collisional Plasmas: Solar Wind

AbstractWe have studied turbulent plasma as a complex system applying the method known as Horizontal Visibility Graph (HVG) to obtain the Kullback-Leibler Divergence (KLD) as a first approach to characterize the reversibility of the time series of the magnetic fluctuations. For this, we have developed the method on Particle In Cell (PIC) simulations for a magnetized plasma and on solar wind magnetic time series, considering slow and fast wind. Our numerical results show that low irreversibility values are verified for magnetic field time series associated with Maxwellian distributions. In addition, considering the solar wind plasma, our preliminary results seem to indicate that greater irreversibility degrees are reached by the magnetic field associated with slow solar wind.

Visibility graphs of fractional Wu–Baleanu time series

ABSTRACT We study time series generated by the parametric family of fractional discrete maps introduced by Wu and Baleanu, presenting an alternative way of introducing these maps. For the values of the parameters that yield chaotic time series, we have studied the Shannon entropy of the degree distribution of the natural and horizontal visibility graphs associated to these series. In these cases, the degree distribution can be fitted with a power law. We have also compared the Shannon entropy and the exponent of the power law fitting for the different values of the fractionary exponent and the scaling factor of the model. Our results illustrate a connection between the fractionary exponent and the scaling factor of the maps, with the respect to the onset of the chaos.

Multiplex Limited Penetrable Horizontal Visibility Graph from EEG Signals for Driver Fatigue Detection.

Driver fatigue is an important contributor to road accidents, and driver fatigue detection has attracted a great deal of attention on account of its significant importance. Numerous methods have been proposed to fulfill this challenging task, though, the characterization of the fatigue mechanism still, to a large extent, remains to be investigated. To address this problem, we, in this work, develop a novel Multiplex Limited Penetrable Horizontal Visibility Graph (Multiplex LPHVG) method, which allows in not only detecting fatigue driving but also probing into the brain fatigue behavior. Importantly, we use the method to construct brain networks from EEG signals recorded from different subjects performing simulated driving tasks under alert and fatigue driving states. We then employ clustering coefficient, global efficiency and characteristic path length to characterize the topological structure of the networks generated from different brain states. In addition, we combine average edge overlap with the network measures to distinguish alert and mental fatigue states. The high-accurate classification results clearly demonstrate and validate the efficacy of our multiplex LPHVG method for the fatigue detection from EEG signals. Furthermore, our findings show a significant increase of the clustering coefficient as the brain evolves from alert state to mental fatigue state, which yields novel insights into the brain behavior associated with fatigue driving.

Horizontal visibility graphs are uniquely determined by their directed degree sequence

Horizontal visibility graphs (HVGs) are graphs generated from time (or otherwise ordered) series data. Components of the graph reveal key underlying properties of the series data that generated the graph and there are several applications that use HVGs to uncover information about series data. Recently, a bijection between the degree sequence and adjacency matrix of canonical form HVGs was established. This paper extends the relation between the degree sequence and adjacency matrix of HVGs and then presents an algorithm to uniquely construct an HVG from its directed degree sequence.

Comparative study of multiple measures on temporal irreversibility of daily air temperature anomaly variations over China

As a fundamental property of nonlinear time series, temporal irreversibility (TI) is a hot topic in various fields. Among these studies, many methods and measures have been developed. However, only a few studies contribute to the performance comparison of these methods and measures. Taking the daily mean air temperature from observations over China as an example, the performance of five well developed measures was compared in this study. Three of them are directly related to air temperature increments with A defined from sign ratio, G from magnitude ratio and E from skewness, and the other two measures are L2 from directed horizontal visibility graph (DHVG for short) and L1 from consecutive increasing and decreasing steps (CIDS for short). The results show that five measures reach almost consistent conclusions to TI of daily mean air temperature from observations over China, but with different statistical significance and sensitivity to low-frequency trend and measured noise. They also show that the higher TI occurs over southern China, and lower TI over northeast China and southwest China close to Basin of Sichuan and Yungui-Tibetan Plateau. Further studies indicate that this dominated TI behavior is closely related to the number difference between extreme decreasing temperature variations and extreme increasing temperature variations. When conditional threshold exceeds certain critical values, this marked TI can be well quantified by the extreme decreasing but not increasing temperature variations. At last, combining all aspects considered, measure from CIDS is the best among these five TI measures due to its robustness to low-frequency trend and measured noise.

Time series irreversibility analysis using Jensen–Shannon divergence calculated by permutation pattern

An important feature of the time series in the real world is that its distribution has different degrees of asymmetry, which is what we call irreversibility. In this paper, we propose a new method named permutation pattern (PP) to calculate the Kullback–Leibler divergence ( $${D}_{\mathrm{KL}}$$ ) and the Jensen–Shannon divergence ( $${D}_{\mathrm{JS}}$$ ) to explore the irreversibility of time series. Meanwhile, we improve $${D}_{\mathrm{JS}}$$ and obtain a complete mean divergence ( $${D}_{\mathrm{m}}$$ ) through averaging $${D}_{\mathrm{KL}}$$ of a time series and its inverse time series. The variation trend of $${D}_{\mathrm{m}}$$ is similar to $${D}_{\mathrm{JS}}$$ , but the value of $${D}_{\mathrm{m}}$$ is slightly larger and the description of irreversibility is more intuitive. Furthermore, we compare $${D}_{\mathrm{JS}}$$ and $${D}_{\mathrm{m}}$$ calculated by PP with those calculated by the horizontal visibility graph, and discuss their respective characteristics. Then, we investigate the advantages of $$\mathrm{D}_{\mathrm{JS}}$$ and $${D}_{\mathrm{m}}$$ through length variation, dynamic time variation, multiscale and so on. It is worth mentioning that we introduce Score and variance to analyze the practical significance of stock irreversibility.

Detect and evaluate dependencies between aero-engine gas path system variables based on multiscale horizontal visibility graph analysis

Identifying the interaction in aero-engine gas path system is a crucial problem that facilitates the understanding of emerging structures in the complex system. By employing the multiscale horizontal visibility graph method to aero-engine gas path system, the interaction characteristics between gas path system parameters are established. Comparing with normalized mutual information method, the advantage of the multiscale horizontal visibility graph method in detecting interaction is showed. And then, the effect of time scaleson the multiscale horizontal visibility graph analysis is investigated. The results show that the multiscale horizontal visibility graph method is effective to detect the interaction of gas path system parameters.

Tetradic motif profiles of horizontal visibility graphs

Network motif analysis is a useful tool for the investigation of complex networks. We study the profiles of tetradic motifs in horizontal visibility graphs (HVGs) converted from multifractal binomial measures, fractional Gaussian noises, and heartbeat rates. The profiles of tetradic motifs contain the spatial information (visibility) and temporal information (relative magnitude) among the data points in the corresponding time series. For multifractal binomial measures, the occurrence frequencies of the tetradic motifs are determined, which converge to a constant vector (2/3, 0, 8/99, 8/33, 1/99, 0). For fractional Gaussian noises, the motif occurrence frequencies are found to depend nonlinearly on the Hurst exponent and the length of time series. These findings suggest the potential ability of tetradic motif profiles in distinguishing different types of time series. Finally, we apply the tetradic motif analysis to heartbeat rates of healthy subjects, congestive heart failure (CHF) subjects, and atrial fibrillation (AF) subjects. Different subjects can be distinguished from the occurrence frequencies of tetradic motifs.

Differential temporal asymmetry among different temperature variables’ daily fluctuations

As one of the most important indicators of nonlinear time series, temporal asymmetric (TA) or temporal irreversible behaviors of daily fluctuations from four temperature variables, including mean temperature (Tmean), maximum temperature (Tmax), minimum temperature (Tmin) and diurnal temperature range (DTR, DTR = Tmax − Tmin), have been quantified through both observations and reanalysis over China by two TA measures. One is L2 from directed horizontal visibility graph and the other is L1 from consecutive increasing and decreasing steps. The results show that there are differential TA features among daily temperature fluctuations. Firstly, there are marked differences in TA strengths among different temperature variables. It is found that dominated uniformly significant TA (larger than the threshold from corresponding surrogates) emerges in almost all observed Tmean fluctuations. However, this kind of uniformly significant TA can’t be found in Tmax, Tmin and DTR for both observed and reanalysis data sets. Since both TA measures L1 and L2 quantify the temporal structures in the given series, this distinguishable TA strengths found in different temperature variables indicates that there are distinct temporal structures in the different temperature variables’ variations. Secondly, the TA in each temperature variable is region dependent. The TA strength for each temperature variable is spatially non-uniform with some strong and weak TA regional patterns and these strong and weak TA regional patterns may depend on local weather or climate conditions. Moreover, comparison studies of the same temperature variable reveal that time irreversible features are distinguishable between observations and reanalysis, and this differential feature can be taken as an index to evaluate the quality of reanalysis.

Analysing the Dynamics of Interbeat Interval Time Series Using Grouped Horizontal Visibility Graph

Horizontal visibility graph (HVG) motifs have been recently introduced to analyze the dynamical information encoded by biological signals. However, the result of the analysis strongly depends on the selected window size of the motifs. Different sizes ranging from 3 to 5 have been previously used, but such small window sizes are insufficient to cope with the complexity of biological systems and often fail to extract salient features of the encoded information. It is known that larger window size increases the total number of possible motifs, and it leads to the distribution of the statistics into too many motifs, which causes each individual motif to contain too little information and make it even more difficult to reliably detect system dynamics. To resolve this problem, we group the motifs based on the number of edges. Using the grouped motifs, we propose grouped horizontal visibility entropy (GHVE) to quantify the complexity based on the probability distribution of the observations within these groups. We apply GHVE to quantify the complexity of simulated white and 1/f noise. The results reveal that the 1/f noise time series exhibits a higher complexity than white noise time series, which indicates that the 1/f noise is structurally more complex than white Gaussian noise. We apply the method for analyzing interbeat intervals time series. The results show that the proposed GHVE measure is more accurate in distinguishing healthy and pathological subjects than its non-grouped counter-part HVG. It is, therefore, better suited to detect changes in aging, disease severity, and activity levels (sleep and wake period).

A novel method of heart rate variability measurement

The complex fluctuation of heart rate variability reflects the autonomous regulation function of the heart. In this paper, a novel method of measuring the heart rate variability is proposed. Firstly, the heart rate variability signal is decomposed by the improved complete ensemble empirical mode decomposition with adaptive noise method, and the multiple intrinsic mode functions are obtained, and the bubble entropy of each intrinsic mode function is calculated to obtain an entropy value vector. Then, the vector is mapped to a network based on a limited penetrable horizontal visibility graph method. By calculating various characteristic parameters of the network, the coupling relationship between the nonlinear features of heart rate variability in different time-frequency scale states are studied. The characteristic parameters include mean value of aggregation coefficient (MC), the characteristic path length (CL), the topological entropy of network (TE), the network level weighted bubble value (WB), and the pseudo mean value of node weight (PW). Firstly, the heart rate variabilities of 29 patients with congestive heart failure and 29 normal sinus heart rhythm subjects are analyzed by time domain, frequency domain and ICBN analysis method, the <i>T</i> test is used for statistical analysis, and Fisher discriminant method is used for classification. The results show that the time domain triangular index HRVTI, frequency domain index LF/HF, WB, PW and CL in ICBN have statistical differences. The accuracy rate of recognition model based on WB, CL, frequency domain index LF/HF and Fisher discriminant method is 89.66%. Secondly, the heart rate variabilities of 43 patients with atrial fibrillation arrhythmia and another 43 normal sinus heart rhythm subjects are analyzed by the same methods, including the time domain analyzed method, frequency domain analyzed method, and ICBN analyzed method. Then, the T test is also used for statistical analysis, and Fisher discriminant method is used for classification. The results show that using the time domain index pNN5 and RMSSD, frequency index LF/HF, ICBN index WB and PW as the feature vectors, and the Fisher discriminant mode as the classifier, the accuracy rate of recognition for atrial fibrillation arrhythmia is 91.86%. From these results it is concluded that the ICBN method provides a new idea for the heart rate variability measurement.

Creation of language networks based on texts with using visibility graphs algorithms

A method to constructing language networks is proposed. Key words and concepts from the set of documents which describe some subject domain are retrieved. Numeric values are assigned to each word using a TF-IDF metric, that is intended to reflect how important a word is to a document in a collection or corpus. As the result a time series are constructed. A tool in time series analysis – the visibility graph algorithm is used for constructing the graph of subject domain. In this article two actual subject domains (“Space” and “Computer graphic”) are considered for example. The proposed method is used for the set of documents, which are related with “Space” and “Computer graphic”. A network of connections between terms and concepts, which go into textual documents is builded. Building networks of words, the nodes of which are elements of the text, enables to reveal key components of the text. At the same time, the task of determining the important structural elements of the text which are also informationally important, is actual. As a result of the research, it was found that such words as “uranium”, “nuclear”, “waste”, “Jupiter”, “Mercury”, “Moon”, “Earth”, “comet”, “space” and others are key for the subject area “Space”. This article shows that applying only a TF metric is more expedient compared with the TF-IDF metric in case when the set of documents describe one subject domain. Also the results of applying the visibility graphs algorithm and the compactified horizontal visibility graph algorithm are compared. It was found that in some case using the compactified horizontal visibility graph algorithm gives a network of words with more quantity of connections between concepts compared with using the visibility graphs algorithm. An open-source visualization and exploration software for all kinds of graphs and networks Gephi and an original package of specially developed Python modules are used for simulation and visualization as an additional tool. The proposed method can be used for visualization some subject domain, and also for information decision support systems, enabling to reveal key components of a subject domain. Also the results of this article can be used for building UI of information retrieval systems, enabling to make a process of search a relevant information easier.