Multiscale Cross-sample Entropy Research Articles

Multiscale cross-sample entropy based on visibility graph for quantifying time series irreversibility

In this paper, we propose VGMCSE method to obtain multiscale time series irreversibility and gain more information from time irreversibility analysis. The validity of VGMCSE method is illustrated by numerical simulations on synthesized data and then VGMCSE method is applied on traffic time series to give better understanding about the traffic system from the view of time series irreversibility. By means of the VGMCSE planes for all the simulated time series, it can be found that ARFIMA and FGN series are all time reversible, while Logistic map & Hénon map are time irreversible. The simulated series can be classified into two groups: (1) ARFIMA & FGN, (2) Logistic map & Hénon map by their time series irreversibility. It can be found that the relationship between asynchrony of ingoing degree sequence and outgoing degree sequence for the simulated series with different parameter is consistent with the complexity and autocorrelation behavior in the corresponding definition of the time series. However, it can be revealed that whether the series is long-term correlated or no matter how strong the persistence or anti-persistence is, the ARFIMA and FGN series are all time reversible, indicating time series irreversibility might have no direct connection with long-range dependence and stationarity of series. For the empirical analysis, the traffic time series recorded by detectors can be classified into three groups: (1) detector 2037, 2038, (2) detector 3055, (3) detector 2036, 2040, 3054, 3056, 3057, 4051, 4052, suggesting that there is more similar time irreversible behavior for speed and volume time series recorded by detectors with nearer positions, while the traffic time series of some special locations present different time irreversible behavior and these locations should be paid more attention and enhanced monitoring. VGMCSE method can not only show the asynchrony of ingoing degree sequence and outgoing degree sequence, but also detect the time series irreversibility for time series of complex system.

Damage identification of thin plate-like structures combining improved singular spectrum analysis and multiscale cross-sample entropy (ISSA-MCSEn)

In this paper, a new method integrating the improved singular spectrum analysis and the multiscale cross-sample entropy Improved Singular Spectrum Analysis and Multiscale Cross-Sample Entropy, (ISSA-MCSEn) is developed to identify the size of early damages in thin plate-like structures. In the algorithm, with the help of improved singular spectrum analysis (ISSA), the principal components relevant to the reference and damage-induced signals are successfully extracted, and then the components related to the damage are reconstructed for damage size detection. Lastly, the multiscale cross-sample entropy (MCSEn) of the reconstructed signal is computed as a new damage index to evaluate the size of the damage. To validate the proposed ISSA-MCSEn algorithm, two different experiments are conducted on aluminum and composite fiber reinforced polymer (CFRP) plates to detect simulated crack and through-hole, respectively. Comparative performance analysis of ISSA and singular spectrum analysis (SSA) demonstrates that the total increment of the normalized MCSEn by using ISSA is 30%–81% while the one by using SSA is only 6.5%–9%, which demonstrates that the performance of the proposed ISSA is much better than SSA. The experimental results also show that the average of the normalized MCSEn of the proposed algorithm increases by over 77% and 28% as the size of the two damages in CFRP and aluminum plates changes from 0 to 8 mm and 0 to 1.2 mm, respectively. Moreover, the relationship between the normalized MCSEn and damages’ size is well linear, and the Pearson’s coefficient of their fitting curves is more than 0.99, which demonstrates that this linear relationship can be employed for damage size detection in both CRFP and aluminum plates. The linear relationship between the damage size and normalized MCSEn is used for damage detection, and the relative error between the actual and detected size is 1.64%–6.92%. In addition, the performance comparison of ISSA-MCSEn and SSA-FuzzyEn shows that the total increment of the ISSA-MCSEn algorithm due to the damage is 30%–81% while the one of SSA-FuzzyEn is only 4%–15%, which indicates that the proposed ISSA-MCSEn is more sensitive to the damage than SSA-FuzzyEn and it is more suitable for detection of small-size damages.

Modified multiscale sample entropy and cross-sample entropy based on horizontal visibility graph

As a crucial method of the feature extraction, the complexity measurement has a wide range of applications in the field of nonlinear time series research. This paper presents an innovative multiscale sample entropy for measuring the complexity of time series based on the horizontal visibility graph. The modified multiscale sample entropy has been proven to be robust on two artificial time series, and is capable of reducing the undefined entropy generated as a result of the increase in scale. We apply the modified multiscale sample entropy to the diagnosis of epilepsy. Using a novel data processing algorithm that combines frequency bands with decomposition, feature vectors are constructed for Electroencephalography (EEG) signals through the proposed entropy calculation algorithm, and different classes of subjects are categorized based on K-Nearest Neighbors, Support Vector Machine, and Artificial Neural Network. In the meantime, we propose an improved horizontal visibility graph-based multiscale cross-sample entropy method to measure the synchronization between two time series. It shows robustness in artificial data and decreases the appearance of undefined entropy to a certain extent. It is possible to extract the characteristics of sleep EEG signals and divide the subjects’ sleep stages using this method. Furthermore, this paper introduces the surrogate data test and the proposed methods have the ability to detect the nonlinearity and synchronization in simulations and in real-world experiments. Experimental results demonstrate that the two proposed frameworks are effective in monitoring human health and in assessing physical status through EEG signals.

Multiscale entropy and asynchronies of percolation-based price model and Chinese stock market

In recent years, the concept of entropy is widely used to measure the degree of uncertainty or complexity of dynamics system. In our work, we utilize the composite multiscale entropy (CMSE) and the composite multiscale cross-sample entropy (CMSCE) which are two modified algorithms of SampEn and Cross-SampEn by considering multiscale factors, to, respectively, investigate the multiscale complexities and asynchronies (correlations) in the Chinese stock market (SSZ, SZSE and HSI) as well as in our established financial stock price model. The price model is given based on a greatly important statistical physics system, the two-dimensional continuum percolation system. In the model, the fluctuations of stock price changes are assumed to be attributed to the market information interactions among the traders, and the percolation cluster is taken to represent the traders holding the same investment attitude. The empirical CMSE and CMSCE results display and meanwhile make comparisons of abundant complexities and correlations properties about Chinese stock indices and simulation data on both overall and the upwards, downwards trend of their returns.

Composite Multiscale Cross-Sample Entropy Analysis for Long-Term Structural Health Monitoring of Residential Buildings.

This study proposesd a novel, entropy-based structural health monitoring (SHM) system for measuring microvibration signals generated by actual buildings. A structural health diagnosis interface was established for demonstration purposes. To enhance the reliability and accuracy of entropy evaluation at various scales, composite multiscale cross-sample entropy (CMSCE) was adopted to increase the number of coarse-grained time series. The degree of similarity and asynchrony between ambient vibration signals measured on adjacent floors was used as an in-dicator for structural health assessment. A residential building that has been monitored since 1994 was selected for long-term monitoring. The accumulated database, including both the earthquake and ambient vibrations in each seismic event, provided the possibility to evaluate the practicability of the CMSCE-based method. Entropy curves obtained for each of the years, as well as the stable trend of the corresponding damage index (DI) graphs, demonstrated the relia-bility of the proposed SHM system. Moreover, two large earthquake events that occurred near the monitoring site were analyzed. The results revealed that the entropy values may have been slightly increased after the earthquakes. Positive DI values were obtained for higher floors, which could provide an early warning of structural instability. The proposed SHM system is highly stable and practical.

Damage quantification of 3D-printed structure based on composite multiscale cross-sample entropy

This study combined three-dimensional (3D) printing and composite multiscale cross-sample entropy (CMSCE) in structural health monitoring (SHM) and explored a quantification criterion for single-story structural damage index (DI). By quantifying the DI, the study established a SHM system suitable for real-world structures. A numerical model of a seven-story 3D printed structure was first created. Through the establishment of various bracing conditions as failure modes, damage to the structure was simulated properly, and CMSCE was used to effectively indicate the location of damage. Moreover, the DI was used to shorten the assessment time and improve system accuracy. The DI quantification facilitated observation of the effects of various degrees of damage on the analysis results. Based on the results, an experiment involving a 3D-printed structure was conceived. First, an experiment involving a seven-story structure with severe, moderate, and marginal single-story damage was conducted. The signals obtained from these structures were used to perform CMSCE analysis. Structural damage was detected using entropy curves and DI figures to determine the location and degree of damage as well as to quantify the DI. Thus, the study developed a reliable method by combining emerging 3D printing technology with the CMSCE DI to explore the feasibility of practical application.

Forcasting of energy futures market and synchronization based on stochastic gated recurrent unit model

Energy futures market has occupied an extremely significant position in financial markets, attracting a large amount of scholars to search out its price formation mechanism. To predict the price of energy futures has become a pivotal issue. For the sake of enhancing the forecasting accuracy of energy futures prices, a novel model ST-GRU is proposed by embedding stochastic time intensity function into gated recurrent unit model (GRU). ST-GRU, GRU, LSTM, WNN and BPNN models are applied to predict the daily closing prices of West Texas Intermediate crude oil, Brent crude oil, Natural gas and Heating oil respectively. In error assessment, the prediction effects of various models are compared by general benchmarks. Then composite multiscale cross-sample entropy (CMSCE) algorithm is utilized to analyze the synchronization between the predicted value and the real value. In order to further predict the volatility of futures closing price and demonstrate the superiority of the ST-GRU model, the above-mentioned five models are used to predict and analyze 5-day, 10-day and 20-day moving average logarithmic return (MALR) series of four energy indexes. Finally, comparative experiments indicate that ST-GRU model has the highest prediction precision and the best learning performance.

Composite Multiscale Partial Cross-Sample Entropy Analysis for Quantifying Intrinsic Similarity of Two Time Series Affected by Common External Factors.

In this paper, we propose a new cross-sample entropy, namely the composite multiscale partial cross-sample entropy (CMPCSE), for quantifying the intrinsic similarity of two time series affected by common external factors. First, in order to test the validity of CMPCSE, we apply it to three sets of artificial data. Experimental results show that CMPCSE can accurately measure the intrinsic cross-sample entropy of two simultaneously recorded time series by removing the effects from the third time series. Then CMPCSE is employed to investigate the partial cross-sample entropy of Shanghai securities composite index (SSEC) and Shenzhen Stock Exchange Component Index (SZSE) by eliminating the effect of Hang Seng Index (HSI). Compared with the composite multiscale cross-sample entropy, the results obtained by CMPCSE show that SSEC and SZSE have stronger similarity. We believe that CMPCSE is an effective tool to study intrinsic similarity of two time series.

Prediction model of energy market by long short term memory with random system and complexity evaluation

Due to the frequent and violent fluctuation of energy futures prices, the investment risk of energy investors is increased. Forecasting energy futures prices has progressively become the focus of research. However, traditional prediction model only conducts forecasting based on historical data without considering the behavior of the market, resulting in poor accuracy. In this paper, the random time effective function that considers the timeliness of historical data and the random change of market environment is applied to the long short term memory model to establish a novel prediction model, which is denoted by long short term memory with random time effective function model (LSTMRT). LSTM model has the characteristics of selective memory and the internal influence of time series, which is very suitable for the prediction of price time series. Random time effective function can give different weights to historical data. Furthermore, using multiscale cross-sample entropy (MCSE) as an innovative method to reveal the performance of prediction. Finally, comparing with other models selected in this paper, error evaluations and statistical comparisons are utilized to demonstrate the advantages and superiority of the proposed model. LSTMRT model has the effect of random movement and keeps the trend fluctuation of the original nonlinear data, which makes the prediction more accurate and more credible.

Integration of Refined Composite Multiscale Cross-Sample Entropy and Backpropagation Neural Networks for Structural Health Monitoring

This study developed a structural health monitoring (SHM) system based on refined composite multiscale cross-sample entropy (RCMCSE) and an artificial neural network for monitoring structures under ambient vibrations. RCMCSE was applied to enhance the reliability of entropy estimations. First, RCMCSE was implemented to extract damage features, and finite element analysis software was then used to generate training samples, which included stiffness reductions to achieve various damage patterns. A neural network model was constructed and trained using entropy values for these damage patterns. An experiment was conducted on a seven-story steel benchmark structure to validate the performance of the proposed system. Additionally, a confusion matrix was established to evaluate the performance of the proposed system. The results obtained for a scaled-down benchmark structure indicated that 89.8% of the floors were accurately classified, and 90% of the practical damaged floors were correctly diagnosed. The performance evaluation demonstrated that the proposed SHM system exhibited increased damage location accuracy.

Cross-correlation complexity and synchronization of the financial time series on Potts dynamics

The Potts dynamic system is investigated to reproduce and analyze the fluctuation and noises of the financial markets in the present paper. A random financial agent-based model is developed by applying the stochastic dynamic systems to uncover the cross-correlation complexity and synchronization between different markets. In attempt to verify the rationally of the established model, the descriptive statistical analysis and cross-correlation of return behaviors between different series are exhibited by comparing with the actual market returns, SSE, SZSE, HSI, N225, GDAX and IXIC. Furthermore, the multiscale cross-sample entropy (MSCE) analysis and the multiscale generalized complexity synchronization (MGCS) analysis are employed to measure the cross-correlation between different series at multiple scales, where the MGCS method is firstly introduced to verify the synchrony behaviors between simulative series and actual series. Empirical results reveal that the feasibility and effectiveness of the proposed financial model, and exhibit the cross-correlation and synchronization behaviors among the return series from the perspective of complexity analysis.

Investigation of Ensemble Empirical Mode Decomposition Applied for Composite Multiscale Cross-Sample Entropy Analysis

Entropy is an approach for computing uncertainty. Over recent decades, entropy has been developed to measure complexity in biological signals. With the development of entropy, the diversity of its application has increased. In civil engineering, entropy has been combined with structural health monitoring (SHM) for damage detection. Moreover, as algorithms for entropy have continued to be developed, multiscale entropy (MSE), composite multiscale entropy (CMSE), and composite multiscale cross-sample entropy (CMSCE) have been proposed in succession. The aim of this study was to optimize CMSCE to enhance SHM performance. To reduce the influence of ambient noise, ensemble empirical mode decomposition (EEMD) can be used to filter structural dynamic signals. Therefore, the first mode of structure was extracted by using EEMD for entropy analysis and evaluation of damage assessment performance. A numerical simulation of a seven-story steel structure was run to verify the efficacy of EEMD and calculate damage indices for detection of damaged locations. Through this simulation, signals with and without EEMD were compared. As the result, it can be observed that the performance of damage identification was improved in low floors by CMSCE with EEMD.

A Novel Multiscale Cross-Entropy Method Applied to Navigation Data Acquired with a Bike Simulator.

We present herein a new approach - the so-called time-shift multiscale cross-distribution entropy (TSMCDE) - to quantify the complexity between two sequences. By analyzing biomedical data, we reveal that TSMCDE over-performs other cross-entropy measures. Thus, TSMCDE, multiscale cross-sample entropy (MCSE), multiscale cross-distribution entropy (MCDE), and time-shift multiscale cross-sample entropy (TSMCSE) were applied to handlebar angle and speed time series recorded from a bike simulator. Twenty-four subjects divided into two groups (12 subjects each) participated to the study. The first group corresponds to young healthy subjects. The second group corresponds to older adults with loss of autonomy. Our results show that a link may exist between complexity and the age and physical state of a population. Moreover, TSMCDE leads to a better differentiation of the two groups than MCSE, MCDE, and TSMCSE. TSMCDE should now be tested on other types of data and on larger datasets to prove its usefulness and its efficiency.

Performance Evaluation of an Entropy-Based Structural Health Monitoring System Utilizing Composite Multiscale Cross-Sample Entropy.

The aim of this study was to develop an entropy-based structural health monitoring system for solving the problem of unstable entropy values observed when multiscale cross-sample entropy (MSCE) is employed to assess damage in real structures. Composite MSCE was utilized to enhance the reliability of entropy values on every scale. Additionally, the first mode of a structure was extracted using ensemble empirical mode decomposition to conduct entropy analysis and evaluate the accuracy of damage assessment. A seven-story model was created to validate the efficiency of the proposed method and the damage index. Subsequently, an experiment was conducted on a seven-story steel benchmark structure including 15 damaged cases to compare the numerical and experimental models. A confusion matrix was applied to classify the results and evaluate the performance over three indices: accuracy, precision, and recall. The results revealed the feasibility of the modified structural health monitoring system and demonstrated its potential in the field of long-term monitoring.

Analysis of financial stock markets through the multiscale cross-distribution entropy based on the Tsallis entropy

In this paper, we propose a multiscale cross-distribution entropy (MCDE) method based on the Tsallis entropy to analyze financial stock markets. In order to evaluate the effectiveness of this method, we employ it into ARFIMA models. Then, applying the proposed method to analyze financial time series, we conclude that it provides us more exact, detailed and clearer information about the relationships between pairs of financial stock markets in comparison with the multiscale cross-sample entropy. Furthermore, the results show that the embedding dimension m has little influence on MCDE. The stability of financial time series could be affected by the order q. Moreover, the MCDE results of $$q > 0$$ tend to be more stable than the results obtained by $$q < 0$$ , and it may arise from producing amounts of entropy as $$q < 0$$ . The larger the entropy is, the more active the financial time series is.

Entropy-Based Structural Health Monitoring System for Damage Detection in Multi-Bay Three-Dimensional Structures.

In this paper, a structural health monitoring (SHM) system based on multi-scale cross-sample entropy (MSCE) is proposed for detecting damage locations in multi-bay three-dimensional structures. The location of damage is evaluated for each bay through MSCE analysis by examining the degree of dissimilarity between the response signals of vertically-adjacent floors. Subsequently, the results are quantified using the damage index (DI). The performance of the proposed SHM system was determined in this study by performing a finite element analysis of a multi-bay seven-story structure. The derived results revealed that the SHM system successfully detected the damaged floors and their respective directions for several cases. The proposed system provides a preliminary assessment of which bay has been more severely affected. Thus, the effectiveness and high potential of the SHM system for locating damage in large and complex structures rapidly and at low cost are demonstrated.

Multiscale synchrony behaviors of paired financial time series by 3D multi-continuum percolation

Multiscale synchrony behaviors and nonlinear dynamics of paired financial time series are investigated, in an attempt to study the cross correlation relationships between two stock markets. A random stock price model is developed by a new system called three-dimensional (3D) multi-continuum percolation system, which is utilized to imitate the formation mechanism of price dynamics and explain the nonlinear behaviors found in financial time series. We assume that the price fluctuations are caused by the spread of investment information. The cluster of 3D multi-continuum percolation represents the cluster of investors who share the same investment attitude. In this paper, we focus on the paired return series, the paired volatility series, and the paired intrinsic mode functions which are decomposed by empirical mode decomposition. A new cross recurrence quantification analysis is put forward, combining with multiscale cross-sample entropy, to investigate the multiscale synchrony of these paired series from the proposed model. The corresponding research is also carried out for two China stock markets as comparison.

Return volatility duration analysis of NYMEX energy futures and spot

Return volatility plays a key role in quantifying risk, optimizing the portfolio and pricing modelling of financial market. The study focusing on the return volatility of energy market can help greatly understand the energy fluctuating behaviors. In this paper, we introduce a concept of volatility duration into the analysis of the New York Mercantile Exchange (NYMEX) energy market, where the daily closing prices of the futures and spot for the crude oil, natural gas, heating oil and propane are adopted. The volatility duration is defined as the shortest passage time that the future's volatility intensity takes to go beyond or below the current volatility intensity which is time-varying and considered as the basic intensity reference. Then, two main aspects of the statistical properties analysis for the energy volatility duration time series are focused on: one is about the empirical probability distributions and their scaling behaviors are observed; another is about the complexity properties of the energy volatility durations, which are discussed by the entropy measures of the composite multiscale entropy (CMSE) and the composite multiscale cross-sample entropy (CMSCE) approaches.

Multiscale sample entropy and cross-sample entropy based on symbolic representation and similarity of stock markets

A modified multiscale sample entropy measure based on symbolic representation and similarity (MSEBSS) is proposed in this paper to research the complexity of stock markets. The modified algorithm reduces the probability of inducing undefined entropies and is confirmed to be robust to strong noise. Considering the validity and accuracy, MSEBSS is more reliable than Multiscale entropy (MSE) for time series mingled with much noise like financial time series. We apply MSEBSS to financial markets and results show American stock markets have the lowest complexity compared with European and Asian markets. There are exceptions to the regularity that stock markets show a decreasing complexity over the time scale, indicating a periodicity at certain scales. Based on MSEBSS, we introduce the modified multiscale cross-sample entropy measure based on symbolic representation and similarity (MCSEBSS) to consider the degree of the asynchrony between distinct time series. Stock markets from the same area have higher synchrony than those from different areas. And for stock markets having relative high synchrony, the entropy values will decrease with the increasing scale factor. While for stock markets having high asynchrony, the entropy values will not decrease with the increasing scale factor sometimes they tend to increase. So both MSEBSS and MCSEBSS are able to distinguish stock markets of different areas, and they are more helpful if used together for studying other features of financial time series.

Three-dimensional structural health monitoring based on multiscale cross-sample entropy

Three-dimensional structural health monitoring based on multiscale cross-sample entropy