Entropy Profiling: A Reduced—Parametric Measure of Kolmogorov—Sinai Entropy from Short-Term HRV Signal
Abstract
:1. Introduction
- An explanation of the parametric dependency of KS-entropy measures and its impact on short-term HRV analysis.
- A summary of approaches proposed in the last two decades to address parametric dependency and a summary of persisting issues.
- Explaining the concept, formulation and geometric meaning of entropy profiling as a solution to persisting issues in the context of KS entropy for short-term HRV analysis.
- Summary of significant research directions based on the provided entropy profiling solution.
2. Kolmogorov Entropy Methods
- Form vectors of length m each, given by
- Similarly, form vectors of length each, given by
- Take each vector of step 1 as a template vector and find its distance from every vector of , where the distance is given by
- Let be the count of the number of vectors of which lie within a distance r of the vector .
- The probability of a vector to lie within a distance r of the vector is given by
- Finally, we take the average of the natural logarithms of these probabilities for as
- Repeating steps 3 to 6 for the dimension, we get
2.1. Approximate Entropy (ApEn)
2.2. Sample Entropy (SampEn)
3. Limitations of KS-Entropy Methods
3.1. Tolerance r
3.2. Impact of r Choice in Short-Term HRV Analysis
- and are known to produce misleading regularity results owing to their parametric restrictions, particularly the r parameter. A wrong choice of r may give completely unreal regularity estimates for a signal. Thus, the current scheme of r selection in entropy methods makes regularity estimation unreliable.
- A lack of logical flexibility in choosing r places its share of restrictions on signal length as well. For the currently available mode of r selection, at least 1000 data points (≃15 min data) from the incoming signal are required for analysis. Thus, is or incapable of handling signals below 1000 data points? This cannot be explored or concluded unless the lameness in r selection procedure is addressed. Thus, how do we make entropy methods suitable for use on short-length (<15 min) data?
3.3. HRV Complexity Analysis
4. How Has “r-Dependence” Been Addressed by Other Researchers?
4.1. Fuzzy Entropy ()
4.2. Normalized Fuzzy Measure Entropy (Normalized )
5. Entropy Profiling as a Solution
5.1. Signal Irregularity
- Insufficient information: It is first of all necessary to know how many r values should be in the set, what should be the minimum and maximum values of r in the set and so on. In other words, the range and resolution of r values in the set should be defined. We know that previous studies have used a general r range of of signal and highest r resolution of for the purpose. However, these recommended values do not hold any proof of uniform credibility across all data sets [9,18,25,28]. Moreover, these choices have mostly been random. Since our approach aims at generating a complete profile for a given signal, we must ensure that all significant changes in r and thereby entropy are captured. This cannot be achieved if we go with random parametric selection once again. Data driven choices become indispensable here.
- Computational expense involved: Even if we are successful in listing all potential r values for the given signal, the next challenge is to estimate entropy at each of these r values. With available expertise, the only way to do this is by repetitively estimating entropy at each r. The process could be complex and time consuming for high dimensional signals. A computationally efficient approach also becomes indispensable here.
5.2. Signal Complexity
5.3. Formulation of Entropy Profiling
- Geometric interpretations of entropy profiling [49]:
- Form vectors of length m each, given by
- Similarly, form vectors of length each, given by
- Take each vector of step 1 as a template vector and find its distance from every vector , where the distance is given by
- Let D be the matrix containing all elements of and . Then, we define as the set of all unique elements of D, sorted in ascending order. Additionally, let be the number of elements in (shown in Figure 6).
- From the distance vector , the cumulative distribution function is calculated as:
- Repeat step 5 for all : to calculate the entire cumulative distribution matrix for the embedding dimension m as shown:
- Similarly for embedding dimension the cumulative distribution matrix can be obtained by repeating step 4 for all .
- Now, for each column of and , we calculate the average of the probabilities, which is represented as and respectively.
- which is an approximation of the conditional probability of two segments matching at a length if they match at m, can be defined from and as . In this case, we will have number of values since and are calculated over (shown in Figure 8). This can be expressed as , where .
5.4. New Measures of Irregularity from a Profile
- Total sample entropy ():
- Average of sample entropy (:)
6. Case Studies
6.1. Age Based HRV Classification
6.2. Disease Based HRV Classification
6.3. HRV Complexity Analysis
7. Discussion
- Reduced parametric dependence:The idea of entropy profiling with respect to the parameter r makes entropy a function of just N and m. r is completely eliminated from the list of input parameters that need the user’s discretion to be chosen. This is in contrast to the traditional case where entropy is a function of and r. In short, now becomes . Thus, parametric dependence of traditional formulations is significantly reduced. The CHM algorithm used for r-based entropy profiling is completely data driven. It automatically computes all potential r values for the given data. This relieves the user from having to choose a logical or appropriate value for r to be used, as in the case of traditional entropy estimation.
- Short-term analysis and enhanced information retrieval:An irresolute r choice has placed its share of limitations over the use of data length N. With the available generalized mode of r-selection, entropy algorithms could be used only on data lengths —this being approximately 15 min of HRV data, is surely considered short-term data. However, the inability of these algorithms to work on (highly preferred for ease of data acquisition) is majorly due to lameness in r-selection scheme, which can therefore surely be rectified. By generating a complete set of r values and thereby all possible entropy estimates specific to the given signal (of any N), CHM does not place any kind of restrictions on data length. With a signal of any given N, CHM unconditionally tries and retrieves as much information from it as possible.The novel approach of entropy profiling is superior to traditional estimations as: (i) it eliminates the ambiguity that existed in choosing the right value of r for entropy estimations and (ii) it enhances the quality and reliability of signal information retrieved, specially from a short-length signal.An entropy profile contains accurate and complete information about signal irregularity. Different forms of the information can be derived from the generated profile and are called secondary measures of irregularity information. The proficiency of secondary measures such as and to classify physiological data has been demonstrated in this work. Any such secondary statistics of analysis can be obtained from a given entropy profile to suit the purpose of application.In general, estimation is bound by data length limitations. The procedure cannot accurately handle signals of length below 1000 samples. As a consequence, multiscale analysis, that further constricts data length as part of its formulation, remains inapplicable on short-length physiological signals. When augmented with entropy profiling, we see that the process of multiscaling becomes equipped to handle even the short-length signals.
- Computational efficiency:Traditional , calculated at a single r value, takes an execution time of [69]. As an attempt to compare this with the execution time of the cumulative histogram method of profiling, we design the following experiment. Consider the age-based HRV data set used in this study. HRV signal of length 300 from each of the 40 subjects (20 elderly and 20 young) is taken. On each signal, first the CHM is applied. Here, CHM will provide us (a) the data specific range and resolution of r and (b) the complete profile of the signal. The time taken for CHM execution/profile generation per signal is noted down and tabulated in columns 4 and 7 of Table 3. Additionally, from the r range and resolution obtained, we get to know the number of unique r values to be used for profile generation. This number is denoted as n and is tabulated in columns 2 and 5 of Table 3.Now, using n estimated from the above procedure, we try and generate profiles following the traditional approach. In other words, estimation will be repeated n times, each time with a unique r. The time taken for this execution is tabulated in columns 3 and 6 of Table 3.It can be seen that the execution times involving CHM are much less compared to those involving the traditional approach. This is true for both the elderly and young populations of HRV data. For the elderly population, the of execution times are s and s respectively for traditional approach and CHM. In the young population— s and s. The traditional way of generating a complete profile is far more computationally expensive than the CHM.To give a more comprehensible explanation, say we need to compute at “n” number of r values, in order to produce a “complete” profile. The traditional approach will do this by repeating a run of estimation n number of times, each time using a unique r. This implies the total execution time will be . Now, the higher the value of n, the greater the execution time. CHM on the other hand, will do this by simultaneously generating corresponding to all r values in a single run of the algorithm (as explained in Section 5.3). This is irrespective of the value of n (small or large). The total execution time will therefore remain .CHM involves matrix operations and the concept of cumulative distribution to generate an entropy profile. This makes the approach computationally more efficient than the traditional approaches. We see that traditionally, looped operations have been used to compute entropy at multiple r values, increasing computational load.It is to be noted that the computational benefit demonstrated above may vary with the computing platform used. Our results have been validated in MATLAB where the kernel loops are optimized compared to traditional loops.
8. Conclusions
- Future Research Directions
- Data driven selection of embedding dimension m: Similarly to the case of r, embedding dimension m is yet another parameter that limits entropy methods in the analysis of short-term data. Though not as critical as r, m-selection also requires a certain extent of discretion when used in entropy estimations. A perfectly appropriate m value can greatly help improve information retrieval from short-length HRV signals. Since the right choice of m is as ambiguous as that of r, we will need methods to help us choose the appropriate m. Thinking along the same lines as dealing with r, it is best if we are able to completely eliminate the need to choose m from entropy estimations. The prospect of eliminating embedding dimension (m) from entropy formulations is the primary future target. Following an elaborate survey on the selection, use and impacts of varying embedding dimension (m) on entropy, a geometrical explanation of the parameter’s properties and behavior is obtainable. This helps formulate a relation between data and their potential range of m values. Then the demand is for a data-driven algorithm capable of generating these potential m values automatically. In this way, once the complete entropy profile is obtained with respect to m, beneficial measures of regularity or complexity can be extracted from it. The physiological relevance of m-selection in the context of HRV analysis can also be another stream of research here.
- Extending the method for any given physiological signal: A significant extension of our work will be meant to make our methods compatible for use with any kind of physiological signal, not just HRV. Now, an important aspect to be considered here is the frequency of the input signals. For instance, some physiological signals, such as the PPG (photoplethysmographic) signals, are low-frequency in nature, while ones such as EEG (electroencephalographic) signals are of more high frequency components. It is essential to study how our entropy profiling method responds to changes in input signal frequencies. Since the method is entirely dependent on signal dynamics, a frequency-based study makes an indispensable future goal.
- Cross entropy analysis on multi-variate signals: Looking for correlations in bi-variate or multi-variate physiological time series is a research domain that has picked up speed in recent years. Finding synchrony between two different physiological signals (coming from the same system) may reveal significant information about the underlying physiological system. For this purpose, it has been a common practice to employ cross-entropy versions of approximate and sample entropy. Implementing cross-entropy profiling instead of estimations could prove beneficial here too. Using CHM and thereby entropy profiling to perform multi-variate signal analysis will improve prospects of analysis on short-length multivariate signals. This could be an important direction of future experiments and validation attempts.
- Simulation trials using different models: HRV is a non-stationary process, with long-range and short-range dependencies. Entropy profiling assumes stationarity in the signal, when evaluating short segments of the data. However, if it is to be used on longer lengths of data, signal attributes and assumptions may have to change. Signals may become more complex. To test the robustness of the method, it will be significant to run simulations on complex auto-regressive models such as ARFIMA (autoregressive fractionally integrated moving average) [70] and GARCH (generalized autoregressive conditional heteroskedasticity) [27].
Author Contributions
Funding
Conflicts of Interest
References
- Estela, K.B.; Mark, R.; Paul, F.; Joseph, R. Heart Rate Variability in Health and Disease. Scand. J. Work Environ. Health 1995, 21, 85–95. [Google Scholar]
- Huikuri, H.V.; Makikallio, T.; Airaksinen, K.; Mitrani, R.; Castellanos, A.; Myerburg, R.J. Measurement of heart rate variability: A clinical tool or a research toy? J. Am. Coll. Cardiol. 1999, 34, 1878–1883. [Google Scholar] [CrossRef] [Green Version]
- Sandercock, G.R.; Bromley, P.D.; Brodie, D.A. Review: The reliability of short-term measurements of heart rate variability. Int. J. Cardiol. 2005, 103, 238–247. [Google Scholar] [CrossRef]
- Malik, M.; Bigger, J.T.; Camm, A.J.; Kleiger, R.E.; Malliani, A.; Moss, A.J.; Schwartz, P.J. Heart rate variability: Standards of measurement, physiological interpretation and clinical use. Task Force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology. Circulation 1996, 93, 1043–1065. [Google Scholar] [CrossRef]
- Goldberger, A.L.; West, B.J. Applications of nonlinear dynamics to clinical cardiology. Ann. N. Y. Acad. Sci. 1987, 504, 195–213. [Google Scholar] [CrossRef]
- Voss, A.; Schulz, S.; Schroeder, R.; Baumert, M.; Caminal, P. Methods Derived from Nonlinear Dynamics for Analysing Heart Rate Variability. Philos. Trans. Math. Phys. Eng. Sci. 2009, 367, 277–296. [Google Scholar] [CrossRef]
- Voss, A.; Schroeder, R.; Heitmann, A.; Peters, A.; Perz, S. Short Term Heart Rate Variability Influence of Gender and Age in Healthy Subjects. PLoS ONE 2015, 10, e0118308. [Google Scholar] [CrossRef] [Green Version]
- Chen, W.; Zhuang, J.; Yu, W.; Wang, Z. Measuring complexity using FuzzyEn, ApEn, and SampEn. Med. Eng. Phys. 2009, 31, 61–68. [Google Scholar] [CrossRef] [PubMed]
- Lu, S.; Chen, X.; Kanters, J.K.; Solomon, I.C.; Chon, K.H. Automatic Selection of the Threshold Value for Approximate Entropy. IEEE Trans. Biomed. Eng. 2008, 55, 1966–1972. [Google Scholar] [PubMed]
- Li, P.; Liu, C.; Li, K.; Zheng, D.; Liu, C.; Hou, Y. Assessing the complexity of short-term heartbeat interval series by distribution entropy. Med. Biol. Eng. Comput. 2015, 53, 77–87. [Google Scholar] [CrossRef] [PubMed]
- Grassberger, P.; Procaccia, I. Measuring the strangeness of strange attractors. Physica D 1983, 9, 189–208. [Google Scholar] [CrossRef]
- Eckmann, J.P.; Ruelle, D. Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 1985, 57, 617–656. [Google Scholar] [CrossRef]
- Pincus, S.M. Approximate entropy as a measure of system complexity. Proc. Natl. Acad. Sci. USA 1991, 88, 2297–2301. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Pincus, S.M.; Gladstone, I.M.; Ehrenkranz, R.A. A regularity statistic for medical data analysis. J. Clin. Monit. 1991, 7, 335–345. [Google Scholar] [CrossRef] [PubMed]
- Al-Angari, H.M.; Sahakian, A.V. Use of sample entropy approach to study heart rate variability in obstructive sleep apnea syndrome. IEEE Trans. Biomed. Eng. 2007, 54, 1900–1904. [Google Scholar] [CrossRef] [PubMed]
- Richman, J.; Moorman, J. Physiological time-series analysis using approximate entropy and sample entropy. Am. J. Physiol. 2000, 278, H2039–H2049. [Google Scholar] [CrossRef] [Green Version]
- Xie, H.B.; He, W.X.; Liu, H. Measuring time series regularity using nonlinear similarity-based sample entropy. Phys. Lett. A 2008, 372, 7140–7146. [Google Scholar] [CrossRef]
- Xie, H.B.; Chen, W.T.; He, W.X.; Liu, H. Complexity analysis of the biomedical signal using fuzzy entropy measurement. Appl. Soft Comput. J. 2011, 11, 2871–2879. [Google Scholar] [CrossRef]
- Costa, M.; Goldberger, A.L.; Peng, C.K. Multiscale entropy analysis of complex physiologic time series. Phys. Rev. Lett. 2002, 89, 068102. [Google Scholar] [CrossRef] [Green Version]
- Udhayakumar, R.K.; Karmakar, C.; Palaniswami, M. Multiscale Entropy Profiling to Estimate Complexity of Heart Rate Dynamics. Phys. Rev. E 2019, 100, 012405. [Google Scholar] [CrossRef]
- Udhayakumar, R.; Karmakar, C.; Palaniswami, M. Cross Entropy Profiling to Test Pattern Synchrony in Short-Term Signals. In Proceedings of the 2019 41st Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Berlin, Germany, 23–27 July 2019. [Google Scholar]
- Amigó, J.M. The equality of Kolmogorov–Sinai entropy and metric permutation entropy generalized. Phys. D Nonlinear Phenom. 2012, 241, 789–793. [Google Scholar] [CrossRef]
- Pincus, S.M. Irregularity and asynchrony in biologic network signals. Methods Enzymol. 2000, 321, 149–182. [Google Scholar] [PubMed]
- Liu, C.; Liu, C.; Shao, P.; Li, L.; Sun, X.; Wang, X.; Liu, F. Comparison of different threshold values r for approximate entropy: Application to investigate the heart rate variability between heart failure and healthy control groups. Physiol. Meas. 2011, 32, 167. [Google Scholar] [CrossRef] [PubMed]
- Mayer, C.C.; Bachler, M.; Hörtenhuber, M.; Stocker, C.; Holzinger, A.; Wassertheurer, S. Selection of entropy-measure parameters for knowledge discovery in heart rate variability data. BMC Bioinform. 2014, 15, S2. [Google Scholar] [CrossRef] [Green Version]
- Yentes, J.M.; Hunt, N.; Schmid, K.K.; Kaipust, J.P.; McGrath, D.; Stergiou, N. The appropriate use of approximate entropy and sample entropy with short data sets. Ann. Biomed. Eng. 2013, 41, 349–365. [Google Scholar] [CrossRef]
- Liu, L.Z.; Qian, X.Y.; Lu, H.Y. Cross-sample entropy of foreign exchange time series. Phys. A Stat. Mech. Its Appl. 2010, 389, 4785–4792. [Google Scholar] [CrossRef]
- Castiglioni, P.; Di Rienzo, M. How the threshold r influences approximate entropy analysis of heart-rate variability. In Proceedings of the 2008 Computers in Cardiology, Bologna, Italy, 14–17 September 2008; pp. 561–564. [Google Scholar]
- Castiglioni, P.; Zurek, S.; Piskorski, J.; Kosmider, M.; Guzik, P.; Ce, E.; Rampichini, S.; Merati, G. Assessing Sample Entropy of physiological signals by the norm component matrix algorithm: Application on muscular signals during isometric contraction. In Proceedings of the 2013 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Osaka, Japan, 3–7 July 2013; p. 5053. [Google Scholar]
- Boskovic, A.; Loncar-Turukalo, T.; Japundzic-Zigon, N.; Bajic, D. The flip-flop effect in entropy estimation. In Proceedings of the 2011 IEEE 9th International Symposium on Intelligent Systems & Informatics (SISY), Subotica, Serbia, 8–10 September 2011; p. 227. [Google Scholar]
- Lake, D.E.; Moorman, J.R. Accurate estimation of entropy in very short physiological time series: The problem of atrial fibrillation detection in implanted ventricular devices. Am. J. Physiol. 2011, 300, 319–325. [Google Scholar] [CrossRef]
- Liu, C.; Oster, J.; Reinertsen, E.; Li, Q.; Zhao, L.; Nemati, S.; Clifford, G.D. A comparison of entropy approaches for AF discrimination. Physiol. Meas. 2018, 39, 074002. [Google Scholar] [CrossRef]
- Cysarz, D.; Lange, S.; Matthiessen, P.F.; van Leeuwen, P. Regular heartbeat dynamics are associated with cardiac health. Am. J. Physiol. Regul. Integr. Comp. Physiol. 2007, 292, R368–R372. [Google Scholar] [CrossRef]
- Kumar, M.; Pachori, R.B.; Acharya, U.R. Use of Accumulated Entropies for Automated Detection of Congestive Heart Failure in Flexible Analytic Wavelet Transform Framework Based on Short-Term HRV Signals. Entropy 2017, 19, 92. [Google Scholar] [CrossRef] [Green Version]
- Fleisher, L.A.; Pincus, S.M.; Rosenbaum, S.H. Approximate entropy of heart rate as a correlate of postoperative ventricular dysfunction. Anesthesiology 1993, 78, 683–692. [Google Scholar] [CrossRef] [PubMed]
- Signorini, M.G. Nonlinear Analysis of Heart Rate Variability Signal: Physiological Knowledge and Diagnostic Indications. In Proceedings of the Annual International Conference—IEEE Engineering in Medicine and Biology Society, San Francisco, CA, USA, 1–5 September 2004; Volume 7, pp. 5407–5410. [Google Scholar]
- Liu, G.; Wang, Q.; Chen, S.; Zhou, G.; Chen, W.; Wu, Y. Robustness evaluation of heart rate variability measures for age gender related autonomic changes in healthy volunteers. Australas. Phys. Eng. Sci. Med. 2014, 37, 567–574. [Google Scholar] [CrossRef] [PubMed]
- Voss, A.; Heitmann, A.; Schroeder, R.; Peters, A.; Perz, S. Short-term heart rate variability–age dependence in healthy subjects. Physiol. Meas. 2012, 33, 1289–1311. [Google Scholar] [CrossRef] [PubMed]
- Voss, A.; Schroeder, R.; Fischer, C.; Heitmann, A.; Peters, A.; Perz, S. Influence of age and gender on complexity measures for short term heart rate variability analysis in healthy subjects. In Proceedings of the 2013 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Osaka, Japan, 3–7 July 2013; pp. 5574–5577. [Google Scholar]
- Lake, D.E.; Richman, J.S.; Griffin, M.P.; Moorman, J.R.; Mosconi, G.; Carnevali, O.; Habibi, H. Sample entropy analysis of neonatal heart rate variability. Am. J. Physiol. 2002, 283, 789–797. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Xinnian, C.; Solomon, I.; Chon, K. Comparison of the Use of Approximate Entropy and Sample Entropy: Applications to Neural Respiratory Signal. In Proceedings of the 2005 IEEE Engineering in Medicine and Biology 27th Annual Conference, Shanghai, China, 17–18 January 2006; p. 4212. [Google Scholar]
- Costa, M.; Peng, C.K.; Goldberger, A.L.; Hausdorff, J.M. Multiscale entropy analysis of human gait dynamics. Physica A 2003, 330, 53–60. [Google Scholar] [CrossRef]
- Costa, M.; Goldberger, A.L.; Peng, C.K. Multiscale entropy analysis of biological signals. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 2005, 71, 021906. [Google Scholar] [CrossRef] [Green Version]
- Acharya, U.R.; Joseph, K.P.; Kannathal, N.; Lim, C.M.; Suri, J.S. Heart rate variability: A review. Med. Biol. Eng. Comput. 2006, 44, 1031–1051. [Google Scholar] [CrossRef]
- Thuraisingham, R.A.; Gottwald, G.A. On multiscale entropy analysis for physiological data. Physica A 2006, 366, 323–332. [Google Scholar] [CrossRef]
- Liu, C.; Li, K.; Zhao, L.; Liu, F.; Zheng, D.; Liu, C.; Liu, S. Analysis of heart rate variability using fuzzy measure entropy. Comput. Biol. Med. 2013, 43, 100–108. [Google Scholar] [CrossRef]
- Ji, L.; Li, P.; Li, K.; Wang, X.; Liu, C. Analysis of short-term heart rate and diastolic period variability using a refined fuzzy entropy method. Biomed. Eng. Online 2015, 14, 64. [Google Scholar] [CrossRef] [Green Version]
- Liu, Y.; Lin, Y.; Wang, J.; Shang, P. Refined generalized multiscale entropy analysis for physiological signals. Physica A 2018, 490, 975–985. [Google Scholar] [CrossRef]
- Udhayakumar, R.; Karmakar, C.; Palaniswami, M. Understanding Irregularity Characteristics of Short-term HRV Signals using Sample Entropy Profile. IEEE Trans. Biomed. Eng. 2018, 65, 2569–2579. [Google Scholar] [CrossRef] [PubMed]
- Udhayakumar, R.K.; Karmakar, C.; Palaniswami, M. Approximate entropy profile: A novel approach to comprehend irregularity of short-term HRV signal. Nonlinear Dyn. 2016, 88, 823–837. [Google Scholar] [CrossRef]
- Pincus, S.M.; Goldberger, A.L. Physiological time-series analysis: What does regularity quantify? Am. J. Physiol. 1994, 266, H1643–H1656. [Google Scholar] [CrossRef]
- Pincus, S.M.; Viscarello, R.R. Approximate entropy: A regularity measure for fetal heart rate analysis. Obstet. Addit. Gynecol. 1992, 79, 249–255. [Google Scholar]
- Pincus, S. Approximate entropy (ApEn) as a complexity measure. Chaos 1995, 5, 110. [Google Scholar] [CrossRef]
- Ryan, S.M.; Goldberger, A.L.; Pincus, S.M.; Mietus, J.; Lipsitz, L.A. Gender- and age-related differences in heart rate dynamics: Are women more complex than men? J. Am. Coll. Cardiol. 1994, 24, 1700–1707. [Google Scholar] [CrossRef] [Green Version]
- Porta, A.; Gnecchi-Ruscone, T.; Tobaldini, E.; Guzzetti, S.; Furlan, R.; Montano, N. Progressive decrease of heart period variability entropy-based complexity during graded head-up tilt. J. Appl. Physiol. 2007, 103, 1143–1149. [Google Scholar] [CrossRef]
- Madalena, D.C.; Chung-Kang, P.; Ary, L.G. Multiscale Analysis of Heart Rate Dynamics: Entropy and Time Irreversibility Measures. Cardiovasc. Eng. Int. J. 2008, 8, 88–93. [Google Scholar]
- Ahmed, M.U.; Mandic, D.P. Multivariate multiscale entropy: A tool for complexity analysis of multichannel data. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 2011, 84, 061918. [Google Scholar] [CrossRef] [Green Version]
- Costa, M.; Goldberger, A.L.; Peng, C.K. Multiscale entropy to distinguish physiologic and synthetic RR time series. Comput. Cardiol. 2002, 29, 137–140. [Google Scholar] [PubMed]
- Goldberger, A.L. Is the normal heartbeat chaotic or homeostatic? Physiology 1991, 6, 87–91. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Chang, Y.C.; Tsao, J.H.; Wu, H.T.; Chen, H.R.; Liu, A.B.; Yeh, J.J.; Lo, M.T.; Tang, C.J.; Tsai, I.T.; Sun, C.K. Application of a Modified Entropy Computational Method in Assessing the Complexity of Pulse Wave Velocity Signals in Healthy and Diabetic Subjects. Entropy 2014, 16, 4032–4043. [Google Scholar] [CrossRef] [Green Version]
- Wu, S.D.; Wu, C.W.; Lee, K.Y.; Lin, S.G. Modified multiscale entropy for short-term time series analysis. Physica A 2013, 392, 5865–5873. [Google Scholar] [CrossRef]
- Udhayakumar, R.; Karmakar, C.; Li, P.; Wang, X.; Palaniswami, M. Modified Distribution Entropy as a Complexity Measure of Heart Rate Variability (HRV) Signal. Entropy 2020, 22, 1077. [Google Scholar] [CrossRef]
- Freedman, D.; Diaconis, P. On the histogram as a density estimator:L2 theory. Z. Wahrscheinlichkeitstheorie Verwandte Geb. 1981, 57, 453–476. [Google Scholar] [CrossRef] [Green Version]
- Udhayakumar, R.; Karmakar, C.; Palaniswami, M. Secondary measures of regularity from an entropy profile in detecting Arrhythmia. In Proceedings of the 2017 39th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Seogwipo, Korea, 11–15 July 2017; p. 3485. [Google Scholar]
- Udhayakumar, R.; Karmakar, C.; Palaniswami, M. Entropy Profiling to Detect ST Change in Heart Rate Variability Signals. In Proceedings of the 2019 41st Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Berlin, Germany, 23–27 July 2019. [Google Scholar]
- Goldberger, A.L.; Amaral, L.A.N.; Glass, L.; Hausdorff, J.M.; Ivanov, P.C.; Mark, R.G.; Mietus, J.E.; Moody, G.B.; Peng, C.K.; Stanley, H.E. PhysioBank, PhysioToolkit, and PhysioNet: Components of a New Research Resource for Complex Physiologic Signals. Circulation 2000, 101, e215–e220. [Google Scholar] [CrossRef] [Green Version]
- Iyengar, N.; Peng, C.K.; Morin, R.; Goldberger, A.L.; Lipsitz, L.A. Age-related alterations in the fractal scaling of cardiac interbeat interval dynamics. Am. J. Physiol. 1996, 271, 1078–1084. [Google Scholar] [CrossRef] [Green Version]
- Moody, G.B.; Mark, R.G. The impact of the MIT-BIH arrhythmia database. IEEE Eng. Med. Biol. Mag. 2001, 20, 45–50. [Google Scholar] [CrossRef]
- Pan, Y.H.; Wang, Y.H.; Liang, S.F.; Lee, K.T. Fast computation of sample entropy and approximate entropy in biomedicine. Comput. Methods Programs Biomed. 2011, 104, 382–396. [Google Scholar] [CrossRef]
- Contreras-Reyes, J.; Palma, W. Statistical Analysis of Autoregressive Fractionally Integrated Moving Average Models in R. Comput. Stat. 2013, 28, 2309–2331. [Google Scholar] [CrossRef] [Green Version]
N | ||||||||
---|---|---|---|---|---|---|---|---|
p Value | AUC | p Value | AUC | p Value | AUC | p Value | AUC | |
50 | NA | NA | NS | 0.62 | 0.00051 | 0.82 | 0.0012 | 0.80 |
100 | NA | NA | NS | 0.68 | 0.00022 | 0.84 | 0.0106 | 0.74 |
150 | NS | 0.54 | NS | 0.64 | 0.00022 | 0.84 | 0.0179 | 0.72 |
200 | NS | 0.62 | 0.0315 | 0.70 | 0.00004 | 0.88 | 0.0077 | 0.75 |
250 | NS | 0.65 | 0.0193 | 0.72 | 0.00004 | 0.88 | 0.0084 | 0.75 |
300 | NS | 0.68 | 0.0098 | 0.74 | 0.00005 | 0.88 | 0.0207 | 0.72 |
N | ||||||||
---|---|---|---|---|---|---|---|---|
p Value | AUC | p Value | AUC | p Value | AUC | p Value | AUC | |
50 | NA | NA | NS | 0.61 | 1.88 × 10−8 | 0.95 | 0.0001 | 0.81 |
100 | NA | NA | NS | 0.65 | 1.73 × 10−8 | 0.95 | 0.0005 | 0.78 |
150 | NS | 0.60 | 0.0098 | 0.71 | 8.82 × 10−9 | 0.96 | 0.0007 | 0.77 |
200 | NS | 0.61 | 0.0016 | 0.75 | 1.34 × 10−8 | 0.96 | 0.0002 | 0.80 |
250 | NS | 0.65 | 0.0022 | 0.75 | 2.85 × 10−8 | 0.95 | 0.0034 | 0.74 |
300 | NS | 0.66 | 0.0021 | 0.75 | 1.46 × 10−8 | 0.96 | 0.0058 | 0.72 |
Healthy “Elderly” | Healthy “Young” | |||||
---|---|---|---|---|---|---|
Signal | Execution Time (s) | Execution Time (s) | ||||
Traditional Approach | CHM | Traditional Approach | CHM | |||
1 | 127 | 1.08 | 0.08 | 280 | 2.53 | 0.10 |
2 | 143 | 1.17 | 0.09 | 352 | 3.14 | 0.11 |
3 | 127 | 1.10 | 0.08 | 137 | 1.24 | 0.09 |
4 | 323 | 2.75 | 0.12 | 340 | 3.01 | 0.12 |
5 | 31 | 0.29 | 0.07 | 275 | 2.41 | 0.10 |
6 | 78 | 0.74 | 0.08 | 314 | 2.76 | 0.12 |
7 | 91 | 0.77 | 0.09 | 508 | 4.44 | 0.16 |
8 | 119 | 1.15 | 0.08 | 209 | 1.91 | 0.08 |
9 | 356 | 3.23 | 0.12 | 135 | 1.11 | 0.08 |
10 | 144 | 1.23 | 0.08 | 197 | 1.72 | 0.09 |
11 | 460 | 4.10 | 0.14 | 201 | 1.84 | 0.09 |
12 | 265 | 2.36 | 0.10 | 384 | 3.37 | 0.13 |
13 | 62 | 0.63 | 0.07 | 123 | 1.08 | 0.09 |
14 | 133 | 1.18 | 0.09 | 96 | 1.12 | 0.08 |
15 | 64 | 0.61 | 0.08 | 407 | 3.53 | 0.12 |
16 | 186 | 1.65 | 0.09 | 251 | 2.32 | 0.11 |
17 | 99 | 0.87 | 0.08 | 321 | 2.89 | 0.12 |
18 | 267 | 2.24 | 0.10 | 415 | 3.71 | 0.12 |
19 | 396 | 3.49 | 0.12 | 559 | 5.26 | 0.16 |
20 | 97 | 0.78 | 0.08 | 424 | 3.75 | 0.12 |
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Karmakar, C.; Udhayakumar, R.; Palaniswami, M. Entropy Profiling: A Reduced—Parametric Measure of Kolmogorov—Sinai Entropy from Short-Term HRV Signal. Entropy 2020, 22, 1396. https://doi.org/10.3390/e22121396
Karmakar C, Udhayakumar R, Palaniswami M. Entropy Profiling: A Reduced—Parametric Measure of Kolmogorov—Sinai Entropy from Short-Term HRV Signal. Entropy. 2020; 22(12):1396. https://doi.org/10.3390/e22121396
Chicago/Turabian StyleKarmakar, Chandan, Radhagayathri Udhayakumar, and Marimuthu Palaniswami. 2020. "Entropy Profiling: A Reduced—Parametric Measure of Kolmogorov—Sinai Entropy from Short-Term HRV Signal" Entropy 22, no. 12: 1396. https://doi.org/10.3390/e22121396