Double symbolic joint entropy in nonlinear dynamic complexity analysis
aa r X i v : . [ phy s i c s . d a t a - a n ] F e b Double symbolic joint entropy in nonlinear dynamic complexity analysis
Wenpo Yao
School of Telecommunications and Information Engineering,Nanjing University of Posts and Telecommunications, Nanjing 210003, China
Jun Wang ∗ School of Geography and Biological Information, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
Symbolizations, the base of symbolic dynamic analysis, are classified as global static and localdynamic approaches which are combined by joint entropy in our works for nonlinear dynamic com-plexity analysis. Two global static methods, symbolic transformations of Wessel N. symbolic entropyand base-scale entropy, and two local ones, namely symbolizations of permutation and differentialentropy, constitute four double symbolic joint entropies that have accurate complexity detections inchaotic models, logistic and Henon map series. In nonlinear dynamical analysis of different kinds ofheart rate variability, heartbeats of healthy young have higher complexity than those of the healthyelderly, and congestive heart failure (CHF) patients are lowest in heartbeats’ joint entropy values.Each individual symbolic entropy is improved by double symbolic joint entropy among which thecombination of base-scale and differential symbolizations have best complexity analysis. Test resultsprove that double symbolic joint entropy is feasible in nonlinear dynamic complexity analysis.
I. INTRODUCTION
Heart rate variability (HRV), the variation in beat-to-beat intervals represented by RR or NN interval[1], dis-plays irregular and non-stationary behaviors whose non-linear dynamics provide valuable information for cardiacscientific and clinical researches[2, 3]. To measure itsnonlinear dynamical features, some complexity param-eters, such as fractal dimensions, Lyapunov exponents,geometric and entropy methods et al., are proposed[4–6]. Symbolic dynamic analysis, a kind of fast, simpleand efficient method, provides rigorous ways to analyzenonlinear dynamics[7].Symbolic time series analysis consists of symbolizationand statistical analysis to the symbolic series, and it haseffective applications in physiological signal analysis[8, 9].Symbolization involves in transforming infinite-value se-ries into symbol sequence on basis of a given alphabet[10],so it greatly reduce demands on the data and bring con-venience to series analysis[2, 11]. These symbolic trans-formations are classified into two groups, global staticand local dynamic methods[12]. Symbolic transforma-tion in works of Wessel N. et al.[13, 14] and base-scaleentropy[15] belong to global static approaches, and sym-bolizations in permutation entropy[16] and differentialentropy are typical local dynamic ways, and they all em-ploy Shannon entropy for symbolic series analysis. Ourobjective is to take both global and local dynamical in-formation into account to make comprehensive analysisof nonlinear dynamic complexity. There are some fea-sible ideas to combine the two symbolic methods suchas multi-dimension theory[15, 16]. These attempts, how-ever, are more appropriate to be described as compro-mises of, on the one hand, maintaining flexibility of global ∗ [email protected] static methods and of, on the other hand, extractingsufficient local dynamic information. In order to makeefficient use of the two types of symbolizations, we ap-ply joint entropy to combine them for nonlinear dynamiccomplexity.In our contributions to combine the two kinds of sym-bolic transformations, we conduct global static and localdynamic symbolic transformation simultaneously, andapply the two kinds of symbolizations’ joint entropy tononlinear dynamic analysis of classical nonlinear chaoticmodels and three kinds of real-world HRV. II. SYMBOLIC TRANSFORMATION
Symbolization is a course of coarse-graining or re-duction, and its basic idea is to transform series X L = { x , x , . . . , x L } into symbolic sequence S N = { s , s , . . . , s N } whose element s i is a finite number ofsymbols (letters from some alphabet). Global staticmethods perform symbolization according to different se-quences intervals which are identified by several parame-ters obtained from the whole sequence, and local dynam-ical approaches, on other hands, take contribution of lo-cal adjacent elements’ relationships to carry on symbolicrepresentation. Both types of symbolization, targetingdifferent types of nonlinear dynamical information, haveeffective applications in complexity detections. A. Wessel N. Symbolization
To make physiology-connected symbolization which isrelatively easy to interpret, Wessel N. et al. developa four symbols context-dependent pragmatic symbolstransformation[13, 14, 17]. The symbolic transformation,referring to three given levels, namely (1 − α ) µ , µ , and(1 + α ) µ , performs as Eq. (1) s i ( x i ) = µ < x i ≤ (1 + α ) µ α ) µ < x i < ∞ − α ) µ < x i ≤ µ < x i ≤ (1 − α ) µ (1)where µ is the series mean and α is special controllingparameter which is recommended as from 0.03 to 0.07according to tests and does not significantly differ result-ing symbol sequences in nonlinear forecasting of cardiacarrhythmias features. B. Base-scale Symbolization
Symbolization in base-scale entropy[18] is a kindof four-symbol global method, which employs multi-dimensional vector reconstruction firstly as Eq. (2) andmakes symbolic transformation in each vector. X m ( i ) = { x ( i ) , x ( i + τ ) , . . . , x ( i + ( m − τ ) } (2)In Eq.(2), m is embedding dimension and τ is delaytime. And then base scale, the root-mean square of thedifferences between every two contiguous values in a vec-tor, of each reconstructed vector is calculated as Eq. (3). BS m ( i ) = s P m − j =1 [ x ( i + j ) − x ( i + j − m − s i ( x i ) = µ m < x i ≤ µ m + α × BS m x i > µ m + α × BS m µ m − α × BS m < x i ≤ µ m x i ≤ µ m − α × BS m (4)where µ m represents the mean of m-dimension vector X m ( i ) and α describes controlling parameter which couldbe chosen from 0.1 to 2 accordingly. Multi-dimensionalprocedure brings adaptability and flexibility as well assome local dynamical information. In our works, there-fore, we perform symbolic transformation on the wholetime series as a vector to extract global nonlinear infor-mation. C. Permutated Symbolic Transformation
Permutation entropy, with advantages of simplicity,fast calculation and robustness, carries on typical localdynamic symbolization[19, 20]. By comparing neigh-boring values and mapping time series onto symbolssequences[21], permutation entropy is a classical com-plexity parameter. Multi-dimensional procedure, same as base-scale entropy, is needed to transform series intosymbolic sequences. Accordingly to the values’ sizes, se-ries are reorganized in for example ascending order ineach reconstructed vector as x i +( j − τ ≤ x i +( j − τ ≤· · · ≤ x i +( j m − τ . π j = { j , j , · · · , j m } is a new sequence consisting ofthe elements’ original positions, and there are m ! permu-tations considering all possibilities. Permutation entropyis Shannon entropy of all permutations’ probabilities as H ( m ) = − P p ( π i ) log p ( π i ), where p ( π i ) = 0 . D. Symbolization in Differential Entropy
Taking differences between adjacent elements into ac-count, we proposed differential entropy as a dynamiccomplexity measure. This symbolization attributes itscomplexity detection to detailed local dynamic informa-tion. The differences between current element and itsforward and backward ones are D = x ( i + τ ) − x ( i )and D = x ( i ) − x ( i − τ ) where τ is the delay time. Afour-symbols transformation are carried on as Eq. (5) S i ( x i ) = dif f ≥ α · var ≤ dif f < α · var − α · var < dif f <
03 : dif f ≥ − α · var (5)where dif f = k D k − k D k , and var = p ( D + D ) / α in could be adjusted from 0.3 to 0.6.Code series C ( i ), whose formation is the next step fol-lowing symbolization, is constructed by m-bit encodingof symbolic sequences, and measurements for the codeseries involve classical statistics and information theory,such as Shannon entropy. Taking symbols ’abc’ as exam-ple, coding procedure could be c ( i ) = a ∗ n + b ∗ n + c where n should not be smaller than the amount of sym-bols’ types, and code forms do not make significant dif-ferences to symbolic analysis. 3-bit encoding is applied inour following symbolic dynamic analysis to all symbolicsequences. III. DOUBLE SYMBOLIZED DYNAMICANALYSIS
Global static symbolic transformations flexibly selectthe number and size of partitions according to signals’characteristics, and local dynamic symbolizations effec-tively extract local detailed dynamic information. To ob-tain both global static and local dynamical informationis the main concern of this section.Multi-dimension[15, 16, 22] vector reconstruction, avery attractive theoretical problem, is used in the base-scale entropy and permutation entropy. Through vectorreconstruction, global static symbolizations is carried outin each individual vector, making base-scale entropy moreadaptable to signal changes. Symbolizations of differentvectors are independent from each other, and it is help-ful to improve the flexibility of transformation and ex-tract some local dynamic information. And in extremecases, when reconstructed vector is small enough thateach vector contains only 3 or even 2 elements, the globalstatic methods are almost equivalent to local dynamicones. In the multi-dimension method, the selection ofvector length and the setting of delay factor are worthyof further and in-depth researches. Multi-dimension pro-cessing takes account of the two kinds symbolic trans-formations, but it is still a compromising method andcannot give fully comprehensive considerations to bothsides. Another try to combine the two different symbol-izations is to directly integrate the two symbolic seriesinto new sequences. For example, the global static se-quence is ’0123’ and local dynamic orders in ’1230’, andtheir combinated symbols are ’01 12 23 30’(symbols of theglobal static method is in the front of each 2-bit recom-bination). The disadvantage of this symbolic combina-tion is increasing amount of symbols (if symbols amountsof the two symbolizations are N and M, there are N*Msymbols in the combination), but it is worth making suchattempts for some unforeseen achievements.A feasible solution is to combine the two kinds nonlin-ear dynamical information by joint entropy. The Shan-non seminal work rationalized and initiated early effortsinto information theory, which is the most influential con-tribution to entropy[23]. The information contents of two(sub)systems are illustrated in Fig.1, and these relation-ships apply to two kinds of symbolizations as well. H (cid:3) X ( ) H (cid:3) Y ( ) I X ( ; Y ) H (cid:3) , X Y ( ) H (cid:3) X (cid:3) / Y ( ) H (cid:3) / Y X ( )
FIG. 1. The relationships between information entropy of two(sub)systems
Joint entropy, in Eq. (6) or (7), is used to measure thecombined amount of information. H ( X, Y ) =
X X p ( x i , y j ) I ( x i , y j )= − X X p ( x i , y j ) logp ( x i , y j ) (6) H ( X, Y ) = H ( X ) + H ( Y ) − I ( X, Y )= − X p ( x ) logp ( x ) − X p ( y ) logp ( y ) + X logp ( x, y )(7)In combination of two different symbolizations, globalstatic and local dynamical symbolic series are obtainedsimultaneously as X G and X L whose joint entropy are TABLE I. Double symbolic joint entropy in Henon map anal-ysis.R WN-PE JEn WN-DE JEn BS-PE JEn BS-DE JEn0.9 7.7519 9.4581 7.3213 9.02751 8.4924 9.9976 7.9587 9.4639 calculated as H ( X G , X L ). In our following double sym-bolic dynamic analysis, ’BS-PE JEn’ describes the jointentropy of base-scale and permutated symbolization, and’BS-DE JEn’, ’WN-PE JEn’ and ’WN-DE JEn’ are otherthree combinations of base-scale and differential joint en-tropy, Wessel N. et al.’s and permutated joint entropy,Wessel N. et al.’s and differential joint entropy. To an-alyze the relationships between each individual symbolicentropy and four double joint entropies, we do not nor-malize all kinds of symbolic entropy methods. IV. DOUBLE SYMBOLIC JOINT ENTROPYANALYSIS OF CHAOTIC MODELS
The four double symbolic joint entropy methods aretested by logistic and Henon map series. Delay time inthe four symbolizations are set to 1. We refer to choicesof controlling parameters in their original works and theirperformances in logistic map analysis, and set α in base-scale entropy to 0.2 and differential entropy to 0.5, whilevalue to Wessel N. symbolic entropy is 0.3.The canonical form of logistic difference equation, X i +1 = r · x i (1 − x i ), is attractive by virtue of its extremesimplicity[24] and is widely applied in chaotic and non-linear dynamical analysis. Its bifurcation diagram andchaotic detections of four double symbolic joint entropyare shown in Fig.2.Logistic map shows its chaotic characteristics whenr is larger than the cut-off point r*=3.567, exactly ≈ x i +1 = ry i + 1 − . x i , y i +1 = 0 . rx i , where r is a controllingparameter. As r increase from 0.9 to 1, Henon systemexhibits more chaotic behaviors, and its four double sym-bolic joint entropy are listed in Table I.From Tab 1, as Henon series chaotic behaviors increase,four joint entropy methods have corresponding increase,verifying their effective nonlinear complexity detections. W N − PE J E n r* br W N − D E J E n r* cr BS − PE J E n r* r d BS − D E J E n r* er FIG. 2. Logistic equations for varying control parameter from3.5 to 4. (a) Bifurcation diagram. (b) WN-PE JEn. (c) WN-DE JEn. (d) BS-PE JEn. (e) BS-DE JEn.
V. DOUBLE SYMBOLIC JOINT ENTROPY INHRV ANALYSIS
Three kinds of heartbeat intervals (derived from ECG)from Physionet Database[26] are applied in our works.Firstly, 15 subjects with severe congestive heart failure(CHF), NYHA class 3-4[27], including 11 men aged 22to 71 and 4 women aged 54 to 63. Secondly, 20 young(21 to 34 years old) and 20 elderly (68 to 85 years old)underwent 2 hours of continuous data collecting[28] in aresting state in sinus rhythm.We firstly apply four double symbolic joint entropymethods and individual symbolic entropy approaches tothe three kinds of HRV, and analysis results are illus-trated in FIG.3a and 3b. In this part, controlling param-eters are all set to 0.55.From Fig.3a, the four double symbolic joint entropiesdistinguish the three kinds of HRV and share coincident D oub l e sy m bo li c j o i n t en t r op y CHFElderlyYoung aWN−PE JEn WN−DE JEn BS−PE JEn BS−DE JEn I nd i v i dua l sy m bo li c en t r op y CHFElderlyYoung bDE PEBSEWNSE
FIG. 3. Symbolic entropy analysis of three groups of heart-beats. (a) Double symbolic joint entropy. (b) Individualsymbolic entropy (’WNSE’ represents the symbolic entropyin works of Wessel N. et al., ’BSE’ accounts for the base-scale entropy, ’PE’ stands for permutation entropy and ’DE’denotes differential entropy.) distinctions which are consistent with the ’complexity-loss’ theory of aging and disease in relevant researches[2,29–31] that joint entropies of healthy young volunteers’heart rates are higher than those of healthy elderly ones,and CHF patients have the lowest joint entropy values.Healthy young subjects have better cardiac states andtheir heartbeats present more complex processes in na-ture than those of the elderly and CHF patients. Thehealthy elderly subjects represent slight weakness in car-diac function due to aging, and therefore their dynamicfeatures are less than the young ones. CHF group, hav-ing profound abnormalities in cardiac function and severedamage to the cardiac control system, largely lose theirHRV dynamic features, therefore their group have thelowest dynamic complexity.T tests for the four double symbolic joint entropy anal-ysis of three different kinds of heartbeats are carried out,and p values are listed in Table II.From Table II, differences between each two kinds ofheart signals’ complexity extracted by WN-DE JEn andBS-DE JEn are significant ( p < .
05) that the two DE-jointed methods achieve satisfied nonlinear distinctionsamong three groups of HRV and BS-DE joint entropyshow optimal nonlinear dynamic complexity detections.WN-PE JEn and BS-PE JEn effectively separate cardiacrhythms of CHF patients and two kinds of healthy sub-jects while they both fail to significantly distinguish the
TABLE II. p values of four double symbolic joint entropy inthree kinds of heart rates analysis (’0.000’ should be read as p < . elderly and young volunteers’ heartbeats (p=0.083 and0.086, larger than 0.05). The failures of two PE-jointedentropy in distinguishing two kinds of healthy HRV maylie in the misleadings of permutation entropy in theseheart signals nonlinear analysis.In Figure 3b, permutation entropy has different re-sults from other three symbolic entropy that nonlinearcomplexities of the three kinds of HRV show oppositions.Heartbeats of CHF patients have biggest entropy of 5.372and those of elderly persons have permutation entropy of5.335 while complexity of healthy young subjects’ heartrates, 5.121, are lowest. This paradox phenomenon, weguess, may be involved in multi-scale theory that highercomplexity for certain pathologic processes, such as CHF,than for healthy dynamics in permutation entropy anal-ysis lies in that it fails to account for the multi-scaleinformation[9, 29, 31, 32]. Multi-scale concept is toconstruct coarse-grained series y τj = 1 /τ Σ jτi =( j − τ +1 x i ,1 ≤ j ≤ N/τ and for scale 1, { y j } is the original se-ries { x i } . The single-scale permutation entropy maybe related to this inconsistency in our works . Exceptfor permutation entropy, the other three individual sym-bolic entropy methods effectively distinguish the differentheart signals and their results are not inconsistent withprevious ’complexity-loss’ theory. Independent samplest tests for other three symbolic entropy methods showthat they all effectively distinguish three different kindsof heart rate variability in nonlinear dynamic analysis(CHF-Elderly p values of WNSE, BSE and DE are 0.031,0.001 and 0.019, the Elderly-Young correspondings are0.002, 0.024 and 0.042, and CHF-Young p values are all0.000).Through the above analysis, we find that double sym-bolic joint entropy improves nonlinear complexity ex-traction of individual symbolic entropy. The two DE-combined joint entropies, reducing p values of differentkinds of heartbeats of each individual symbolic entropy,improve distinctions among the three groups of heart sig-nals and more effectively identify different kinds of HRV.The two PE-combined joint entropies, having correct dis-tinctions of the three HRV complexity shown in Fig.3a,overcome the unenviable situations of permutation en-tropy and effectively distinguish HRV of CHF patientsand two healthy groups.In this part, we observe impacts of data length on jointentropy analysis. Data length increases from 300 to 4500with step size of 300 for researches on the four joint en- tropy analysis of HRV, and results are shown in Fig.4. W N − PE J E n Data Length
CHFElderlyYoung a W N − D E J E n Data Length
CHFElderlyYoung b BS − PE J E n Data Length
CHFElderlyYoung c BS − D E J E n Data Length
CHFElderlyYoung d FIG. 4. Joint entropies of the three kinds of HRV with theincrease of data length. (a) WN-PE JEn. (b) WN-DE JEn.(c) BS-PE JEn. (d) BS-DE JEn.
Illustrated by Fig 4a and 4b, in the beginning datalength of two WN-joint entropy, heartbeats of CHF pa-tients have higher nonlinear complexity than that of thehealthy elderly and their relationships change when datalength become to and larger than 2000. In Fig 4c and4d, BS-joint entropy values of three groups of HRV areconsistent with normal relationships in previous analysisand become to stable when data length comes to 2000.In the Fig 4, in beginning parts entropy values of twohealthy heartbeats undergo increasing trends while thoseof CHF patients have first-increasing and then-decreasingchanges.In the four subplots, healthy young people maintainhigher entropy to the healthy elderly ones, which are notaffected by the data length. The differences in charts aremainly reflecting in CHF patients entropy trends, reasonsfor this we suppose are that CHF patients HRV signalsare in poor stability that contributes to these fluctua-tions.From Fig.4, the relationships between the three HRVjoint entropies change at the beginning parts and tendto converge as data lengths increase to about 2000 andlarger. And we come to the point that the four dou-ble symbolic joint entropies have certain requirementsfor data length in HRV analysis.
VI. DISCUSSIONS
To take flexibility of global static transformations anddetailed dynamic information extraction of local dynamicmethods into account, we introduce joint entropy to com-bine characteristics of the two kinds of symbolizations.Among the four individual symbolic transformations,there are three with controlling parameter whose adjust-ments play important role in nonlinear dynamic analysis.Referring to choice ranges of controlling parameters givenin their original works, we make adjustment in nonlinearcomplexity detection of chaotic models and physiologicalsignals accordingly. In logistic and Henon map analysis, α is set 0.3 to Wessel N. symbolic entropy, 0.2 to base-scale entropy and 0.5 to differential entropy while in non-linear complexity extraction of heartbeats α are adjustedto 0.55 to all three symbolizations to achieve satisfied re-sults. We find that the choices controlling parameter ofWessel N. symbolization, whether 0.3 in chaotic modelscomplexity detections or 0.55 in different HRV nonlineardynamic analysis, are not in recommended range of 0.03to 0.07 in the original literatures[13, 14]. It seems wecannot find optimal controlling parameters for all differ-ent kinds of data. The reasons, we guess, account forthis lie in differences of structural information or of dy-namical complexity in different types of signals, so theparameters should be adjusted accordingly. And it needto be validated that whether our findings apply to othernonlinear signals.Both global static and local dynamic symbolizationscontain irreplaceable dynamic information about series,and there is not too much redundant information in twotypes of symbolic transformations. Taking heartbeats ofthe first CHF patient ’chf01’ as an example, four com- bined symbolic joint entropies are 7.0001, 6.9738, 8.8819and 8.8555 which are approximately equal to the sum ofeach two individual entropy which are 1.6959 to WNSE,3.5776 to BSE, 5.2779 to DE and 5.3042 to PE. The sameresults are true for those of the healthy elderly personsand CHF patients. Joint entropy values are close to thesum of each symbolization entropy, proving that the twosymbolic transformation approaches extract series non-linear dynamic information from different perspectives,and degree of repetition of the two different symboliza-tions is very low. VII. CONCLUSIONS
The above analysis and tests show that it is feasibleand effective to use joint entropy of global static and lo-cal dynamic symbolizations for series nonlinear dynamiccomplexity detection. Double symbolic joint entropy isbeneficial to improve final nonlinear complexity detec-tions of each individual symbolic entropy in our nonlinearanalysis of heart rate variability. And the ’complexity-loss’ theory of aging and disease is validated in our con-tributions.
VIII. ACKNOWLEDGMENTS
The work is supported by Project supported by theNational Natural Science Foundation of China (GrantNos. 61271082, 61401518, 81201161), Jiangsu Provin-cial Key R & D Program (Social Development) (GrantNo.BE2015700), the Natural Science Foundation ofJiangsu Province (Grant No. BK20141432), NaturalScience Research Major Programmer in Universities ofJiangsu Province (Grant No.16KJA310002), Postgradu-ate Research & Practice Innovation Program of JiangsuProvince (KYCX17-0788). [1] U. R. Acharya, K. P. Joseph, N. Kannathal, C. M. Lim,and J. S. Suri, “Heart rate variability: a review,” Medical& Biological Engineering & Computing , 1031–1051(2006).[2] M. T. Lo, Y. C. Chang, C. Lin, H. W. Young, Y. H. Lin,Y. L. Ho, C. K. Peng, and K. Hu, “Outlier-resilient com-plexity analysis of heartbeat dynamics,” Scientific Re-ports , 8836 (2015).[3] R. K. Udhayakumar, C. Karmakar, and M. 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