An integrated approach based on uniform ...

CHAOS 17, 015117 共2007兲

An integrated approach based on uniform quantization for the evaluation of complexity of short-term heart period variability: Application to 24 h Holter recordings in healthy and heart failure humans A. Portaa兲 Dipartimento di Scienze Precliniche, LITA di Vialba, Universita’ degli Studi di Milano, Laboratorio di Modellistica di Sistemi Complessi, Via G.B. Grassi 74, 20157, Milan, Italy

L. Faes and M. Masé Dipartimento di Fisica, Universita’ di Trento, Trento, Italy

G. D’Addio Fondazione S. Maugeri, IRCCS, Istituto di Riabilitazione di Telese, Italy

G. D. Pinna and R. Maestri Fondazione S. Maugeri, IRCCS, Istituto di Riabilitazione di Montescano, Italy

N. Montano and R. Furlan Dipartimento di Scienze Cliniche “L. Sacco,” Universita’ degli Studi di Milano, Milan, Italy and Medicina Interna II, Ospedale L. Sacco, Milan, Italy

S. Guzzetti Medicina Interna II, Ospedale L. Sacco, Milan, Italy

G. Nollo Dipartimento di Fisica, Universita’ di Trento, Trento, Italy

A. Malliani Dipartimento di Scienze Cliniche “L. Sacco,” Universita’ degli Studi di Milano, Milan, Italy and Medicina Interna II, Ospedale L. Sacco, Milan, Italy

共Received 24 July 2006; accepted 24 October 2006; published online 30 March 2007兲 We propose an integrated approach based on uniform quantization over a small number of levels for the evaluation and characterization of complexity of a process. This approach integrates information-domain analysis based on entropy rate, local nonlinear prediction, and pattern classification based on symbolic analysis. Normalized and non-normalized indexes quantifying complexity over short data sequences 共⬃300 samples兲 are derived. This approach provides a rule for deciding the optimal length of the patterns that may be worth considering and some suggestions about possible strategies to group patterns into a smaller number of families. The approach is applied to 24 h Holter recordings of heart period variability derived from 12 normal 共NO兲 subjects and 13 heart failure 共HF兲 patients. We found that: 共i兲 in NO subjects the normalized indexes suggest a larger complexity during the nighttime than during the daytime; 共ii兲 this difference may be lost if non-normalized indexes are utilized; 共iii兲 the circadian pattern in the normalized indexes is lost in HF patients; 共iv兲 in HF patients the loss of the day-night variation in the normalized indexes is related to a tendency of complexity to increase during the daytime and to decrease during the nighttime; 共v兲 the most likely length L of the most informative patterns ranges from 2 to 4; 共vi兲 in NO subjects classification of patterns with L = 3 indicates that stable patterns 共i.e., those with no variations兲 are more present during the daytime, while highly variable patterns 共i.e., those with two unlike variations兲 are more frequent during the nighttime; 共vii兲 during the daytime in HF patients, the percentage of highly variable patterns increases with respect to NO subjects, while during the nighttime, the percentage of patterns with one or two like variations decreases. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2404630兴 This study integrates information-domain analysis based on entropy rate, local nonlinear prediction, and pattern classification based on symbolic analysis under the unifying framework provided by uniform quantization. This integrated approach provides a list of normalized and

non-normalized indexes useful to quantify and typify complexity over short data series (È300 samples). The application to 24 h Holter recordings of heart period variability both in healthy and pathological populations proves that this approach is reliable and helpful even when carried out over data obtained in nonstandardized conditions.

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I. INTRODUCTION

The assessment of heart period variability over temporal scales ranging from seconds to few minutes allows the indirect investigation of the short-term cardiovascular regulatory mechanisms.1,2 Short-term cardiovascular control system can be seen as the aggregate of several regulatory subsystems 共i.e., central and peripheral self-sustained oscillators and reflex loops兲3 that interact and even compete with each other to prevent that controlled variables 共e.g., arterial pressure兲 assume nonphysiological values via modification of target variables 共e.g., heart period兲. This complexity of the short-term cardiovascular control system results in complexity of the short-term heart period dynamics. The measurement of heart period complexity is important for two main reasons: 共i兲 in complex control systems complexity of the dynamics of a target variable decreases as a consequence of the reduced interaction and competition among subsystems4 and this is the typical situation occurring in cardiovascular control system due to aging and disease;5–10 共ii兲 complexity of the shortterm heart period dynamics depends on the state of the autonomic nervous system 共i.e., it decreases in presence of an increased sympathetic modulation兲,11–13 thus prompting for the noninvasive evaluation of the sympathetic modulation through the quantification of the short-term heart period complexity. The aim of this study is to present an integrated approach for the evaluation and characterization of the complexity of short heart period variability 共⬃300 cardiac beats兲. Short term complexity is evaluated through the calculation of the amount of information carried by the current heart period given that the L − 1 previous values are known in terms of entropy rate11 and local nonlinear predictability.12 Short-term complexity is typified using symbolic analysis14 by classifying patterns that, with their dominant presence, are responsible for the reduction of complexity.15 The study shows that information-domain analysis based on entropy rate, local nonlinear prediction, and pattern classification based on symbolic dynamics 共here jointly applied兲 can be integrated under the unifying framework provided by uniform quantization 共UQ兲 and supports the necessity to introduce both normalized and non-normalized indexes of complexity. This approach was applied to 24 h Holter recordings of heart period variability derived from healthy subjects and heart failure patients to verify its applicability even over data obtained in non-standardized conditions. The proposed complexity indexes were compared with more conventional measures of complexity of short-term heart period variability such as the approximate entropy 共ApEn兲16 and local nonlinear predictability based on k nearest neighbors 共KNN兲.17,18 II. EVALUATION OF SHORT-TERM COMPLEXITY: AN INTEGRATED APPROACH BASED ON UQ A. Definition of pattern

Given the stationary series x = 兵x共i兲 , i = 1 , . . . , N其 let us define as a pattern the ordered sequence of L samples xL共i兲 = x共i兲 , x共i − 1兲 , x共i − 2兲 , . . . , x共i − L + 1兲. This pattern is actually a point in the L-dimensional phase space reconstructed with the technique of the lagged coordinates 共delay ␶ = 1兲

proposed by Takens.19 This sequence xL共i兲 is constituted by the current sample x共i兲 and by the pattern xL−1共i − 1兲 containing L − 1 past samples of x 关i.e., xL−1共i − 1兲 = x共i − 1兲 , . . . , x共i − L + 1兲兴. Therefore, when helpful, in the following, xL共i兲 will be indicated as xL共i兲 = x共i兲 , xL−1共i − 1兲, thus stressing that the current value x共i兲 may be conditioned by past values xL−1共i − 1兲. In the following we need to set a coarse-graining procedure necessary to approximate distributions and conditional distributions and actually calculate entropy, entropy rate, and local nonlinear predictability. Coarse graining approach sets even the symbolism exploited in symbolic dynamics analysis.20 B. Coarse graining procedure based on UQ

A coarse graining based on UQ implies that the dynamics of x is uniformly spread over ␰ quantization levels of amplitude ␧=

xmax − xmin , ␰

共1兲

where xmax and xmin represent the maximum and the minimum values of x, respectively. The UQ procedure produces a quantized series x␰ = 兵x␰共i兲 , i = 1 , . . . , N其, whose values are integer ranging from 0 to ␰ − 1 共x␰ is actually a sequence of symbols兲 and quantized patterns xL␰ = 兵xL␰ 共j兲 = x␰共j兲 , x␰共j − 1兲 , x␰共j − 2兲 , . . . , x␰共j − L + 1兲 , j = L , . . . , N其. UQ actually imposes a uniform partition of the L-dimensional space into ␰L disjoint hypercubes of size ␧. While increasing ␰, the partition becomes finer and the coarse graining is less crude. After UQ all the patterns xL共j兲 belonging to the same hypercube of xL共i兲 with i ⫽ j are actually undistinguishable from it 关i.e., xL␰ 共j兲 = xL␰ 共i兲兴. As an effect of UQ the quantized pattern ␰ 共i − 1兲 can be coded 共base 10兲 as15 xL−1 L−2

␰ 关xL−1 共i

␰ − 1兲兴10 = 兺 x␰共i − 1 − k兲 · ␰L−2−k = hL−1 共i − 1兲,

共2兲

k=0

␰ thus univocally mapping the pattern xL−1 共i − 1兲 into the inte␰ ger value hL−1共i − 1兲 ranging from 0 to ␰L−1 − 1 and the ␰ 共i − 1兲 into the bidiL-dimensional pattern xL␰ 共i兲 = x␰共i兲 , xL−1 ␰ ␰ mensional one: x 共i兲 , hL−1共i − 1兲. This transformation is helpful to provide a graphical representation of complexity of a series and speed calculation of entropy and predictability 共see Secs. V A, V B, and VI A兲. When dealing with local nonlinear prediction, it may be worth considering the actual value of the first component, x共i兲, instead of considering the quantized one, x␰共i兲, thus defining the L-dimensional pattern ␰ 共i − 1兲 that can be mapped in the bidimensional as x共i兲 , xL−1 ␰ 共i − 1兲. point x共i兲 , hL−1

C. Shannon and conditional entropies based on UQ

The Shannon entropy 共SE兲 associated with the probability distribution of xL␰ , P共xL␰ 兲, measures the amount of information carried by the L-dimensional quantized patterns xL␰ as SE共xL␰ 兲 = − 兺 p关xL␰ 共i兲兴log兵p关xL␰ 共i兲兴其,

共3兲

where p关xL␰ 共i兲兴 = n关xL␰ 共i兲兴 / 共N − L + 1兲 with n关xL␰ 共i兲兴 is the number of times that xL␰ 共i兲 is detected in xL␰ and the sum is ex-


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Complexity of heart period variability

tended to all different patterns found in xL␰ . In the following, SE共xL␰ 兲 will be indicated as SE共L , ␰兲. SE is actually expressed in bits if the base of logarithm is 2 共in our application we utilized natural logarithm indicated with log, thus the units are nats兲. Lower and upper limits of 共3兲 are 0 and L log ␰, corresponding to one-peak and fully flat P共xL␰ 兲, respectively. Analogously, SE associated with the probability distribution of x␰ given that the previous L − 1 quantized samples are ␰ ␰ 共i − 1兲, P关x␰ / xL−1 共i − 1兲兴, measures the information carxL−1 ried by the values of x␰ when the previous L − 1 samples are ␰ 共i − 1兲 as xL−1 ␰ ␰ 共i − 1兲兴共i − 1兲兴 = − 兺 p关x␰共i兲/xL−1 SE关x␰/xL−1 ␰ 共i − 1兲兴其, ⫻log兵p关x␰共i兲/xL−1

共4兲

21

where

␰ ␰ ␰ p关x␰共i兲/xL−1 共i − 1兲兴 = p关x␰共i兲,xL−1 共i − 1兲兴/p关xL−1 共i − 1兲兴

共5兲

and the sum is extended to all different values x␰共i兲 given ␰ 共i − 1兲. that the previous L − 1 quantized samples are xL−1 Lower and upper limits of 共4兲 are 0 and log ␰, corresponding ␰ 共i − 1兲兴, respectively. The to one-peak and fully flat P关x␰ / xL−1 conditional entropy 共CE兲, measuring the amount of information carried by the most recent sample of patterns xL␰ when the previous L − 1 samples are known is defined as ␰ ␰ 共i − 1兲兴SE关x␰/xL−1 共i − 1兲兴, CE共xL␰ 兲 = 兺 p关xL−1

共6兲

i.e., the sum of the uncertainty carried by x␰ when the previ␰ ␰ ous L − 1 values are xL−1 共i − 1兲兵quantified by SE关x␰ / xL−1 共i ␰ − 1兲兴其, weighed by the probability of the pattern xL−1共i − 1兲, ␰ ␰ 共i − 1兲兴, and extended to all different patterns in xL−1 . p关xL−1 Equations 共4兲 and 共6兲 have the same lower and upper limits. In the following, CE共xL␰ 兲 will be indicated as CE共L , ␰兲. By exploiting 共5兲, it can be easily proved that CE共L, ␰兲 = SE共L, ␰兲 − SE共L − 1, ␰兲.

共7兲

After defining CE共1 , ␰兲 as the Shannon entropy of x␰, CE共1 , ␰兲, the normalized CE 共NCE兲11 NCE共L, ␰兲 =

CE共L, ␰兲 CE共1, ␰兲

共8兲

is dimensionless and independent of the shape of the probability distribution of x␰, P共x␰兲. NCE ranges from 0 to 1, indicating the maximum and the minimum amount of uncertainty associated to one sample of x when the previous L − 1 ones are known. CE decreases as a function of L toward 0, when the knowledge of past values is actually helpful to reduce the uncertainty associated to future samples. D. Bias in the evaluation of CE and definition of the corrected CE ␰ consider a pattern xL␰ 共i兲 = x␰共i兲 , xL−1 共i − 1兲 such that ␰ occurs only once in xL−1. For this pattern

Let us ␰ 共i − 1兲 xL−1 ␰ 共i − 1兲兴 = 1 关it is the result of a unique appearance p关x␰共i兲 / xL−1 ␰ ␰ 共i − 1兲 in xL−1 兴. In this case, as of the conditioning pattern xL−1 ␰ log兵p关x␰共i兲 / xL−1共i − 1兲兴其 = 0, the contribution of the pattern ␰ ␰ 共i兲 occurring only once in xL−1 to the CE is null. It is xL−1 ␰ 共i − 1兲 is only worth noting that the unique appearance of xL−1

an effect of the shortness of the data sequence; indeed, if the sequence could be enlarged, several conditions patterns iden␰ ␰ tical to xL−1 共i − 1兲 might be found and a p关x␰共i兲 / xL−1 共i − 1兲兴 ␰ ⫽ 1 might be estimated. If xL−1共i − 1兲 is found only once in ␰ ␰ , xL␰ 共i兲 = x␰共i兲 , xL−1 共i − 1兲 will be found once in xL␰ indepenxL−1 ␰ dently of x 共i兲. Therefore, the number of patterns found only once cannot diminish with L. In addition, due to the spreading of the points xL␰ with L, new patterns found only once are created. Thus, assigned the series length N, the number of patterns found only once increases monotonically as a function of L towards N − L + 1 共only if the series is perfectly predictable, it remains to 0兲. The main consequence of this statement is that CE decreases to 0 with L regardless of the type of dynamics 共e.g., even in case of a white noise兲, thus giving the false impression that uncertainty may be reduced while increasing the length of the conditioning pattern.22 In order to counteract this bias Porta et al.11,22 defined the corrected conditional entropy 共CCE兲 as CCE共L, ␰兲 = CE共L, ␰兲 + CE共1, ␰兲 · perc共L, ␰兲

共9兲

and the normalized CCE 共NCCE兲 as NCCE共L, ␰兲 = NCE共L, ␰兲 + perc共L, ␰兲,

共10兲

where perc共L , ␰兲 is fraction of L-dimensional quantized patterns found only once in xL␰ 关0 艋 perc共L , ␰兲艋 1兴. It was shown that CCE 共or NCCE兲: 共i兲 remains constant in case of white noise; 共ii兲 decreases to zero in case of fully predictable signals; 共iii兲 exhibits a minimum if the repetitive pattern is embedded in noise. The minimum is taken as a complexity index and termed as CI, in the case of the CCE minimum, and NCI, in the case of the NCCE minimum 共the larger the index, the more complex and less regular the series兲. E. Local nonlinear prediction based on UQ

Local nonlinear approach23,24 defines as the best predictor of x共i兲 based on the L − 1 previous samples xL−1共i − 1兲 a suitable statistics over all those x共j兲 values such that their previous L − 1 values, xL−1共j − 1兲, are similar to xL−1共i − 1兲. Using the median as statistics, the predictor xˆ 关i/xL−1共i − 1兲兴 = median关x共j兲/xL−1共j − 1兲 is similar to xL−1共i − 1兲兴.

共11兲

12,13

After UQ, 共11兲 becomes

␰ ␰ ␰ xˆ 关i/xL−1 共i − 1兲兴 = median关x共j兲/xL−1 共j − 1兲 = xL−1 共i − 1兲兴, 共12兲

i.e., the median over all those samples x共j兲 following a quan␰ ␰ 共j − 1兲 equal to xL−1 共i − 1兲. After defining the tized pattern xL−1 prediction error 共PE兲 based on L − 1 past samples as the difference between the original and the predicted values ␰ ␰ 共i − 1兲兴 = x共i兲 − xˆ 关i/xL−1 共i − 1兲兴, e关i/xL−1

共13兲

the mean squared PE 共MSPE兲 N

1 ␰ 共i − 1兲兴 MSPE共L, ␰兲 = 兺 e2关i/xL−1 N − L + 1 i=L

共14兲

is utilized to evaluate the goodness of prediction. MSPE共1 , ␰兲 is set to the mean squared deviation 共MSD兲 from


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the median of x 共i.e., the best predictor when no past samples are utilized兲. MSPE= 0 means perfect prediction, while a large MSPE means that prediction is poor. We normalize MSPE as12 NMSPE共L, ␰兲 =

MSPE共L, ␰兲 , MSPE共1, ␰兲

共15兲

thus obtaining a dimensionless function making MSPE independent of the variance of the signal. NMSPE ranges from 0 共perfect predictability兲 to 1 共perfect unpredictability兲. NMSPE decreases as a function of L toward 0, when past values are actually helpful to predict future values. F. Bias in the “in-sample” evaluation of MSPE and definition of the corrected MSPE

The “in-sample” procedure evaluates the predictor over the same data utilized to set it 关i.e., x共i兲 belongs to the same ␰ 共i − 1兲兴兲. The inset of data utilized to calculate xˆ关i / xL−1 sample procedure cannot prevent overfitting 共i.e., the perfect adequacy of prediction to data while increasing the number L of past samples utilized to predict兲 Indeed, let us consider ␰ ␰ 共i − 1兲 such that xL−1 共i − 1兲 occurs again a pattern x共i兲 , xL−1 ␰ ␰ only once in xL−1. For this pattern, xˆ关i / xL−1共i − 1兲兴 coincides with x共i兲. In this case, as e共i兲 = 0, the contribution of the ␰ 共i − 1兲 to the MSPE is null. Therefore, analopattern xL−1 gously to CE in Sec. II D MSPE decreases with L regardless of the type of dynamics, thus giving a false impression that predictability increases as a function of the length of the conditioning pattern.12 In order to prevent overfitting prediction is usually validated “out-of sample.”23,24 The out-of-sample procedure divides the entire series into two halves: the first half 共the “learning” set兲 is utilized to set the predictor and the second one 共the “test” set兲 to test it. However, when the series are very short 共they cannot be divided in two parts long enough to guarantee to be reliable learning and test sets兲 and cannot be enlarged without becoming nonstationary 共the two halves would have different statistical properties兲, the in-sample evaluation becomes mandatory. Therefore, instead of the in sample evaluation of MSPE, Porta et al.12 proposed the in sample evaluation of the corrected MSPE 共CMSPE兲 defined as CMSPE共L, ␰兲 = MSPE共L, ␰兲 + MSPE共1, ␰兲 · perc共L, ␰兲共16兲 or the normalized CMSPE 共NCMSPE兲 NCMSPE共L, ␰兲 = NMSPE共L, ␰兲 + perc共L, ␰兲,

共17兲

where perc共L , ␰兲 has the same definition as in 共9兲 and 共10兲. Since perc共L , ␰兲 grows monotonically with L toward 1, CMSPE and NCMSPE do not decrease toward 0, thus preventing overfitting. If the series is neither completely predictable nor fully unpredictable, CMSPE 共or NCMSPE兲 shows a minimum over L. The minimum is taken as an unpredictability index and termed UPI, in the case of the CMSPE minimum, and NUPI, in the case of the NCMSPE minimum 共the larger the index, the less predictable the series兲. In order to distinguish UPI and NUPI based on UQ from UPI and NUPI

derived from different coarse graining approaches, the subscript UQ will be adopted, thus becoming UPIUQ and NUPIUQ. G. Optimization of the length of the conditioning pattern and pattern classification

The most important advantage in the calculation of CI 共or NCI兲 and UPI 共or NUPI兲 is that the length of the conditioning pattern is not a priori assigned. In addition, the minimization procedure provides the length of the pattern that allows the reduction of the uncertainty about the future to the highest degree 共termed as LMIN in the following兲. As quan␰ tized patterns with length LMIN 共i.e., xLMIN 兲 are the most informative features about future values, we propose to decom␰ can pose the dynamics of x according to them. Patterns xLMIN ␰ be very easily classified by ranking hLMIN and by counting ␰ how many times the same integer value hLMIN 共i兲 appear in ␰ ␰ ␰ 共i兲 is hLMIN 兵i.e., n关hLMIN共i兲兴其. The rate of occurrence of xLMIN ␰ n关hLMIN共i兲兴 / 共N − LMIN + 1兲 multiplied by 100. Unfortunately, since the number of possible patterns is ␰LMIN, classification may become unmanageable even with small LMIN. Therefore, it may be useful to introduce procedures of redundancy reduction to group patterns into a smaller number of families, thus monitoring the rate of occurrence of few families instead of that of numerous patterns.15 An example of procedure of redundancy reduction has been proposed by Porta et al.15 and exploited in Guzzetti et al.:25 all the patterns with L = 3 are grouped without loss into four families according to the number and types of variations from one symbol to the next one. The pattern families are: 共i兲 patterns with no variation 共0V, all the symbols are equal兲; 共ii兲 patterns with one variation 共1V, two consecutive symbols are equal and the remaining one is different兲; 共iii兲 patterns with two like variations 共2LV, the three symbols form an ascending or descending ramp兲; 共iv兲 patterns with two unlike variations 共2UV, the three symbols form a peak or a valley兲. The rates of occurrence of these patterns will be indicated as 0V%, 1V%, 2LV%, and 2UV% in the following. III. QUANTIFICATION OF SHORT-TERM COMPLEXITY BASED ON TRADITIONAL APPROACHES A. ApEn

Similarly to CE, ApEn16 belongs to the family of the entropy rates. Therefore, ApEn is calculated using an equation formally similar to 共7兲 in which CE is substituted with ApEn and SE is approximated by 1 ⌽共L,r兲 = − N−L+1

N−L+1

兺 i=1

log Ci共L,r兲,

共18兲

where Ci共L , r兲 represents the number of points that can be found at a distance smaller than r from xL共i兲 divided by N − L + 1 共i.e., the estimate of the probability of finding a point xL共j兲 at distance less than r from xL共i兲兲. Therefore, ApEn is actually calculated using a coarse graining approach based on overlapping cells with constant size.13 Here we used the Euclidean norm to evaluate distance. In 共18兲, r has the same meaning of ␧ and fixes the level of coarse graining of the


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dynamics. ApEn was calculated with L − 1 = 2 and r = 20% of the standard deviation as in Pincus et al.16 Normalized ApEn 共NApEn兲 was calculated as well by dividing ApEn by ⌽共1 , r兲. B. Local nonlinear prediction based on KNN

The predictor based on KNN17,24 is again set by 共11兲, but the definition of similarity is different. Indeed, the k points nearest to xL−1共i − 1兲 are considered to be similar to xL−1共i − 1兲 independently of the actual distance from it. This means that, instead of coarse graining the L − 1 dimensional phase space with disjoint cells of equal size, as occurs in case of UQ, this approach uses overlapping cells of variable size.13 A weighed mean is utilized instead of median in 共11兲 simply because the KNN of xL−1共i − 1兲 cannot be considered equivalent in determining xˆ关i / xL−1共i − 1兲兴关some KNNs may be even very far from xL−1共i − 1兲兴. Usually17 the weights are the inverse of the distance between the KNN and xL−1共i − 1兲, thus increasing the importance of the points actually close to xL−1共i − 1兲. To prevent overfitting, MSPE was calculated “out-of-sample.”17 NMSPE was calculated by dividing MSPE by the variance estimated on the second half of the series 共i.e., the test set兲. MSPE and NMSPE were calculated with L − 1 = 3 and k = 6 as in Kanters et al.:18 these values will be indicated as UPIKNN and NUPIKNN. IV. EXPERIMENTAL PROTOCOL AND DATA ANALYSIS A. Experimental protocol

We studied 12 normal 共NO兲 humans 共age: median= 43, range= 34– 55兲 and 13 age-matched heart failure 共HF兲 patients 共2 out of 13 are in NYHA class I, 2 out of 13 in NYHA class III, and the remaining in class II, age: median= 37, range= 33– 56; ejection fraction: median= 25%, range = 13% – 30%兲. They underwent 24 h Holter recordings 共Oxford Medilog System兲 during usual everyday activities. Sampling frequency was 250 Hz. The position of the QRS complex was automatically located and labeled as normal or aberrant by the Holter analysis software and was then carefully edited by an expert analyst. Heart period was approximated as the temporal distance between two consecutive R peaks. Only recordings with at least 50% of the analyzed periods in sinus rhythm during both nighttime and daytime were considered. Twenty-four hour beat-to-beat RR series were preprocessed according to the following criteria: 共1兲 RR intervals associated with single ectopic beats were replaced by their mean value; 共2兲 artifacts and runs of tachycardic beats were replaced by M values equal to the mean RR, such a way that M times mean RR was less than or equal to the substituted value; 共3兲 RR values differing from the preceding one more than 30% 共absolute value兲 were replaced as for artifacts. The mean RR was computed as a moving average centered on the beat to correct, with a buffer of ±3 beats labeled as normal. B. Data analysis over 24 h Holter recordings

The 24 h heart rate variability series were analyzed during daytime 共from 9:00 am to 7:00 pm兲 and during nighttime

共from 0:00 to 5:00 am兲. Methods for the quantification and characterization of complexity were applied to sequences of 300 cardiac beats with a 50% overlap. Only in the case of local nonlinear prediction based on KNN and as a consequence of the out-of-sample procedure, the sequence length was 600 cardiac beats 共the first 300 heart periods formed the learning set and the second 300 samples the test set兲 and the overlap was 75%. Series were linearly detrended. Methods based on UQ were carried out with ␰ = 6. Both during the daytime and nighttime, we constructed the distribution of parameters based on entropy rates 共i.e., CI, NCI, ApEn, NApEN兲, local nonlinear prediction 共UPIUQ, NUPIUQ, UPIKNN, and NUPIKNN兲 and symbolic analysis 共0V%, 1V%, 2LV%, 2UV%兲 and the distribution of LMIN. The median of the distribution was extracted for successive statistical analysis. Results were reported as mean± standard deviation 共SD兲 of the median values. C. Statistical analysis

A paired t-test 共or Wilcoxon signed rank test when appropriate兲 was utilized to test the difference between parameters derived during daytime and nighttime inside the same population 共NO or HF兲. An unpaired t-test 共or MannWhitney rank sum test when appropriate兲 was carried out to check the difference between parameters derived from NO and HF populations inside the same period of analysis 共daytime or nighttime兲. A p ⬍ 0.05 was considered statistically significant. V. RESULTS A. Evaluating CE based on UQ

Figure 1 shows an example of short beat-to-beat heart period series 共⬃300 cardiac beats兲 recorded during the daytime 关Fig. 1共a兲兴 and nighttime 关Fig. 1共b兲兴 in a healthy subject. By visual inspection, the series appears less complex 共i.e., more regular兲 during the daytime than during the nighttime.

FIG. 1. Examples of short beat-to-beat RR interval series 共⬃300 cardiac beats兲 extracted during the daytime 共a兲 and nighttime 共b兲. The raw series are uniformly quantized over ␰ = 6 共c兲 and 共d兲, thus becoming series of integers that preserve the main dynamical features of the original series.


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FIG. 2. Graphical visualization of complexity of the RR series during the ␰ ␰ daytime depicted in Fig. 1共a兲. 共a兲 Probability distribution of hL−1 , P共hL−1 兲 representing the probability distribution of the 共L − 1兲-dimensional quantized ␰ ␰ patterns xL−1 , P共xL−1 兲. 共b兲 Scatter plot of the one-step-ahead value x共i兲 ver␰ ␰ sus their previous L − 1 quantized samples xL−1 共i − 1兲 coded by hL−1 共i − 1兲. ␰ ␰ Two examples of conditional distributions P关x / hL−1 共i − 1兲兴 with hL−1 共i − 1兲 ␰ = 43 共c兲, and hL−1 共i − 1兲 = 173 共d兲. All the graphs are derived with L − 1 = 3 and ␰ = 6.

After UQ with ␰ = 6, the quantized series 关Figs. 1共c兲 and 1共d兲兴 preserve the main dynamical features of the original series. Figure 2 aims at giving an immediate graphical represen␰ and of tation of the dispersion of the quantized patterns xL−1 the one-step-ahead samples x given that the previous L − 1 ␰ 共i − 1兲 relevant to the RR series quantized samples are xL−1 during daytime depicted in Fig. 1共a兲. Figure 2共a兲 shows the probability distribution of the 共L − 1兲-dimensional quantized ␰ ␰ 兲关or P共hL−1 兲 according to the transformation patterns, P共xL−1 ␰ of xL−1 given by 共2兲兴. It is worth noting that, as a result of the partition imposed by UQ that allows the univocal mapping of ␰ ␰ into hL−1 , the distribution of 共L − 1兲-dimensional quanxL−1 ␰ 兲, can be easily visualized indepentized patterns, P共xL−1 dently of the pattern length L 共here, L − 1 = 3兲. Figure 2共b兲 ␰ 共i − 1兲 as open circles 关actually, represents the pairs x␰共i兲 , hL−1 the original value x共i兲 is plotted instead of its quantized version x␰共i兲兴. This type of representation has the advantage to give the immediate representation of the dispersion of x ␰ 共i − 1兲 given that the quantized L − 1 previous samples are xL−1 ␰ coded by hL−1共i − 1兲. The tight strip of the open circles along of the main diagonal indicates that the conditional distribu␰ 共i − 1兲关two examples are reported in tions of x assigned xL−1 Figs. 2共c兲 and 2共d兲兴 are tight around median and suggests a large predictability of the series. CE共L , ␰兲 is a way to quantify the dispersion of the open circles along the main diago-

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FIG. 3. Graphical visualization of complexity of the RR series during the ␰ ␰ nighttime depicted in Fig. 1共b兲. 共a兲 Probability distribution of hL−1 , P共hL−1 兲 representing the probability distribution of the 共L − 1兲-dimensional quantized ␰ ␰ patterns xL−1 , P共xL−1 兲. 共b兲 Scatter plot of the one-step-ahead value x共i兲 vs ␰ ␰ their previous L − 1 quantized samples xL−1 共i − 1兲 coded by hL−1 共i − 1兲. Three ␰ ␰ examples of conditional distributions P关x / hL−1 共i − 1兲兴 with hL−1 共i − 1兲 = 23 ␰ ␰ 共c兲, hL−1 共i − 1兲 = 53 共d兲, and hL−1 共i − 1兲 = 172 共e兲. All the graphs are derived with L − 1 = 3 and ␰ = 6.

nal 共the width of the strip兲. Indeed, according to 共6兲, CE共L , ␰兲 can be simply calculated by summing the SE of all the con␰ 共i − 1兲兴 reported in ditional distributions weighed by p关hL−1 Fig. 2共a兲. Figure 3 has the same structure of Fig. 2. Figure 3 aims at describing the dispersion of the quantized patterns ␰ 共i − 1兲 and of one-step-ahead samples x given that the xL−1 ␰ 共i − 1兲 relevant to previous L − 1 quantized samples are xL−1 the RR series during nighttime shown in Fig. 1共b兲. As in Fig. 2 L − 1 is 3. Figure 3 shows the probability distribution of the ␰ ␰ , P共xL−1 兲关or 共L − 1兲-dimensional quantized patterns xL−1 ␰ equivalently P共hL−1兲; Fig. 3共a兲兴, the scatter plot of value x共i兲 ␰ 共i − 1兲 versus their previous L − 1 quantized samples xL−1 ␰ coded by hL−1共i − 1兲关Fig. 3共b兲兴 and three examples of condi␰ ␰ 共i − 1兲兴 with hL−1 共i − 1兲 = 23, tional distributions P关x / hL−1 ␰ ␰ hL−1共i − 1兲 = 53 and hL−1共i − 1兲 = 172 关Figs. 3共c兲–3共e兲兴. The comparison between Figs. 2 and 3 points out that the prob␰ is more complex during nighttime ability distribution of xL−1 than during daytime 共i.e., SE共L − 1 , ␰兲 is larger in Fig. 3共a兲 than in Fig. 2共a兲 and the open circles in Fig. 3共b兲 are more scattered along the main diagonal than in Fig. 2共b兲, thus indicating that L − 1 previous samples are less useful to reduce the uncertainty about future values during nighttime than during daytime 关i.e., CE共L , ␰兲 is larger in Fig. 3共b兲 than ␰ 共i − 1兲兴 in Fig. 2共b兲兴. The conditional distribution P关x / hL−1 shown in Fig. 3共c兲 indicates that, given the pattern


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FIG. 4. NCCE calculated over the RR series during the daytime and nighttime depicted in Figs. 1共a兲 and 1共b兲. The NCCE functions exhibit a clear, although different from 0, minimum 共i.e., NCI兲, thus indicating that the knowledge of past samples are useful to reduce 共although not completely兲 the uncertainty associated to future values. During the daytime, the minimum is deeper.

␰ hL−1 共i − 1兲 = 23, its successive value x共i兲 can be indicated without uncertainty. This certainty is not the result of several ␰ 共i − 1兲 = 23 whose next values, x共i兲, are similar, patterns hL−1 but it is the result of an unique appearance of the pattern ␰ ␰ 共i − 1兲 = 23 关in correspondence with hL−1 共i − 1兲 = 23, there hL−1 is only one open circle in Fig. 3共b兲兴. Conditional distribu␰ 共i − 1兲 = 23 are responsible tions such as that associated to hL−1 for the CE decrease toward 0 with L 共see Sec. II D兲. Figure 4 shows NCCE calculated over the series depicted in Figs. 1共a兲 and 1共b兲. The NCCE minimum is deeper during the daytime 关Fig. 4共a兲兴 than during the nighttime 关Fig. 4共b兲兴, thus indicating a smaller complexity during the daytime 关NCI is 0.48 in Fig. 4共a兲 and 0.81 in Fig. 4共b兲兴. LMIN is 3 in both series. Figure 5 shows the mean 共+SD兲 of NCI 关Fig. 5共a兲兴 and CI 关Fig. 5共b兲兴 derived from 24 h Holter recordings during the daytime 共solid bar兲 and nighttime 共open bar兲 in NO subjects and HF patients. In NO subjects NCI 关Fig. 5共a兲兴 was significantly smaller during the daytime 共0.56± 0.06兲 than during nighttime 共0.63± 0.04兲, whereas the day-night variation of NCI was not observed in HF patients 共0.61± 0.07 during the daytime and 0.59± 0.07 during the nighttime兲. No significant NCI difference was observed between NO subjects and HF patients, even though there was a tendency in HF patients towards larger NCI values during the daytime and smaller NCI values during the nighttime 关Fig. 5共a兲兴. CI 关Fig. 5共b兲兴 did not exhibit any day-night variation both in NO subjects 共0.82± 0.08 nats during the daytime and 0.86± 0.06 nats during nighttime兲 and in HF patients 共0.89± 0.13 nats during

FIG. 5. Complexity analysis of short-term heart period variability series 共300 cardiac beats兲 derived from 24 h Holter recordings in NO subjects and HF patients during daytime 共solid bar兲 and nighttime 共open bar兲: the bars represent the mean 共+SD兲 of NCI 共a兲 and of CI 共b兲. The symbol *** indicates p ⬍ 0.001.

␰ FIG. 6. Scatter plot of the pairs x共i兲 , hL−1 共i − 1兲关as shown in Fig. 2共b兲, open ␰ ␰ circles兴 with superposed the scatter plot of the pairs xˆ关i / hL−1 共i − 1兲兴, hL−1 共i ␰ ˆ − 1兲共solid circles兲, where x关i / hL−1共i − 1兲兴 is the local nonlinear prediction of ␰ x共i兲 given that the previous L − 1 samples are xL−1 共i − 1兲 coded by the integer ␰ hL−1 共i − 1兲. The plot is constructed over the RR series during the daytime, depicted in Fig. 1共a兲. The graph is derived with L − 1 = 3 and ␰ = 6.

daytime and 0.82± 0.14 nats during nighttime兲 and any remarkable difference between NO subjects and HF patients. Both in NO subjects and HF patients, the most likely values of LMIN were 3 and 4 during daytime and nighttime, respectively. B. Evaluating MSPE based on UQ ␰ Figures 6 and 7 report the pairs x共i兲 , hL−1 共i − 1兲关open circles, as in Figs. 3共b兲 and 4共b兲, respectively兴 with super␰ ␰ 共i − 1兲兴 , hL−1 共i − 1兲共solid circles兲, posed the pairs xˆ关i / hL−1 ␰ where xˆ关i / hL−1共i − 1兲兴 is the best prediction of x共i兲 given that ␰ ␰ 共i − 1兲 coded by hL−1 共i − 1兲. the previous L − 1 samples are xL−1 While Fig. 6 is relevant to the RR series during daytime depicted in Fig. 1共a兲, Fig. 7 is relevant to that during nighttime shown in Fig. 1共b兲. Figures 6 and 7 clearly indicate that the dispersion of x共i兲 around the best prediction 兵i.e., ␰ 共i − 1兲兴其 is larger during nighttime 共Fig. 7兲 than durxˆ关i / hL−1 ing daytime 共Fig. 6兲. In Fig. 7, the unique open circle present ␰ 共i − 1兲 = 23 关see Fig. 3共b兲 at in correspondence with hL−1 ␰ hL−1共i − 1兲 = 23兴 is completely masked by the solid circle representing its best prediction. Since x共i兲 and its best prediction ␰ 共i − 1兲 = 23 coincide, e共i兲 = 0. Therefore, the unique apat hL−1

␰ FIG. 7. Scatter plot of the pairs x共i兲 , hL−1 共i − 1兲, as shown in Fig. 3共b兲, open ␰ ␰ circles, with superposed the scatter plot of the pairs xˆ关i / hL−1 共i − 1兲兴,hL−1 共i ␰ − 1兲共solid circles兲, where xˆ关i / hL−1 共i − 1兲兴 is the local nonlinear prediction of ␰ x共i兲 given that the previous L − 1 samples are xL−1 共i − 1兲 coded by the integer ␰ hL−1 共i − 1兲. The plot is constructed over the RR series during the nighttime depicted in Fig. 1共b兲. The graph is derived with L − 1 = 3 and ␰ = 6.


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FIG. 8. NCMSPE calculated over the RR series during daytime and nighttime depicted in Figs. 1共a兲 and 1共b兲. The NCMSPE functions exhibit a clear, although different from 0, minimum 共i.e., NUPI兲, thus indicating that the RR series is neither completely predictable nor fully unpredictable. During daytime the minimum is deeper.

␰ pearance of the pattern xL−1 共i − 1兲 is responsible for the decrease of the MSPE 共as well as of the CE兲 toward 0 with L 共see Sec. II F兲. Figure 8 shows NCMSPE calculated over the series depicted in Figs. 1共a兲 and 1共b兲. The NCMSPE minimum is deeper during the daytime 关Fig. 8共a兲兴 than during the nighttime 关Fig. 8共b兲兴, thus indicating a larger predictability of the RR series during the daytime 关NUPI is 0.17 in Fig. 8共a兲 and 0.67 in Fig. 8共b兲兴. LMIN is 2 during the daytime and 3 during the nighttime. Figure 9 shows the mean 共+SD兲 of NUPIUQ 关Fig. 9共a兲兴 and UPIUQ 关Fig. 9共b兲兴 derived from 24 h Holter recordings during the daytime 共solid bar兲 and nighttime 共open bar兲 in NO subjects and HF patients. In NO subjects NUPIUQ 关Fig. 9共a兲兴 was significantly smaller during the daytime than during the nighttime 共0.26± 0.07 versus 0.36± 0.09兲, whereas the day-night variation of NUPIUQ was not observed in HF patients 共0.34± 0.10 during daytime and 0.32± 0.09 during nighttime兲. During daytime, NUPIUQ was significantly larger in HF patients than in NO subjects 关Fig. 9共a兲兴. The day-night variation of UPIUQ 关Fig. 9共b兲兴 was significant both in NO subjects 共616± 518 ms2 during the daytime and 1347± 1258 ms2 during the nighttime兲 and in HF patients 共320± 288 ms2 during the daytime and 593± 476 ms2 during the nighttime兲. This result was correlated with the increase of the RR variance during the nighttime observable in both populations. During the nighttime, UPIUQ was significantly smaller in HF patients than in NO subjects 关Fig. 9共b兲兴. This result was correlated with the reduction of the RR variance in HF patients. In NO subjects the most likely value of LMIN was 2 both during daytime and nighttime, whereas it was 3 in HF patients.

FIG. 9. Complexity analysis of short-term heart period variability series 共300 cardiac beats兲 derived from 24 h Holter recordings in NO subjects and HF patients during the daytime 共solid bar兲 and nighttime 共open bar兲: the bars represent the mean 共+SD兲 of NUPIUQ 共a兲 and UPIUQ 共b兲. The symbols ***, **, and * indicate p ⬍ 0.001, p ⬍ 0.01, and p ⬍ 0.05, respectively.

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FIG. 10. Rates of occurrence of the families 0V, 1V, 2LV, and 2UV expressed as a percentage derived from the RR series during the daytime and nighttime depicted in Figs. 1共a兲 and 1共b兲.

C. Pattern classification based on UQ

Figure 10 shows the rate of occurrence of patterns with L = 3 derived from the RR series depicted in Figs. 1共a兲 and 1共b兲. During the nighttime 关Fig. 10共b兲兴, 0V% strongly decreases and 2LV% and 2UV% increase with respect to the daytime 关Fig. 10共a兲兴. The percentage of the 1V patterns is similar in Figs. 10共a兲 and 10共b兲. Figure 11 shows the mean 共+SD兲 of 0V% 关Fig. 11共a兲兴, 1V% 关Fig. 11共b兲兴, 2LV% 关Fig. 11共c兲兴 and 2UV% 关Fig. 11共d兲兴 derived from 24 h Holter recordings during the daytime 共solid bar兲 and nighttime 共open bar兲 in NO subjects and HF patients. In NO subjects 0V% 关Fig. 11共a兲兴 was significantly larger during the daytime than during the nighttime 共45± 7 versus 36± 8兲, whereas the reverse situation was observed in case of 2UV% 关Fig. 11共d兲 8 ± 4 versus 14± 7兴. None of the variables 0V%, 1V%, 2LV%, and 2UV% exhibited any significant day-night variation in HF patients. During the daytime, 2LV% 关Fig. 11共c兲兴 was significantly smaller in HF patients than in NO subjects 共2 ± 2 versus 4 ± 2兲 and, conversely, 2UV% 关Fig. 11共d兲兴 was significantly larger 共19± 10 versus 8 ± 4兲. During nighttime 1V% 关Fig. 11共b兲兴 and 2LV% 关Fig. 11共c兲兴 were significantly smaller in HF patients than in NO subjects 共38± 6 versus 43± 2 and 3 ± 2 versus 5 ± 1兲.

FIG. 11. Complexity analysis of short-term heart period variability series 共300 cardiac beats兲 derived from 24 h Holter recordings in NO subjects and HF patients during daytime 共solid bar兲 and nighttime 共open bar兲: the bars represent the mean 共+SD兲 of 0V% 共a兲, 1V% 共b兲, 2LV% 共c兲, and 2UV% 共d兲. The symbols ** and * indicate p ⬍ 0.01 and p ⬍ 0.05, respectively.


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FIG. 12. Complexity analysis of short-term heart period variability series 共300 cardiac beats兲 derived from 24 h Holter recordings in NO subjects and HF patients during daytime 共solid bar兲 and nighttime 共open bar兲: the bars represent the mean 共+SD兲 of NApEn 共a兲, ApEn 共b兲, NUPIKNN 共c兲, and UPIKNN 共d兲. The symbols ***, **, and * indicate p ⬍ 0.001, p ⬍ 0.01, and p ⬍ 0.05, respectively.

D. Complexity analysis carried out with traditional approaches

Figure 12 shows the mean 共+SD兲 of NApEn 关Fig. 12共a兲兴, ApEn 关Fig. 12共b兲兴, NUPIKNN 关Fig. 12共c兲兴 and UPIKNN 关Fig. 12共d兲兴 derived from 24 h Holter recordings during daytime 共solid bar兲 and nighttime 共open bar兲 in NO subjects and HF patients. NApEn 关Fig. 12共a兲兴 exhibited a circadian rhythm in NO subjects 共0.49± 0.02 during the daytime and 0.51± 0.01 during the nighttime兲. The same circadian rhythm in NApEn 关Fig. 12共a兲兴 was observed in HF patients 共0.49± 0.03 during daytime and 0.50± 0.02 during nighttime兲. On the contrary, ApEn did not exhibit any day-night variation in both populations 关Fig. 12共b兲兴. No significant NApEn and ApEn difference was observed between NO subjects and HF patients both during daytime and nighttime. The bar-graphs of NUPIKNN and UPIKNN 关Fig. 12共c兲 and 12共d兲兴 followed exactly those of NUPIUQ and UPIUQ 关Figs. 9共a兲 and 9共b兲兴. NUPIKNN was 0.25± 0.09 during the daytime and 0.43± 0.10 during the nighttime in NO subjects and 0.36± 0.12 during the daytime and 0.36± 0.09 during the nighttime in HF patients, while UPIKNN was 561± 483 ms2 during the daytime and 1419± 1280 ms2 during the nighttime in NO subjects and 328± 291 ms2 during the daytime and 624± 479 ms2 during the nighttime in HF patients. VI. DISCUSSION A. An integrated approach based on UQ for the evaluation of the short-term complexity

We propose an integrated approach for the evaluation of complexity of the short-term heart period variability based on UQ. The proposed approach utilizes a coarse graining completely different from more traditional approaches measuring the short-term heart period complexity in terms of ApEn16 and of local nonlinear prediction based on KNN.17,18,26 Indeed, UQ imposes a partition of the L-dimensional phase space into a finite number of disjoint cells, while ApEn and local nonlinear prediction based on

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KNN use intersecting cells of constant and variable size, respectively.13 The imposition of a partition allows the univocal labeling of each cell 共and of all the patterns inside the cell兲, thus permitting the easy visualization of the current sample conditioned by L − 1 previous values in the plane ␰ 共x共i兲 , hL−1 共i − 1兲兲关see Figs. 2共b兲, 3共b兲, 6, and 7兴 and giving a graphical perception of the complexity of the series. In addition, it renders very fast the calculation of CE using 共7兲 ␰ through the application of sorting algorithms applied to hL−1 ␰ ␰ and hL and the calculation of xˆ关i / xL−1共i − 1兲兴 using 共12兲 ␰ through the application of sorting algorithms applied to hL−1 ␰ ␰ to group together all x共j兲 with hL−1共j − 1兲 = hL−1共i − 1兲, thus allowing the reduction of the computation time from O共N2兲 to O共N log N兲 in both cases. The proposed approach quantifies complexity as the information carried by the most recent sample when the previous L − 1 samples are known in terms of entropy rate 共quantified by CE兲 and local nonlinear prediction 共quantified by in-sample MSPE兲. An ad-hoc correction term12,22 prevents the decrease of CE and in-sample MSPE to 0, thus avoiding the a priori selection of the pattern length L and identifying the optimal pattern length LMIN as that minimizing the uncertainty about future samples 共assigned N = 300兲. As the most likely LMIN ranges from 2 to 4 the choice of classifying patterns with L = 3 seems to be reasonable.15 This pattern length was justified in a more empirical way in Porta et al.,15 i.e., given a series of N = 300 samples and assigned ␰ = 6, the largest L that produces a number of theoretical pattern ␰L smaller than N − L + 1 共i.e., the number of patterns actually extracted from N samples兲 is L = 3 共␰L = 216兲, thus being the largest L that maintains statistical robustness of the classification. Although the number of possible patterns is not very high 共at least with L = 3兲, it is helpful to apply a redundancy reduction criterion to group all the patterns into a smaller number of families. A viable and useful criterion for redundancy reduction without loss is based on the frequency content of the pattern.15 This criterion classifies four pattern families 共i.e., 0V, 1V, 2LV, and 2UV兲 from the most stable 共i.e., 0V兲 to the most variable 共2UV兲 and the rate of occurrence of these patterns has been correlated with the modulation of the autonomic nervous system activity.25 The present study points out that normalized and nonnormalized indexes of complexity provide complementary information and need to be jointly explored. Indeed, the normalization of CI and ApEn parameters by the Shannon entropy of the series to derive NCI and NApEn reduces the dependence of the entropy-based complexity indexes on the shape of the static distribution, thus quantifying only the “dynamical” complexity. The normalization of the UPIUQ and UPIKNN parameters by the variance of the series reduces the dependence of the unpredictability-based complexity indexes on the absolute levels of variability 共e.g., in the case of heart period series on the amplitude of the respiratory sinus arrhythmia兲. The proposed indexes of complexity provide complementary information with respect to more traditional indexes like variance or those based on second-order statistics 共e.g., autocorrelation function兲. Indeed, while variance dramatically decreases in HF patients,30 complexity when measured, e.g., in terms of normalized entropy rate does not


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significantly vary. In addition, as nonlinear dynamics are likely 共mostly during nighttime when respiration is deeper and slower12,13兲 indexes based on second-order statistics cannot fully account for the complexity generated by nonlinear mechanisms. B. Complexity of short-term heart period variability derived from 24 h Holter recordings in NO and HF populations

Previous studies suggests that in NO subjects the normalized complexity of short-term heart period variability depends on the state of the autonomic nervous system: it is usually reduced during experimental conditions inducing a increase of the sympathetic modulation11–13 such as 80° head-up tilt, infusion of nitroprusside or handgrip. The present study confirms this observation even over 24 h Holter recordings: indeed in NO subjects the normalized complexity is larger during nighttime than during daytime 共nighttime is a period characterized by a sympathetic withdrawal with respect to daytime兲. This result is consistently found by all the normalized indexes of complexity 共NCI, NUPIUQ, NApEn, and NUPIKNN兲 regardless of whether they are based on entropy rate 共NCI and NApEn兲 or local nonlinear prediction 共NUPIUQ and NUPIKNN兲 and regardless of the type of coarse graining. It is worth noting that nonnormalized indexes measuring the absolute level of the entropy rate such as CI and ApEn cannot detect the day-night variation of complexity in NO subjects. The difference between normalized and non-normalized complexity indexes based on entropy rates may explain some puzzling results reported in the literature; e.g., the lack of decrease of complexity 共when measured as ApEn兲 during 60° head-up tilt.27 The increase of complexity during nighttime with respect to daytime is confirmed by non-normalized indexes of complexity based on local nonlinear prediction 共i.e., UPIUQ and UPIKNN兲 as well. However, this result is likely to be influenced by the strong correlation between the variance and both UPIUQ and UPIKNN. 共heart period variance is larger during nighttime兲. The correlation of UPIUQ and UPIKNN with variance explains even the decrease of UPIUQ and UPIKNN observed in HF patients with respect to NO subjects. All the normalized complexity indexes indicate that the day-night variation is lost in HF patients 共only NApEn shows a significant circadian variation兲. In HF patients the loss of the daynight variation is a direct consequence of the increase of the normalized complexity indexes during the daytime. This result corroborates the association between normalized complexity indexes and sympathetic modulation: indeed, in HF patients sympathetic modulation is likely to be reduced in presence of a high dominant sympathetic tone.28,29 C. Pattern classification in short-term heart period variability derived from 24 h Holter recordings in NO and HF populations

The present study confirms the ability of an approach based on symbolic dynamics to gauge cardiac neural modulation25 and the importance of classifying patterns with L = 3 when short-term heart period variability is analyzed.15

Indeed, the association between the presence of stable patterns 共i.e., 0V兲 and sympathetic modulation and between the presence of instable patterns 共i.e., 2UV兲 and vagal modulation is suggested by the clear day-night variation of 0V% and 2UV% with 0V% decreasing and 2UV% increasing during the nighttime. During the daytime in HF patients the importance of instable patterns 共i.e., 2UV%兲 increases with respect to NO subjects. As this result cannot be interpreted as an increase of the parasympathetic modulation 共in HF patients parasympathetic modulation should be smaller than in NO subjects兲, it suggests that in this pathological population the pattern 2UV might uncover cardiac electrical instabilities that may induce runs of short-long-short 共or long-short-long兲 heart periods characterized by negligible power 共the overall RR variance is decreased in HF patients兲. These data suggest the important presence of episodes of “microscopic heart period alternans” in HF patients that may be worth investigating with specific experimental protocols both in human and animal models. It is worth observing that the 2UV pattern is the only variable pattern the importance of which increases in HF patients: indeed, both 1V% and 2LV% decrease in HF patients with respect to NO subjects according to the decrease of variance. VII. CONCLUSIONS

The application of the proposed approach to 24 h Holter recordings of heart period variability confirms the utility of the evaluation and the characterization of short-term complexity even under uncontrolled experimental conditions and during daily activities. Indeed, indexes both quantifying complexity 共mostly normalized complexity兲 and typifying complexity through symbolic analysis identify two epochs characterized by different cardiac autonomic regulations in NO humans 共i.e., daytime and nighttime兲 and differentiate NO subjects from HF patients. The results suggest the potential efficacy of these indexes in 24 h Holter analysis of heart rate variability and prompt for further applications to larger databases including patients with different cardiac pathologies.31 1

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