Entropy measures, entropy estimators, and their performance in quantifying complex dynamics: Effects of artifacts, nonstationarity, and long-range correlations

Entropy measures are widely applied to quantify the complexity of dynamical systems in diverse fields. However, the practical application of entropy methods is challenging, due to the variety of entropy measures and estimators and the complexity of real-world time series, including nonstationarities and long-range correlations (LRC). We conduct a systematic study on the performance, bias, and limitations of three basic measures (entropy, conditional entropy, information storage) and three traditionally used estimators (linear, kernel, nearest neighbor). We investigate the dependence of entropy measures on estimator- and process-specific parameters, and we show the effects of three types of nonstationarities due to artifacts (trends, spikes, local variance change) in simulations of stochastic autoregressive processes. We also analyze the impact of LRC on the theoretical and estimated values of entropy measures. Finally, we apply entropy methods on heart rate variability data from subjects in different physiological states and clinical conditions. We find that entropy measures can only differentiate changes of specific types in cardiac dynamics and that appropriate preprocessing is vital for correct estimation and interpretation. Demonstrating the limitations of entropy methods and shedding light on how to mitigate bias and provide correct interpretations of results, this work can serve as a comprehensive reference for the application of entropy methods and the evaluation of existing studies.

Figure 1

Computation of entropy measures for a stationary binary order-1 Markov process. Since this process takes discrete values, entropies are computed using the base 2 logarithm and measured in bits. (a) Dependence of the transition probabilities and the entropy measures on the internal coupling parameter $\delta$. When $\delta$ rises from 0 to 1, the probability that a state transition keeps the process in the same state ($p(X_n=x|X_{n-1}=x)$) moves from $1/2$ to 0; accordingly, conditional entropy decreases from 1 to 0 (green dotted line with diamond) and information storage increases from 0 to 1 (purple dashed line with down-triangle); note that, since for this process the marginal probabilities are unaffected by $\delta$, the entropy of the process does not change (red solid line with up-triangle). The values of entropy measures for $\delta =0,0.5,1$ are marked with full symbols. (b), (c), (d) Entropy analysis for representative parameter values, showing a realization of the process (solid line with full circles), the total and present system information as a function of time, and the Venn diagram of the entropy measures (present information, red oval; past information, blue oval). For this stationary process, the present information is the same at all times and measures the entropy of the process ($E$), while the total information increases at a constant rate measured by the conditional entropy ($C$); the information storage ($S$) is the information shared between the present and the past (overlap of ovals), while the $C$ is the part of the present information not shared with the past. When the process is fully random ($\delta =0$), the total information increases at the maximum rate ($C=1$) and there is no stored information ($S=0$). On the contrary, a fully predictable process ($\delta =1$) does not produce new information ($C=0$) and stores the whole information contained in its present state ($S=1$). Any intermediate parameter configuration ($0<\delta <1$) yields a partially predictable process with presence of both information production and information storage ($0<S<1, 0<C<1$).

Figure 2

Schematic illustration of the linear estimation of entropy measures. In this paradigmatic example in which a time series $\{x_1,...,x_8\}$ is considered as a realization of the process $X$, and the past of the process is approximated with $m=1$ lags ($X_n^{-}\approx X_n^m=X_{n-1}$), seven realizations of $(X_n,X_{n-1})$ are found and used to fill a two-dimensional space. Then, the linear regression of $X_n$ on $X_{n-1}$ is performed yielding the regression line $X_n=a{X}_{n-1}$ (red line) and, for a given observation $x_n$ of $X_n$, the prediction error $u_n$ is taken as the difference between the true and the predicted value, $u_n=x_n-a{x}_{n-1}$. The estimates $\{u_2,...,u_8\}$ are finally used to compute the variance of $U_n$ and the conditional entropy according to Eq. (12).

Figure 3

Schematic illustration of the kernel estimation of entropy measures. In this example, the past of the process $X$ is approximated using $m=1$ samples, and the Heaviside kernel with fixed threshold $r$ is used. In the $(m+1)$-dimensional space spanned by the realizations of $(X_n,X_{n-1})$, the probability of a given reference point $(x_n,x_{n-1})$ (red dot) is estimated as the fraction of points whose distance are less than $r$ (gray dots) from it: $p(x_n,x_{n-1})=2/7$. The distance between two points is computed as the maximum between the horizontal and the vertical distance between the two points. Similarly, in the $m$-dimensional space spanned by the realizations of $X_{n-1}$, the probability of $x_{n-1}$ is approximated using the same threshold $r$, yielding $p\left(x_{n-1}\right)=3/7$. This procedure is repeated varying the reference point and the conditional entropy is estimated by Eq. (16).

Figure 4

Schematic illustration of the knn estimation of entropy measures. In this example, the past of the process $X$ is approximated with $m=1$ samples and $k=2$ neighbors are used in the search for neighbors. In the $(m+1)$-dimensional space spanned by the realizations of $(X_n,X_{n-1})$, a neighbor search is performed using the maximum norm to find the $k\mathrm{th}$ nearest neighbor (blue dot) of the assigned reference point (red dot). Then, the distance between these two points, $0.5{\varepsilon }_n$ is used in the projected one-dimensional space spanned by the realizations of of $X_n^1=X_{n-1}$ as threshold distance to find the neighbors of the reference point $x_{n-1}$; in this example, $N_{X_n^1}=3$ neighbors are counted. This procedure is repeated varying the reference point and the obtained values of ${\varepsilon }_n$ and $N_{X_n^1}$ are then used in Eq. (24) to compute the conditional entropy.

Figure 5

Autoregressive processes: characterization of a stationary order-2 AR process for different values of the AR amplitude $\rho$ and frequency $f$. (a)–(c) Exemplary realizations of the process obtained varying $\rho$ (columns) and $f$ (rows). Results: When $f$ is fixed, the process appears more regular for higher values of $\rho$. (d) Autocorrelation of the process as a function of the lag $\tau$ for the realizations in (c). (e) 2D phase plots of $(X_n,X_{n-1})$ and $(X_n,X_{n-2})$, and 3D phase plots of $(X_n,X_{n-1},X_{n-2})$, for the realizations in (c). When $\rho$ is fixed, the process exhibits shorter memory and weaker dependence of the present on the past for intermediate frequency $f$.

Figure 6

Processes with long-range correlations: characterization of a fractionally integrated process with long-range power-law correlations for different values of the differencing parameter $d$, which controls the correlations. (a) Exemplary realizations of the process with $d=-0.5,0,0.5$. (b) Results of detrended fluctuation analysis (DFA) applied to longer realizations ($2^{20}$ data points) of the process in (a). For $d\in [-0.5,0.5]$, the DFA exponent is $\alpha =(2d+1)/2$. (c) 2D phase plots of $(X_n,X_{n-1})$ and $(X_n,X_{n-2})$, and 3D phase plots of $(X_n,X_{n-1},X_{n-2})$, for the realizations in (a). Results: For fixed modulus of the differencing parameter, positively correlated processes exhibits stronger dependence of the present on the past than anti-correlated processes.

Figure 7

Autoregressive processes with long-range correlations: characterization of a stationary fractionally integrated AR process with both autoregression and power-law long-range correlations for different values of the differencing (correlation) parameter $d$ and fixed values of the AR parameters $\rho$ and $f$. (a) Exemplary realizations of the process with $d=-0.5,0,0.5$ and fixed $\rho =0.8,f=0.25$. (b) 2D phase plots of $(X_n,X_{n-1})$ and $(X_n,X_{n-2})$, and 3D phase plots of $(X_n,X_{n-1},X_{n-2})$ for the realizations in (a). Results: The fractionally integrated AR process displays weaker dependence of the present on the past than the corresponding pure autoregressive process with the same AR parameters in the presence of positive long-range correlations and stronger dependence in the presence of negative long-range correlations.

Figure 8

Performance of entropy estimators and entropy measures for stationary AR processes. (a)–(c) Exemplary realizations of AR processes generated with fixed $f=0.25$ and varying $\rho =0,0.6,0.9$, with corresponding probability distributions reported on the right; note that if $\rho =0$ the process is a white noise $\forall f$. (d)–(f) Dependence of entropy measures and entropy estimates, obtained for AR processes with fixed amplitude and frequency ($\rho =0.6,f=0.25$), on the length $N$ of the time series generated as process realizations. (g)–(i) Dependence of entropy measures and entropy estimates on the AR amplitude $\rho$ with fixed AR frequency ($f=0.25$) and time series length ($N=300$). (j)–(l) Dependence of entropy measures and entropy estimates on the AR frequency with fixed AR amplitude $\rho =0.6$ and time series length ($N=300$). Panels (d)–(l) report the theoretical values (black solid line) and the estimated distributions (mean and $25\%–75\%$ percentiles over 100 realizations) of entropy (d), (g), (j), conditional entropy (e), (h), (k), and information storage (f), (i), (l) obtained with the linear estimator (green squares), the kernel estimator implemented with threshold $r=0.2$ (red circles), and the knn estimator implemented with $k=10$ neighbors (blue triangles). Results: The expected values of all entropy measures do not change with the realization length $N$. Moreover, for these normalized time series the theoretical values of entropy are unaffected by the AR parameters. The conditional entropy decreases with the increase of $\rho$ when $f$ is fixed, and increases with increasing $f$ for $f\in (0,0.25]$ and decreases for $f\in (0.25,0.5)$ when $\rho$ is fixed. The theoretical behavior of information storage is the opposite of that of the conditional entropy. The estimates obtained with the linear and knn estimators are close to the theoretical values for all entropy measures, while the estimates of entropy and conditional entropy obtained with the kernel estimator are strongly biased and exhibit high variance for short time series.

Figure 9

Dependence of entropy estimators on estimator parameters for stationary AR processes. Plots depict the theoretical values (black solid line) and the estimated distributions (mean and $25\%–75\%$ percentiles over 100 realizations, colored symbols and error bars) of entropy (a), (d), (g), conditional entropy (b), (e), (h), and information storage (c), (f), (i) computed using the linear estimator (a)–(c), the kernel estimator (g)–(i) implemented with threshold $r=0.1$ (green open squares), $r=0.2$ (blue full triangles), and $r=0.5$ (red open circles), and the knn estimator (g), (h) implemented with $k=5$ (green open squares), $k=10$ (blue full triangles), and $k=30$ (red open circles) neighbors. Estimates are computed over realizations of length $N=300$, generated with fixed AR frequency $f=0.25$ and varying the AR amplitude in the range $\rho \in \{0,0.4,0.6,0.8,0.9\}$. Results: The kernel estimates of entropy and conditional entropy are strongly dependent on the parameter $r$ setting the kernel threshold, whereas the knn estimates are much less sensitive to the parameter $k$ setting the number of neighbors. Since the linear estimator assumes the form of the probability distributions, it has no free parameters.

Figure 10

Dependence of entropy measures and entropy estimators on AR process parameters. Plots depict the theoretical values (lines) and the estimated distributions (mean and $25\%–75\%$ percentiles over 100 realizations lasting $N=300$ samples, colored symbols and error bars) of entropy (a), (d), (g), conditional entropy (b), (e), (h) and information storage (c), (f), (i) computed using the linear estimator (a)–(c), the kernel estimator implemented with threshold $r=0.2$ (d)–(f), and the knn estimator implemented with $k=10$ neighbors (g)–(i). Each measure is computed as a function of the AR frequency varying in the range $f\in (0–0.5)$ for different values of the AR amplitude ($\rho =0$, red crosses and solid lines; $\rho =0.6$, blue long-dashed lines and open squares; $\rho =0.8$, pink short-dashed lines and full triangles; $\rho =0.9$, green dotted lines and open circles). Results: The linear estimates of entropy measures are close to the theoretical values regardless of the values of the AR amplitude $\rho$ and frequency $f$. The kernel and knn estimates exhibit a bias that is more evident for high values of the AR amplitude ($\rho \ge 0.8$), and for very low or very high values of the AR frequency ($f\le 0.1,f\ge 0.4$).

Figure 11

Performance comparison for AR signals with and without sinusoidal trends: alteration of signal properties and entropy estimates. (a) and (b) show exemplary realizations of a stationary AR process with amplitude $\rho =0.6$ and frequency $f=0.25$ before and after superposition of a sinusoidal trend with amplitude $A=1$ and period $T=100$; the corresponding probability distributions are shown on the right. Signals are normalized to zero mean and unit variance. (c), (d) 2D phase plots of ($X_n,X_{n-2}$) derived from the time series in (a), (b). The generating equation of this AR process with $\rho =0.6$ and $f=0.25$ is $X_n=-0.36X_{n-2}+U_n$, which yields the theoretical temporal relation between $X_n$ and $X_{n-2}$ shown by the solid black line; the estimated temporal relation obtained through linear least-squares fit of the two clouds of points is shown by the red dashed lines. (e), (f) Entropy (shaded bars), conditional entropy (white bars) and information storage (gray bars) expressed as theoretical values computed for the stationary AR process without trends and estimated values computed for the time series in (a), (b). Estimations are performed using the linear estimator, the kernel estimator with threshold $r=0.2$, and the knn estimator with $k=10$ neighbors. Results: The presence of a sinusoidal trend superimposed to a realization of the AR process alters the probability distribution of $X_n$ and distorts the temporal relation between $X_n$ and $X_{n-2}$. This results in a significant decrease of the conditional entropy and in a significant increase of the information storage for all estimators.

Figure 12

Effects of nonstationarity due to sinusoidal trends on the estimation of entropy measures. Plots depict the behavior of entropy (a), (d), (g), conditional entropy (b), (e), (h) and information storage (c), (f), (i) computed as a function of the amplitude $\rho$ of an AR process with fixed frequency $f=0.25$, expressed as theoretical values computed for the original process without trends (black solid lines), and estimated distributions (mean and $25\%–75\%$ percentiles) computed over 100 realizations of $N=300$ samples of the process, each corrupted with an additive sinusoidal trend of period $T$ and amplitude $A$ and normalized to zero mean and unit variance (colored symbols and error bars: $T=100,A=1$, red crosses and solid lines; $T=100,A=5$, blue open squares and long-dashed lines; $T=400,A=1$, pink full triangles and short-dashed lines; $T=400,A=5$, green open circles and dotted lines). Estimates are performed using the linear estimator (a)–(c), the kernel estimator implemented with threshold $r=0.2$ (d)–(f), and the knn estimator implemented with $k=10$ neighbors (g)–(i). Results: The presence of trends impairs the ability of all estimators to quantify the changes of conditional entropy and information storage induced by variations in the AR amplitude $\rho$. Moreover, trends induce an estimation bias bias proportional to the trend amplitude $A$.

Figure 13

Performance comparison for AR signals with and without random spikes: alteration of signal properties and entropy estimates. (a), (b) Exemplary realizations of a stationary AR process with amplitude $\rho =0.8$ and frequency $f=0.25$ before and after superposition of spikes with amplitude $A=5$ to 5% of the time series points; the corresponding probability distributions are shown on the right. Signals are normalized to zero mean and unit variance. (c), (d) 2D phase plots of ($X_n,X_{n-2}$) derived from the time series in (a), (b). The generating equation of this AR process with $\rho =0.8$ and $f=0.25$ is $X_n=-0.64X_{n-2}+U_n$, which yields the theoretical temporal relation between $X_n$ and $X_{n-2}$ shown by the solid black line; the estimated temporal relation obtained through linear least-squares fit of the two clouds of points is shown by the red dashed lines. (e)–(f) Entropy (shaded bars), conditional entropy (white bars), and information storage (gray bars) expressed as theoretical values computed for the stationary AR process without spikes and estimated values computed for the time series in (a), (b). Estimations are performed using the linear estimator, the kernel estimator with threshold $r=0.2$, and the knn estimator with $k=10$ neighbors. Results: The presence of spikes superimposed to a realization of the AR process concentrates the probability distribution of the process and adds random outliers, which blurs the detection of the temporal relation between between $X_n$ and $X_{n-2}$. This results in the inability of the linear estimator to detect the information storage in the process, while the kernel and knn estimators are less affected.

Figure 14

Effects of nonstationarity due to random spikes on the estimation of entropy measures. Plots depict the behavior of entropy (a), (d), (g), conditional entropy (b), (e), (h), and information storage (c), (f), (i) computed as a function of the amplitude $\rho$ of an AR process with fixed frequency $f=0.25$, expressed as theoretical values computed for the original process without spikes (black solid lines), and estimated distributions (mean and $25\%–75\%$ percentiles) computed over 100 realizations of $N=300$ samples of the process, each corrupted with additive random spikes of amplitude $A$ occurring with probability $P$ and normalized to zero mean and unit variance (colored symbols and error bars: $P=5\%,A=1$, red crosses and solid lines; $P=5\%,A=5$, blue open squares and long-dashed lines; $P=20\%,A=1$, pink full triangles and short-dashed lines; $P=20\%,A=5$, green open circles and dotted lines). Estimates are performed using the linear estimator (a)–(c), the kernel estimator implemented with threshold $r=0.2$ (d)–(f), and the knn estimator implemented with $k=10$ neighbors (g)–(i). Results: The presence of spikes partially impairs the ability to quantify the changes of conditional entropy and information storage induced by variations in the AR amplitude $\rho$; the impairment is more evident for the linear estimator and for high percentages of spikes. Moreover, spikes induce an estimation bias proportional to both the amplitude and the percentage of spikes.

Figure 15

Performance comparison for AR signals with and without segments of high variance: alteration of signal properties and entropy estimates in the presence of local changes in signal variance. (a), (b) Exemplary realizations of a stationary AR process with amplitude $\rho =0.9$ and frequency $f=0.25$ before and after inflating random segments by an amplification factor $\sigma =5$ (a total of $P=20\%$ of the time series points are inflated); the corresponding probability distributions are shown on the right. Signals are normalized to zero mean and unit variance. (c), (d) 2D phase plots of ($X_n,X_{n-2}$) derived from the time series in (a), (b). The generating equation of this AR process with $\rho =0.9$ and $f=0.25$ is $X_n=-0.81X_{n-2}+U_n$, which yields the theoretical temporal relation between $X_n$ and $X_{n-2}$ shown by the solid black line; the estimated temporal relation obtained through linear least-squares fit of the two clouds of points is shown by the red dashed lines. (e), (f) Entropy (shaded bars), conditional entropy (white bars), and information storage (gray bars) expressed as theoretical values computed for the stationary AR process without changes in variance and estimated values computed for the time series in (a), (b). Estimations are performed using the linear estimator, the kernel estimator with threshold $r=0.2$, and the knn estimator with $k=10$ neighbors. Results: The presence of segments with higher variance concentrates the probability distribution of the process and disperses a portion of the points without distorting their temporal relation. This results in a mild decrease of the conditional entropy and increase of the information storage for the kernel and knn estimators, while opposite changes are appreciated for the linear estimator.

Figure 16

Effects of nonstationarity due to local changes in the signal variance on the estimation of entropy measures. Plots depict the behavior of entropy (a), (d), (g), conditional entropy (b), (e), (h), and information storage (c), (f), (i) computed as a function of the amplitude $\rho$ of an AR process with fixed frequency $f=0.25$, expressed as theoretical values computed for the original process without changes in variance (black solid lines), and estimated distributions (mean and $25\%–75\%$ percentiles) computed over 100 realizations of $N=300$ samples of the process, each corrupted by randomly distributed inflated segments and normalized to zero mean and unit variance. Each inflated segment lasts 20 points and is generated by magnifying original data points by a factor of $\sigma$, with the percentage of inflated points to the total signal length being $P\%$ (colored symbols and error bars: $P=5\%,\sigma =5$, red crosses and solid lines; $P=5\%,\sigma =10$, blue open squares and long-dashed lines; $P=20\%,\sigma =5$, pink full triangles and short-dashed lines; $P=20\%,\sigma =10$, green open circles and dotted lines). Estimates are performed using the linear estimator (a)–(c), the kernel estimator implemented with threshold $r=0.2$ (d)–(f), and the knn estimator implemented with $k=10$ neighbors (g)–(i). Results: The presence of segments with high variance does not impair significantly the ability to quantify the changes of conditional entropy and information storage induced by variations in the AR amplitude $\rho$. However, local changes in the signal variance induce an estimation bias proportional to the percentage of inflated points and—to a lower extent—to the amplitude of inflation.

Figure 17

Performance of entropy estimators and entropy measures for fractionally integrated processes with long-range power-law correlations. Plots depict the theoretical values (black solid lines) and the estimated distributions (mean and $25\%–75\%$ percentiles over 100 realizations of length $N=100$ (red crosses and solid lines), $N=300$ (blue open squares and long-dashed lines), $N=500$ (pink solid triangles with short-dashed lines), and $N=2000$ (green open circles with dotted lines)) of entropy (a), (d), (g), conditional entropy (b), (e), (h), and information storage (c), (f), (i) computed as a function of the differencing parameter $d$. Estimates are obtained using the linear estimator (a)–(c), the kernel estimator implemented with threshold $r=0.2$ (d)–(f), and the knn estimator implemented with $k=10$ neighbors (g)–(i). Results: Signals with positive correlations present lower conditional entropy and higher information storage than signals with anticorrelation of the same strength. All estimators reflect the changes in entropy measures with the type and strength of power-law correlation, with a higher accuracy for longer time series length, but exhibit a bias which increases with the correlation strength.

Figure 18

Exemplary realizations of fractionally integrated autoregressive processes for varying AR amplitude $\rho$ and differencing (correlation) parameter $d$. Plots depict signals with (a) weak autoregression ($\rho =0.3$) and strong anti-correlation ($d=-0.5$), (b) weak autoregression ($\rho =0.3$) and strong positive correlation ($d=0.5$), (c) strong autoregression ($\rho =0.9$) and strong anticorrelation ($d=-0.5$), (d) strong autoregression ($\rho =0.9$) and strong positive correlation ($d=0.5$). Slow trends in the signal are present for strong positive long-range correlation, and less evident for anticorrelated signals or in the presence of a strong AR component.

Figure 19

Performance of entropy estimators and entropy measures for fractionally integrated autoregressive processes. Plots depict the theoretical values (lines) and the estimated distributions (mean and $25\%–75\%$ percentiles over 100 realizations lasting $N=300$ samples, colored symbols and error bars) of entropy (a, d, g), conditional entropy (b), (e), (h), and information storage (c), (f), (i) computed using the linear estimator (a)–(c), the kernel estimator implemented with threshold $r=0.2$ (d)–(f), and the knn estimator implemented with $k=10$ neighbors (g)–(i). Each measure is computed as a function of the AR amplitude varying in the range $\rho \in (0,0.9)$ for fixed AR frequency ($f=0.25$) and different values of the differencing parameter ($d=-0.5$, red crosses and solid lines; $\rho =-0.25$, blue long-dashed lines and open squares; $\rho =0$, pink short-dashed lines and full triangles; $\rho =0.25$, green dotted lines and open circles; $\rho =0.5$, gray dash-dotted lines and open circles). Results: The presence of strong positive correlation alters markedly the dependence of conditional entropy and information storage on the AR amplitude, whereas anticorrelation induces only a shift in the measures without affecting substantially the dependence on $\rho$. All estimators are biased in approximating conditional entropy and information storage for fractionally correlated AR signals. The bias is more evident for positive than negative correlations and for stronger than weaker correlations.

Figure 20

Exemplary signals of consecutive heart beat intervals for one healthy subject (a), (c) and one CHF subject (b), (d), during day (a), (b) and night (c), (d). The original signals of healthy subjects typically exhibit larger variability and amplitude than those of CHF subjects.

Figure 21

Exemplary HRV time series of 300 points measured from the same CHF subject during day shown in Fig. 20 under different pre-processing procedures. Panels depict the original signal (a), the same signal in (a) normalized to zero mean (b), the same signal in (a) normalized to zero mean and unit variance (c), the same signal in (a) with slow trends removed by a linear high pass filter and normalized to zero mean and unit variance.

Figure 22

Performance of entropy estimators and entropy measures for heart rate variability signals of healthy subjects and congestive heart failure (CHF) patients measured during wake (W) and sleep (S) states. Panels contain the behavior of entropy (a), (d), (g), conditional entropy (b), (e), (h), and information storage (c), (f), (i) computed for signals normalized to zero mean within each window (shaded bar), signals normalized to zero mean and unit standard deviation within each window (open bar), and signals detrended by a linear high-pass filter and normalized to zero mean and unit standard deviation within each window (solid bar) for healthy subjects (left side with white background and red bars) and CHF subjects (right side with gray background and blue bars) during wake (W) and sleep (S). Barplots depict the mean + standard deviation across subjects of the median value of entropy measures computed for consecutive windows of 300 data points with a 150 points overlap using $m=2$ lagged components and implemented with the linear estimator (a)–(c), the kernel estimator with threshold $r=0.2$ (d)–(f), and the knn estimator with $k=10$ neighbors (g)–(i). Symbols denote statistical significance ($p<0.05$) of the differences between W and S ($*$, paired $t$-test) or between healthy and CHF ($\#$, unpaired $t$-test). Results: Compared to healthy subjects, CHF patients exhibit lower entropy values reflecting depressed HRV, and higher conditional entropy and lower information storage reflecting higher dynamical complexity. Healthy subjects display a day-to-night increase of the dynamical complexity (higher conditional entropy and lower information storage during S than during W) that is not observed in CHF patients.