# Day and Night Changes of Cardiovascular Complexity: A Multi-Fractal Multi-Scale Analysis

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}) for scales shorter than 16 beats and a long-term coefficient (α

_{2}) for longer scales [3]. The original bi-scale method was then extended in two ways. One way was to provide a multiscale spectrum of self-similarity coefficients, a function of the scale n in beats, α(n) [4,5,6]. Another way was to provide a multifractal spectrum of self-similarity coefficients, a function of the moment order q, α(q) [7,8]. The multifractal spectrum includes q = 2—the second-order moment used in the original DFA method for monofractal series—and allows detecting multifractality when α(q) differs substantially between positive and negative q orders. The multiscale and the multifractal methods were finally combined in the multifractal-multiscale DFA, a versatile approach that describes multifractal structures localized over specific scales and that provides a surface of scale coefficients, α(q,n) [9]. Recent works demonstrated the capability of the multifractal-multiscale DFA of heart rate variability to classify different types of cardiac patients [10] and to describe alterations in the heart rate complexity due to an impaired integrative autonomic control in paraplegic individuals [11].

## 2. Materials and Methods

#### 2.1. Subjects and Data Collection

#### 2.2. Multifractal-Multiscale Detrended Fluctuation Analysis

_{i}of length L beats, we calculated its cumulative sum, y

_{i}. We split y

_{i}into M maximally overlapped blocks of n beats (two consecutive blocks have n-1 beats in common). We detrended each block with least-square polynomial regression and calculated the variance of the residuals in each k-th block, σ

^{2}

_{n}(k). The variability function F

_{q}(n) is the q-th moment of σ

^{2}

_{n}[7]:

_{B}(q,n), calculating the derivative of log F

_{q}(n) vs. log n [17]. This was done for detrending polynomials of order 1 and 2 (see examples of the corresponding F

_{q}(n) estimates in Figure 1. Previous empirical analyses suggested that the second-order polynomial overfits block sizes shorter than 12 beats, but at the same time, it appears to more efficiently remove long-term trends [17,18,19]. Therefore, we estimated a single α

_{B}(q,n) function combining the estimates after detrending of order 1 and 2 with a weighted average which weights more the order one at the shorter scales as proposed in [17].

^{2}

_{n}, which corresponds to the second-order moment, or q = 2. If the series is monofractal, all the moment orders q provide the same slope α. By contrast, for multifractal series positive moment order q weight more the contribution of the fractal components with greater amplitude, negative moment order q weight more the contribution of the fractal components with lower amplitude.

_{IBI}

_{S}(q) and long-term coefficient α

_{L}(q).

#### 2.3. Nonlinearity Index

^{i,j}(q,τ) with 1 ≤ I ≤ 100, to be compared with the coefficients of the original series j, α

^{O,j}(q,τ). For the comparison, we calculated π

^{j}(q,τ), defined at each q and τ as the percentile of the distribution of 100 surrogate α

^{i,j}(q,τ) coefficients in which was the original α

^{O,j}(q,τ) coefficient (to apply a 2-tail statistics, percentiles greater than 50% were transformed into their complement to 1 as in [23]). π

^{j}(q,τ) may range between 50% and 0%: the lower its value, the more significant the deviation of the original scale coefficient α

^{O,j}from the distribution of the 100 surrogate coefficients α

^{i}

^{,j}. Large deviations from the surrogates distribution are suggestive of nonlinear components in the original series. Therefore, we defined a short-term nonlinearity index at each moment order q, NL

_{S}(q), by calculating the percentage of scales in the range 8 ≤ τ ≤ 16 s, where π

^{j}(q,τ) was ≤1%. Similarly, we calculated the percentage of scales with π

^{j}(q,τ) ≤ 1% for 16 < τ ≤ 512 s to define the long-term nonlinearity index NL

_{L}(q). Both NL

_{S}(q) and NL

_{L}(q) may range between 0% and 100%. Their higher values indicate moment orders q that better detect the presence of nonlinear components.

#### 2.4. Spectral Analysis

#### 2.5. Statistical Analysis

_{S}(q) and α

_{L}(q), and nonlinearity indices, NL

_{S}(q) and NL

_{L}(q), were also compared between Day and Night at each q by the Wilcoxon signed-rank test. IBI, SBP, and DBP levels and power spectra were compared between Day and Night by the paired t-test, after log-transformation of the spectral indices to remove the skewness of their distribution [24]. The threshold for statistical significance was set at 5% with a two-sided alternative hypothesis. All the tests were performed with “R: A Language and Environment for Statistical Computing” software package (R Core Team, R Foundation for Statistical Computing, Vienna, Austria, 2019).

## 3. Results

#### 3.1. Day vs. Night

_{S}(q) and α

_{L}(q) indices corresponds on average to 20.7 beats in the Day and 15.5 beats in the Night period, and the α

_{L}(q) upper scale at τ = 512 s corresponds to 661.2 beats and 495.3 beats in the Day and Night periods, respectively.

#### 3.2. Nonlinearity

_{S}(q) and NL

_{L}(q). The highest degree of nonlinearity is detected at Night by NL

_{S}(q), which is close to 100% for all the cardiovascular series between q = −2 and q = +4, with the notable exception of q = 2. In fact, at q = 2 NL

_{S}falls to 0% for all the signals. NL

_{S}tends to be higher at night with significant differences at some q < 0 for IBI and DBP and at q > 2 for IBI. Long-term nonlinear components are mainly present in IBI at night. In fact, NL

_{L}(q) of IBI is greater than 50% during night-time at all q but q = 2. Furthermore, it is significantly greater at night for all q ≠ 2. NL

_{L}too is close to 0% at q = 2, both during Day and Night, for heart rate and blood pressure.

## 4. Discussion

#### 4.1. Day vs. Night

_{L}(q) is larger for q < 0 (Figure 4b). We may associate this night/day oscillation to an endogenous circadian rhythm previously described in the heart rate by a monofractal DFA exponent (i.e., for q = 2) estimated over scales between 20 and 400 beats [29]. This endogenous rhythm was hypothesized to contribute to the period of the cardiac vulnerability reported in epidemiological studies. Our work suggests that this night/day rhythm (1) is highlighted by a multifractal approach that assesses negative moment orders and (2) is better quantified in a narrower range of scales, between 128 s and 256 s. Therefore, our results may prove to be of clinical importance by allowing designing new tools for the complexity analysis of heart rate that better stratify the cardiovascular risk. Interestingly, our study also provides evidence that a night/day modulation with greater daytime values is present at the same scales in blood pressure too, suggesting that a common physiological mechanism is at the origin of the circadian oscillation in the heart rate and the blood pressure self-similarity coefficients.

_{1}for scales between 4 and 16 beats and a long-term coefficient α

_{2}for scales between 16 and 64 beats. The study in [30] used the scale n = 11 beats to separate α

_{1}from α

_{2}and reported a significant decrease in α

_{1}at night. We did not consider scales short as in this study because at τ < 8 s the multifractal estimates can be affected by large estimation bias for negative q orders. However, α

_{1}and the LF/HF powers ratio of the heart rate are correlated [13] and the reduction in the LF/HF powers ratio we reported at night in Table 1 is coherent with the night reduction of α

_{1}in [30]. These authors, however, also showed a significant increase of α

_{2}at night, which appears in contrast with the night decrease of the long-term scale coefficients reported both in [29] and in our work. To correctly interpret the results of the three studies, we should consider carefully the scale ranges where the coefficients are estimated. To illustrate this point, Figure 8 plots the coefficients we calculated as the derivative of log F

_{q}(n) vs. log n in Equation (1), i.e., α

_{B}(q,n), for q = 2. The scale n is expressed in beats to facilitate the comparison with previous studies [29,30]. As the estimation bias is negligible for q = 2, α

_{B}is plotted from n = 6 beats. The night/day comparison shows a significant nocturnal decrease of α

_{B}at scales < 11 beats, in line with the α

_{1}results in [30], and greater night-time values at scales where α

_{2}was estimated in [30]. These greater values correspond to the small area of statistical significance that appears in our Figure 3 at scales τ ≤ 16 s and at orders q ≥ 2. The α coefficient calculated in [29] between 20 and 400 beats overlaps partially with α

_{2}but covers a much wider range of longer scales, which includes the band between 128 and 256 beats where we found a significant night decrease of α

_{B}. Therefore our study and the studies in [29,30] provide coherent results if the correct scale ranges are considered. The comparison of Figure 8 also highlights the importance to provide estimates of the scale coefficients as a continuous function of the scale n to correctly identify phenomena which may occur in nearby scales with different characteristics.

#### 4.2. Nonlinearity

_{S}(q) and NL

_{L}(q), that indicate the moment orders and the scale ranges, where α(q,τ) provides information on nonlinear dynamics. These indices are close to 0% for q = 2, supporting previous theoretical speculations indicating that the monofractal DFA and the power spectrum provide similar information [12,13]. However, we also found clear nonlinear components for q between −2 and +4 at the short scales, more pronounced at night, both for the heart rate and the blood pressure. Furthermore, at night, substantial nonlinear components appear in heart rate at the longer scales. It should be noted that higher nonlinear components at night have been previously demonstrated by a noise titration procedure applied to Volterra/Wiener models fitting 24-h heart rate series [30]. The similarity of results obtained with so different approaches supports the evidence that nonlinearity prevails at night.

_{1}, and by the standard spectral methods are correlated [31]. This inevitably reduces the additional prediction power of α

_{1}compared to the spectral method. The traditional bi-scale monofractal model is based on the second-order moment, q = 2. By showing that DFA coefficients evaluated for q ≠ 2 provide information substantially different from that of the spectral powers, particularly at night, our study suggests that the multifractal multiscale DFA approach might effectively integrate the information of traditional spectral methods, possibly improving the clinical value of DFA.

## 5. Limitations and Conclusions

_{q}(n) vs. log n. This axis transformation is similar to mapping the “cycles/beat” in “Equivalent Hz” in the spectral analysis of cardiovascular series [34]. However, if results obtained with scales expressed as τ in seconds are discussed in relation to other studies based on scales defined in number of beats, readers should be aware that discrepancies may arise because possible differences in the heart rate level between conditions or groups may change the ranges of scales that define short-term and long-term DFA coefficients.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Bak, P.; Tang, C.; Wiesenfeld, K. Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett.
**1987**, 59, 381–384. [Google Scholar] [CrossRef] [PubMed] - Nagy, Z.; Mukli, P.; Herman, P.; Eke, A. Decomposing Multifractal Crossovers. Front. Physiol.
**2017**, 8, 8. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Peng, C.K.; Havlin, S.; Hausdorff, J.M.; E Mietus, J.; Stanley, H.; Goldberger, A.L. Fractal mechanisms and heart rate dynamics. Long-range correlations and their breakdown with disease. J. Electrocardiol.
**1995**, 28. [Google Scholar] [CrossRef] - Echeverria, J.C.; Woolfson, M.S.; Crowe, J.A.; Hayes-Gill, B.R.; Croaker, G.D.H.; Vyas, H. Interpretation of heart rate variability via detrended fluctuation analysis and alpha-beta filter. Chaos
**2003**, 13, 467–475. [Google Scholar] [CrossRef] - Castiglioni, P.; Parati, G.; Civijian, A.; Quintin, L.; Di Rienzo, M. Local Scale Exponents of Blood Pressure and Heart Rate Variability by Detrended Fluctuation Analysis: Effects of Posture, Exercise, and Aging. IEEE Trans. Biomed. Eng.
**2008**, 56, 675–684. [Google Scholar] [CrossRef] - Castiglioni, P.; Parati, G.; Di Rienzo, M.; Carabalona, R.; Cividjian, A.; Quintin, L. Scale exponents of blood pressure and heart rate during autonomic blockade as assessed by detrended fluctuation analysis. J. Physiol.
**2010**, 589, 355–369. [Google Scholar] [CrossRef] - Kantelhardt, J.W.; Zschiegner, S.A.; Koscielny-Bunde, E.; Havlin, S.; Bunde, A.; Stanley, H. Multifractal detrended fluctuation analysis of nonstationary time series. Phys. A Stat. Mech. its Appl.
**2002**, 316, 87–114. [Google Scholar] [CrossRef] [Green Version] - Makowiec, D.; Rynkiewicz, A.; Wdowczyk-Szulc, J.; Żarczyńska-Buchowiecka, M.; Gała̧ska, R.; Kryszewski, S. Aging in autonomic control by multifractal studies of cardiac interbeat intervals in the VLF band. Physiol. Meas.
**2011**, 32, 1681–1699. [Google Scholar] [CrossRef] - Gierałtowski, J.; Żebrowski, J.; Baranowski, R. Multiscale multifractal analysis of heart rate variability recordings with a large number of occurrences of arrhythmia. Phys. Rev. E
**2012**, 85, 021915. [Google Scholar] [CrossRef] - Kokosińska, D.; Gierałtowski, J.; Zebrowski, J.J.; Orłowska-Baranowska, E.; Baranowski, R. Heart rate variability, multifractal multiscale patterns and their assessment criteria. Physiol. Meas.
**2018**, 39, 114010. [Google Scholar] [CrossRef] - Castiglioni, P.; Merati, G.; Parati, G.; Faini, A. Decomposing the complexity of heart-rate variability by the multifractal-multiscale approach to detrended fluctuation analysis: An application to low-level spinal cord injury. Physiol. Meas.
**2019**, 40, 084003. [Google Scholar] [CrossRef] [PubMed] - Heneghan, C.; McDarby, G. Establishing the relation between detrended fluctuation analysis and power spectral density analysis for stochastic processes. Phys. Rev. E
**2000**, 62, 6103–6110. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Willson, K.; Francis, D.P. A direct analytical demonstration of the essential equivalence of detrended fluctuation analysis and spectral analysis of RR interval variability. Physiol. Meas.
**2002**, 24, N1–N7. [Google Scholar] [CrossRef] [PubMed] - Mancia, G.; Ferrari, A.; Gregorini, L.; Parati, G.; Pomidossi, G.A.; Bertinieri, G.; Grassi, G.; Di Rienzo, M.; Pedotti, A.; Zanchetti, A. Blood pressure and heart rate variabilities in normotensive and hypertensive human beings. Circ. Res.
**1983**, 53, 96–104. [Google Scholar] [CrossRef] [Green Version] - Di Rienzo, M.; Castiglioni, P.; Parati, G. Arterial Blood Pressure Processing. In Wiley Encyclopedia of Biomedical Engineering; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2006; pp. 98–109. [Google Scholar]
- Task Force of the European Society of Cardiology the North American Society of Pacing Electrophysiology. Heart Rate Variability. Circulation
**1996**, 93, 1043–1065. [Google Scholar] [CrossRef] [Green Version] - Castiglioni, P.; Faini, A. A Fast DFA Algorithm for Multifractal Multiscale Analysis of Physiological Time Series. Front. Physiol.
**2019**, 10, 115. [Google Scholar] [CrossRef] [Green Version] - Kantelhardt, J.W.; Koscielny-Bunde, E.; Rego, H.H.; Havlin, S.; Bunde, A. Detecting long-range correlations with detrended fluctuation analysis. Phys. A Stat. Mech. Appl.
**2001**, 295, 441–454. [Google Scholar] [CrossRef] [Green Version] - Bunde, A.; Havlin, S.; Kantelhardt, J.W.; Penzel, T.; Peter, J.-H.; Voigt, K. Correlated and Uncorrelated Regions in Heart-Rate Fluctuations during Sleep. Phys. Rev. Lett.
**2000**, 85, 3736–3739. [Google Scholar] [CrossRef] [Green Version] - Castiglioni, P.; Parati, G.; Lombardi, C.; Quintin, L.; Di Rienzo, M. Assessing the fractal structure of heart rate by the temporal spectrum of scale exponents: A new approach for detrended fluctuation analysis of heart rate variability. Biomed. Tech. Eng.
**2011**, 56, 175–183. [Google Scholar] [CrossRef] - Gautama, T. Surrogate Data; MATLAB Central File Exchange; MATLAB: Natick, MA, USA, 2005. [Google Scholar]
- Schreiber, T.; Schmitz, A. Surrogate time series. Phys. D Nonlinear Phenom.
**2000**, 142, 346–382. [Google Scholar] [CrossRef] [Green Version] - Castiglioni, P.; Parati, G.; Faini, A. Can the Detrended Fluctuation Analysis Reveal Nonlinear Components of Heart Rate Variabilityƒ. In Proceedings of the 2019 41st Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Berlin, Germany, 23–27 July 2019; IEEE: Berlin, Germany, 2019; Volume 2019, pp. 6351–6354. [Google Scholar]
- Castiglioni, P.; Parati, G.; Omboni, S.; Mancia, G.; Imholz, B.P.; Wesseling, K.H.; Di Rienzo, M. Broad-band spectral analysis of 24 h continuous finger blood pressure: Comparison with intra-arterial recordings. Clin. Sci.
**1999**, 97, 129–139. [Google Scholar] [PubMed] - Di Rienzo, M.; Castiglioni, P.; Mancia, G.; Parati, G.; Pedotti, A. 24 h sequential spectral analysis of arterial blood pressure and pulse interval in free-moving subjects. IEEE Trans. Biomed. Eng.
**1989**, 36, 1066–1075. [Google Scholar] [CrossRef] [PubMed] - Parati, G.; Castiglioni, P.; Di Rienzo, M.; Omboni, S.; Pedotti, A.; Mancia, G. Sequential spectral analysis of 24-hour blood pressure and pulse interval in humans. Hypertension
**1990**, 16, 414–421. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Castiglioni, P.; Di Rienzo, M.; Veicsteinas, A.; Parati, G.; Merati, G. Mechanisms of blood pressure and heart rate variability: An insight from low-level paraplegia. Am. J. Physiol. Integr. Comp. Physiol.
**2007**, 292, R1502–R1509. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Japundžić, N.; Grichois, M.-L.; Zitoun, P.; Laude, D.; Elghozi, J.-L. Spectral analysis of blood pressure and heart rate in conscious rats: Effects of autonomic blockers. J. Auton. Nerv. Syst.
**1990**, 30, 91–100. [Google Scholar] [CrossRef] - Hu, K.; Ivanov, P.C.; Hilton, M.F.; Chen, Z.; Ayers, R.T.; Stanley, H.E.; Shea, S. Endogenous circadian rhythm in an index of cardiac vulnerability independent of changes in behavior. Proc. Natl. Acad. Sci. USA
**2004**, 101, 18223–18227. [Google Scholar] [CrossRef] [Green Version] - Vandeput, S.; Verheyden, B.; Aubert, A.; Van Huffel, S. Nonlinear heart rate dynamics: Circadian profile and influence of age and gender. Med. Eng. Phys.
**2012**, 34, 108–117. [Google Scholar] [CrossRef] - Sassi, R.; Cerutti, S.; Lombardi, F.; Malik, M.; Huikuri, H.V.; Peng, C.-K.; Schmidt, G.; Yamamoto, Y.; Gorenek, B.; Lip, G.Y.; et al. Advances in heart rate variability signal analysis: Joint position statement by the e-Cardiology ESC Working Group and the European Heart Rhythm Association co-endorsed by the Asia Pacific Heart Rhythm Society. Europace
**2015**, 17, 1341–1353. [Google Scholar] [CrossRef] - Schmidt, T.F.H.; Wittenhaus, J.; Steinmetz, T.F.; Piccolo, P.; Lüpsen, H. Twenty-Four-Hour Ambulatory Noninvasive Continuous Finger Blood Pressure Measurement with PORTAPRES. J. Cardiovasc. Pharmacol.
**1992**, 19, S117. [Google Scholar] [CrossRef] - Constant, I.; Laude, D.; Murat, I.; Elghozi, J.L. Pulse rate variability is not a surrogate for heart rate variability. Clin. Sci.
**1999**, 97, 391–397. [Google Scholar] [CrossRef] [Green Version] - Baselli, G.; Cerutti, S.; Civardi, S.; Liberati, D.; Lombardi, F.; Malliani, A.; Pagani, M. Spectral and cross-spectral analysis of heart rate and arterial blood pressure variability signals. Comput. Biomed. Res.
**1986**, 19, 520–534. [Google Scholar] [CrossRef]

**Figure 1.**Multifractal variability functions F

_{q}(n) for inter-beat interval (IBI) with different orders of detrending polynomials: average over the group of participants. The F

_{q}(n) functions are plotted in blue for q > 0, in black for q = 0, and in red for q < 0; the dashed line is q = 2, second-order moment of the traditional monofractal detrended fluctuation analysis (DFA). Upper panels: F

_{q}(n) estimated with 1st order (linear) detrending during (

**a**) Day and (

**b**) Night. Lower panels: F

_{q}(n) with 2nd-order (quadratic) detrending during (

**c**) Day and (

**d**) Night.

**Figure 2.**Surfaces of multifractal multiscale DFA coefficients, α(q,τ), during Day and Night periods. Average over 14 participants, for scales τ between 8 and 2048 s and moment orders q between −5 and +5; IBI = inter-beat-interval; SBP = systolic blood pressure; DBP = diastolic blood pressure.

**Figure 3.**Day–Night comparison of cross sections of multifractal multiscale DFA coefficients. (

**a**) Cross sections of α(q,τ) of IBI for scales τ between 8 and 2048 s and moment orders q between −5 and +5: average over the group of 14 participants in the Day subperiod; q < 0 in red, q > 0 in blue, q = 0 in black; the dotted line is α for q = 2 (second order moment of the monofractal DFA); (

**b**) α(q,τ) of IBI as in panel (

**a**) for the Night subperiod; (

**c**) color map representing the statistical significance (p value) of the Day vs. Night comparison of IBI scale coefficients calculated at each τ and q after the Wilcoxon signed rank test; (

**d**) α(q, τ) of SBP in the Day subperiod represented as in panel (

**a**); (

**e**) α(q, τ) of SBP in the Night subperiod represented as in panel (

**a**); (

**f**) color map of the Day vs. Night statistical significance for SBP scale coefficients; (

**g**) α(q, τ) of DBP during Day represented as in panel (

**a**); (

**h**) α(q, τ) of DBP during Night represented as in panel (

**a**); (

**i**) color map of the Day vs. Night statistical significance for DBP coefficients.

**Figure 4.**Day–Night comparison of multifractal short- and long-term coefficients. (

**a**) Short-term coefficients, α

_{S}(q), for IBI in Day (open circles) and Night (solid circles) periods and for −5 ≤ q ≤ +5: median ±standard error of the median over N = 14 participants; the * indicates Day vs. Night differences significant at p < 0.05; (

**b**) long-term coefficients, α

_{L}(q), of IBI represented as in panel (

**a**); (

**c**) short-term coefficients of SBP and (

**d**) long-term coefficients of SBP, represented as in panel (

**a**); (

**e**) short-term coefficients and (

**f**) long-term coefficients of DBP, represented as in panel (

**a**).

**Figure 5.**Assessment of nonlinearity during daytime. Upper panels refer to IBI: (

**a**) α(q,τ) coefficients for the original series (average over N = 14 participants, see panel (

**a**) for line colors); (

**b**) α(q,τ) for the corresponding phase-randomized surrogate series; (

**c**) color map of the percentile of the distribution of surrogate estimates in which is the original estimate (average over N = 14 participants). Mid panels refer to SBP: (

**d**) α(q,τ) for the original series; (

**e**) α(q,τ) for the corresponding surrogate series; (

**f**) color map of percentiles. Lower panels refer to DBP: (

**g**) α(q,τ) for the original series; (

**h**) α(q,τ) for the corresponding surrogate series; (

**i**) color map of percentiles.

**Figure 6.**Assessment of nonlinearity during night-time. Upper panels refer to IBI: (

**a**) α(q,τ) for the original series (average over N = 14 participants, see panel (

**a**) for of line colors), (

**b**) α(q,τ) for the corresponding phase-randomized surrogate series, and (

**c**) color map of the percentile of the distribution of surrogate estimates in which is the original estimate (average over N = 14 participants). Mid panels refer to SBP: (

**d**) α(q,τ) for the original series; (

**e**) α(q,τ) for the surrogate series; (

**f**) color map of percentiles. Lower panels refer to DBP: (

**g**) α(q,τ) for the original series; (

**h**) α(q,τ) for the surrogate series; (

**i**) color map of percentiles.

**Figure 7.**Day vs. Night comparison of short-term and long-term indices of nonlinearity. (

**a**) Short-term index, NL

_{S}(q), for IBI in Day (open circles) and Night (solid circles) periods and for −5 ≤ q ≤ +5: median ±standard error of the median over N = 14 participants; the * indicates Day vs. Night differences significant at p < 0.05; (

**b**) long-term nonlinearity index, NL

_{L}(q), of IBI; (

**c**) short-term and (

**d**) long-term nonlinearity index of SBP; (

**e**) short-term and (

**f**) long-term nonlinearity index of DBP.

**Figure 8.**Day–night comparison of the multiscale monofractal DFA coefficients of IBI plotted vs. the block size n in beats. (

**a**) α

_{B}(q,n) calculated for q = 2 (second-order moment of the monofractal DFA) during daytime (red) and nighttime (blue): mean +/− sem over the group of N = 14 participants; the arrows indicate the scale ranges for estimating α

_{1}and α

_{2}as defined by Vanderput et al. in [31] and for estimating α as defined by Hu et al. in [30]. (

**b**) W statistics for the day-night difference in α

_{B}(2,n); when W is above the red horizontal line, the difference at the corresponding scale is significant at p < 5%.

Day | Night | p Value | |
---|---|---|---|

IBI | |||

mean (ms) | 774.4 (97.3) | 1033.7 (174.1) | <0.01 |

total power (ms^{2}) | 11,217 (10,569) | 11,751 (7313) | 0.57 |

VLF power (ms^{2}) | 5885 (5763) | 5905 (3599) | 0.62 |

LF power (ms^{2}) | 1453 (1219) | 2083 (1946) | 0.25 |

HF power (ms^{2}) | 538 (576) | 1219 (1036) | <0.01 |

LF/HF powers ratio | 3.56 (1.4) | 2.21 (1.5) | <0.01 |

SBP | |||

mean (mmHg) | 123.7 (12.8) | 108.6 (17.5) | <0.01 |

total power (mmHg^{2}) | 134.7 (98) | 58.4 (35.4) | <0.01 |

VLF power (mmHg^{2}) | 65.0 (49.9) | 29.3 (19.4) | <0.01 |

LF power (mmHg^{2}) | 22.8 (13.4) | 9.7 (6.2) | <0.01 |

HF power (mmHg^{2}) | 7.3 (4) | 4.0 (2.3) | <0.01 |

DBP | |||

mean (mmHg) | 70.2 (8.9) | 60.2 (10.1) | <0.01 |

total power (mmHg^{2}) | 53.5 (22.6) | 30.4 (17.9) | <0.01 |

VLF power (mmHg^{2}) | 25.8 (12.7) | 15.3 (9.5) | <0.01 |

LF power (mmHg^{2}) | 10.3 (4) | 5.6 (3.5) | <0.01 |

HF power (mmHg^{2}) | 2.8 (1.1) | 1.8 (1.1) | <0.01 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Castiglioni, P.; Omboni, S.; Parati, G.; Faini, A.
Day and Night Changes of Cardiovascular Complexity: A Multi-Fractal Multi-Scale Analysis. *Entropy* **2020**, *22*, 462.
https://doi.org/10.3390/e22040462

**AMA Style**

Castiglioni P, Omboni S, Parati G, Faini A.
Day and Night Changes of Cardiovascular Complexity: A Multi-Fractal Multi-Scale Analysis. *Entropy*. 2020; 22(4):462.
https://doi.org/10.3390/e22040462

**Chicago/Turabian Style**

Castiglioni, Paolo, Stefano Omboni, Gianfranco Parati, and Andrea Faini.
2020. "Day and Night Changes of Cardiovascular Complexity: A Multi-Fractal Multi-Scale Analysis" *Entropy* 22, no. 4: 462.
https://doi.org/10.3390/e22040462