# Ordinal Patterns in Heartbeat Time Series: An Approach Using Multiscale Analysis

## Abstract

**:**

## 1. Introduction

## 2. Terminology and Notation: Ordinal Patterns

**Definition**

**1**

**Definition**

**2.**

**Definition**

**3.**

## 3. The Data

## 4. Numerical Analysis

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

c1rr | c2rr | c3rr | c4rr | c5rr | ||||||

s | Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ |

1 | $0.565761$ | $0.0266769$ | $0.559309$ | $0.0290653$ | $0.567659$ | $0.0230525$ | $0.544987$ | $0.0357687$ | $0.540677$ | $0.0379$ |

25 | $0.565925$ | $0.0263007$ | $0.559512$ | $0.0288102$ | $0.567692$ | $0.0233685$ | $0.54487$ | $0.0355832$ | $0.540604$ | $0.0382577$ |

50 | $0.56497$ | $0.0270585$ | $0.559621$ | $0.0286887$ | $0.567367$ | $0.023753$ | $0.544424$ | $0.0354137$ | $0.544424$ | $0.0354137$ |

75 | $0.56529$ | $0.0264484$ | $0.559033$ | $0.0284339$ | $0.567506$ | $0.0236121$ | $0.544782$ | $0.0354761$ | $0.53852$ | $0.0398734$ |

100 | $0.564318$ | $0.0265197$ | $0.559584$ | $0.029156$ | $0.567742$ | $0.02381$ | $0.543665$ | $0.036225$ | $0.541077$ | $0.0363404$ |

n1rr | n2rr | n3rr | n4rr | n5rr | ||||||

s | Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ |

1 | $0.543486$ | $0.036892$ | $0.544803$ | $0.0366675$ | $0.54408$ | $0.0362984$ | $0.554415$ | $0.0322511$ | $0.528802$ | $0.040533$ |

25 | $0.543122$ | $0.0371364$ | $0.54011$ | $0.037032$ | $0.545062$ | $0.036156$ | $0.554613$ | $0.01318338$ | $0.528907$ | $0.040142$ |

50 | $0.543282$ | $0.0373946$ | $0.54417$ | $0.037435$ | $0.544899$ | $0.0359118$ | $0.554543$ | $0.0317886$ | $0.52937$ | $0.0405128$ |

75 | $0.543006$ | $0.038461$ | $0.54417$ | $0.0382501$ | $0.545281$ | $0.035161$ | $0.554957$ | $0.0314647$ | $0.52946$ | $0.0405214$ |

100 | $0.543139$ | $0.0380009$ | $0.542742$ | $0.0385849$ | $0.544892$ | $0.0355305$ | $0.553622$ | $0.0321451$ | $0.528615$ | $0.040817$ |

a1rr | a2rr | a3rr | a4rr | a5rr | ||||||

s | Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ |

1 | $0.573109$ | $0.0199455$ | $0.574673$ | $0.0185836$ | $0.575351$ | $0.0180058$ | $0.574673$ | $0.01885836$ | $0.574673$ | $0.0185836$ |

25 | $0.573017$ | $0.0196502$ | $0.574934$ | $0.0181619$ | $0.575338$ | $0.0179431$ | $0.574934$ | $0.0181619$ | $0.574934$ | $0.0181619$ |

50 | $0.573126$ | $0.0196991$ | $0.575128$ | $0.0174678$ | $0.575286$ | $0.0178132$ | $0.575128$ | $0.0174678$ | $0.575128$ | $0.0174678$ |

75 | $0.572824$ | $0.0199265$ | $0.574989$ | $0.0190529$ | $0.575573$ | $0.0172598$ | $0.574989$ | $0.0190529$ | $0.574989$ | $0.0190529$ |

100 | $0.573328$ | $0.0188351$ | $0.574735$ | $0.0175133$ | $0.575098$ | $0.0183671$ | $0.574735$ | $0.0175133$ | $0.574735$ | $0.0175133$ |

**Table A2.**Mean and standard deviation for Rényi entropy taking $m=3$ and $1\le \tau \le 100$ for different values of $\alpha $.

Rényi Entropy$\mathit{m}=\mathbf{3}$, $\mathbf{1}\le \tau \le \mathbf{100}$ | ||||||||||

$\alpha =\frac{1}{2}$ | $\alpha =\frac{1}{3}$ | $\alpha =\frac{1}{5}$ | $\alpha =\frac{1}{10}$ | $\alpha =\frac{1}{100}$ | ||||||

Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | |

c1 | 0.99375 | 0.0016742 | 0.995901 | 0.00108809 | 0.997573 | 0.000639407 | 0.998799 | 0.000314675 | 0.999881 | 0.0000310164 |

c2 | 0.992373 | 0.00440852 | 0.994975 | 0.00288105 | 0.997013 | 0.00170015 | 0.998518 | 0.00083924 | 0.999853 | 0.0000829394 |

c3 | 0.995505 | 0.00135784 | 0.997034 | 0.000876451 | 0.998235 | 0.000512168 | 0.999123 | 0.000250985 | 0.999913 | 0.0000246431 |

c4 | 0.997599 | 0.00231847 | 0.998417 | 0.00151459 | 0.999059 | 0.000893696 | 0.999533 | 0.000441177 | 0.999954 | 0.0000436064 |

c5 | 0.996863 | 0.00300583 | 0.997932 | 0.00197033 | 0.998771 | 0.00116577 | 0.99939 | 0.000576648 | 0.999939 | 0.0000570993 |

n1 | 0.998866 | 0.00217053 | 0.999249 | 0.00143686 | 0.999552 | 0.000856727 | 0.999777 | 0.000426199 | 0.999978 | 0.0000424152 |

n2 | 0.998122 | 0.00255504 | 0.99876 | 0.00168197 | 0.999262 | 0.000998495 | 0.999633 | 0.000495129 | 0.999963 | 0.0000491351 |

n3 | 0.99842 | 0.00220677 | 0.998955 | 0.00145612 | 0.999377 | 0.000865973 | 0.99969 | 0.000429973 | 0.999969 | 0.0000427177 |

n4 | 0.99882 | 0.00108625 | 0.999219 | 0.000717013 | 0.999535 | 0.000426684 | 0.999768 | 0.000211997 | 0.999977 | 0.000021077 |

n5 | 0.998207 | 0.00337506 | 0.998817 | 0.00221474 | 0.999296 | 0.00131093 | 0.99965 | 0.000648507 | 0.999965 | 0.0000642098 |

a1 | 0.999883 | 0.0000746988 | 0.999922 | 0.0000496687 | 0.999953 | 0.0000297382 | 0.999977 | 0.0000148454 | 0.999998 | $1.48239*{10}^{-6}$ |

a2 | 0.99982 | 0.0000692071 | 0.99988 | 0.0000460782 | 0.999928 | 0.0000276183 | 0.999964 | 0.0000137984 | 0.999996 | $1.37888*{10}^{-6}$ |

a3 | 0.999918 | 0.0000260503 | 0.999945 | 0.0000173427 | 0.999967 | 0.0000103941 | 0.999984 | $5.19273\times {10}^{-6}$ | 0.999998 | $5.18886\times {10}^{-7}$ |

a4 | 0.999821 | 0.0000319789 | 0.999881 | 0.0000213546 | 0.999929 | 0.0000128298 | 0.999964 | $6.42139\times {10}^{-6}$ | 0.999996 | $6.42724\times {10}^{-7}$ |

a5 | 0.999809 | 0.0000419525 | 0.999873 | 0.0000279582 | 0.999924 | 0.0000167702 | 0.999962 | $8.38334\times {10}^{-6}$ | 0.999996 | $8.38178\times {10}^{-7}$ |

$\alpha =2$ | $\alpha =3$ | $\alpha =5$ | $\alpha =10$ | $\alpha =100$ | ||||||

Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | Mean | $\sigma $ | |

c1 | 0.971488 | 0.00810825 | 0.954373 | 0.0131568 | 0.919262 | 0.022572 | 0.857684 | 0.0325554 | 0.782297 | 0.0321941 |

c2 | 0.96679 | 0.0200377 | 0.948698 | 0.0309635 | 0.915286 | 0.049456 | 0.864263 | 0.072046 | 0.800394 | 0.0849372 |

c3 | 0.980388 | 0.00696758 | 0.969188 | 0.0117617 | 0.945976 | 0.0220604 | 0.90024 | 0.0384958 | 0.82708 | 0.0428283 |

c4 | 0.989489 | 0.0106662 | 0.983489 | 0.0166264 | 0.970898 | 0.0267743 | 0.941862 | 0.0390777 | 0.871617 | 0.0415113 |

c5 | 0.986294 | 0.0134194 | 0.978577 | 0.0205494 | 0.962802 | 0.0320426 | 0.928468 | 0.0437329 | 0.85563 | 0.0435773 |

n1 | 0.995255 | 0.00885824 | 0.992754 | 0.0129209 | 0.987708 | 0.0191147 | 0.975157 | 0.0261347 | 0.918917 | 0.0280182 |

n2 | 0.991919 | 0.0109831 | 0.987451 | 0.0164899 | 0.978208 | 0.0251411 | 0.955622 | 0.0341087 | 0.887612 | 0.0347691 |

n3 | 0.993263 | 0.00924895 | 0.989563 | 0.0136436 | 0.981805 | 0.0203264 | 0.961995 | 0.0276005 | 0.896366 | 0.0297364 |

n4 | 0.994969 | 0.00465057 | 0.992169 | 0.00711745 | 0.98615 | 0.011609 | 0.970086 | 0.0179958 | 0.907732 | 0.0205302 |

n5 | 0.992308 | 0.0144336 | 0.988136 | 0.0209432 | 0.979637 | 0.0296429 | 0.958767 | 0.0378302 | 0.892409 | 0.0385716 |

a1 | 0.999529 | 0.00030556 | 0.99929 | 0.000464605 | 0.998802 | 0.000792727 | 0.997539 | 0.00164017 | 0.980157 | 0.00615212 |

a2 | 0.999274 | 0.000280116 | 0.998906 | 0.000423513 | 0.998161 | 0.000716948 | 0.996254 | 0.00148206 | 0.974081 | 0.00573953 |

a3 | 0.999669 | 0.000105529 | 0.9995 | 0.000159653 | 0.999158 | 0.00027073 | 0.998272 | 0.00056533 | 0.984414 | 0.00439242 |

a4 | 0.999277 | 0.000126117 | 0.998908 | 0.000187545 | 0.998155 | 0.000307843 | 0.996192 | 0.000600026 | 0.972335 | 0.00283459 |

a5 | 0.999225 | 0.000168423 | 0.998827 | 0.000253341 | 0.998015 | 0.000424909 | 0.995882 | 0.000865272 | 0.970802 | 0.00409266 |

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**Figure 1.**The ordinal pattern $(3,1,2)$ is represented by two different vectors, $(2,3,1)$ (left) and $(2.5,3,2)$ (right). The distance between the values is clearly different.

**Figure 2.**Heart beat interval series n1 (

**left**), a1 (

**middle**), and c1 (

**right**). In each figure, heart beat (RR) intervals (in seconds) are plotted.

**Figure 3.**In the left, the permutation entropy (PE) for $m=3$ and $s=1$ is represented for each subject. The blue color represents the congestive heart failure (CHF) group, green has been used for the healthy (H) group, and red for the atrial fibrillation (AF) group. The delay parameter $\tau $ has been represented in the $OX$ axis. In the middle, the same situation has been represented using box plots, grouping the subjects belonging to the same group. Finally, in the right, we have taken the mean with respect to $\tau $ when $1\le \tau \le 100$ for each subject and have represented the corresponding box plot.

**Figure 4.**Average of Rényi entropy and average weighted Rényi entropy for the CHF group and $\alpha =\frac{1}{2}$. We have fixed $m=3$ (dark blue), $m=4$ (light blue), and $m=5$ (green). The top row is for the Rényi permutation entropy (RPE) and the bottom one for the weighted Rényi entropy (WRPE). The projection of the values (contour plot) of the entropy for each fixed embedding dimension for the CHF group have been drawn in 2D graphs.

**Figure 5.**Average Rényi entropy (RPE) (blue) and average weighted Rényi entropy (WRPE) (red) for the CHF group with $\alpha =\frac{1}{2}$, $m=3$, and $s=1,2,3$, and 5. Observe that for $s=1$, the WRPE is higher than the RPE for most values of $\tau $. Nevertheless, for $s\ge 3$ the situation changes.

**Figure 6.**Average Rényi entropy in the top row and average weighted Rényi entropy in the bottom row for the H group and $\alpha =\frac{1}{2}$. We have fixed $m=3$ (dark blue), $m=4$ (light blue), and $m=5$ (green). The projections of the entropy for each fixed embedding dimension for the H group have been drawn in 2D graphs.

**Figure 7.**Average Rényi entropy (blue) and average weighted Rényi entropy (red) for the H group and $\alpha =\frac{1}{2}$, $m=3$, and $s=1,2,3$, and 5.

**Figure 8.**Average Rényi entropy in the top row and average weighted Rényi entropy in the bottom row for the AF group and $\alpha =\frac{1}{2}$. We have fixed $m=3$ (dark blue), $m=4$ (light blue), and $m=5$ (green). The contour of the entropy for each fixed embedding dimension for the AF group has been drawn In 2D graphs.

**Figure 9.**Average Rényi entropy (RPE) (blue) and average weighted Rényi entropy (WRPE) (red) for the AF group and $\alpha =\frac{1}{2}$, $m=3$, and $s=1,2,3$, and 5.

**Figure 10.**Average Rényi entropy in the top row and average weighted Rényi entropy in the bottom row for the CHF group and $\alpha =2$. We have fixed $m=3$ (dark blue), $m=4$ (light blue), and $m=5$ (green). The projections of the entropy for each fixed embedding dimension for the CHF group have been drawn in 2D graphs.

**Figure 11.**Average Rényi entropy (RPE, blue) and average weighted Rényi entropy (WRPE, red) for the CHF group and $\alpha =2$, $m=3$, and $s=1,2,3$, and 5. Observe that for $s=1$, the WRPE is higher for most values of $\tau $; nevertheless, for $s=5$ the situation changes.

**Figure 12.**Average Rényi entropy and average weighted Rényi entropy for the H group and $\alpha =2$. We have fixed $m=3$ (dark blue), $m=4$ (light blue), and $m=5$ (green). The projections of the entropy for each fixed embedding dimension for the H group have been drawn in 2D graphs.

**Figure 13.**Average Rényi entropy (RPE, blue) and average weighted Rényi entropy (WRPE, red) for the H group and $\alpha =2$, $m=3$, and $s=1,2,3$, and 5.

**Figure 14.**Average Rényi entropy in the top row and average weighted Rényi entropy in the bottom row for the AF group and $\alpha =2$. We have fixed $m=3$ (dark blue), $m=4$ (light blue), and $m=5$ (green). The contour plots of the entropy for each fixed embedding dimension for the AF group have been drawn in 2D graphs.

**Figure 15.**Average of Rényi entropy (RPE, blue) and average of weighted Rényi entropy (WRPE, red) for the AF group and $\alpha =\frac{1}{2}$, $m=3$, and $s=1,2,3$, and 5.

**Figure 16.**For $\alpha =\frac{1}{2}$, values of s and $\tau $ for which the three groups are separated by the multiscale Rényi permutation entropy (MRPE). In the 3D graphs, the OX axis represents the parameter s, where $1\le s\le 20$, in the OY axis is represented the parameter $\tau $, $1\le \tau \le 100$, and finally in the OZ axis is represented the difference between the three groups, when the groups are differentiated by the MRPE.

**Figure 17.**Fixing $\alpha =2$, values of s and $\tau $ for which the three groups are separated by the MRPE are shown. In the 3D graphs, the OX axis represents the parameter s, where $1\le s\le 20$, in the OY axis is represented the parameter $\tau $, $1\le \tau \le 100$, and finally in the OZ axis is represented the difference between the three groups, when the groups are differentiated by the MRPE.

**Figure 18.**The top row shows the RPE for $s=8$, $\tau =1$ and $m=3$ (left), $m=4$ (middle), and $m=5$ (right). In red is the AF group, in green is the H group, and the CHF results are in blue. The OX axis represents $\alpha \in [2,100]$. and the RPE is represented in the OY axis. The bottom graph shows the results for $m=3$, $s=8$, and $\tau =1$ for $\alpha \in (0,0.5)$.

**Figure 19.**Rényi entropy for $\alpha =\frac{1}{2}$ and embedding dimension $m=3$. We have drawn the Rényi entropy for the CHF group in blue, for the healthy group in green, and for the AF group in red, on the left. A zoomed-in view of the right figure is shown in the middle, where the differences between the three groups when $\tau $ runs over the range can be appreciated. Finally, box plots for each group have been represented on the right.

**Figure 20.**Rényi entropy for $\alpha =2$ and embedding dimension $m=3$. We have drawn the Rényi entropy for the CHF group in blue, for the healthy group in green, and for the AF group in red on the left.

**Figure 21.**Rényi entropy for $\alpha =100$ and embedding dimension $m=3$. We have drawn the Rényi entropy for the CHF group in blue, for the healthy group in green, and for the AF group in red on the left. A zoomed-in view of the right figure is shown in the middle, where the differences between the three groups when $\tau $ runs over the range can be appreciated. Finally, box plots for each group have been represented on the right.

**Table 1.**For $m=3$ and $\alpha =\frac{1}{2}$, values of s and $\tau $ for which the three groups are disjoint.

s | $\mathit{\tau}$ |
---|---|

1 | 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100 |

2 | 3, 4, 5, 6, 7, 8, 9, 10, 11, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62 |

3 | 2, 3, 4, 5, 6, 19, 20, 23, 24, 25, 26, 27, 28, 29 |

4 | 2, 3, 4, 5, 19, 20, 21 |

5 | 2, 3, 4, 15, 16, 17, 79 |

6 | 1, 2, 3, 13 |

7 | 1, 2, 11 |

8 | 1, 2, 10 |

9 | 1, 2 |

10 | 1, 2 |

11 | 1 |

12 | 1 |

13 | 1 |

14 | 1 |

15 | 1 |

16 | 1 |

17 | 1 |

18 | 1 |

**Table 2.**Fixing $\alpha =\frac{1}{2}$, values for the parameters $m,s,\tau \in \mathbb{N}$ in the ranges $3\le m\le 5$, $1\le s\le 20$, and $1\le \tau \le 100$ such that the CHF group, H group, and AF group are disjoint. The difference is included, as well as the order relation between them.

m | s | $\mathit{\tau}$ | Difference | Relation |
---|---|---|---|---|

3 | 1 | 3 | 0.000111612 | $\mathrm{CHF}>\mathrm{AF}>\mathrm{H}$ |

4 | 1 | 3 | 0.000111612 | $\mathrm{CHF}>\mathrm{AF}>\mathrm{H}$ |

5 | 1 | 3 | 0.000111612 | $\mathrm{CHF}>\mathrm{AF}>\mathrm{H}$ |

5 | 11 | 1 | 0.0018635 | $\mathrm{AF}>\mathrm{H}>\mathrm{CHF}$ |

5 | 6 | 2 | 0.00249138 | $\mathrm{AF}>\mathrm{H}>\mathrm{CHF}$ |

**Table 3.**Fixing $\alpha =2$, values for the parameters $m,s,\tau \in \mathbb{N}$ in the ranges $3\le m\le 5$, $1\le s\le 20$, and $1\le \tau \le 100$ such that the CHF group, H group, and AF group are disjoint. The difference and the order relation between them are also included.

m | s | $\mathit{\tau}$ | Difference | Relation |
---|---|---|---|---|

3 | 1 | 3 | 0.000435339 | $\mathrm{CHF}>\mathrm{AF}>\mathrm{H}$ |

4 | 11 | 1 | 0.00213423 | $\mathrm{AF}>\mathrm{H}>\mathrm{CHF}$ |

4 | 6 | 2 | 0.00461844 | $\mathrm{AF}>\mathrm{H}>\mathrm{CHF}$ |

4 | 1 | 3 | 0.000435339 | $\mathrm{CHF}>\mathrm{AF}>\mathrm{H}$ |

5 | 11 | 1 | 0.00213423 | $\mathrm{AF}>\mathrm{H}>\mathrm{CHF}$ |

5 | 6 | 2 | 0.00461844 | $\mathrm{AF}>\mathrm{H}>\mathrm{CHF}$ |

5 | 9 | 1 | 0.0105166 | $\mathrm{AF}>\mathrm{H}>\mathrm{CHF}$ |

5 | 10 | 1 | 0.00585264 | $\mathrm{AF}>\mathrm{H}>\mathrm{CHF}$ |

5 | 5 | 2 | 0.00935592 | $\mathrm{AF}>\mathrm{H}>\mathrm{CHF}$ |

5 | 1 | 3 | 0.000435339 | $\mathrm{CHF}>\mathrm{AF}>\mathrm{H}$ |

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**MDPI and ACS Style**

Muñoz-Guillermo, M.
Ordinal Patterns in Heartbeat Time Series: An Approach Using Multiscale Analysis. *Entropy* **2019**, *21*, 583.
https://doi.org/10.3390/e21060583

**AMA Style**

Muñoz-Guillermo M.
Ordinal Patterns in Heartbeat Time Series: An Approach Using Multiscale Analysis. *Entropy*. 2019; 21(6):583.
https://doi.org/10.3390/e21060583

**Chicago/Turabian Style**

Muñoz-Guillermo, María.
2019. "Ordinal Patterns in Heartbeat Time Series: An Approach Using Multiscale Analysis" *Entropy* 21, no. 6: 583.
https://doi.org/10.3390/e21060583