# Amplitude- and Fluctuation-Based Dispersion Entropy

^{*}

## Abstract

**:**

## 1. Introduction

**s**is $-{\sum}_{k=1}^{N}Pr\{{s}_{k}\}log(Pr\{{s}_{k}\})$, where $Pr\{{s}_{k}\}$ is the probability of occurrence of pattern ${s}_{k}$ [11]. When all the probability values are equal, the maximum entropy occurs, while if one probability is certain and the others are impossible, minimum entropy is achieved [9,11].

## 2. Methods

#### 2.1. Dispersion Entropy (DispEn) with Different Mapping Techniques

- (1)
- First, ${x}_{j}(j=1,2,\text{}\dots \text{},N)$ are mapped to $\mathit{c}$ classes with integer indices from 1 to $\mathit{c}$. The classified signal is ${u}_{j}(j=1,2,\text{}\dots \text{},N)$. A number of linear and nonlinear mapping techniques, introduced in Section 2.3, can be used in this step.
- (2)
- Time series ${\mathbf{u}}_{i}^{m,c}$ are made with embedding dimension m and time delay d according to ${\mathbf{u}}_{i}^{m,c}=\{{u}_{i}^{c},{u}_{i+d}^{c},\text{}\dots \text{},{u}_{i+(m-1)d}^{c}\}$, $i=1,2,\dots ,N-(m-1)d$ [9,10]. Each time series ${\mathbf{u}}_{i}^{m,c}$ is mapped to a dispersion pattern ${\pi}_{{v}_{0}{v}_{1}\text{}\dots \text{}{v}_{m-1}}$, where ${u}_{i}^{c}={v}_{0}$, ${u}_{i+d}^{c}={v}_{1}$,..., ${u}_{i+(m-1)d}^{c}={v}_{m-1}$. The number of possible dispersion patterns assigned to each vector ${\mathbf{u}}_{i}^{m,c}$ is equal to ${c}^{m}$, since the signal ${\mathbf{u}}_{i}^{m,c}$ has m elements and each can be one of the integers from 1 to c [9].
- (3)
- For each of ${c}^{m}$ potential dispersion patterns ${\pi}_{{v}_{0}\text{}\dots \text{}{v}_{m-1}}$, relative frequency is obtained as follows:$$\begin{array}{c}\hfill p({\pi}_{{v}_{0}\dots {v}_{m-1}})=\frac{\#\{i\left(\right)open="|"\; close>i\le N-(m-1)d,{\mathbf{u}}_{i}^{m,c}\phantom{\rule{4.pt}{0ex}}\mathrm{has}\mathrm{type}{\pi}_{{v}_{0}\dots {v}_{m-1}}}{\}}N-(m-1)d,\end{array}$$
- (4)
- Finally, based on the Shannon’s definition of entropy, the DispEn value is calculated as follows:$$\mathrm{DispEn}(\mathbf{x},m,c,d)=-\sum _{\pi =1}^{{c}^{m}}p({\pi}_{{v}_{0}\dots {v}_{m-1}})\xb7ln\left(\right)open="("\; close=")">p({\pi}_{{v}_{0}\dots {v}_{m-1}})$$

**x**. For simplicity, we set $d=1$, $m=2$, and $c=3$. The ${3}^{2}=9$ potential dispersion patterns are depicted on the right of Figure 1. ${x}_{j}$ ($j=1,2,\dots ,10$) are linearly mapped into three classes with integer indices from 1 to 3, as can be seen in Figure 1. Next, a window with length 2 (embedding dimension) moves along the signal and the number of each of the dispersion patterns is counted. The relative frequency is shown on the bottom left of Figure 1. Finally, using Equation (2), the DispEn value of

**x**is equal to $-(\frac{2}{9}ln(\frac{2}{9})+\frac{2}{9}ln(\frac{2}{9})+\frac{2}{9}ln(\frac{2}{9})+\frac{1}{9}ln(\frac{1}{9})+\frac{1}{9}ln(\frac{1}{9})+\frac{1}{9}ln(\frac{1}{9}))=1.7351$.

#### 2.2. Fluctuation-Based Dispersion Entropy (FDispEn)

**x**is equal to $-(\frac{1}{8}ln(\frac{1}{8})+\frac{1}{8}ln(\frac{1}{8})+\frac{2}{8}ln(\frac{2}{8})+\frac{2}{8}ln(\frac{2}{8})+\frac{2}{8}ln(\frac{2}{8}))=1.5596$.

#### 2.3. Mapping Approaches Used in DispEn and FDispEn

**x**, respectively.

**x**is calculated as follows:

^{th}element of the classified signal and rounding involves either increasing or decreasing a number to the next digit [9]. It is worth noting that DispEn with NCDF and DispEn with linear mapping were compared by the use of several synthetic time series and four biomedical and mechanical datasets [9]. The results illustrated the superiority of DispEn with NCDF over DispEn with linear mapping.

## 3. Parameters of DispEn and FDispEn

#### 3.1. Effect of Number of Classes, Embedding Dimension, and Signal Length on DispEn and FDispEn

#### 3.2. Effect of Number of Classes and Noise Power on DispEn and FDispEn

## 4. Evaluation of Mapping Approaches for DispEn and FDispEn

## 5. Univariate Entropy Methods vs. Changes from Periodicity to Non-Periodic Nonlinearity

## 6. Comparison Between SampEn, PerEn and Its Improvements, and Newly Developed DispEn and FDispEn

#### 6.1. SampEn vs. DispEn and FDispEn

- SampEn values for short signals are either undefined or unreliable, as in its algorithm, the number of matches whose differences are smaller than a defined threshold is counted. When the time series length is too small, this number may be 0, leading to undefined values [16,53]. However, the results obtained by DispEn, FDispEn, and PerEn are always defined. To illustrate this issue, we created 40 realizations of white noise with length 50 sample points. The mean and median of DispEn, FDispEn, PerEn, and SampEn values for the 40 realizations are shown in Figure 8. The results show that SampEn, unlike DispEn, FDispEn, and PerEn, yield undefined values. Note that we set $m=2$ for SampEn, DispEn, and FDispEn, and $m=3$ for PerEn, as advised before.

#### 6.2. PerEn and Its Improvements vs. DispEn and FDispEn

- PerEn considers only the order of amplitude values, and, thus, some information regarding the amplitude values themselves may be ignored [18]. For example, the embedded vectors $\{1,10,2\}$ and $\{1,3,2\}$ have similar permutations, leading to the same motif (0,2,1) ($m=3$) because the extent of the differences between sequential samples is not considered in the original definition of PerEn. To alleviate this deficiency, modified PerEn (MPerEn) based on mapping equal values into the same symbol was developed [17]. However, the second and third shortcomings were not addressed by MPerEn. Amplitude-aware PerEn (AAPerEn) deals with the problem with adding a variable contribution, depending on amplitude, instead of a constant number to each level in the histogram representing the probability of each motif [7]. It was also addressed by the use of modified ordinal patterns [56]. Mapping data to a number of classes based on their amplitude values makes DispEn and FDispEn deal with this issue as well.
- When there are equal values in the embedded vector, Bandt and Pompe [10] proposed ranking the possible equalities based on their order of emergence or solving this condition by adding noise. Considering the first alternative, for instance, the permutation pattern for both the embedded vectors $\{1,2,4\}$ and $\{1,4,4\}$ are (0,1,2) ($m=3$). As another example, assume $\mathbf{z}\mathbf{1}=\{1,2,2,2\}$ and $\mathbf{z}\mathbf{2}=\{1,2,3,4\}$. The PerEn with $m=3$ of $\mathbf{z}\mathbf{1}$ is exactly the same as $\mathbf{z}\mathbf{2}$, both equalling 0 although, unlike $\mathbf{z}\mathbf{1}$, $\mathbf{z}\mathbf{2}$ is strictly ascending. Adding noise may not lead to a precise answer because, for example, the embedded vector $\{1,5,5\}$ has two possible permutation patterns as (0,1,2) and (0,2,1) and there are not any differences between them. It should be noted that this issue is particularly relevant for digitized signals with large quantization steps. Fadlallah et al. have recently proposed weighted PerEn (WPerEn) to weight the motif counts by statistics derived from the time series patterns [8]. However, WPerEn does not take into account the first and third alleviations of PerEn. It was addressed in AAPerEn [7] as well. Assigning close amplitude values to an equal class, FDispEn and DispEn deal with this deficiency.
- PerEn is sensitive to noise (even when the SNR of a signal is high), since a small change in amplitude value may vary the order relations among amplitudes. For instance, noise on $\mathbf{z}\mathbf{3}=\{1,2,2.01\}$ may alter the motif from (0,1,2) to (0,2,1). This problem is present for WPerEn, MPerEn, AAPerEn, and the approach developed in [56]. However, DispEn and FDispEn address the problem with mapping data into a few classes and, thus, a small change in amplitude will probably not alter the (index of) class.

## 7. Computation Cost of DispEn, FDispEn, and PerEn

## 8. Forbidden Amplitude- and Fluctuation-Based Dispersion Patterns

- Step 1: Null hypothesis. We have all the dispersion patterns, while the permutation pattern $({\ell}_{1},{\ell}_{2},\dots ,{\ell}_{m})$ does not exist for the signal
**x**. - Step 2: Rejection of null hypothesis. As the permutation pattern $({\ell}_{1},{\ell}_{2},\dots ,{\ell}_{m})$ does not exist, we do not have any dispersion patterns sorted as $({\ell}_{1},{\ell}_{2},\dots ,{\ell}_{m})$. This is in contradiction with the fact that we have all the dispersion patterns for
**x**. Hence, the null hypothesis is rejected. - Step 3: Conclusion. When we have all the dispersion patterns, all the permutation patterns are present too. It confirms the fact that a forbidden permutation pattern leads to several forbidden dispersion patterns. Thus, if a signal is deterministic, and so does not have several permutation patterns, there are a number of forbidden dispersion patterns. Consequently, lack of dispersion patterns, like permutation patterns [57,58], reflects the deterministic behavior of a signal.

**x**, we do not have the following dispersion patterns: (2,3,1), (2,4,1), (2,5,1), (2,6,1), (3,4,1), (3,5,1), (3,6,1), (4,5,1), (4,6,1), (5,6,1), (3,4,2), (3,5,2), (3,6,2), (4,5,2), (4,6,2), (5,6,2), (4,5,3), (4,6,3), (5,6,3), and (5,6,4); and fluctuation-based dispersion patterns: (1,−2), (2,−3), (3,−4), (4,−5), (1,−3), (2,−4), (3,−5), (1,−4), (2,−5), (1,−5), (1,−2), (2,−3), (3,−4), (1,−3), (2,−4), (1,−4), (1,−2), (2,−3), (1,−3), and (1,−2). This demonstrates that lack of a permutation pattern results in lack of several dispersion and fluctuation-based dispersion patterns. Accordingly, as permutation patterns are used to discriminate deterministic from stochastic series based on lack of permutation patterns [57,58], dispersion and fluctuation-based patterns are able to be utilized as well.

## 9. Applications of DispEn and FDispEn to Biomedical Time Series

#### 9.1. Blood Pressure in Rats

#### 9.2. Gait Maturation Database

## 10. Conclusions

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Illustration of the DispEn algorithm using linear mapping of $\mathbf{x}=\{3.6,4.2,1.2,3.1,4.2,2.1,3.3,4.6,6.8,8.4\}$ with the number of classes 3 and embedding dimension 2.

**Figure 2.**Mean and SD of results obtained by the DispEn and FDispEn with logsig and different values of embedding dimension and number of classes for 40 realizations of univariate white noise. Logarithm scale for both of the axes is used.

**Figure 3.**Average and SD of $NrmEntN=\frac{\mathrm{entropy}\mathrm{of}\mathrm{a}\mathrm{series}\mathrm{with}\mathrm{noise}}{\mathrm{entropy}\mathrm{of}\mathrm{a}\mathrm{series}\mathrm{without}\mathrm{noise}}$ values obtained by the DispEn using logsig with a different number of classes computed from the logistic map with additive 40 independent realizations of WGNs with different noise power. NrmEntN compares the sensitivity of DispEn to WGN with different SNRs.

**Figure 4.**Average and SD of $NrmEntN=\frac{\mathrm{entropy}\mathrm{of}\mathrm{a}\mathrm{series}\mathrm{with}\mathrm{noise}}{\mathrm{entropy}\mathrm{of}\mathrm{a}\mathrm{series}\mathrm{without}\mathrm{noise}}$ values obtained by the PerEn, and DispEn and FDispEn with different mapping techniques computed from the logistic map with additive 40 independent realizations of WGNs with different noise power. NrmEntN compares the sensitivity of each method to WGN with different SNRs.

**Figure 5.**Mean and SD of entropy values obtained by DispEn and FDispEn with different mapping techniques and PerEn, computed from 40 different white noise (red colour), pink noise (blue colour), and brown noise (black colour).

**Figure 6.**Logistic map with parameter $\alpha $ changing from 3.5 to 3.99 and entropy values of the logistic map to understand better SampEn, PerEn, DispEn, and FDispEn.

**Figure 7.**Average and SD of entropy values obtained by the DispEn, FDispEn, and SampEn with different numbers of classes (for DispEn and FDispEn) and different threshold values (SampEn) using a MIX process evolving from randomness to periodic oscillations. We used a window with length 1500 samples moving along the MIX process (temporal window).

**Figure 8.**Mean and median of results obtained by (

**b**) DispEn; (

**c**) FDispEn with logsig; (

**d**) PerEn and (

**e**) SampEn, for 40 realizations of (

**a**) white noise.

**Figure 9.**Mean and median of results obtained by (

**b**) DispEn; (

**c**) FDispEn with logsig; (

**d**) PerEn and (

**e**) SampEn, for 20 realizations of (

**a**) ${x}_{i}=sin(i/20)+0.3\eta $.

**Figure 10.**Mean and SD of the normalized number of forbidden amplitude- and fluctuation-based dispersion and permutation patterns ($\frac{\mathrm{number}\mathrm{of}\mathrm{forbidden}\mathrm{patterns}}{\mathrm{potential}\mathrm{number}\mathrm{of}\mathrm{patterns}}$) as functions of the signal length.

**Figure 11.**Mean and median of results obtained by PerEn, LZC, SampEn, and DispEn and FDispEn with logsig from salt-sensitive (SS) vs. salt protected (SP) rats’ blood pressure signals.

**Figure 12.**Mean and median of results obtained by PerEn, LZC, SampEn, and DispEn and FDispEn with logsig for young and elderly children’s stride-to-stride recordings.

**Table 1.**CVs of DispEn and FDispEn with logsig, and SampEn values for the MIX process with $p=0.5$ and length 1000 samples.

Method | $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}2$ | $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}4$ | $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}6$ | $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}8$ | $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}10$ |

DispEn | 0.0021 | 0.0034 | 0.0045 | 0.0041 | 0.0048 |

$c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}2$ | $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}4$ | $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}6$ | $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}8$ | $c\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}10$ | |

FDispEn | 0.0078 | 0.0064 | 0.0040 | 0.0043 | 0.0049 |

$r\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.1\phantom{\rule{3.33333pt}{0ex}}\times $ SD | $r\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.2\phantom{\rule{3.33333pt}{0ex}}\times $ SD | $r\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.3\phantom{\rule{3.33333pt}{0ex}}\times $ SD | $r\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.4\phantom{\rule{3.33333pt}{0ex}}\times $ SD | $r\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}0.5\phantom{\rule{3.33333pt}{0ex}}\times $ SD | |

SampEn | 0.0604 | 0.0342 | 0.0224 | 0.0174 | 0.0150 |

**Table 2.**Comparison between DispEn and FDispEn and SampEn, PerEn, and AAPerEn in terms of ability to characterize short signals, sensitivity to noise, type of entropy, and computational cost.

Characteristics | DispEn | FDispEn | AAPerEn | PerEn | SampEn |
---|---|---|---|---|---|

Short signals | reliable | reliable | reliable | reliable | undefined |

Sensitivity to noise | no | no | yes | yes | no |

Type of entropy | ShEn | ShEn | ShEn | ShEn | ConEn |

Computational cost | O(N) | O(N) | O(N) | O(N) | O(${N}^{2}$) |

**Table 3.**Computational time of DispEn and FDispEn with logsig, SampEn, and PerEn with different embedding dimension values and signal lengths.

Number of Samples | 300 | 1000 | 3000 | 10,000 | 30,000 | 100,000 |
---|---|---|---|---|---|---|

DispEn ($m=2$) | 0.0022 s | 0.0022 s | 0.0025 s | 0.0057 s | 0.0080 s | 0.0225 s |

DispEn ($m=3$) | 0.0028 s | 0.0035 s | 0.0076 s | 0.0115 s | 0.0284 s | 0.0888 s |

DispEn ($m=4$) | 0.0084 s | 0.0094 s | 0.0205 s | 0.0505 s | 0.1422 s | 0.4752 s |

FDispEn ($m=2$) | 0.0022 s | 0.0025 s | 0.0028 s | 0.0034 s | 0.0062 s | 0.0175 s |

FDispEn ($m=3$) | 0.0025 s | 0.0031 s | 0.0038 s | 0.0062 s | 0.0150 s | 0.0490 s |

FDispEn ($m=4$) | 0.0054 s | 0.0064 s | 0.0120 s | 0.0284 s | 0.0699 s | 0.2535 s |

SampEn ($m=2$) | 0.0023 s | 0.0208 s | 0.1841 s | 1.8478 s | 16.8394 s | 193.1970 s |

SampEn ($m=3$) | 0.0022 s | 0.0206 s | 0.1808 s | 1.8337 s | 16.9200 s | 189.4041 s |

SampEn ($m=4$) | 0.0019 s | 0.0193 s | 0.1631 s | 1.8322 s | 16.5596 s | 189.1037 s |

PerEn ($m=2$) | 0.0014 s | 0.0015 s | 0.0016 s | 0.0020 s | 0.0034 s | 0.0099 s |

PerEn ($m=3$) | 0.0014 s | 0.0016 s | 0.0016 s | 0.0024 s | 0.0043 s | 0.0115 s |

PerEn ($m=4$) | 0.0015 s | 0.0016 s | 0.0019 s | 0.0026 s | 0.0054 s | 0.0113 s |

**Table 4.**Differences between results for SS vs. SSBN13 Dahl rats (blood pressure data), and for elderly vs. young children (gait maturation dataset) obtained by DispEn and FDispEn with logsig, LZC, SampEn, and PerEn based on the Hedges’ g effect size.

Dataset | DispEn | FDispEn | PerEn | LZC | SampEn |
---|---|---|---|---|---|

Blood pressure | 1.35 (very large) | 0.46 (medium) | 0.31 (small) | 1.74 (huge) | 0.84 (large) |

Gait maturation | 0.74 (large) | 0.75 (large) | 0.63 (medium) | 0.16 (small) | 0.79 (large) |

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**MDPI and ACS Style**

Azami, H.; Escudero, J.
Amplitude- and Fluctuation-Based Dispersion Entropy. *Entropy* **2018**, *20*, 210.
https://doi.org/10.3390/e20030210

**AMA Style**

Azami H, Escudero J.
Amplitude- and Fluctuation-Based Dispersion Entropy. *Entropy*. 2018; 20(3):210.
https://doi.org/10.3390/e20030210

**Chicago/Turabian Style**

Azami, Hamed, and Javier Escudero.
2018. "Amplitude- and Fluctuation-Based Dispersion Entropy" *Entropy* 20, no. 3: 210.
https://doi.org/10.3390/e20030210