# Entropy Measures as Descriptors to Identify Apneas in Rheoencephalographic Signals

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Entropy Definitions

#### 2.1.1. Shannon Entropy

#### 2.1.2. Corrected Conditional Entropy

#### 2.1.3. Approximate Entropy

^{m}dimensional space are then created:

_{i}

^{m}(r) is computed as:

^{m}and 30

^{m}points [46,47].

^{m}criteria. Finally, chosen r values covered the range of 0.05 to 0.3 times the standard deviation of the input signal.

#### 2.1.4. Sample Entropy

_{1}) is higher than the value obtained with another signal (x

_{2}) for a pair of m and r values, a new m-r pair would still provide higher SampEn values for the signal x

_{1}[51]. Nonetheless, Castiglioni et al. [50] detected inconsistencies in SampEn calculations when studying mechanomyographic signals for certain m values, and Yentes et al. [28] published similar findings for some r choices, suggesting that under certain conditions SampEn can also be affected by inconsistencies.

#### 2.1.5. Fuzzy Entropy

^{m}is calculated as:

#### 2.2. Experimental Protocol

#### 2.3. Data Analysis

- Maximum amplitude (Max)
- Minimum amplitude (Min)
- Amplitude range (Range)
- Slope of the increasing edge (α)
- Area under the curve of each cardiac cycle (Area)
- Time between two consecutive maximums (Δtmax)
- Time between two consecutive minimums (Δtmin)
- Time between a minimum and its consecutive maximum (Δtmin-max)

## 3. Results

#### 3.1. Parameters Selection for Each Entropy Metric

#### 3.2. Final Parameter and Entropy Values

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Corrected conditional entropy (CCE(N, m, ε)) values of apnea and baseline recordings as a function of (

**a**) the quantification intervals (ε), (

**b**) the embedding dimension (m) and (

**c**) the signal length (N). The corresponding statistical significance (p-value) of the differences between apnea and baseline recordings is presented in (

**d**), (

**e**) and (

**f**), respectively.

**Figure 3.**The influence of the signal length and the number of quantization levels in the regularity index (ρ) is analyzed in: (

**a**) values of ρ as a function of the number of quantification intervals (ε) and (

**b**) values of ρ as a function of the signal length (N). The results of the statistical analysis (p-value) comparing apnea and baseline signals using this entropy indexes are shown in: (

**c**) p-values versus the number of quantification intervals and (

**d**) p-values versus the signal length.

**Figure 4.**Values of the entropy CCE (ε = 20; N = 2000) as a function of the embedding dimension m for all apnea (

**a**) and baseline (

**b**) recordings, including their median values (thick black line).

**Figure 5.**Values of the entropies (

**a**) ApEn, (

**b**) SampEn and (

**c**) FuzzyEn as a function of the number of samples (N) and the dimension (m) for apnea (solid line) and baseline segments (dashed line).

**Figure 6.**Entropy values of ApEn(2,r,2000), SampEn(2,r,2000) and FuzzyEn(2,2,r,2000) as a function of r for apnea and baseline recordings (

**a**–

**c**) and the corresponding p-values (

**d**–

**f**).

**Figure 7.**(

**a**) Median FuzzyEn values as a function of n, including the 25–75 interquartile range (colored area); (

**b**) standard deviation of FuzzyEn as a function of n; (

**c**) p-value obtained comparing FuzzyEn values in apnea and baseline groups as a function of n.

**Figure 8.**Standard deviation of ApEn(2, r, 2000), SampEn(2, r, 2000) and FuzzyEn(2, 2, r, 2000) as a function of r comparing baseline and apnea segments.

**Figure 9.**Receiver operating characteristic (ROC) curves of all entropy metrics providing statistically significant differences between apnea and baseline recordings.

**Figure 10.**Boxplot of all selected entropy metrics, showing the median values (horizontal red lines) and outliers (red crosses).

Signal Length (N) (Samples) | Embedding Dimension (m) | Filtering Level (r) | Quantization Intervals (ε) | Fuzzy Function Gradient (n) | |
---|---|---|---|---|---|

Shannon Entropy | 1000 to 4000 | 2 to 4 | - | 10 to 50 | - |

Corrected Conditional Entropy | 1000 to 4000 | 2 to 4 * | - | 10 to 50 | - |

Approximate Entropy | 1000 to 4000 | 2 to 4 | 0.05 to 0.3 | - | - |

Sample Entropy | 1000 to 4000 | 2 to 4 | 0.05 to 0.3 | - | - |

Fuzzy entropy | 1000 to 4000 | 2 to 4 | 0.05 to 0.3 | - | 2 to 10 |

**Table 2.**Statistical significance values of the entropies ApEn, SampEn and FuzzyEn for apnea detection as a function of the embedding dimension (m) and the signal length (N).

N = 1000 | N = 2000 | N = 3000 | N = 4000 | |
---|---|---|---|---|

ApEn | ||||

m = 2 | 0.0044 | 0.0006 | 0.0006 | 0.0004 |

m = 3 | 0.0131 | 0.0014 | 0.0013 | 0.0004 |

m = 4 | 0.6379 | 0.4915 | 0.5376 | 0.3391 |

SampEn | ||||

m = 2 | 0.048 | 0.014 | 0.017 | 0.017 |

m = 3 | 0.166 | 0.195 | 0.145 | 0.136 |

m = 4 | 0.387 | 0.280 | 0.183 | 0.172 |

FuzzyEn | ||||

m = 2 | 0.00076 | 0.00013 | 0.00014 | 0.00012 |

m = 3 | 0.00086 | 0.00018 | 0.00016 | 0.00014 |

m = 4 | 0.00329 | 0.00042 | 0.00022 | 0.00021 |

**Table 3.**Mean values and standard deviation (std) of all entropy metrics when comparing apnea and baseline recordings. The values of the set of parameters that best describe these entropies are included. Statistics as p-value, area under the curve (AUC) and accuracy (acc) are provided to assess the ability of the entropy metrics to distinguish between apnea and baseline.

Entropy Measure | Parameters | Apnea Mean ± std | Baseline Mean ± std | p-Value | AUC | acc (%) |
---|---|---|---|---|---|---|

ApEn | r = 0.25 m = 2 N = 2000 | 0.155 ± 0.045 | 0.118 ± 0.035 | 0.0003 | 0.789 | 69.8 |

SampEn | r = 0.25 m = 2 N = 2000 | 0.111 ± 0.031 | 0.092 ± 0.022 | 0.0132 | 0.698 | 60.4 |

FuzzyEn | r = 0.25 m = 2 N = 2000 n = 2 | 0.021 ± 0.009 | 0.015 ± 0.006 | 0.0001 | 0.809 | 69.8 |

CCE | ε = 20 m = 2 N = 2000 | 0.581 ± 0.063 | 0.518 ± 0.075 | 0.0016 | 0.744 | 67.9 |

ρ | ε = 20 N = 2000 | 0.838 ± 0.024 | 0.854 ± 0.017 | 0.0084 | 0.713 | 62.3 |

**Table 4.**Mean values and standard deviation (std) of all the features extracted from the linear time series and p_value statistics illustrating their ability to distinguish between apnea and baseline signals.

Parameter | Units | Apnea Mean ± std | Baseline Mean ± std | p-Value |
---|---|---|---|---|

Max | Ω | 0.041 ± 0.014 | 0.045 ± 0.017 | 0.356 |

Min | Ω | −0.051 ± 0.017 | −0.054 ± 0.018 | 0.523 |

Range | Ω | 0.092 ± 0.028 | 0.099 ± 0.033 | 0.376 |

Δtmax | samples | 238.7 ± 22.1 | 254.9 ± 43.2 | 0.084 |

Δtmin | samples | 242.11 ± 23.2 | 248.6 ± 38.8 | 0.455 |

Δtmin-max | samples | 52.88 ± 27.36 | 60.56 ± 24.76 | 0.217 |

α | a.u. | 0.002 ± 0.001 | 0.002 ± 0.001 | 0.406 |

Area | Ω.s | 12.453 ± 4.766 | 13.471 ± 4.856 | 0.446 |

δmax | Ω/s | 0.006 ± 0.002 | 0.005 ± 0.002 | 0.272 |

δrange | Ω/s | 0.007 ± 0.002 | 0.007 ± 0.002 | 0.145 |

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**MDPI and ACS Style**

González, C.; Jensen, E.; Gambús, P.; Vallverdú, M.
Entropy Measures as Descriptors to Identify Apneas in Rheoencephalographic Signals. *Entropy* **2019**, *21*, 605.
https://doi.org/10.3390/e21060605

**AMA Style**

González C, Jensen E, Gambús P, Vallverdú M.
Entropy Measures as Descriptors to Identify Apneas in Rheoencephalographic Signals. *Entropy*. 2019; 21(6):605.
https://doi.org/10.3390/e21060605

**Chicago/Turabian Style**

González, Carmen, Erik Jensen, Pedro Gambús, and Montserrat Vallverdú.
2019. "Entropy Measures as Descriptors to Identify Apneas in Rheoencephalographic Signals" *Entropy* 21, no. 6: 605.
https://doi.org/10.3390/e21060605