# Compensation of Measurement Uncertainty in a Remote Fetal Monitor

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## Abstract

**:**

## 1. Introduction

## 2. Uncertainty Estimation Model

- Sample size, $n$
- Distribution, ${F}_{X}$
- The degree of probabilistic dependence between samples, in addition to the fact that only linear or correlation dependence is considered ${\mathsf{\rho}}_{{\mathrm{X}}_{\mathrm{i}}{\mathrm{X}}_{\mathrm{j}}}$.

## 3. Uncertainty Estimation Approach

- System model: device model and perturbation model.
- Distribution function estimation of the perturbations and correlation function.
- Estimators properties analysis using the Monte-Carlo method.
- Correction factors calculation to compensate estimators.
- Confidence intervals for the $R$ estimation.

#### 3.1. System Model

#### 3.1.1. Device Model

#### 3.1.2. Perturbation Model

#### 3.2. Distribution Function Estimation of the Perturbations and Correlation Function

#### 3.3. Estimators Properties Analysis Using the Monte-Carlo Method

#### 3.4. Correction Factors Calculation to Compensate Estimators

- Additive factor for ${\overline{\mathrm{Y}}}_{\mathrm{k}}$ $\left({\left[{\mathsf{\Phi}}^{\mathrm{T}}\mathsf{\Phi}\right]}^{-1}{\mathsf{\Phi}}^{\mathrm{T}}\mathsf{\Xi}\right)$
- Multiplicative factor for ${\mathrm{S}}_{\mathrm{k}}$ (${\mathrm{C}}_{\mathrm{k}}$)

#### 3.5. Confidence Intervals for the $R$ Estimation

#### 3.6. Remote Fetal Monitoring Equipment

#### 3.7. Experimental Clinical Studies

## 4. Results and Discussion

^{®}for building the model, calculating the compensated uncertainty, and evaluating the reliability.

#### 4.1. Fetal Monitor Model

#### 4.2. Stochastic Model

#### 4.3. Stochastic Model Validation

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Single-input and single-output (SISO) system: (

**a**) With white noise to the input; (

**b**) With additive noise to the output.

**Figure 4.**Simulation procedure with the Monte-Carlo method (MMC) to determine the variance ${s}_{k}^{2}$.

**Figure 7.**Map showing the location of the first four patients from Table 1.

**Figure 10.**Corrected and uncorrected reliability (R) calculations; (

**a**) with two clinical studies; (

**b**) with five clinical studies.

Patient Name | Age (Year) | Studies Number | Place | Distance from OMC |
---|---|---|---|---|

Subject 1 | 37 | 42 | Toliman | 81.7 Km |

Subject 2 | 41 | 45 | Jalpan de Serra | 187 Km |

Subject 3 | 35 | 40 | Querétaro | 9.1 Km |

Subject 4 | 26 | 43 | Amealco de Bonfil | 71.1 Km |

Subject 5 | 43 | 32 | Querétaro | 9.4 Km |

Subject 6 | 18 | 34 | San Luis Potosí | 215 Km |

Subject 7 | 28 | 36 | Querétaro | 10.7 km |

Subject 8 | 31 | 38 | Querétaro | 18.3 Km |

Subject 9 | 19 | 43 | Colima | 576 Km |

Subject 10 | 21 | 47 | Querétaro | 13.5 Km |

Subject 11 | 39 | 51 | Querétaro | 30.4 Km |

Subject 12 | 42 | 53 | Querétaro | 13.1 Km |

Subject 13 | 29 | 54 | El Marqués | 7.1 Km |

Subject 14 | 33 | 57 | El Marqués | 6.8 Km |

Subject 15 | 36 | 58 | Salamanca | 89.7 Km |

Subject 16 | 25 | 58 | Puebla | 329 Km |

Subject 17 | 42 | 58 | Guadalajara | 378 Km |

Subject 18 | 45 | 61 | Querétaro | 2.1 Km |

Subject 19 | 38 | 75 | Querétaro | 17.4 Km |

Patient Name | $\mathit{\sigma}$ | $\mathit{n}$ |
---|---|---|

Subject 1 | 7.75 | 12731 |

Subject 2 | 10.98 | 13821 |

Subject 3 | 12.77 | 9882 |

Subject 4 | 13.97 | 12062 |

Subject 5 | 9.19 | 13472 |

Subject 6 | 8.54 | 12778 |

Subject 7 | 10.52 | 13417 |

Subject 8 | 9.44 | 13125 |

Subject 9 | 11.13 | 12156 |

Subject 10 | 12.75 | 10243 |

Subject 11 | 8.64 | 13454 |

Subject 12 | 8.68 | 13357 |

Subject 13 | 9.61 | 13399 |

Subject 14 | 12.97 | 12410 |

Subject 15 | 6.88 | 13316 |

Subject 16 | 7.70 | 11677 |

Subject 17 | 7.49 | 13481 |

Subject 18 | 11.93 | 12010 |

Subject 19 | 8.07 | 14070 |

Patient Name | $\mathit{\sigma}$ | ${\overline{\mathit{y}}}_{5}$ | ${\mathit{s}}_{2}$ | ${\mathit{c}}_{2}{\mathit{s}}_{2}$ | ${\mathit{s}}_{3}$ | ${\mathit{c}}_{3}{\mathit{s}}_{3}$ | ${\mathit{s}}_{4}$ | ${\mathit{c}}_{4}{\mathit{s}}_{4}$ | ${\mathit{s}}_{5}$ | ${\mathit{c}}_{5}{\mathit{s}}_{5}$ |
---|---|---|---|---|---|---|---|---|---|---|

Subject 1 | 7.75 | 148.6 | 3.70 | 7.29 | 4.51 | 7.65 | 4.85 | 7.54 | 5.15 | 7.52 |

Subject 2 | 10.98 | 142.9 | 5.10 | 10.23 | 6.44 | 10.98 | 7.23 | 11.27 | 7.75 | 11.37 |

Subject 3 | 12.77 | 157.9 | 6.23 | 12.45 | 7.75 | 13.24 | 8.67 | 13.49 | 9.14 | 13.36 |

Subject 4 | 13.97 | 140.7 | 6.77 | 13.37 | 8.31 | 14.12 | 9.41 | 14.61 | 9.89 | 14.48 |

Subject 5 | 9.19 | 139.4 | 4.36 | 8.68 | 5.48 | 9.32 | 6.02 | 9.38 | 6.35 | 9.32 |

Subject 6 | 8.54 | 140.6 | 3.99 | 7.95 | 4.99 | 8.51 | 5.50 | 8.57 | 6.05 | 8.86 |

Subject 7 | 10.52 | 131.6 | 5.08 | 10.17 | 6.31 | 10.77 | 7.02 | 10.93 | 7.38 | 10.82 |

Subject 8 | 9.44 | 138.3 | 4.48 | 8.92 | 5.57 | 9.49 | 6.31 | 9.83 | 6.68 | 9.80 |

Subject 9 | 11.13 | 139.5 | 5.25 | 10.37 | 6.65 | 11.27 | 7.29 | 11.26 | 7.89 | 11.43 |

Subject 10 | 12.75 | 132.9 | 6.26 | 12.67 | 7.78 | 13.32 | 8.68 | 13.56 | 9.30 | 13.63 |

Subject 11 | 8.64 | 123.7 | 4.05 | 8.03 | 5.17 | 8.77 | 5.68 | 8.84 | 6.17 | 9.02 |

Subject 12 | 8.68 | 135.7 | 3.96 | 7.92 | 5.12 | 8.70 | 5.70 | 8.87 | 6.07 | 8.87 |

Subject 13 | 9.61 | 148.0 | 4.51 | 9.03 | 5.67 | 9.65 | 6.40 | 9.93 | 6.82 | 9.95 |

Subject 14 | 12.97 | 141.3 | 6.39 | 12.72 | 8.10 | 13.83 | 8.96 | 13.95 | 9.50 | 13.92 |

Subject 15 | 6.88 | 129.7 | 3.14 | 6.22 | 3.92 | 6.68 | 4.36 | 6.80 | 4.58 | 6.70 |

Subject 16 | 7.70 | 134.2 | 3.52 | 7.07 | 4.42 | 7.55 | 4.99 | 7.76 | 5.38 | 7.86 |

Subject 17 | 7.49 | 141.5 | 3.40 | 6.82 | 4.27 | 7.30 | 4.72 | 7.34 | 5.03 | 7.35 |

Subject 18 | 11.93 | 132.8 | 6.04 | 11.84 | 7.37 | 12.49 | 8.11 | 12.58 | 8.59 | 12.51 |

Subject 19 | 8.07 | 150.6 | 3.66 | 7.29 | 4.67 | 7.96 | 5.29 | 8.26 | 5.62 | 8.25 |

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**MDPI and ACS Style**

Arciniega-Montiel, S.; Ronquillo-Lomeli, G.; Salas-Zúñiga, R.; Salgado-Jiménez, T.; Barriga-Rodríguez, L.
Compensation of Measurement Uncertainty in a Remote Fetal Monitor. *Appl. Sci.* **2020**, *10*, 3274.
https://doi.org/10.3390/app10093274

**AMA Style**

Arciniega-Montiel S, Ronquillo-Lomeli G, Salas-Zúñiga R, Salgado-Jiménez T, Barriga-Rodríguez L.
Compensation of Measurement Uncertainty in a Remote Fetal Monitor. *Applied Sciences*. 2020; 10(9):3274.
https://doi.org/10.3390/app10093274

**Chicago/Turabian Style**

Arciniega-Montiel, Sadot, Guillermo Ronquillo-Lomeli, Roberto Salas-Zúñiga, Tomás Salgado-Jiménez, and Leonardo Barriga-Rodríguez.
2020. "Compensation of Measurement Uncertainty in a Remote Fetal Monitor" *Applied Sciences* 10, no. 9: 3274.
https://doi.org/10.3390/app10093274