# Compensation of Measurement Uncertainty in a Remote Fetal Monitor
## 4 Results and Discussion
### Stochastic Model Validation  
From the experimental data set (data for validation), the correction coefficients \(c_{n}\)(calculated during the modeling process) and the algorithm proposed in Figure 5, the corrected standard deviation and the expected fetal heart rate were estimated. For the validation process, \(N=5\) clinical studies with \(M=900\) samples per clinical study and \(\beta\) defined with \(k=1\) were considered to determine the minimum percentage of samples that fall within one standard deviation. Table 3 presents the results of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Patient Name** & \(\sigma\) & \(n\) \\ \hline 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 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Estimated variance and data number used in stochastic modeling. Figure 9: Modeled distribution functions of the patients. the modeled standard deviation \(\sigma\), estimated \(s_{n}\), and corrected \(c_{n}s_{n}\), and the expected value and \(\overline{y}_{n}\) with samples from five clinical studies for all patients. The algorithm in Figure 5, additionally, allows the evaluation of the reliability improvement using the corrected and uncorrected standard deviation (uncertainty). To verify the reliability behavior, the proposed algorithm was executed using two bands level: \(\beta=k^{2}c_{n}\) to obtain the corrected reliability value and \(\beta=k^{2}\) to obtain the reliability value without correction. Figure 10 shows the results of the corrected and uncorrected reliability for two clinical studies \(n=\sqrt{2}\), Figure 10a, and for five clinical studies \(n=5\), Figure 10b. The reliability increases when considering the corrected standard deviation (uncertainty). In the case of total ignorance of the system, the Chebyshev inequality indicates that at least \(1-\frac{1}{k^{2}}\) of the sample, data must be within \(k\) standard deviations of the mean, for \(k=\sqrt{2}\) for at least \(50\%\) of the data, see Figure 10. However, having the distribution function models allow us, in addition to estimating the compensated uncertainty, to know the number of measurements that were within \(\sqrt{2}\) standard deviations, which was close to \(90\%\). The parameter estimator behavior in traditional models [12; 13; 14], was performed considering some theoretical distributions [3], and their behavior was evaluated considering finite samples. The problems that arose were that the distributions used were not representative of the applications considered in this work, and the estimator properties within finite time windows depended on the distributions. The method proposed in this work allows defining distributions in practical applications with few samples; that solve the problems presented in classical models. Reliability determination with few samples could be useful due to the visual analysis of fetal heart rate tracings being subjective and inconsistent [5]. There is disagreement in the clinical decision in fetal heart rate (FHR) classification [7]. A simpler and more objective set of guidelines could provide better reliability [10]. Given the inherent limitations of electronic fetal monitoring technology and of the ability of human beings to characterize fetal heart rate parameters and patterns, the low-reliability degree may represent the best-case scenario of the visual fetal heart rate interpretation reliability [6].