# Compensation of Measurement Uncertainty in a Remote Fetal Monitor
## 2 Uncertainty Estimation Model  
One of the main problems in metrology is the estimation of measurement uncertainty. According to [19], \"estimated standard deviation associated to output, or measurement estimation is called combined standard deviation, and it is determined from the estimated standard deviation associated for each estimation of output, called standard uncertainty\". In general, the process to be measured is ignored, and regularly, there are only just a few samples available. Measurements have defects that lead to an error in the result of such measurements. Although it is not possible to compensate for the random error in a measurement result, its effect can generally be reduced by increasing the number of observations until the expectations or expected value becomes zero. Modeling of stochastic systems has been thoroughly studied, considering the hypothesis of the great numbers law. However, a measurement system is a stochastic system, with a limited or finite number of samples. The remote fetal monitor is a stochastic measurement system with a limited number of samples. Below an uncertainty estimation model for a measurement system with limited samples is presented. A probability space \((\Omega,\mathcal{F}_{0},P)\) for a random variable \(X:\Omega\rightarrow\mathbb{R}^{n}\) where the conditional expectation \(E[(\cdot),\mathcal{F}_{0}]\) concerning to sigma-algebra \(\mathcal{F}_{0}=\left\{\phi,\ \Omega\right\}\)[20] may be expressed as  
\[E[X|\mathcal{F}_{0}]=E[X] \tag{1}\]  
This means that the best estimation for the random variable, when there is no a priori knowledge of the process, is the mathematical expectation, and the associated standard uncertainty (\(u_{c}\)) may be obtained from the probability density function (PDF) of the random variable [21]. Therefore, the mean is proposed to estimate the following sample, and standard uncertainty is obtained from the standard deviation estimator. Let \(A\) be the event where a device does not fail, the reliability is by definition \(P(A)\) if the indicator function is considered \(\chi_{A}\),  
\[\chi_{A}(\omega)=\left\{ \begin{array}{ll}1,&\omega\in A\\ 0,&\omega\
otin A\end{array} \right. \tag{2}\]  
This means that the problem of calculating reliability is changed for the problem of estimating the expected value, for which the use of the Monte-Carlo method in this work is necessary. In the estimation of the mathematical expectation, the equation regularly used is  
\[\overline{X}_{n}=\frac{1}{n}\ \sum_{k=1}^{n}X_{k}, \tag{3}\]  
being \(n\) the sample number. In the normal case with independent samples, the mean estimator is optimal in the sense of the mean square error. In addition, the large number theorem is consistent for large samples, and its distribution is known by the central limit theorem [20], and is efficient [22]. In the general case, the estimator \(\overline{X}_{n}\) is a random variable that depends on:  
1. Sample size, \(n\)
2. Distribution, \(F_{X}\)
3. The degree of probabilistic dependence between samples, in addition to the fact that only linear or correlation dependence is considered \(\rho_{X_{i}X_{i}}\). To measure the quality of this, estimate the standard deviation is used (called uncertainty in the metrology area). This standard deviation is also estimated; this is done with \(S_{n}\) defined as  
\[S_{n}:=\sqrt{S_{n}^{2}}, \tag{4}\]  
\[S_{n}^{2}:=\frac{1}{n-1}\sum_{k=1}^{n}{(x_{k}-\overline{x}_{n})^{2}}, \tag{5}\]  
for independent samples with the same distribution we get,  
\[E\Big{(}S_{\overline{X}_{n}}^{2}\Big{)}=\frac{E\Big{\{}S_{X}^{2}\Big{\}}}{n}= \frac{\sigma_{X}^{2}}{n}. \tag{6}\]  
So, the problem of determining the quality of the estimate focuses on determining \(\sigma_{X}\). However, we must identify the restrictions in the estimation.