# Conditional Dependencies and Position Fixing
## 7 Normalizing the Pool of Data
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(\mathcal{S}_{H}\)\({}^{3}\) & \(\mathcal{S}_{W}\) [m] \({}^{4}\) & \(\mathcal{\Theta}_{H}\)\({}^{5}\) & \(\mathcal{\Theta}_{W}\)\({}^{6}\) & \(\mathcal{\Theta}_{R\!i}\)\({}^{7}\) & \(\mathcal{\mathcal{\hat{d}}}_{i}^{M}\) [m] \({}^{8}\) \\ \hline \(HM_{1}\)\({}^{1}\) & 5.34 & 11.6 & 0.061 & 0.157 & 0.208 & 14.509 \\ \(HM_{2}\)\({}^{1}\) & 6.88 & 15.8 & 0.050 & 0.214 & 0.253 & 23.981 \\ \(HM_{3}\)\({}^{1}\) & 3.58 & 7.4 & 0.074 & 0.100 & 0.167 & 8.502 \\ \hline \(HM^{ref}\)\({}^{2}\) & \(\mathcal{\hat{S}}_{H}^{ref}\)\({}^{9}\) & \(\mathcal{\hat{S}}_{W}^{ref}\)\({}^{10}\) & \(\mathcal{\hat{\Theta}}_{H}^{ref}\)\({}^{11}\) & \(\mathcal{\hat{\Theta}}_{W}^{ref}\)\({}^{12}\) & \(\mathcal{\hat{\Theta}}_{R}^{ref}\)\({}^{13}\) & \\ \hline \(HM_{i}\) -th histogram; \({}^{2}\)_HMref_[FOOTNOTE:]Footnote 1: \(HM_{i}\)-th histogram; \({}^{2}\)_HMref_ logical reference structure; \({}^{3}\)_S\({}_{H}\)_ bin height measure for \(i\)-th histogram; \({}^{4}\)_S\({}_{W}\)_ bin width for \(i\)-th histogram; \({}^{5}\)_G\({}_{H}\)_ calculated with Equation (1) uncertainty regarding bin height measure for set third histogram, in which case a relation is made to \(HM^{ref}\); \(\mathcal{\hat{\Theta}}_{W}\)_ calculated with Equation (2) uncertainty regarding bin width measure for best third histogram, in which case a relation is made to \(HM^{ref}\); \({}^{7}\)_\(\mathcal{\hat{\Theta}}_{R}\)_ calculated with Equation (3) uncertainty for \(i\)-th histogram; \({}^{8}\)_\(d_{i}^{M}\)_ modified due to uncertainty in 0.5-cut range of membership functions and calculated with the Equation (6) measure related to the best third histogram extension, in which a relation is made to \(\mathcal{\hat{S}}_{W}^{ref}\); \({}^{9}\)_\(\mathcal{\hat{S}}_{H}^{ref}=\max\limits_{i}\mathcal{\hat{S}}_{Hi}\)_ reference bin height measure for set of histograms; \({}^{10}\)_S\({}^{ref}_{W}=\min\limits_{i}\mathcal{\hat{S}}_{Wi}\)_ reference bin width measure for set of histograms; \({}^{11}\)_\(\mathcal{\hat{\Theta}}_{H}^{ref}\)_ reference bin height measure for set of histograms; \({}^{12}\)_\(\mathcal{\hat{\Theta}}_{W}^{ref}\)_ reference bin width measure for set of histograms; \({}^{13}\)_\(\mathcal{\hat{\Theta}}_{R}^{ref}\)_ combined reference uncertainty measure, calculated with Equation (3).[ENDFOOTNOTE] & \(\mathcal{\hat{S}}_{H}^{ref}\) & \(\mathcal{\hat{S}}_{H}^{ref}\) & \(\mathcal{\hat{S}}_{H}^{ref}\) & \(\mathcal{\hat{S}}_{H}^{ref}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Selected features of example pools of three histograms. where \(x_{ic}\) is the mean of the \(i\)-th bin; \(x_{il}=x_{ic}-d_{i}^{M}/2\) is the leftmost limit of the \(i\)-th bin; \(x_{ir}=x_{ic}+d_{i}^{M}/2\) is the rightmost limit of the \(i\)-th bin; \(a_{i}\) is the steepness of the sigmoidal membership function. The probability assignment consists of a pair of figures:  
\[\{a,\ b\}=\bigg{\{}C*\sqrt{P\left(H_{i}.x\big{|}\ o_{j}\right)^{2}+P\left(H_{i }.y\big{|}\ o_{j}\right)^{2}},\ \ \Theta_{XYj}\ \bigg{\}}. \tag{8}\]  
The probability assignment engages the \(j\)-th piece of the two-dimensional evidence and the \(i\)-th hypothesis object with \(x\) and \(y\) attributes, and it takes the form of a two-element set that is called the simple belief assignment. In Equation (8), \(C\) is a constant and is assumed to be equal to one, and it refers to the width of an area at approximately the \(i\)-th hypothesis point. Note that \(g_{Xi}\) and \(g_{Yi}\) are density functions: \(P(H_{i}.x\big{|}\ o_{j})=f(g_{Xi}(H_{i}.x))\) is the conditional dependency for the \(H_{i}.x\) (attribute \(x\) in hypothesis object \(H_{i}\)) included in the \(o_{j}\) evidence (\(j\)-th observation), and \(P(H_{i}.y\big{|}\ o_{j})=f(g_{Yi}(H_{i}.x))\) is the conditional dependency for the \(H_{i}.y\) (attribute \(y\) in hypothesis object \(H_{i}\)) included in the \(o_{j}\) evidence (\(j\)-th observation). \(\Theta_{XYj}=\max(\Theta_{Xj},\ \Theta_{Yj})\) is the uncertainty of the \(j\)-th two-dimensional piece of evidence.