The outcome of this step is as follows: a scalar value named parameter_2, referred to as \(r_{inner}\); moreover, \(m\times k\) matrix named parameter_3, referred to as the neighbors’ summary; each row in the neighbors’ summary corresponds to an observation in __ds; and the number of \(k\) columns is equal to the number of classes in __ds. Parameters 2 and 3 are calculated in sequence as follows:
3.1.2.1. Fundamentally, a vector d of the Euclidean distances between each observation and other observations in __ds is calculated; the distance is the square root of the squared difference between two observations of size \(n\), and it is formulated as follows:
\(\begin{equation} \label{eq:2}
\sqrt{〖(X_j-X_e)〗^2}
\end{equation}\)
Next, the value of parameter_2 is captured; that is, the midrange of the vector \(d\) is tuned by the value \(0.01\le l\le0.99\). The calculation of parameter_2 is formulated as follows:
\(\begin{equation} \label{eq:1}
\frac{\min(d)+\max(d)}{2}\times l
\end{equation} \)
3.1.2.2. Finally, if the Euclidean distance in Equation (\ref{eq:2}) is less than or equal to parameter_2 in Equation (\ref{eq:1}), the comparison is considered inside the n-sphere; the final result is a matrix, outlined above in (\ref{103702}), named parameter_3. Fig. (\ref{326434}) illustrates the outcome from this step using the showcase example in (\ref{993524}).