$0$ to $1$ and also takes the value 1, when either of the observations are missing.

However, to effectively apply the KNN imputation approach, a challenging issue is the optimal value of $k$, and the other is selecting neighbours. The optimal $k$-value can be selected using only non-missing parts [@kim2004missing]. This $k$-value estimating procedure considers some elements of the non-missing parts as artificial missing values, and finds an expected $k$-value that produces the best estimation ability for the artificial missing values. This method does not perform well when there are large amounts of missing data. In the proposed approach we determine this parameter optimally using cross validation [@stone1974cross].

### KNN Imputation {#knn}

Suppose the $j$-th input feature of $X_i$ is missing (i.e., $D_{ij} = 0$ from ) and has to be imputed. After the distances (calculated by the HEOM defined in ) from $X_i$ to all other training instances ($\{X_k\}_{k=1,k \neq i}^n$) are computed, its $k$-nearest neighbours are chosen from the training set. In our notation, $\mathscr{V}_{X_i} = \{v_\ell\}_{\ell=1}^k$ represents the set of $k$-nearest neighbors of $X_i$ arranged in increasing order of its distance as defined by -. The $k$-closest cases are selected after instances with missing entries in the 