the $j$-th feature is continuous and observed for both instances, is $$\label{32}
    GRC\;(X_{aj},X_{bj}) = \frac{\Delta_{\min}\; + \;\rho\;\Delta_{\max}}{|X_{aj} - X_{bj}|\; + \;\rho\;\Delta_{\max}}\;,$$ where $\Delta_{\min} = \min_c \min_k |X_{ak} - X_{ck}|$, $\Delta_{\max} = \max_c \max_k |X_{ak} - X_{ck}|$, $\rho \in [0,1]$ (usually $\rho = 0.5$ is taken [@ju1982control]), $b, c \in \{1,2,...,n\}$, and, $k, j \in \{1,2,...,p\}$ and for categorical feature $j$, $GRC(X_{aj},X_{bj})$ is 1 if they have the same values, 0 otherwise. If either $X_{aj}$ or $X_{bj}$ is missing, then $GRC(X_{aj},X_{bj})$ is 0. The Grey Relational Gradient (GRG) between the instances is defined as: $$\label{33}
    GRG (X_a,X_b) = \frac{1}{p} \sum _{j=1}^p GRC(X_{aj},X_{bj}),$$ where $a \in \{1,2,...,n\}$. We note that if $GRG(X_a,X_b)$ is larger than $GRG(X_a,X_c)$ then the difference between $X_a$ and $X_b$ is less than the difference between $X_a$ and $X_c$, which is the opposite of the Heterogenous Euclidean Overlap Metric (HEOM) defined in Section \[heom\]. The Grey Relational Gradient satisfies the following axioms which makes it a distance metric [@ju1982control]:

1.  Normality: The value of $GRG(X_a,X_b)$ is between 0 and 1.

2.  Dual Symmetry: Given only two observations $X_a$ and $X_b$ in the relational space, then $GRG(X_a,X_b) = GRG(X_b,X_a)$.

3.  Wholeness: If 3 or more observations are made in the 