Algorithm 1: Theoretical k-means algorithm for Pareto I and II distributions

Step 1: For a given pdf $p\left(x\right)$, the number of MSE-RPs m, initial iteration $t=0$, and tolerance $ϵ$, input an initial set of number-theoretic methods representative points (NTM-RPs) [16] $y_1^{\left(t\right)}<y_2^{\left(t\right)}<\dots <y_m^{\left(t\right)}$. Define a partition of $\mathbb{R}$ as: $$I_i^{\left(t\right)}=\left(a_i^{\left(t\right)},a_{i+1}^{\left(t\right)}\right], i=1,\dots ,m-1, I_m^{\left(t\right)}=\left(a_{m-1}^{\left(t\right)},a_m^{\left(t\right)}\right),$$ where $$a_1^{\left(t\right)}=-\infty , a_i^{\left(t\right)}={\displaystyle \frac{y_{i-1}^{\left(t\right)}+y_i^{\left(t\right)}}{2}}, i=2,\dots ,m, a_m^{\left(t\right)}=\infty .$$ Step 2: Calculate probabilities: $$p_j^{\left(t\right)}={\int }_{I_j^{\left(t\right)}}p\left(x\right) x, j=1,\dots ,m;$$ Step 3: Calculate conditional means: For $j=1,\cdots ,m-1$: $$y_j^{(t+1)}={\displaystyle \frac{{\int }_{I_j^{\left(t\right)}}xp\left(x\right) dx}{{\int }_{I_j^{\left(t\right)}}p\left(x\right) dx}}={\displaystyle \frac{{\int }_{I_j^{\left(t\right)}}xp\left(x\right) dx}{p_j^{\left(t\right)}}}, j=1,\dots ,m-1.$$ For $P\left(I\right)(1,\alpha )$: $$y_m^{(t+1)}={\displaystyle \frac{\alpha }{\alpha -2}}{y}_{m-1}^{(t+1)}.$$ For $P\left(II\right)(0,1,\alpha )$: $$y_m^{(t+1)}={\displaystyle \frac{\alpha y_{m-1}^{(t+1)}+2}{\alpha -2}}.$$ Step 4: Sort $\{y_1^{(t+1)},\dots ,y_m^{(t+1)}\}$ from smallest to largest. Step 5: Calculate $|y_m^{(t+1)}-b|$, where $$b={\displaystyle \frac{{\int }_{I_m^{\left(t\right)}}xp\left(x\right) dx}{p_m^{\left(t\right)}}}.$$ Step 6: If $|y_m^{(t+1)}-b|<ϵ$, the process stops, and $\left\{y_j^{\left(t\right)}\right\}$ are delivered as the MSE-RPs of the distribution with probabilities $\left\{p_j^{\left(t\right)}\right\}$. Otherwise, let $t:=t+1$ and return to Step 1.