Algorithm 2: Parametric k-means algorithm

Step 1: For a given pdf $p\left(x\right)$, the number of MSE-RPs m, and initial iteration $t=0$, input an initial set of points $y_1^{\left(t\right)}<y_2^{\left(t\right)}<\dots <y_m^{\left(t\right)}$. 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: $$p_j^{\left(t\right)}={\int }_{I_j^{\left(t\right)}}p\left(x\right)dx, j=1,\dots ,m;$$ Step 3: Calculate conditional means: $$y_j^{(t+1)}={\displaystyle \frac{{\int }_{I_j^{\left(t\right)}}xp\left(x\right)dx}{{\int }_{I_j^{\left(t\right)}}p\left(x\right)dx}}={\displaystyle \frac{{\int }_{I_j^{\left(t\right)}}xp\left(x\right)dx}{p_j^{\left(t\right)}}},j=1,\cdots ,m.$$ Step 4: If $\left\{y_j^{\left(t\right)}\right\}$ and $\left\{y_j^{(t+1)}\right\}$ are identical, 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.