Algorithm 1 Gibbs sampling for the proposed method

Input: Datasets ${\{y_t,{\mathit{z}}_t\}}_{t=1}^T$; hyperparameters a, b, $\alpha$, ${\{{\mu }_j\}}_{j=1}^q$, ${\{{\beta }_j\}}_{j=1}^q$; number of truncating components L; number of all samples N; initial sample ${}_{\left(0\right)}\mathit{\varphi },{}_{\left(0\right)}\mathit{\pi },{}_{\left(0\right)}\mathit{\lambda },{}_{\left(0\right)}\mathit{\omega },{}_{\left(0\right)}\mathit{x},{}_{\left(0\right)}\mathit{k}$. Output: All posterior samples ${\{{}_{\left(n\right)}\mathit{\varphi },{}_{\left(n\right)}\mathit{\pi },{}_{\left(n\right)}\mathit{\lambda },{}_{\left(n\right)}\mathit{\omega },{}_{\left(n\right)}\mathit{x},{}_{\left(n\right)}\mathit{k}\}}_{n=1}^N$ 1:for$n=1$ to N do 2:    For each one string $(s_1,\cdots ,s_q)$, collect a sample ${\varphi }_{s_1,\cdots ,s_q}$ from its multinomial full conditional $$p({\varphi }_{s_1,\cdots ,s_q}=l\mid \mathit{\xi })\propto {\pi }_l\prod _{c=1}^{C_0}{\{{\lambda }_l\left(c\right)\}}^{n_{s_1,\cdots ,s_q}\left(c\right)}$$    where $n_{s_1,\cdots ,s_q}\left(c\right)={\sum }_1^T\mathbf{1}\{x_{1,t}=s_1,\cdots ,x_{q,t}=s_q,y_t=c\}$. 3:    For $l=1,\cdots ,L$, update ${\pi }_l$ by the following rules $$\begin{array}{c}{V}_l\mid \mathit{\xi }\sim \mathbf{Beta}(1-b+n_l,a+lb+\sum _{k>l}{n}_k),l<L\\ V_L=1,{\pi }_l=V_l\prod _{k=1}^{l-1}(1-V_k)\end{array}$$    where $n_l={\sum }_{(s_1,\cdots ,s_q)}\mathbf{1}\{{\varphi }_{s_1,\cdots ,s_q}=l\}$. 4:    For $l=1,\cdots ,L$, collect samples ${\mathit{\lambda }}_l$ from their respective Dirichlet full conditionals $${\mathit{\lambda }}_l\mid \mathit{\xi }\sim \mathbf{Dir}\{\alpha +n_l\left(1\right),\cdots ,\alpha +n_l\left(C_0\right)\}$$    where $n_l\left(c\right)={\sum }_{(s_1,\cdots ,s_q)}\mathbf{1}\{{\varphi }_{s_1,\cdots ,s_q}=l\}{n}_{s_1,\cdots ,s_q}\left(c\right)$. 5:    For $j=1,\cdots ,q$, for $c=1,\cdots ,C_j$, collect samples $${\mathit{\omega }}^{\left(j\right)}\left(c\right)\mid \mathit{\xi }\sim \mathbf{Dir}\{{\beta }_j+n_{j,c}\left(1\right),\cdots ,{\beta }_j+n_{j,c}\left(k_j\right)\}$$    where $n_{j,c}\left(s_j\right)={\sum }_{t=1}^T\mathbf{1}\{x_{j,t}=s_j,z_{j,t}=c\}$. 6:    For $j=1,\cdots ,q$, for $t=1,\cdots ,T$, collect samples $x_{j,t}$ from their corresponding multinomial full conditionals $$p(x_{j,t}=s\mid \mathit{\xi },x_{i,t}=s_i,i\ne j)\propto {\omega }_s^{\left(j\right)}\left(z_{j,t}\right){\mathit{\lambda }}_{{\varphi }_{s_1,\cdots ,s,\cdots ,s_q}}\left(y_t\right)$$ 7:    For $j=1,\cdots ,q$, collect samples $k_j$ from their respective multinomial full conditionals $$p(k_j=k\mid \mathit{\xi })\propto \exp (-{\mu }_{j}k)\prod _{c=1}^{C_j}{n}_{j,c}^{-k{\beta }_j},k_j=\underset{t}{max}\{x_{j,t}\},\cdots ,C_j$$    where $n_{j,c}={\sum }_{t=1}^T\mathbf{1}\{z_{j,t}=c\}$. 8: end for