## 3 Materials and Methods
### Regional Topic Model and Its Online Inference Algorithm
In other words, motion region label set \(R=\left\{\,\mathbf{\tau}_{i}\right\}_{i=1}^{I}\) obeys a multinomial distribution of \(R\) dimension whose parameter is \(\mathbf{\rho}\), \(R\sim\text{Mul}(\mathbf{\rho})\)  
\[\log p\left(R\,|\mathbf{\rho}\right)\ =\sum_{\sigma=1}^{R}I_{r}\log\rho_{r} \tag{3}\]  
Likewise, global region weight \(\mathbf{\rho}\) obeys a symmetric Dirichlet prior distribution \(\mathbf{\rho}\sim\text{Dir}(\beta)\) whose parameter is \(\beta\)  
\[\log p\left(\mathbf{\rho}|\beta\right)\ =\log\Gamma(R\beta)-R\log\Gamma(\beta)+ \sum_{\sigma=1}^{R}(\beta-1)\log\rho_{r} \tag{4}\]  
Under the known motion topic label \(\mathbf{\mathrm{t}}_{jn}=t\) and known motion region label \(\mathbf{\mathrm{r}}_{jn\in i}=r\) of sample \(\mathbf{\mathrm{x}}_{jn}\), the sample subset \(X_{jtr}=\left\{\mathbf{\mathrm{x}}_{jn}|\mathbf{\mathrm{t}}_{jn}=t,\ \mathbf{\mathrm{r}}_{jn\in i}=r \right\}_{n=1}^{N_{j}}\) allocated by document \(j\) obeys a \(R\) dimension multinomial distribution \(X_{jtr}\sim\text{Mul}(\mathbf{\mathrm{\theta}}_{rt})\) whose parameter is \(\mathbf{\mathrm{\theta}}_{rt}=\left\{\theta_{\omegart}\right\}_{\omega=1}^{\Omega}\)  
\[\log p\left(X_{jtr}|\mathbf{\mathrm{\theta}}_{rt}\right)\ =\sum_{\omega=1}^{\Omega}N_{j \omegart}\log\theta_{\omegart} \tag{5}\]  
The hybrid parameter \(\mathbf{\mathrm{\theta}}_{rt}\) obeys a symmetric Dirichlet prior distribution \(\mathbf{\mathrm{\theta}}_{rt}\sim\text{Dir}(\lambda)\) whose parameter is \(\lambda\)  
\[\log p\left(\mathbf{\mathrm{\theta}}_{rt}|\lambda\right)\ =\log\Gamma(\Omega\lambda)- \Omega\log\Gamma(\lambda)+\sum_{\omega=1}^{\Omega}(\lambda-1)\log\theta_{ \omegart} \tag{6}\]  
The construction of RTM is summarized as that: the number of motion topics is \(T\); the number of motion regions is \(R\); the video sequence contains \(J\) documents; the observed sample set is \(X_{j}=\left\{\mathbf{\mathrm{x}}_{jn}\right\}_{n=1}^{N_{j}}=\left\{\left(i, \omega\right)_{jn}\right\}_{n=1}^{N_{j}}\); the corresponding latent variables set is \(Z_{j}=\left\{\,T_{j},\ R\,\right\}=\left\{\mathbf{\mathrm{t}}_{jn},\ \mathbf{\mathrm{r}}_{jn\in i}\right\}_{n=1}^{N_{j}}\). Then, the generative process of RTM is as follows, the corresponding graphical model is shown in Figure 2.  
Generate \(\mathbf{\rho}\sim\text{Dir}(\beta)\)  
For each sample \(\mathbf{\mathrm{x}}_{jn}\)  
Generate a regional label \(\mathbf{\mathrm{r}}_{i}\sim\mathbf{\rho}\) for its location \(i\)  
For each video clip \(j\)  
Generate a motion topic weight vector \(\mathbf{\pi}_{j}\sim\text{Dir}(\alpha)\)  
For each sample \(\mathbf{\mathrm{x}}_{jn}\)  
Generate motion topic label \(\mathbf{\mathrm{t}}_{jn}\ \sim\mathbf{\pi}_{j}\)  
Generate \(\mathbf{\mathrm{\theta}}_{rt}\sim\text{Dir}(\lambda)\)  
Generate \(\mathbf{\mathrm{x}}_{jn}\sim Mult(.|\mathbf{\mathrm{\theta}}_{rt})\)  
Figure 2: Graphical model of RTM. As the generative process of RTM described above, the unknown parameters to be estimated are \(\mathbf{\pi}_{j}=\left\{\pi_{jt}\right\}_{t=1}^{T}\), \(\mathbf{\rho}=\left\{\rho_{r}\right\}_{r=1}^{R}\) and \(\theta_{rt}=\left\{\theta_{urt}\right\}_{\omega=1}^{\Omega}\); the known data are the observed samples \(X_{j}=\left\{\mathbf{x}_{jn}\right\}_{n=1}^{N_{j}}\) and their joint distribution. As shown in Equation (7),  
\[