## 3 Materials and Methods
### Regional Topic Model and Its Online Inference Algorithm
```
Initialize global variational parameters \(\varsigma_{ir}^{(0)}\), \(\
u_{r}^{(0)}\) and \(\mu_{\omega rt}^{(0)}\) while the number of random iterations \(t=0:t_{\max}\) do  Update iteration step: \(\rho_{t}\triangleq a(\tau_{0}+t)^{-\kappa}\)  Import a small batch documents \(B_{t}\)  Initialize \(T_{j}^{(0)}\) from \((1,\ldots,T)\), \(R_{j}^{(0)}\) from \((1,\ldots,R)\) for\(\in B_{t}\) do  For a sample \(n=1:N_{j}\) of document \(j\) local MCMC inference is adopted  end_for \(N\) times iteration in burn-in time is not processed  For converged Markov chain \(s=1:S\) do \(\widehat{N}_{(B_{t})\omega rt}=\mathbb{E}_{\eta_{(B_{t})\omega rt}}\Bigl{[}N_{ (B_{t})\omega rt}\Bigr{]}\approx(1/S)\sum_{s=1}^{S}N_{B_{t}\omega rt}^{(s)}\) end_for \(\left\{ \begin{array}{ll}\log\varsigma_{ir}^{(t)}&=(1-\rho_{t})\log\varsigma_ {ir}^{(t-1)}~{}+~{}\rho_{t}\log\varsigma_{ir}^{(t)}\\ \
u_{r}^{(t)}&=(1-\rho_{t})\
u_{r}^{(t-1)}~{}+~{}\rho_{t}\
u_{r}^{(t)}\\ \mu_{\omega rt}^{(t)}&=(1-\rho_{t})\mu_{\omega rt}^{(t-1)}~{}+~{}\rho_{t}\theta _{ir}^{(t)}\end{array} \right.\), where \(\left\{\begin



## 4 Results

### Evaluation Criterion

In the text mining and statistic inference of nature language, perplexity is always used to evaluate the performance of model, which is computed by \(perp(X^{\text{test}}|\hat{\mathbf{\varphi}})=\exp\{-\log p(X^{\text{test}}|\hat{\mathbf{ \varphi}})/N^{\text{test}}\}\). \(\hat{\mathbf{\varphi}}\) denotes the parametric estimation of trained model, \(X^{\text{test}}\) and \(N^{\text{test}}\) are test dataset and observed samples respectively. Perplexity is the negative log likelihood (NLL) - log \(p(X^{\text{test}}|\hat{\mathbf{\varphi}})\) divided by the number of observed samples \(N^{\text{test}}\). As described in above, the computation of perplexity is mainly the NLL computation of trained model, and NLL denotes the cross entropy of trained model and unknown testing data. Perplexity represents the uncertainty of trained model for unknown test set's estimation. Therefore, the lower the NLL value is, the better the model performance is. In our RTM, NLL can be computed for motion topic and region of test video clip. The parameters estimation of trained model and a test video clip are learning in RTM, the learned local motion topic weight estimation \(\widetilde{\pi}_{t}\) and global parameter estimation \(\widetilde{\varsigma}_{ir}\), \(\widetilde{\theta}_{out}\) of original trained model is computed for NLL of test clips. \[ \begin{array}{ll}\text{rLDA\_NLL}\Big{(}X^{\text{test}}\Big{|}\widetilde{\pi }_{t},\widetilde{\varsigma}_{ir},\widetilde{\theta}_{out}\Big{)}&=-\sum_{n=1}^ {N^{\text{test}}}\log\Big{(}\sum_{r=1}^{R}\widetilde{\varsigma}_{ir}\sum_{t=1 }^{T}\widetilde{\pi}_{t}\widetilde{\theta}_{x^{\text{test}}=\omega,rt}^{\text {test}}\Big{)}\\ &=-N_{(\omega,i)}\log\Big{(}\sum_{r=1}^{R}\widetilde{\varsigma}_{ir}\sum_{t=1 }^{T}\widetilde{\pi}_{t}\widetilde{\theta}_{out}\Big{)}\end{array}  \tag{24}\]

Besides, t_NLL and r_NLL are computed by:

\[ \begin{array}{ll}\text{t\_NLL}\Big{(}X^{\text{test}}\Big{|}\widetilde{ \varsigma}_{ir},\widetilde{\theta}_{out}\Big{)}&=-\sum_{n=1}^{N^{\text{test}}} \log\Big{(}\mathbb{I}(\textbf{t}_{n}=t)\sum_{r=1}^{R}\widetilde{\varsigma}_{ir }\widetilde{\theta}_{x^{\text{test}}_{n}=\omega,rt}\Big{)}\\ &=-N_{(\omega,i)}\log\Big{(}\sum_{r=1}^{R}\widetilde{\varsigma}_{ir}\widetilde {\theta}_{out}\Big{)}\end{array}  \tag{25}\]

\[ \begin{array}{ll}\text{r\_NLL}\Big{(}X^{\text{test}}\Big{|}\widetilde{\pi}_ {t},\widetilde{\theta}_{out}\Big{)}&=-\sum_{n=1}^{N^{\text{test}}}\log\Big{(} \mathbb{I}(\textbf{t}_{n}=r)\sum_{t=1}^{T}\widetilde{\pi}_{t}\widetilde{\theta }_{x^{\text{test}}_{n}=\omega,rt}\Big{)}\\ &=-N_{(\omega,r)}\log\Big{(}\sum_{t=1}^{T}\widetilde{\pi}_{t}\widetilde{\theta }_{out}\Big{)}\end{array}  \tag{26}\]

t_NLL is used to evaluate the performance of learning motion topic by our model, and r_NLL is used to evaluate the performance of learning motion region.