Poisson-Gamma Dynamical Systems with Non-Stationary Transition Dynamics

Thanks for the reviewer's constructive comments, our answers for the questions are as follows:

1. In practice, users can leverage the prior knowledge about the specific task to set the length of sub-intervals, or treat the length of sub-interval as a hyper-parameter, tuning it with part of the time series data. For the varying length of sub-intervals, as we mentioned in the conclusion section, we plan to adopt temporal point process for change point detection, and thus we can partition the whole time interval via the change points. We consider to simultaneously learn a set of change points to capture the irregulaly-spaced sub-intervals behind non-stationary sequential counts, and to learn the short dynamics underlying each sub-interval. To the end, we consider to work on an EM-type algorithm to maximize the model’s log-likelihood, in which the change points are formulated as latent variables. In principle, we can adopt Gaussian Process with Polya-Gamma augmentation technique to construct continuously changing transition kernel, and this will be our future work.

2. For the complexity of the three proposed transition kernels, the complexity of Dir-Dir construction is $\mathcal{O}\left(TK/M \right)$ and the complexity of Dir-Gam-Dir and PR-Gam-Dir constructions is $\mathcal{O}\left( TK^2/M\right)$. We believe it is worthy to proposed Dir-Gam-Dir and PR-Gam-Dir constructions though they are more complicated. First, in contrast to Dir-Dir construction, Dir-Gam-Dir and PR-Gam-Dir chains explicitly model the interactions among components and thus can improve the flexibility of the proposed model. And as shown in Fig.6 in our paper, the Dir-Gam-Dir and PR-Gam-Dir chains indeed capture more complicated non-stationary dynamics. Furthermore, the PR-Gam-Dir construction can induce sparse patterns. Besides, it is hard to find dataset that perfectly meet the models' assumption, hence the results of different models may be indistinguishable. As we claimed in the conclusion part, we consider to generalize Dirichlet belief networks by incorporating the proposed Dirichlet Markov chain constructions, and we also consider to capture non-stationary interaction dynamics among individuals over online social networks in the future research. For more complicated transition dynamics, the difference among the Dirichlet Markov chains will be amplified.

3. We will carefully revise our paper to merge or reduce some overlap contents in sec.2 and sec.5 to improve the readability of our paper for the final version.

We sincerely appreciate the reviewer's valuable time, positive comments for our manuscript.

We thank the reviewer's constructive comments. We clarify that in contrast to the reviewer's concern, our proposed methods outperform PGDS not only in data smoothing task. As shown in Table 1 in our paper, the proposed three models outperform the baselines in most tasks, even if we only consider NS-PGDS(Dir-Dir), it still outperforms PGDS in almost all tasks. The experimental results are computed via 10 random initialization points, and the means and standard deviations are listed in Table 1 of our paper.

As for the hyper-parameter $K$, we set $K=100$ for the baselines, in fact, because of the sparsity nature of Dirichlet process, the rank of the model will be determined automatically and is not a big concern. We adopt relative small $K$ for our proposed methods because the time-varying transition matrices will increase the number of parameters of the models. Therefore, we adopt relatively small $K$ to reduce the computational burden. Besides, experimental results show that NS-PGDS can outperform baselines even with less parameters (much small $K$). Therefore, we believe it is a fair comparison between NS-PGDS and baselines.

Answers to Questions:

1. In principle, allowing transition matrix changes at every time step is more natural than what we have done, however, as the reviewer pointed out, this approach will significantly increase the computational burden of the model. More importantly, if we assume the transition matrices change at every time step, then each matrix will be estimated via data point at only one time step, the estimation error will be huge, and the model can not converge.

2. For all baselines, we set $K=100$, and for PGDS, we set $\tau_0=1$, $\gamma_0=50$, $\eta_0=\epsilon_0=0.1$. For GP-DPFA, we wet $\gamma=1$, $c=1$, and $\theta_0=0.01$. For GMC-RATE, we set $\alpha=1$, $\beta=1$. For GMC-HIER, we set $\alpha_z=\beta_z=1$ and $\alpha_{\theta}=\beta_{\theta}=1$. For BGAR, we set $\rho=0.9$, $\alpha=1$ and $\beta=1$. And we will provide these information in the appendix of the final version.

3. We conducted Student's t-test to test the statistical significance. We evaluated the statistical significance of experimental results between PGDS and NS-PGDS(Dir-Dir) and the p-values are listed in below table. The results show that the proposed NS-PGDS significantly outperforms PGDS in forecasting task on four datasets. However, for data smoothing task, NS-PGDS only outperform PGDS marginally. That may because that data smoothing task is a much simpler task compared with data forecasting, PGDS is competent to this task very well, and thus it is hard for NS-PGDS to exceed PGDS by a large margin for smoothing task.

4. It is common for topic models to infer similar latent factors because topic models define a topic (latent factor) as frequency of words, there no reason for a word to appear in only one topic. The order means the frequency of words for a latent factor, for example, “image-sparse-matrix" means the top three frequent words of this topic are 'image', 'sparse' and 'matrix'.

The authors thank the reviewer's valuable feedback. The main contributions of this paper are: (i) We extend state-of-the-art PGDS model such that the transition matrix of PGDS can evolve over time and thus better fit non-stationary environment. (ii) In order to model the time-varying transition matrices, we propose three Dirichlet Markov chains for capturing the complex transition dynamics behind sequential count data. (iii) We leverage Dirichlet-Multinomial-Beta augmentation technique to design the Gibbs sampler for the proposed Dirichlet Markov chains, which is not trivial.

For the experimental results, we conducted Student's t-test to test the statistical significance. We evaluated the statistical significance of experimental results between PGDS and NS-PGDS(Dir-Dir) and the p-values are listed in below table. The results show that the proposed NS-PGDS significantly outperforms PGDS in forecasting task on four datasets. However, for data smoothing task, NS-PGDS only outperforms PGDS marginally. That may because that data smoothing task is a much simpler task compared with data forecasting, PGDS is competent to this task very well, and thus it is hard for NS-PGDS to exceed PGDS by a large margin for smoothing task. Besides, note that as shown in Fig 5 in our paper, the time-varying transition kernels indeed discover some interesting information about the time-varying interactions of research topics in NeurIPS conference, which could not be discovered via a constant transition matrix.

Scalability with the number of sub-intervals: The complexity of the Gibbs inference algorithms for the three proposed methods scale linearly with the number of sub-intervals.

How to choose the number of sub-intervals: For many real-world applications, the user will have prior knowledge about the specific application, and the prior knowledge can be leveraged to choose the length of each sub-interval. For ICEWS dataset, it contains international relations event of a year, and we assume the transition matrix is stationary within a month. Therefore we set $M=30$ for ICEWS. Similarly, we assume the interactions of research topics is stationary within 5 years for NeurIPS conference and the transition dynamics of COVID-19 in U.S. is stationary within 20 days. For USEI dataset, because $T \approx 340$, we heuristically split it into 10 sub-intervals and set $M=34$. In general, user can treat the length of sub-interval as a hyper-parameter, and set it via the user's prior knowledge or tuning it with part of the time series data.

Adaptability: Indeed, to allow the proposed model determines the length of sub-interval adaptively is of great significance. As we mentioned in the conclusion section, we plan to adopt temporal point process for change point detection, We consider to simultaneously learn a set of change points to capture the irregulaly-spaced subintervals behind non-stationary sequential counts, and to learn the short dynamics underlying each subinterval. To the end, we consider to work on an EM-type algorithm to maximize the model’s log-likelihood, in which the change points are formulated as latent variables.

Extra training time: The training time for PGDS and NS-PGDS are listed in below table, we set $K=100$ for all models. Below table shows that the proposed models achieve better results with little extra training time.