A Dirichlet Mixture Model of Hawkes Processes for Event Sequence Clustering
How to cluster event sequences generated via different point processes is an interesting and important problem in statistical machine learning. To solve this problem, we propose and discuss an effective model-based clustering method based on a novel Dirichlet mixture model of a special but significant type of point processesHawkes process. The proposed model generates the event sequences with different clusters from the Hawkes processes with different parameters, and uses a Dirichlet distribution as the prior distribution of the clusters. We prove the identifiability of our mixture model and propose an effective variational Bayesian inference algorithm to learn our model. An adaptive inner iteration allocation strategy is designed to accelerate the convergence of our algorithm. Moreover, we investigate the sample complexity and the computational complexity of our learning algorithm in depth. Experiments on both synthetic and real-world data show that the clustering method based on our model can learn structural triggering patterns hidden in asynchronous event sequences robustly and achieve superior performance on clustering purity and consistency compared to existing methods.
In this work, the authors introduce a novel method to cluster sequences of events. This method relies on a Dirichlet mixture model of Hawkes processes to describe the data and assign events to clusters. This differs from previous work using Hawkes Processes for sequential data by focusing directly on clustering the sequences themselves rather than sub-events; in particular, compared to prior work such as by Luo et al., 2015, the clustering is done as part of the modeling itself rather than applied after learning the parameters of the model, which leads to a sizable improvement in experimental results. 
Quality: This paper is of good quality.
Clarity: This paper is overall clear, but contains some grammar and style mistakes - some have been noted below in the detailed comments, but the authors ideally should have their manuscript proof-read for English grammatical and style issues. Figure 2 is too small to be easily readable. 
Originality: This paper introduces a new method to model sequences of events.
Significance: This paper appears to be significant.
Detailed comments:
- In section 2, it would be valuable to provide a formal definition of Hawkes Process; in particular, it would improve readability to put the intensity equation (line 67) in its own equation environment, and then refer to it by number in Section 3 when relevant.
- When comparing line 67 to Eq. 1, it seems that the sum over the event types (c_i) is missing; could you please clarify this?
- Line 156: you state that the runtime of the algorithm is linearly proportional to the number of inner iterations - could you write out the complexity explicitly?
- Along the same lines, could you please state the complexity of MMHP in section 4.4?
- The second paragraph of section 4.2 needs to be made clearer. 
Clarity comments:
- You do not need to define R+ line 101
- line 23: "the previous two"
- line 39: "prior of the clusters"
- line 51: "the problems of overfitting"
- line 61: "A Hawkes"
- line 73: "was first proposed"
- line 77: "was used"
- line 100: "basis"
- line 109: "a lot of work" should be rephrased
- line 125: "goes to infinity"
- line 232: "state of the art"
- line 255: spacing