We propose an effective method to solve the event sequence clustering problems based on a novel Dirichlet mixture model of a special but significant type of point processes --- Hawkes process. In this model, each event sequence belonging to a cluster is generated via the same Hawkes process with specific parameters, and different clusters correspond to different Hawkes processes. The prior distribution of the Hawkes processes is controlled via a Dirichlet distribution. We learn the model via a maximum likelihood estimator (MLE) and propose an effective variational Bayesian inference algorithm. We specifically analyze the resulting EM-type algorithm in the context of inner-outer iterations and discuss several inner iteration allocation strategies. The identifiability of our model, the convergence of our learning method, and its sample complexity are analyzed in both theoretical and empirical ways, which demonstrate the superiority of our method to other competitors. The proposed method learns the number of clusters automatically and is robust to model misspecification. Experiments on both synthetic and real-world data show that our method can learn diverse triggering patterns hidden in asynchronous event sequences and achieve encouraging performance on clustering purity and consistency.
We introduce a new nonparametric clustering model which combines the recently proposed distance-dependent Chinese restaurant process (dd-CRP) and non-linear, spectral methods for dimensionality reduction. Our model retains the ability of nonparametric methods to learn the number of clusters from data. At the same time it addresses two key limitations of nonparametric Bayesian methods: modeling data that are not exchangeable and have many correlated features. Spectral methods use the similarity between documents to map them into a low-dimensional spectral space where we then compare several clustering methods. Our experiments on handwritten digits and text documents show that nonparametric methods such as the CRP or dd-CRP can perform as well as or better than k-means and also recover the true number of clusters. We improve the performance of the dd-CRP in spectral space by incorporating the original similarity matrix in its prior. This simple modification results in better performance than all other methods we compared to. We offer a new formulation and first experimental evaluation of a general Gibbs sampler for mixture modeling with distance-dependent CRPs. Time series clustering has been shown effective in providing useful information in various domains. There seems to be an increased interest in time series clustering as part of the effort in temporal data mining research. To provide an overview, this paper surveys and summarizes previous works that investigated the clustering of time series data in various application domains. The basics of time series clustering are presented, including general-purpose clustering algorithms commonly used in time series clustering studies, the criteria for evaluating the performance of the clustering results, and the measures to determine the similarity dissimilarity between two time series being compared, either in the forms of raw data, extracted features, or some model parameters. The past researchs are organized into three groups depending upon whether they work directly with the raw data either in the time or frequency domain, indirectly with features extracted from the raw data, or indirectly with models built from the raw data. The uniqueness and limitation of previous research are discussed and several possible topics for future research are identified. Moreover, the areas that time series clustering have been applied to are also summarized, including the sources of data used. It is hoped that this review will serve as the steppingstone for those interested in advancing this area of research. Statistical models based on Markov chains, especially mixtures of Markov chains, have recently been studied and demonstrated to be effective in various data mining applications such as tourist flow analysis, animal migration modeling, and transportation administration. Nevertheless, the research so far has mainly focused on analyzing data at individual levels. Due to security and privacy reasons, however, the observations in practice usually consist of coarse-grained statistics of individual data, a.k.a. aggregate data, rendering learning mixtures of Markov chains an even more challenging problem. In this work, we show that this challenging problem, although intractable in its original form, can be solved approximately by posing structural constraints on the transition matrices. The proposed structural constraints include specifying active state sets corresponding to the chains and adding a pairwise sparse regularization term on transition matrices. Based on these two structural constraints, we propose a constrained least-squares method to learn mixtures of Markov chains. We further develop a novel iterative algorithm that decomposes the overall problem into a set of convex subproblems and solves each subproblem efficiently, making it possible to effectively learn mixtures of Markov chains from aggregate data. We propose a framework for generating synthetic data and analyze the complexity of our algorithm. Additionally, the empirical results of the convergence and the robustness of our algorithm are also presented. These results demonstrate the effectiveness and efficiency of the proposed algorithm, comparing with traditional methods. Experimental results on real-world data sets further validate that our algorithm can be used to solve practical problems. This paper presents a test of hypotheses to compare two stationary time series as well as an accompanying classification procedure that uses this test of hypotheses to cluster stationary time series. Our hypotheses testing procedure, which unlike the existing tests, does not require the time series to be independent, is based on the differences between estimated parameters of the autoregressive models that are fitted to the time series. The classification procedure is based on the p-value of the test that is applied to every pair of given time series.  A new method is presented to get an insight into univariate time series data. The problem addressed is how to identify patterns and trends on multiple time scales (days, weeks, seasons) simultaneously. The solution presented is to cluster similar daily data patterns, and to visualize the average patterns as graphs and the corresponding days on a calendar. This presentation provides a quick insight into both standard and exceptional patterns. Furthermore, it is well suited to interactive exploration. Two applications, numbers of employees present and energy consumption, are presented. Point process data are commonly observed in fields like healthcare and the social sciences. Designing predictive models for such event streams is an under-explored problem, due to often scarce training data. In this work we propose a multitask point process model, leveraging information from all tasks via a hierarchical Gaussian process (GP). Nonparametric learning functions implemented by a GP, which map from past events to future rates, allow analysis of flexible arrival patterns. To facilitate efficient inference, we propose a sparse construction for this hierarchical model, and derive a variational Bayes method for learning and inference. Experimental results are shown on both synthetic data and as well as real electronic health-records data.  We propose a Multi-task Multi-dimensional Hawkes Process (MMHP) for modeling event sequences where there exist multiple triggering patterns within sequences and structures across sequences. MMHP is able to model the dynamics of multiple sequences jointly by imposing structural constraints and thus systematically uncover clustering structure among sequences. We propose an effective and robust optimization algorithm to learn MMHP models, which takes advantage of alternating direction method of multipliers (ADMM), majorization minimization and Euler-Lagrange equations. Our experimental results demonstrate that MMHP performs well on both synthetic and real data. This article is concerned with variable selection for cluster analysis. The problem is regarded as a model selection problem in the model-based cluster analysis context. A general model generalizing the model of Raftery and Dean (2006) is proposed to specify the role of each variable. This model does not need any prior assumptions about the link between the selected and discarded variables. Models are compared with BIC. Variables role is obtained through an algorithm embedding two backward stepwise variable selection algorithms for clustering and linear regression. The consistency of the resulting criterion is proved under regularity conditions. Numerical experiments on simulated datasets and a genomics application highlight the interest of the proposed variable selection procedure. In many applications in social network analysis, it is important to model the interactions and infer the influence between pairs of actors, leading to the problem of dyadic event modeling which has attracted increasing interests recently. In this paper we focus on the problem of dyadic event attribution, an important missing data problem in dyadic event modeling where one needs to infer the missing actor-pairs of a subset of dyadic events based on their observed timestamps. Existing works either use fixed model parameters and heuristic rules for event attribution, or assume the dyadic events across actor-pairs are independent. To address those shortcomings we propose a probabilistic model based on mixtures of Hawkes processes that simultaneously tackles event attribution and network parameter inference, taking into consideration the dependency among dyadic events that share at least one actor. We also investigate using additive models to incorporate regularization to avoid overfitting. Our experiments on both synthetic and real-world data sets on international armed conflicts suggest that the proposed new method is capable of significantly improve accuracy when compared with the state-of-the-art for dyadic event attribution. Diffusion network inference and meme tracking have been two key challenges in viral diffusion. This paper shows that these two tasks can be addressed simultaneously with a probabilistic model involving a mixture of mutually exciting point processes. A fast learning algorithms is developed based on mean-field variational inference with budgeted diffusion bandwidth. The model is demonstrated with applications to the diffusion of viral texts in (1) online social networks (e.g., Twitter) and (2) the blogosphere on the Web. Dirichlet process (DP) mixture models are the cornerstone of non- parametric Bayesian statistics, and the development of Monte-Carlo Markov chain (MCMC) sampling methods for DP mixtures has enabled the application of non- parametric Bayesian methods to a variety of practical data analysis problems. However, MCMC sampling can be prohibitively slow, and it is important to ex- plore alternatives. One class of alternatives is provided by variational methods, a class of deterministic algorithms that convert inference problems into optimization problems (Opper and Saad 2001; Wainwright and Jordan 2003). Thus far, varia- tional methods have mainly been explored in the parametric setting, in particular within the formalism of the exponential family (Attias 2000; Ghahramani and Beal 2001; 2003). In this paper, we present a variational inference algorithm for DP mixtures. We present experiments that compare the algorithm to Gibbs sampling algorithms for DP mixtures of Gaussians and present an application to a large-scale image analysis problem. Clusters in document streams, such as online news articles, can be induced by their textual contents, as well as by the temporal dynamics of their arriving patterns. Can we leverage both sources of information to obtain a better clustering of the documents, and distill information that is not possible to extract using contents only? In this paper, we propose a novel random process, referred to as the Dirichlet-Hawkes process, to take into account both information in a unified framework. A distinctive feature of the proposed model is that the preferential attachment of items to clusters according to cluster sizes, present in Dirichlet processes, is now driven according to the intensities of cluster-wise self-exciting temporal point processes, the Hawkes processes. This new model establishes a previously unexplored connection between Bayesian Nonparametrics and temporal Point Processes, which makes the number of clusters grow to accommodate the increasing complexity of online streaming contents, while at the same time adapts to the ever changing dynamics of the respective continuous arrival time. We conducted large-scale experiments on both synthetic and real world news articles, and show that Dirichlet-Hawkes processes can recover both meaningful topics and temporal dynamics, which leads to better predictive performance in terms of content perplexity and arrival time of future documents.