In the framework of model-based clustering, a model, called multi-partitions clustering, allowing several latent class variables has been proposed. This model assumes that the distribution of the observed data can be factorized into several independent blocks of variables, each block following its own mixture model. In this paper, we assume that each block follows a non parametric latent class model, {\it i.e.} independence of the variables in each component of the mixture with no parametric assumption on their class conditional distribution. The purpose is to deduce, from the observation of a sample, the number of blocks, the partition of the variables into the blocks and the number of components in each block, which characterise the proposed model. By following recent literature on model and variable selection in non-parametric mixture models, we propose to discretize the data into bins. This permits to apply the classical multi-partition clustering procedure for parametric multinomials, which are based on a penalized likelihood method (\emph{e.g.} BIC). The consistency of the procedure is obtained and an efficient optimization is proposed. The performances of the model are investigated on simulated data.
We are concerned in clustering continuous data sets subject to non-ignorable missingness. We perform clustering with a specific semi-parametric mixture, under the assumption of conditional independence given the component. The mixture model isused for clustering and not for estimating the density of the full variables (observed and unobserved), thus we do not need other assumptions on the component distribution neither to specify the missingness mechanism. Estimation is performed by maximizing an extension of smoothed likelihood allowing missingness. This optimization is achieved by a Majoriza…
Bayesian model-based clustering is a widely applied procedure for discovering groups of related observations in a dataset. These approaches use Bayesian mixture models, estimated with MCMC, which provide posterior samples of the model parameters and clustering partition. While inference on model parameters is well established, inference on the clustering partition is less developed. A new method is developed for estimating the optimal partition from the pairwise posterior similarity matrix generated by a Bayesian cluster model. This approach uses non-negative matrix factorization (NMF) to prov…
There are two notoriously hard problems in cluster analysis, estimating the number of clusters, and checking whether the population to be clustered is not actually homogeneous. Given a dataset, a clustering method and a cluster validation index, this paper proposes to set up null models that capture structural features of the data that cannot be interpreted as indicating clustering. Artificial datasets are sampled from the null model with parameters estimated from the original dataset. This can be used for testing the null hypothesis of a homogeneous population against a clustering alternative…
Local spectral clustering for heterogeneous clustering structures
School of Economics and Management, Fuzhou University（福州大学经济与管理学院）；School of Economics, Xiamen University（厦门大学经济学院）；The Wang Yanan Institute for Studies in Economics, Xiamen University（厦门大学王亚南经济研究院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Beijing Institute of Mathematical Sciences and Applications（北京应用数学与计算物理研究所）
There is a very rich literature proposing Bayesian approaches for clustering starting with a prior probability distribution on partitions. Most approaches assume exchangeability, leading to simple representations in terms of Exchangeable Partition Probability Functions (EPPF). Gibbs-type priors encompass a broad class of such cases, including Dirichlet and Pitman-Yor processes. Even though there have been some proposals to relax the exchangeability assumption, allowing covariate-dependence and partial exchangeability, limited consideration has been given on how to include concrete prior knowle…
University of Ljubljana（卢布尔雅那大学）；Slovenian Environment Agency（斯洛文尼亚环境署）；Tampere University（坦佩雷大学）
We investigate a novel non-parametric regression-based clustering algorithm for longitudinal data analysis. Combining natural cubic splines with Gaussian mixture models (GMM), the algorithm can produce smooth cluster means that describe the underlying data well. However, there are some shortcomings in the algorithm: high computational complexity in the parameter estimation procedure and a numerically unstable variance estimator. Therefore, to further increase the usability of the method, we incorporated approaches to reduce its computational complexity, we developed a new, more stable variance…
University of Tübingen（图宾根大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Technical University of Munich（慕尼黑工业大学）
Despite the ubiquity of kernel-based clustering, surprisingly few statistical guarantees exist beyond settings that consider strong structural assumptions on the data generation process. In this work, we take a step towards bridging this gap by studying the statistical performance of kernel-based clustering algorithms under non-parametric mixture models. We provide necessary and sufficient separability conditions under which these algorithms can consistently recover the underlying true clustering. Our analysis provides guarantees for kernel clustering approaches without structural assumptions …
Motivated by the problem of accurately predicting gap times between successive blood donations, we present here a general class of Bayesian nonparametric models for clustering. These models allow for prediction of new recurrences, accommodating covariate information that describes the personal characteristics of the sample individuals. We introduce a prior for the random partition of the sample individuals which encourages two individuals to be co-clustered if they have similar covariate values. Our prior generalizes PPMx models in the literature, which are defined in terms of cohesion and sim…
In cluster analysis, it can be useful to interpret the partition built from the data in the light of external categorical variables which were not directly involved to cluster the data. An approach is proposed in the model-based clustering context to select a model and a number of clusters which both fit the data well and take advantage of the potential illustrative ability of the external variables. This approach makes use of the integrated joint likelihood of the data and the partitions at hand, namely the model-based partition and the partitions associated to the external variables. It is n…
The fundamental aim of clustering algorithms is to partition data points. We consider tasks where the discovered partition is allowed to vary with some covariate such as space or time. One approach would be to use fragmentation-coagulation processes, but these, being Markov processes, are restricted to linear or tree structured covariate spaces. We define a partition-valued process on an arbitrary covariate space using Gaussian processes. We use the process to construct a multitask clustering model which partitions datapoints in a similar way across multiple data sources, and a time series mod…
In Bayesian semi-parametric analyses of time-to-event data, non-parametric process priors are adopted for the baseline hazard function or the cumulative baseline hazard function for a given finite partition of the time axis. However, it would be controversial to suggest a general guideline to construct an optimal time partition. While a great deal of research has been done to relax the assumption of the fixed split times for other non-parametric processes, to our knowledge, no methods have been developed for a gamma process prior, which is one of the most widely used in Bayesian survival analy…
An exponential-time exact algorithm is provided for the task of clustering n items of data into k clusters. Instead of seeking one partition, posterior probabilities are computed for summary statistics: the number of clusters, and pairwise co-occurrence. The method is based on subset convolution, and yields the posterior distribution for the number of clusters in O(n * 3^n) operations, or O(n^3 * 2^n) using fast subset convolution. Pairwise co-occurrence probabilities are then obtained in O(n^3 * 2^n) operations. This is considerably faster than exhaustive enumeration of all partitions.
Accurate estimation of the change in crime over time is a critical first step towards better understanding of public safety in large urban environments. Bayesian hierarchical modeling is a natural way to study spatial variation in urban crime dynamics at the neighborhood level, since it facilitates principled ``sharing of information'' between spatially adjacent neighborhoods. Typically, however, cities contain many physical and social boundaries that may manifest as spatial discontinuities in crime patterns. In this situation, standard prior choices often yield overly-smooth parameter estimat…
Systems Biology Ireland（爱尔兰系统生物学）；University College Dublin（都柏林大学）
In many real life problems, objects are described by large number of binary features. For instance, documents are characterized by presence or absence of certain keywords; cancer patients are characterized by presence or absence of certain mutations etc. In such cases, grouping together similar objects/profiles based on such high dimensional binary features is desirable, but challenging. Here, I present a Bayesian non parametric algorithm for clustering high dimensional binary data. It uses a Dirichlet Process (DP) mixture model and simulated annealing to not only cluster binary data, but also…
In climate change study, the infrared spectral signatures of climate change have recently been conceptually adopted, and widely applied to identifying and attributing atmospheric composition change. We propose a Bayesian hierarchical model for spatial clustering of the high-dimensional functional data based on the effects of functional covariates and local features. We couple the functional mixed-effects model with a generalized spatial partitioning method for: (1) producing spatially contiguous clusters for the high-dimensional spatio-functional data; (2) improving the computational efficienc…
State University of New York at Buffalo（纽约州立大学布法罗分校）；University of California, Irvine（加州大学欧文分校）
Current state-of-the-art nonparametric Bayesian text clustering methods model documents through multinomial distribution on bags of words. Although these methods can effectively utilize the word burstiness representation of documents and achieve decent performance, they do not explore the sequential information of text and relationships among synonyms. In this paper, the documents are modeled as the joint of bags of words, sequential features and word embeddings. We proposed Sequential Embedding induced Dirichlet Process Mixture Model (SiDPMM) to effectively exploit this joint document represe…
Laboratoire ERIC, Université de Lyon（里昂大学ERIC实验室）；Technicolor（特艺集团）；Université de Toulouse（图卢兹大学）；CNRS（法国国家科学研究中心）
We present a dual-view mixture model to cluster users based on their features and latent behavioral functions. Every component of the mixture model represents a probability density over a feature view for observed user attributes and a behavior view for latent behavioral functions that are indirectly observed through user actions or behaviors. Our task is to infer the groups of users as well as their latent behavioral functions. We also propose a non-parametric version based on a Dirichlet Process to automatically infer the number of clusters. We test the properties and performance of the mode…
Université de Technologie de Compiègne（贡比涅技术大学）；Shanghai University（上海大学）
Evidential clustering is an approach to clustering in which cluster-membership uncertainty is represented by a collection of Dempster-Shafer mass functions forming an evidential partition. In this paper, we propose to construct these mass functions by bootstrapping finite mixture models. In the first step, we compute bootstrap percentile confidence intervals for all pairwise probabilities (the probabilities for any two objects to belong to the same class). We then construct an evidential partition such that the pairwise belief and plausibility degrees approximate the bounds of the confidence i…
Initialisation of the EM algorithm in model-based clustering is often crucial. Various starting points in the parameter space often lead to different local maxima of the likelihood function and, so to different clustering partitions. Among the several approaches available in the literature, model-based agglomerative hierarchical clustering is used to provide initial partitions in the popular MCLUST R package. This choice is computationally convenient and often yields good clustering partitions. However, in certain circumstances, poor initial partitions may cause the EM algorithm to converge to…
The paper exposes a non-parametric approach to latent and co-latent modeling of bivariate data, based upon alternating minimization of the Kullback-Leibler divergence (EM algorithm) for complete log-linear models. For categorical data, the iterative algorithm generates a soft clustering of both rows and columns of the contingency table. Well-known results are systematically revisited, and some variants are presumably original. In particular, the consideration of square contingency tables induces a clustering algorithm for weighted networks, differing from spectral clustering or modularity maxi…
Hierarchical and k-medoids clustering are deterministic clustering algorithms based on pairwise distances. Using these same pairwise distances, we propose a novel stochastic clustering method based on random partition distributions. We call our method CaviarPD, for cluster analysis via random partition distributions. CaviarPD first samples clusterings from a random partition distribution and then finds the best cluster estimate based on these samples using algorithms to minimize an expected loss. We compare CaviarPD with hierarchical and k-medoids clustering through eight case studies. Cluster…
We investigate the parameter estimation of regression models with fixed group effects, when the group variable is missing while group related variables are available. This problem involves clustering to infer the missing group variable based on the group related variables, and regression to build a model on the target variable given the group and eventually additional variables. Thus, this problem can be formulated as the joint distribution modeling of the target and of the group related variables. The usual parameter estimation strategy for this joint model is a two-step approach starting by …
In comparing clustering partitions, Rand index (RI) and Adjusted Rand index (ARI) are commonly used for measuring the agreement between the partitions. Both these external validation indexes aim to analyze how close is a cluster to a reference (or to prior knowledge about the data) by counting corrected classified pairs of elements. When the aim is to evaluate the solution of a fuzzy clustering algorithm, the computation of these measures require converting the soft partitions into hard ones. It is known that different fuzzy partitions describing very different structures in the data can lead …
A novel elastic time distance for sparse multivariate functional data is proposed and used to develop a robust distance-based two-layer partition clustering method. With this proposed distance, the new approach not only can detect correct clusters for sparse multivariate functional data under outlier settings but also can detect those outliers that do not belong to any clusters. Classical distance-based clustering methods such as density-based spatial clustering of applications with noise (DBSCAN), agglomerative hierarchical clustering, and $K$-medoids are extended to the sparse multivariate f…
A limitation of many clustering algorithms is the requirement to tune adjustable parameters for each application or even for each dataset. Some techniques require an \emph{a priori} estimate of the number of clusters while density-based techniques usually require a scale parameter. Other parametric methods, such as mixture modeling, make assumptions about the underlying cluster distributions. Here we introduce a non-parametric clustering method that does not involve tunable parameters and only assumes that clusters are unimodal, in the sense that they have a single point of maximal density whe…
Multiple imputation (MI) is a popular method for dealing with missing values. However, the suitable way for applying clustering after MI remains unclear: how to pool partitions? How to assess the clustering instability when data are incomplete? By answering both questions, this paper proposed a complete view of clustering with missing data using MI. The problem of partitions pooling is here addressed using consensus clustering while, based on the bootstrap theory, we explain how to assess the instability related to observed and missing data. The new rules for pooling partitions and instability…
We propose a clustering procedure to group K populations into subgroups with the same dependence structure. The method is adapted to paired population and can be used with panel data. It relies on the differences between orthogonal projection coefficients of the K density copulas estimated from the K populations. Each cluster is then constituted by populations having significantly similar dependence structures. A recent test statistic from Ngounou-Bakam and Pommeret (2022) is used to construct automatically such clusters. The procedure is data driven and depends on the asymptotic level of the …
We consider the simultaneous clustering of rows and columns of a matrix and more particularly the ability to measure the agreement between two co-clustering partitions. The new criterion we developed is based on the Adjusted Rand Index and is called the Co-clustering Adjusted Rand Index named CARI. We also suggest new improvements to existing criteria such as the Classification Error which counts the proportion of misclassified cells and the Extended Normalized Mutual Information criterion which is a generalization of the criterion based on mutual information in the case of classic classificat…
Similarity-based Random Partition Distribution for Clustering Functional Data
Graduate School of Economics, The University of Tokyo（东京大学经济学研究科）；Faculty of Economics, Keio University（庆应义塾大学经济学部）；School of Commerce, Meiji University（明治大学商学部）
Density-based clustering relies on the idea of linking groups to some specific features of the probability distribution underlying the data. The reference to a true, yet unknown, population structure allows to frame the clustering problem in a standard inferential setting, where the concept of ideal population clustering is defined as the partition induced by the true density function. The nonparametric formulation of this approach, known as modal clustering, draws a correspondence between the groups and the domains of attraction of the density modes. Operationally, a nonparametric density est…
Cluster analysis is an unsupervised learning strategy that can be employed to identify subgroups of observations in data sets of unknown structure. This strategy is particularly useful for analyzing high-dimensional data such as microarray gene expression data. Many clustering methods are available, but it is challenging to determine if the identified clusters represent distinct subgroups. We propose a novel strategy to investigate the significance of identified clusters by comparing the within- cluster sum of squares from the original data to that produced by clustering an appropriate unimoda…
Nonhierarchical clustering depending on unsupervised algorithms may not retrieve the optimal partition of datasets. Determining if clusters fit ``natural partitions`` can be achieved using cluster validity indices (CVIs). Most existing CVIs consider criteria such as cohesion, separation, and their equivalents. However, these binary relations may provide neither the optimal measure of partition suitability nor reference values corresponding to the worst partition. Moreover, previous CVI studies have been mostly focused on fitting correct partitions according to researchers' a priori assumptions…
Relaxing partition admissibility in Cluster-DAGs: a causal calculus with arbitrary variable clustering
Univ Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔理工学院）；LIG（格勒诺布尔信息实验室）；University of Münster（明斯特大学）；Institute of Medical Informatics（医学信息学研究所）；Université de Lorraine（洛林大学）；CRAN（自动控制研究中心）
We study a non-parametric approach to multivariate density estimation. The estimators are piecewise constant density functions supported by binary partitions. The partition of the sample space is learned by maximizing the likelihood of the corresponding histogram on that partition. We analyze the convergence rate of the sieve maximum likelihood estimator, and reach a conclusion that for a relatively rich class of density functions the rate does not directly depend on the dimension. This suggests that, under certain conditions, this method is immune to the curse of dimensionality, in the sense …
An unsupervised classification method for point events occurring on a network of lines is proposed. The idea relies on the distributional flexibility and practicality of random partition models to discover the clustering structure featuring observations from a particular phenomenon taking place on a given set of edges. By incorporating the spatial effect in the random partition distribution, induced by a Dirichlet process, one is able to control the distance between edges and events, thus leading to an appealing clustering method. A Gibbs sampler algorithm is proposed and evaluated with a sens…
Markov chain Monte Carlo (MCMC) methods are often used in clustering since they guarantee asymptotically exact expectations in the infinite-time limit. In finite time, though, slow mixing often leads to poor performance. Modern computing environments offer massive parallelism, but naive implementations of parallel MCMC can exhibit substantial bias. In MCMC samplers of continuous random variables, Markov chain couplings can overcome bias. But these approaches depend crucially on paired chains meetings after a small number of transitions. We show that straightforward applications of existing cou…
Water pollution is a major global environmental problem, and it poses a great environmental risk to public health and biological diversity. This work is motivated by assessing the potential environmental threat of coal mining through increased sulfate concentrations in river networks, which do not belong to any simple parametric distribution. However, existing network models mainly focus on binary or discrete networks and weighted networks with known parametric weight distributions. We propose a principled nonparametric weighted network model based on exponential-family random graph models and…
