Fortunately, in many realistic scenarios, only a handful of features may be relevant in distinguishing clusters.

In this paper, we advocate a novel measure for the purpose of checking the quality of a cluster partition for a sample into several distinct classes, and thus, determine the unknown value for the true number of clusters prevailing the provided set of data. Our objective leads us to the development of an approach through applying the multinomial distribution to the distances of data members, clustered in a group, from their respective cluster representatives. This procedure is carried out independently for each of the clusters, and the concerned statistics are combined together to design our targeted measure. Individual clusters separately possess the category-wise probabilities which correspond to different positions of its members in the cluster with respect to a typical member, in the form of cluster-centroid, medoid or mode, referred to as the corresponding cluster representative. Our method is robust in the sense that it is distribution-free, since this is devised irrespective of the parent distribution of the underlying sample. It fulfills one of the rare coveted qualities, present in the existing cluster accuracy measures, of having the capability to investigate whether the assigned sample owns any inherent clusters other than a single group of all members or not. Our measure's simple concept, easy algorithm, fast runtime, good performance, and wide usefulness, demonstrated through extensive simulation and diverse case-studies, make it appealing.

Clustering is a technique in machine learning that attempts to find clusters of observations within a dataset. The goal is to find clusters such that the observations within each cluster are quite similar to each other, while observations in different clusters are quite different from each other. Clustering is a form of unsupervised learning because we're simply attempting to find structure within a dataset rather than predicting the value of some response variable. When this information is available, clustering can be used to identify households that are similar and may be more likely to purchase certain products or respond better to a certain type of advertising. One of the most common forms of clustering is known as k-means clustering.

Functional magnetic resonance imaging (fMRI) data have become increasingly available and are useful for describing functional connectivity (FC), the relatedness of neuronal activity in regions of the brain. This FC of the brain provides insight into certain neurodegenerative diseases and psychiatric disorders, and thus is of clinical importance. To help inform physicians regarding patient diagnoses, unsupervised clustering of subjects based on FC is desired, allowing the data to inform us of groupings of patients based on shared features of connectivity. Since heterogeneity in FC is present even between patients within the same group, it is important to allow subject-level differences in connectivity, while still pooling information across patients within each group to describe group-level FC. To this end, we propose a random covariance clustering model (RCCM) to concurrently cluster subjects based on their FC networks, estimate the unique FC networks of each subject, and to infer shared network features. Although current methods exist for estimating FC or clustering subjects using fMRI data, our novel contribution is to cluster or group subjects based on similar FC of the brain while simultaneously providing group- and subject-level FC network estimates. The competitive performance of RCCM relative to other methods is demonstrated through simulations in various settings, achieving both improved clustering of subjects and estimation of FC networks. Utility of the proposed method is demonstrated with application to a resting-state fMRI data set collected on 43 healthy controls and 61 participants diagnosed with schizophrenia.

Clustering, a fundamental activity in unsupervised learning, is notoriously difficult when the feature space is high-dimensional. Fortunately, in many realistic scenarios, only a handful of features are relevant in distinguishing clusters. This has motivated the development of sparse clustering techniques that typically rely on k-means within outer algorithms of high computational complexity. Current techniques also require careful tuning of shrinkage parameters, further limiting their scalability. In this paper, we propose a novel framework for sparse k-means clustering that is intuitive, simple to implement, and competitive with state-of-the-art algorithms. We show that our algorithm enjoys consistency and convergence guarantees. Our core method readily generalizes to several task-specific algorithms such as clustering on subsets of attributes and in partially observed data settings. We showcase these contributions via simulated experiments and benchmark datasets, as well as a case study on mouse protein expression.

Sequencing items in adaptive learning systems typically relies on a large pool of interactive assessment items (questions) that are analyzed into a hierarchy of skills or Knowledge Components (KCs). Educational data mining techniques can be used to analyze students performance data in order to optimize the mapping of items to KCs. Standard methods that map items into KCs using item-similarity measures make the implicit assumption that students performance on items that depend on the same skill should be similar. This assumption holds if the latent trait (mastery of the underlying skill) is relatively fixed during students activity, as in the context of testing, which is the primary context in which these measures were developed and applied. However, in adaptive learning systems that aim for learning, and address subject matters such as K6 Math that consist of multiple sub-skills, this assumption does not hold. In this paper we propose a new item-similarity measure, termed Kappa Learning (KL), which aims to address this gap. KL identifies similarity between items under the assumption of learning, namely, that learners mastery of the underlying skills changes as they progress through the items. We evaluate Kappa Learning on data from a computerized tutor that teaches Fractions for 4th grade, with experts tagging as ground truth, and on simulated data. Our results show that clustering that is based on Kappa Learning outperforms clustering that is based on commonly used similarity measures (Cohen Kappa, Yule, and Pearson).

Homicide mortality is a worldwide concern and has occupied the agenda of researchers and public managers. In Brazil, homicide is the third leading cause of death in the general population and the first in the 15-39 age group. In South America, Brazil has the third highest homicide mortality, behind Venezuela and Colombia. To measure the impacts of violence it is important to assess health systems and criminal justice, as well as other areas. In this paper, we analyze the spatial distribution of homicide mortality in the state of Goias, Center-West of Brazil, since the homicide rate increased from 24.5 per 100,000 in 2002 to 42.6 per 100,000 in 2014 in this location. Moreover, this state had the fifth position of homicides in Brazil in 2014. We considered socio-demographic variables for the state, performed analysis about correlation and employed three clustering algorithms: K-means, Density-based and Hierarchical. The results indicate the homicide rates are higher in cities neighbors of large urban centers, although these cities have the best socioeconomic indicators.

The availability of large microarray data has led to a growing interest in biclustering methods in the past decade. Several algorithms have been proposed to identify subsets of genes and conditions according to different similarity measures and under varying constraints. In this paper we focus on the exclusive row biclustering problem for gene expression data sets, in which each row can only be a member of a single bicluster while columns can participate in multiple ones. This type of biclustering may be adequate, for example, for clustering groups of cancer patients where each patient (row) is expected to be carrying only a single type of cancer, while each cancer type is associated with multiple (and possibly overlapping) genes (columns). We present a novel method to identify these exclusive row biclusters through a combination of existing biclustering algorithms and combinatorial auction techniques. We devise an approach for tuning the threshold for our algorithm based on comparison to a null model in the spirit of the Gap statistic approach [20]. We demonstrate our approach on both synthetic and real-world gene expression data and show its power in identifying large span non-overlapping rows sub matrices, while considering their unique nature. The Gap statistic approach succeeds in identifying appropriate thresholds in all our examples. This was presented at the IEEE International Conference in Data Mining at 2012 [22], and published at the Journal of Computer Science and Technology [23].

Dynamic tensor data are becoming prevalent in numerous applications. Existing tensor clustering methods either fail to account for the dynamic nature of the data, or are inapplicable to a general-order tensor. Also there is often a gap between statistical guarantee and computational efficiency for existing tensor clustering solutions. In this article, we aim to bridge this gap by proposing a new dynamic tensor clustering method, which takes into account both sparsity and fusion structures, and enjoys strong statistical guarantees as well as high computational efficiency. Our proposal is based upon a new structured tensor factorization that encourages both sparsity and smoothness in parameters along the specified tensor modes. Computationally, we develop a highly efficient optimization algorithm that benefits from substantial dimension reduction. In theory, we first establish a non-asymptotic error bound for the estimator from the structured tensor factorization. Built upon this error bound, we then derive the rate of convergence of the estimated cluster centers, and show that the estimated clusters recover the true cluster structures with a high probability. Moreover, our proposed method can be naturally extended to co-clustering of multiple modes of the tensor data. The efficacy of our approach is illustrated via simulations and a brain dynamic functional connectivity analysis from an Autism spectrum disorder study.