Multiple Imputation based Clustering Validation (MIV) for Big Longitudinal Trial Data with Missing Values in eHealth

Web-delivered trials are an important component in eHealth services. These trials, mostly behavior-based, generate big heterogeneous data that are longitudinal, high dimensional with missing values. Unsupervised learning methods have been widely applied in this area, however, validating the optimal number of clusters has been challenging. Built upon our multiple imputation (MI) based fuzzy clustering, MIfuzzy, we proposed a new multiple imputation based validation (MIV) framework and corresponding MIV algorithms for clustering big longitudinal eHealth data with missing values, more generally for fuzzy-logic based clustering methods. Specifically, we detect the optimal number of clusters by auto-searching and -synthesizing a suite of MI-based validation methods and indices, including conventional (bootstrap or cross-validation based) and emerging (modularity-based) validation indices for general clustering methods as well as the specific one (Xie and Beni) for fuzzy clustering. The MIV performance was demonstrated on a big longitudinal dataset from a real web-delivered trial and using simulation. The results indicate MI-based Xie and Beni index for fuzzy-clustering are more appropriate for detecting the optimal number of clusters for such complex data. The MIV concept and algorithms could be easily adapted to different types of clustering that could process big incomplete longitudinal trial data in eHealth services

Parsimonious Time Series Clustering

We introduce a parsimonious model-based framework for clustering time course data. In these applications the computational burden becomes often an issue due to the number of available observations. The measured time series can also be very noisy and sparse and a suitable model describing them can be hard to define. We propose to model the observed measurements by using P-spline smoothers and to cluster the functional objects as summarized by the optimal spline coefficients. In principle, this idea can be adopted within all the most common clustering frameworks. In this work we discuss applications based on a k-means algorithm. We evaluate the accuracy and the efficiency of our proposal by simulations and by dealing with drosophila melanogaster gene expression data

An efficient kk-means-type algorithm for clustering datasets with incomplete records

The $k$-means algorithm is arguably the most popular nonparametric clustering method but cannot generally be applied to datasets with incomplete records. The usual practice then is to either impute missing values under an assumed missing-completely-at-random mechanism or to ignore the incomplete records, and apply the algorithm on the resulting dataset. We develop an efficient version of the $k$-means algorithm that allows for clustering in the presence of incomplete records. Our extension is called $k_m$-means and reduces to the $k$-means algorithm when all records are complete. We also provide initialization strategies for our algorithm and methods to estimate the number of groups in the dataset. Illustrations and simulations demonstrate the efficacy of our approach in a variety of settings and patterns of missing data. Our methods are also applied to the analysis of activation images obtained from a functional Magnetic Resonance Imaging experiment.Comment: 21 pages, 12 figures, 3 tables, in press, Statistical Analysis and Data Mining -- The ASA Data Science Journal, 201

Relational visual cluster validity

The assessment of cluster validity plays a very important role in cluster analysis. Most commonly used cluster validity methods are based on statistical hypothesis testing or finding the best clustering scheme by computing a number of different cluster validity indices. A number of visual methods of cluster validity have been produced to display directly the validity of clusters by mapping data into two- or three-dimensional space. However, these methods may lose too much information to correctly estimate the results of clustering algorithms. Although the visual cluster validity (VCV) method of Hathaway and Bezdek can successfully solve this problem, it can only be applied for object data, i.e. feature measurements. There are very few validity methods that can be used to analyze the validity of data where only a similarity or dissimilarity relation exists – relational data. To tackle this problem, this paper presents a relational visual cluster validity (RVCV) method to assess the validity of clustering relational data. This is done by combining the results of the non-Euclidean relational fuzzy c-means (NERFCM) algorithm with a modification of the VCV method to produce a visual representation of cluster validity. RVCV can cluster complete and incomplete relational data and adds to the visual cluster validity theory. Numeric examples using synthetic and real data are presente

Sparse Subspace Clustering: Algorithm, Theory, and Applications

In many real-world problems, we are dealing with collections of high-dimensional data, such as images, videos, text and web documents, DNA microarray data, and more. Often, high-dimensional data lie close to low-dimensional structures corresponding to several classes or categories the data belongs to. In this paper, we propose and study an algorithm, called Sparse Subspace Clustering (SSC), to cluster data points that lie in a union of low-dimensional subspaces. The key idea is that, among infinitely many possible representations of a data point in terms of other points, a sparse representation corresponds to selecting a few points from the same subspace. This motivates solving a sparse optimization program whose solution is used in a spectral clustering framework to infer the clustering of data into subspaces. Since solving the sparse optimization program is in general NP-hard, we consider a convex relaxation and show that, under appropriate conditions on the arrangement of subspaces and the distribution of data, the proposed minimization program succeeds in recovering the desired sparse representations. The proposed algorithm can be solved efficiently and can handle data points near the intersections of subspaces. Another key advantage of the proposed algorithm with respect to the state of the art is that it can deal with data nuisances, such as noise, sparse outlying entries, and missing entries, directly by incorporating the model of the data into the sparse optimization program. We demonstrate the effectiveness of the proposed algorithm through experiments on synthetic data as well as the two real-world problems of motion segmentation and face clustering

Shape Interaction Matrix Revisited and Robustified: Efficient Subspace Clustering with Corrupted and Incomplete Data

The Shape Interaction Matrix (SIM) is one of the earliest approaches to performing subspace clustering (i.e., separating points drawn from a union of subspaces). In this paper, we revisit the SIM and reveal its connections to several recent subspace clustering methods. Our analysis lets us derive a simple, yet effective algorithm to robustify the SIM and make it applicable to realistic scenarios where the data is corrupted by noise. We justify our method by intuitive examples and the matrix perturbation theory. We then show how this approach can be extended to handle missing data, thus yielding an efficient and general subspace clustering algorithm. We demonstrate the benefits of our approach over state-of-the-art subspace clustering methods on several challenging motion segmentation and face clustering problems, where the data includes corrupted and missing measurements.Comment: This is an extended version of our iccv15 pape