On Bayesian "central clustering": Application to landscape classification of Western Ghats

Landscape classification of the well-known biodiversity hotspot, Western Ghats (mountains), on the west coast of India, is an important part of a world-wide program of monitoring biodiversity. To this end, a massive vegetation data set, consisting of 51,834 4-variate observations has been clustered into different landscapes by Nagendra and Gadgil [Current Sci. 75 (1998) 264--271]. But a study of such importance may be affected by nonuniqueness of cluster analysis and the lack of methods for quantifying uncertainty of the clusterings obtained. Motivated by this applied problem of much scientific importance, we propose a new methodology for obtaining the global, as well as the local modes of the posterior distribution of clustering, along with the desired credible and "highest posterior density" regions in a nonparametric Bayesian framework. To meet the need of an appropriate metric for computing the distance between any two clusterings, we adopt and provide a much simpler, but accurate modification of the metric proposed in [In Felicitation Volume in Honour of Prof. B. K. Kale (2009) MacMillan]. A very fast and efficient Bayesian methodology, based on [Sankhy\={a} Ser. B 70 (2008) 133--155], has been utilized to solve the computational problems associated with the massive data and to obtain samples from the posterior distribution of clustering on which our proposed methods of summarization are illustrated.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS454 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Multiple testing with persistent homology

Multiple hypothesis testing requires a control procedure. Simply increasing simulations or permutations to meet a Bonferroni-style threshold is prohibitively expensive. In this paper we propose a null model based approach to testing for acyclicity, coupled with a Family-Wise Error Rate (FWER) control method that does not suffer from these computational costs. We adapt an False Discovery Rate (FDR) control approach to the topological setting, and show it to be compatible both with our null model approach and with previous approaches to hypothesis testing in persistent homology. By extending a limit theorem for persistent homology on samples from point processes, we provide theoretical validation for our FWER and FDR control methods

Evaluation of second-level inference in fMRI analysis

We investigate the impact of decisions in the second-level (i.e., over subjects) inferential process in functional magnetic resonance imaging on (1) the balance between false positives and false negatives and on (2) the data-analytical stability, both proxies for the reproducibility of results. Second-level analysis based on a mass univariate approach typically consists of 3 phases. First, one proceeds via a general linear model for a test image that consists of pooled information from different subjects. We evaluate models that take into account first-level (within-subjects) variability and models that do not take into account this variability. Second, one proceeds via inference based on parametrical assumptions or via permutation-based inference. Third, we evaluate 3 commonly used procedures to address the multiple testing problem: familywise error rate correction, False Discovery Rate (FDR) correction, and a two-step procedure with minimal cluster size. Based on a simulation study and real data we find that the two-step procedure with minimal cluster size results in most stable results, followed by the familywise error rate correction. The FDR results in most variable results, for both permutation-based inference and parametrical inference. Modeling the subject-specific variability yields a better balance between false positives and false negatives when using parametric inference