Merging KK-means with hierarchical clustering for identifying general-shaped groups

Clustering partitions a dataset such that observations placed together in a group are similar but different from those in other groups. Hierarchical and $K$-means clustering are two approaches but have different strengths and weaknesses. For instance, hierarchical clustering identifies groups in a tree-like structure but suffers from computational complexity in large datasets while $K$-means clustering is efficient but designed to identify homogeneous spherically-shaped clusters. We present a hybrid non-parametric clustering approach that amalgamates the two methods to identify general-shaped clusters and that can be applied to larger datasets. Specifically, we first partition the dataset into spherical groups using $K$-means. We next merge these groups using hierarchical methods with a data-driven distance measure as a stopping criterion. Our proposal has the potential to reveal groups with general shapes and structure in a dataset. We demonstrate good performance on several simulated and real datasets.Comment: 16 pages, 1 table, 9 figures; accepted for publication in Sta

Multi-PeV Signals from a New Astrophysical Neutrino Flux Beyond the Glashow Resonance

The IceCube neutrino discovery was punctuated by three showers with $E_\nu$ ~ 1-2 PeV. Interest is intense in possible fluxes at higher energies, though a marked deficit of $E_\nu$ ~ 6 PeV Glashow resonance events implies a spectrum that is soft and/or cutoff below ~few PeV. However, IceCube recently reported a through-going track event depositing 2.6 $\pm$ 0.3 PeV. A muon depositing so much energy can imply $E_{\nu_\mu} \gtrsim$ 10 PeV. We show that extending the soft $E_\nu^{-2.6}$ spectral fit from TeV-PeV data is unlikely to yield such an event. Alternatively, a tau can deposit this much energy, though requiring $E_{\nu_\tau}$ ~10x higher. We find that either scenario hints at a new flux, with the hierarchy of $\nu_\mu$ and $\nu_\tau$ energies suggesting a window into astrophysical neutrinos at $E_\nu$ ~ 100 PeV if a tau. We address implications, including for ultrahigh-energy cosmic-ray and neutrino origins.Comment: 6 pages, 4 figures + 3 pages Supplementary Material; updated to agree with version published in Physical Review Letter

Extra-matrix Mg\u3csup\u3e2+\u3c/sup\u3e Limits Ca\u3csup\u3e2+\u3c/sup\u3e Uptake and Modulates Ca\u3csup\u3e2+\u3c/sup\u3e Uptake-independent Respiration and Redox State in Cardiac Isolated Mitochondria

Cardiac mitochondrial matrix (m) free Ca2+ ([Ca2+]m) increases primarily by Ca2+ uptake through the Ca2+ uniporter (CU). Ca2+ uptake via the CU is attenuated by extra-matrix (e) Mg2+ ([Mg2+]e). How [Ca2+]m is dynamically modulated by interacting physiological levels of [Ca2+]e and [Mg2+]e and how this interaction alters bioenergetics are not well understood. We postulated that as [Mg2+]e modulates Ca2+ uptake via the CU, it also alters bioenergetics in a matrix Ca2+–induced and matrix Ca2+–independent manner. To test this, we measured changes in [Ca2+]e, [Ca2+]m, [Mg2+]e and [Mg2+]m spectrofluorometrically in guinea pig cardiac mitochondria in response to added CaCl2 (0–0.6 mM; 1 mM EGTA buffer) with/without added MgCl2 (0–2 mM). In parallel, we assessed effects of added CaCl2 and MgCl2 on NADH, membrane potential (ΔΨm), and respiration. We found that \u3e0.125 mM MgCl2 significantly attenuated CU-mediated Ca2+ uptake and [Ca2+]m. Incremental [Mg2+]e did not reduce initial Ca2+uptake but attenuated the subsequent slower Ca2+ uptake, so that [Ca2+]m remained unaltered over time. Adding CaCl2 without MgCl2 to attain a [Ca2+]m from 46 to 221 nM enhanced state 3 NADH oxidation and increased respiration by 15 %; up to 868 nM [Ca2+]m did not additionally enhance NADH oxidation or respiration. Adding MgCl2 did not increase [Mg2+]m but it altered bioenergetics by its direct effect to decrease Ca2+ uptake. However, at a given [Ca2+]m, state 3 respiration was incrementally attenuated, and state 4 respiration enhanced, by higher [Mg2+]e. Thus, [Mg2+]e without a change in [Mg2+]m can modulate bioenergetics independently of CU-mediated Ca2+ transport

Stochastic majorisation: exploding some myths

The analysis of many randomised algorithms involves random variables that are not independent, and hence many of the standard tools from classical probability theory that would be useful in the analysis, such as the Chernoff--Hoeffding bounds are rendered inapplicable. However, in many instances, the random variables involved are, nevertheless {\em negatively related\/} in the intuitive sense that when one of the variables is ``large'', another is likely to be ``small''. (this notion is made precise and analysed in [1].) In such situations, one is tempted to conjecture that these variables are in some sense {\em stochastically dominated\/} by a set of {\em independent\/} random variables with the same marginals. Thereby, one hopes to salvage tools such as the Chernoff--Hoeffding bound also for analysis involving the dependent set of variables. The analysis in [6, 7, 8] seems to strongly hint in this direction. In this note, we explode myths of this kind, and argue that stochastic majorisation in conjunction with an independent set of variables is actually much less useful a notion than it might have appeared