Diffusion Maps, Spectral Clustering and Eigenfunctions of Fokker-Planck operators

This paper presents a diffusion based probabilistic interpretation of spectral clustering and dimensionality reduction algorithms that use the eigenvectors of the normalized graph Laplacian. Given the pairwise adjacency matrix of all points, we define a diffusion distance between any two data points and show that the low dimensional representation of the data by the first few eigenvectors of the corresponding Markov matrix is optimal under a certain mean squared error criterion. Furthermore, assuming that data points are random samples from a density p(\x) = e^{-U(\x)} we identify these eigenvectors as discrete approximations of eigenfunctions of a Fokker-Planck operator in a potential 2U(\x) with reflecting boundary conditions. Finally, applying known results regarding the eigenvalues and eigenfunctions of the continuous Fokker-Planck operator, we provide a mathematical justification for the success of spectral clustering and dimensional reduction algorithms based on these first few eigenvectors. This analysis elucidates, in terms of the characteristics of diffusion processes, many empirical findings regarding spectral clustering algorithms.Comment: submitted to NIPS 200

Numeric and fluid dynamic representation of tornadic double vortex thunderstorms

Current understanding of a double vortex thunderstorm involves a pair of contra-rotating vortices that exists in the dynamic updraft. The pair is believed to be a result of a blocking effect which occurs when a cylindrical thermal updraft of a thunderstorm protrudes into the upper level air and there is a large amount of vertical wind shear between the low level and upper level air layers. A numerical tornado prediction scheme based on the double vortex thunderstorm was developed. The Energy-Shear Index (ESI) is part of the scheme and is calculated from radiosonde measurements. The ESI incorporates parameters representative of thermal instability and blocking effect, and indicates appropriate environments for which the development of double vortex thunderstorms is likely

Asking the experts : developing and validating parental diaries to assess children's minor injuries

The methodological issues involved in parental reporting of events in children's everyday lives are discussed with reference to the development and validation of an incident diary, collecting concurrent data on minor injuries in a community study of children under eight years old. Eighty-two mothers participated in a comparison over nine days of daily telephone interviews and structured incident diaries. Telephone methods resulted in more missing data, and participants in both groups expressed a preference for the diary method. This diary was then validated on a sample of 56 preschool and school-aged children by comparing injury recording by a research health visitor with that of their mothers. Each failed to report some injuries, but there was good agreement overall, and in descriptive data on injuries reported by both. Parental diaries have the potential to provide rich data, of acceptable validity, on minor events in everyday life

The significance of ram breed and ewe breed in crossbreeding for meat production from sheep in an intensive system

International audienc

Intestinal permeability in normally nourished and malnourished children with and without diarrhea

We evaluated the association between diarrhea, malnutrition and intestinal function using the lactulose-mannitol test. Our study showed that a third of all children have abnormal intestinal permeability, there was an expected increase of permeability in children with acute diarrhea and alteration in intestinal permeability was greater in children with concurrent malnutrition and diarrhea

Self-Similar Collisionless Shocks

Observations of gamma-ray burst afterglows suggest that the correlation length of magnetic field fluctuations downstream of relativistic non-magnetized collisionless shocks grows with distance from the shock to scales much larger than the plasma skin depth. We argue that this indicates that the plasma properties are described by a self-similar solution, and derive constraints on the scaling properties of the solution. For example, we find that the scaling of the characteristic magnetic field amplitude with distance from the shock is B \propto D^{s_B} with -1<s_B<=0, that the spectrum of accelerated particles is dn/dE \propto E^{-2/(s_B+1)}, and that the scaling of the magnetic correlation function is \propto x^{2s_B} (for x>>D). We show that the plasma may be approximated as a combination of two self-similar components: a kinetic component of energetic particles and an MHD-like component representing "thermal" particles. We argue that the latter may be considered as infinitely conducting, in which case s_B=0 and the scalings are completely determined (e.g. dn/dE \propto E^{-2} and B \propto D^0). Similar claims apply to non- relativistic shocks such as in supernova remnants, if the upstream magnetic field can be neglected. Self-similarity has important implications for any model of particle acceleration and/or field generation. For example, we show that the diffusion function in the angle \mu of momentum p in diffusive shock acceleration models must satisfy D_{\mu\mu}(p,D) = D^{-1}D'_{\mu\mu}(p/D), and that a previously suggested model for the generation of large scale magnetic fields through a hierarchical merger of current-filaments should be generalized. A numerical experiment testing our analysis is outlined (Abridged).Comment: 16 pages, 1 figure, accepted for publication in Ap