The Beylkin-Cramer Summation Rule and A New Fast Algorithm of Cosmic Statistics for Large Data Sets

Based on the Beylkin-Cramer summation rule, we introduce a new fast algorithm that enable us to explore the high order statistics efficiently in large data sets. Central to this technique is to make decomposition both of fields and operators within the framework of multi-resolution analysis (MRA), and realize theirs discrete representations. Accordingly, a homogenous point process could be equivalently described by a operation of a Toeplitz matrix on a vector, which is accomplished by making use of fast Fourier transformation. The algorithm could be applied widely in the cosmic statistics to tackle large data sets. Especially, we demonstrate this novel technique using the spherical, cubic and cylinder counts in cells respectively. The numerical test shows that the algorithm produces an excellent agreement with the expected results. Moreover, the algorithm introduces naturally a sharp-filter, which is capable of suppressing shot noise in weak signals. In the numerical procedures, the algorithm is somewhat similar to particle-mesh (PM) methods in N-body simulations. As scaled with $O(N\log N)$, it is significantly faster than the current particle-based methods, and its computational cost does not relies on shape or size of sampling cells. In addition, based on this technique, we propose further a simple fast scheme to compute the second statistics for cosmic density fields and justify it using simulation samples. Hopefully, the technique developed here allows us to make a comprehensive study of non-Guassianity of the cosmic fields in high precision cosmology. A specific implementation of the algorithm is publicly available upon request to the author.Comment: 27 pages, 9 figures included. revised version, changes include (a) adding a new fast algorithm for 2nd statistics (b) more numerical tests including counts in asymmetric cells, the two-point correlation functions and 2nd variances (c) more discussions on technic

A Phase Field Model for Continuous Clustering on Vector Fields

A new method for the simplification of flow fields is presented. It is based on continuous clustering. A well-known physical clustering model, the Cahn Hilliard model, which describes phase separation, is modified to reflect the properties of the data to be visualized. Clusters are defined implicitly as connected components of the positivity set of a density function. An evolution equation for this function is obtained as a suitable gradient flow of an underlying anisotropic energy functional. Here, time serves as the scale parameter. The evolution is characterized by a successive coarsening of patterns-the actual clustering-during which the underlying simulation data specifies preferable pattern boundaries. We introduce specific physical quantities in the simulation to control the shape, orientation and distribution of the clusters as a function of the underlying flow field. In addition, the model is expanded, involving elastic effects. In the early stages of the evolution shear layer type representation of the flow field can thereby be generated, whereas, for later stages, the distribution of clusters can be influenced. Furthermore, we incorporate upwind ideas to give the clusters an oriented drop-shaped appearance. Here, we discuss the applicability of this new type of approach mainly for flow fields, where the cluster energy penalizes cross streamline boundaries. However, the method also carries provisions for other fields as well. The clusters can be displayed directly as a flow texture. Alternatively, the clusters can be visualized by iconic representations, which are positioned by using a skeletonization algorithm.

Methods of Hierarchical Clustering

We survey agglomerative hierarchical clustering algorithms and discuss efficient implementations that are available in R and other software environments. We look at hierarchical self-organizing maps, and mixture models. We review grid-based clustering, focusing on hierarchical density-based approaches. Finally we describe a recently developed very efficient (linear time) hierarchical clustering algorithm, which can also be viewed as a hierarchical grid-based algorithm.Comment: 21 pages, 2 figures, 1 table, 69 reference

Mock galaxy catalogs using the quick particle mesh method

Sophisticated analysis of modern large-scale structure surveys requires mock catalogs. Mock catalogs are used to optimize survey design, test reduction and analysis pipelines, make theoretical predictions for basic observables and propagate errors through complex analysis chains. We present a new method, which we call "quick particle mesh", for generating many large-volume, approximate mock catalogs at low computational cost. The method is based on using rapid, low-resolution particle mesh simulations that accurately reproduce the large-scale dark matter density field. Particles are sampled from the density field based on their local density such that they have N-point statistics nearly equivalent to the halos resolved in high-resolution simulations, creating a set of mock halos that can be populated using halo occupation methods to create galaxy mocks for a variety of possible target classes.Comment: 13 pages, 16 figures. Matches version accepted by MNRAS. Code available at http://github.com/mockFactor