A non-parametric and scale-independent method for cluster analysis II: the multivariate case

A general method is described for detecting and analysing galaxy systems. The multivariate geometrical structure of the sample is studied by using an extension of the method which we introduced in a previous paper. The method is based on an estimate of the probability density underlying a data sample. The density is estimated by using an iterative and adaptive kernel estimator. The used kernels have spherical symmetry, however we describe a method in order to estimate the locally optimal shape of the kernels. We use the results of the geometrical structure analysis in order to study the effects that is has on the cluster parameter estimate. This suggests a possible way to distinguish between structure and substructure within a sample. The method is tested by using simulated numerical models and applied to two galaxy samples taken from the literature. The results obtained for the Coma cluster suggest a core-halo structure formed by a large number of geometrically independent systems. A different conclusion is suggested by the results for the Cancer cluster indicating the presence of at least two independent structures both containing substructure. The dynamical consequences of the results obtained from the geometrical analysis will be described in a later paper. Further applications of the method are suggested and are currently in progress.Comment: To appear in Monthly Notices of R.A.S., 50 pages of text, latex file, aasms style, figures are available on request from the Autho

Non-Parametric Probabilistic Image Segmentation

We propose a simple probabilistic generative model for image segmentation. Like other probabilistic algorithms (such as EM on a Mixture of Gaussians) the proposed model is principled, provides both hard and probabilistic cluster assignments, as well as the ability to naturally incorporate prior knowledge. While previous probabilistic approaches are restricted to parametric models of clusters (e.g., Gaussians) we eliminate this limitation. The suggested approach does not make heavy assumptions on the shape of the clusters and can thus handle complex structures. Our experiments show that the suggested approach outperforms previous work on a variety of image segmentation tasks

Flexible parametric bootstrap for testing homogeneity against clustering and assessing the number of clusters

There are two notoriously hard problems in cluster analysis, estimating the number of clusters, and checking whether the population to be clustered is not actually homogeneous. Given a dataset, a clustering method and a cluster validation index, this paper proposes to set up null models that capture structural features of the data that cannot be interpreted as indicating clustering. Artificial datasets are sampled from the null model with parameters estimated from the original dataset. This can be used for testing the null hypothesis of a homogeneous population against a clustering alternative. It can also be used to calibrate the validation index for estimating the number of clusters, by taking into account the expected distribution of the index under the null model for any given number of clusters. The approach is illustrated by three examples, involving various different clustering techniques (partitioning around medoids, hierarchical methods, a Gaussian mixture model), validation indexes (average silhouette width, prediction strength and BIC), and issues such as mixed type data, temporal and spatial autocorrelation

The rich cluster of galaxies ABCG~85. IV. Emission line galaxies, luminosity function and dynamical properties

This paper is the fourth of a series dealing with the cluster of galaxies ABCG 85. Using our two extensive photometric and spectroscopic catalogues (with 4232 and 551 galaxies respectively), we discuss here three topics derived from optical data. First, we present the properties of emission line versus non-emission line galaxies, showing that their spatial distributions somewhat differ; emission line galaxies tend to be more concentrated in the south region where groups appear to be falling onto the main cluster, in agreement with the hypothesis (presented in our previous paper) that this infall may create a shock which can heat the X-ray emitting gas and also enhance star formation in galaxies. Then, we analyze the luminosity function in the R band, which shows the presence of a dip similar to that observed in other clusters at comparable absolute magnitudes; this result is interpreted as due to comparable distributions of spirals, ellipticals and dwarfs in these various clusters. Finally, we present the dynamical analysis of the cluster using parametric and non-parametric methods and compare the dynamical mass profiles obtained from the X-ray and optical data.Comment: accepted for publication in A&

Non-parametric deprojection of NIKA SZ observations: Pressure distribution in the Planck-discovered cluster PSZ1 G045.85+57.71

The determination of the thermodynamic properties of clusters of galaxies at intermediate and high redshift can bring new insights into the formation of large-scale structures. It is essential for a robust calibration of the mass-observable scaling relations and their scatter, which are key ingredients for precise cosmology using cluster statistics. Here we illustrate an application of high resolution $(< 20$ arcsec) thermal Sunyaev-Zel'dovich (tSZ) observations by probing the intracluster medium (ICM) of the \planck-discovered galaxy cluster \psz\ at redshift $z = 0.61$, using tSZ data obtained with the NIKA camera, which is a dual-band (150 and 260~GHz) instrument operated at the IRAM 30-meter telescope. We deproject jointly NIKA and \planck\ data to extract the electronic pressure distribution from the cluster core ($R \sim 0.02\, R_{500}$) to its outskirts ($R \sim 3\, R_{500}$) non-parametrically for the first time at intermediate redshift. The constraints on the resulting pressure profile allow us to reduce the relative uncertainty on the integrated Compton parameter by a factor of two compared to the \planck\ value. Combining the tSZ data and the deprojected electronic density profile from \xmm\ allows us to undertake a hydrostatic mass analysis, for which we study the impact of a spherical model assumption on the total mass estimate. We also investigate the radial temperature and entropy distributions. These data indicate that \psz\ is a massive ($M_{500} \sim 5.5 \times 10^{14}$ M$_{\odot}$) cool-core cluster. This work is part of a pilot study aiming at optimizing the treatment of the NIKA2 tSZ large program dedicated to the follow-up of SZ-discovered clusters at intermediate and high redshifts. (abridged)Comment: 16 pages, 10 figure