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In applications of cluster analysis, one usually needs to determine the number of clusters, K, and the assignment of observations to each cluster. A clustering technique based on recursive application of a multivariate test of bimodality which automatically estimates both K and the cluster assignments is presented.
Cluster analysis aids research in many different fields, from business to biology to aerospace. It consists of using statistical techniques to group objects in large sets of data into meaningful classes. However, this process of ordering data points presents much uncertainty because it involves several steps, many of which are subject to researcher judgment as well as inconsistencies depending on the specific data type and research goals. These steps include the method used to cluster the data, the variables on which the cluster analysis will be operating, the number of resulting clusters, and parts of the interpretation process. In most cases, the number of clusters must be guessed or estimated before employing the clustering method. Many remedies have been proposed, but none is unassailable and certainly not for all data types. Thus, the aim of current research for better techniques of determining the number of clusters is generally confined to demonstrating that the new technique excels other methods in performance for several disparate data types. Our research makes use of a new cluster-number-determination technique based on the clustergram: a graph that shows how the number of objects in the cluster and the cluster mean (the ordinate) change with the number of clusters (the abscissa). We use the features of the clustergram to make the best determination of the cluster-number.
Cluster analysis is a data mining technique that is meant ot simplify the process of classifying data points. The basic clustering process requires an input of data points and the number of clusters wanted. The clustering algorithm will then pick starting C points for the clusters, which can be either random spatial points or random data points. It then assigns each data point to the nearest C point where "nearest usually means Euclidean distance, but some algorithms use another criterion. The next step is determining whether the clustering arrangement this found is within a certain tolerance. If it falls within this tolerance, the process ends. Otherwise the C points are adjusted based on how many data points are in each cluster, and the steps repeat until the algorithm converges,
The luminosity function (LF) of galaxy systems is determined, from single galaxies and small groups to rich clusters. The LF of rich (Abell) clusters is found to be a steeply decreasing function of cluster number density with cluster luminosity, and the groupings LF (that is, groups to rich clusters) can be well represented by a Schechter-type (1976) analytic form. The general LF of all galaxy systems is also found to be a smooth and steeply decreasing function of number density with luminosity, exhibiting an exponential cutoff in the rich-cluster luminosity domain, a power law dependence over most of the intermediate luminosity range, and a flattening of slope below the characteristic galactic luminosity. Comparisons of the LF with the LF of galaxies and X-ray clusters of galaxies are also discussed.
We demonstrate that it is possible to constrain the energy content of the Universe with high accuracy using observations of clusters of galaxies only. The degeneracies in the cosmological parameters are lifted by combining constraints from different observables of galaxy clusters. We show that constraints on cosmological parameters from galaxy cluster number counts as a function of redshift and accurate angular diameter distance measurements to clusters are complementary to each other and their combination can constrain the energy density content of the Universe well. The number counts can be obtained from X-ray and/or SZ (Sunyaev-Zeldovich effect) surveys, the angular diameter distances can be determined from deep observations of the intra-cluster gas using their thermal bremsstrahlung X-ray emission and the SZ effect (X-SZ method). In this letter we combine constraints from simulated cluster number counts expected from a 12 deg2 SZ cluster survey and constraints from simulated angular diameter distance measurements based on using the X-SZ method assuming an expected accuracy of 7% in the angular diameter distance determination of 70 clusters with redshifts less than 1.5. We find that R, can be determined within about 25%, A within 20%, and w within 16%. Any cluster survey can be used to select clusters for high accuracy distance measurements, but we assumed accurate angular diameter distance measurements for only 70 clusters since long observations are necessary to achieve high accuracy in distance measurements. Thus the question naturally arises: How to select clusters of galaxies for accurate diameter distance determinations? In this letter, as an example, we demonstrate that it is possible to optimize this selection changing the number of clusters observed, and the upper cut off of their redshift range. We show that constraints on cosmological parameters from combining cluster number counts and angular diameter distance measurements, as opposed to general expectations, will not improve substantially selecting clusters with redshifts higher than one. This important conclusion allow us to restrict our cluster sample to clusters closer than one, in a range where the observational time for accurate distance measurements are more manageable. Subject headings: cosmological parameters - cosmology: theory - galaxies: clusters: general - X-rays: galaxies: clusters
This study maps and measures assorted properties of new dated crater clusters that formed recently when impactors fragmented in the atmosphere of Mars. We report these statistics for 77 clusters: number of craters, size of cluster, dispersion of cluster, direction (azimuth) from which the impactor approached, and an estimate of the angle from vertical of the impact. Clusters range from a few to hundreds of craters, with most containing tens of craters. They are most commonly dispersed over hundreds of meters, with extents ranging from a few meters to a few kilometers. We find that dispersion generally does not correlate with topographic elevation. However, when the highest elevations are disregarded, clusters are more dispersed at lower elevations, as expected. Impact azimuths are randomly distributed and do not express a clear directionality of incoming meteoroids. Results suggest impacts occur closer to horizontal than expected, which could be due to observational effects. The characteristics we report here provide important constraints for future work in understanding atmospheric fragmentation processes; properties of the impactors themselves, such as density and orbital parameters; and the seismic detectability of impacts. These are critical aspects to understand, as approximately half of current impacts are observed to be clusters.
This study maps and measures assorted properties of new dated crater clusters that formed recently when impactors fragmented in the atmosphere of Mars. We report these statistics for 77 clusters:number of craters, size of cluster, dispersion of cluster, direction (azimuth) from which the impactor approached, and an estimate of the angle from vertical of the impact. Clusters range from a few to hundreds of craters, with most containing tens of craters. They are most commonly dispersed over hundreds of meters,with extents ranging from a few meters to a few kilometers. We find that dispersion generally does not correlate with topographic elevation. However, when the highest elevations are disregarded, clusters are more dispersed at lower elevations, as expected. Impact azimuths are randomly distributed and do not express a clear directionality of incoming meteoroids. Results suggest impacts occur closer to horizontal than expected, which could be due to observational effects. The characteristics we report here provide important constraints for future work in understanding atmospheric fragmentation processes; properties of the impactors themselves, such as density and orbital parameters; and the seismic detectability of impacts. These are critical aspects to understand, as approximately half of current impacts are observed to be clusters.
Direct evidence is reported for the simultaneous occurrence of Ostwald ripening and short-distance cluster mobility during annealing of discontinuous metal films on clean amorphous substrates. The annealing characteristics of very thin particulate deposits of silver on amorphized clean surfaces of single-crystalline thin graphite substrates have been studied by in situ transmission electron microscopy (TEM) under controlled environmental conditions in the temperature range from 25 to 450 C. It was possible to monitor all stages of the experiments by TEM observation of the same specimen area. Slow Ostwald ripening was found to occur over the entire temperature range, but the overriding surface transport mechanism was short-distance cluster mobility. This was concluded from in situ observations of individual particles during annealing and from measurements of cluster size distributions, cluster number densities, area coverages, and mean cluster diameters.
Neutrino dipole moments mu(nu) would increase the core mass of red giants at the helium flash by delta(Mc) = 0.015 solar mass x mu(nu)/10 to the -12th muB (where muB is the Bohr magneton) because of enhanced neutrino losses. Existing measurements of the bolometric magnitudes of the brightest red giants in 26 globular clusters, number counts of horizontal-branch stars and red giants in 15 globular clusters, and statistical parallax determinations of field RR Lyr luminosities yield delta(Mc) = 0.009 + or - 0.012 solar mass, so that conservatively mu(nu) is less than 3 x 10 to the -12th muB.
Observations of 1478 Abell clusters of galaxies with the NRAO 91-m telescope at 1400 MHz are reported. The measured beam shape was deconvolved from the measured source Gaussian fits in order to estimate the source size and position angle. All detected sources within 0.5 corrected Abell cluster radii are listed, including the cluster number, richness class, distance class, magnitude of the tenth brightest galaxy, redshift estimate, corrected cluster radius in arcmin, right ascension and error, declination and error, total flux density and error, and angular structure for each source.
The machine readable version of the compilation, as it is currently being distributed from the Astronomical Data Center, is described. The catalog contains redshifts and velocity dispersions for all Abell clusters for which these data had been published up to 1986 July. Also included are 1950 equatorial coordinates for the centers of the listed clusters, numbers of observations used to determine the redshifts, and bibliographical references citing the data sources.
We here propose a segmentation algorithm of texture image for a computer vision system on a space robot. An improved adaptive resonance theory (ART2) for analog input patterns is adapted to classify the image based on a set of texture image features extracted by a fast spatial gray level dependence method (SGLDM). The nonlinear thresholding functions in input layer of the neural network have been constructed by two parts: firstly, to reduce the effects of image noises on the features, a set of sigmoid functions is chosen depending on the types of the feature; secondly, to enhance the contrast of the features, we adopt fuzzy mapping functions. The cluster number in output layer can be increased by an autogrowing mechanism constantly when a new pattern happens. Experimental results and original or segmented pictures are shown, including the comparison between this approach and K-means algorithm. The system written in C language is performed on a SUN-4/330 sparc-station with an image board IT-150 and a CCD camera.
The number and growth of massive galaxy clusters are sensitive probes of cosmological structure formation. Surveys at various wavelengths can detect clusters to high redshift, but the fact that cluster mass is not directly observable complicates matters, requiring us to simultaneously constrain scaling relations of observable signals with mass. The problem can be cast as one of regression, in which the data set is truncated, the (cosmology-dependent) underlying population must be modeled, and strong, complex correlations between measurements often exist. Simulations of cosmological structure formation provide a robust prediction for the number of clusters in the Universe as a function of mass and redshift (the mass function), but they cannot reliably predict the observables used to detect clusters in sky surveys (e.g. X-ray luminosity). Consequently, observers must constrain observable-mass scaling relations using additional data, and use the scaling relation model in conjunction with the mass function to predict the number of clusters as a function of redshift and luminosity.
In a number of galaxy clusters, Chandra has discovered a new phenomenon, the sharp gas density edges. Depending on the sign of the temperature jump across the edge, these features may either be bow shocks or cold fronts. While the merger origin of the bow shocks is unambiguous, what causes cold fronts is not entirely clear, as they are observed both in mergers and in relaxed clusters. The purpose of the study of A3376, an interesting cluster with density edges, is to understand the origin of cold fronts and to look for possible shocks. This cluster also has a radio halo, and comparison of an X-ray temperature map and radio image may shed light on the nature of the cluster radio halos, which is our secondary goal. The XMM data for A3376 have been obtained, processed and cleaned. Most of the preliminary analysis, including deriving a temperature map, is complete. We also have Chandra data for this cluster, and intend to compare and combine the two datasets. We are well underway toward determining the dynamical state of A3376 and the nature of its gas density discontinuities.
An important question in cluster analysis and pattern recognition is the determination of the number of clusters into which a given population should be divided. Frequently, particularly when certain specific clustering methods are being used, the number of clusters is taken to be equal to the number of modes, or local maxima, in the probability density function underlying the given data set. The use of kernal density estimates in mode estimation is discussed. The test statistic to be used is defined and a bootstrap technique for assessing significance is given. An illustrative application is followed by an examination of the asymptotic behavior of the test statistic.
We present new data on the globular cluster system of the elliptical galaxy NGC 3923 which show that it has the most globular clusters per unit luminosity of any noncluster elliptical yet observed, with S(sub N) = 6.4 +/- 1.4. NGC 3923 is also among the brightest ellipticals outside of a galaxy cluster for which the number of globular clusters has been determined. Our observation of a large number of clusters per unit luminosity (high S(sub N)-value) for a bright elliptical in a sparse environment is consistent with the suggestion of Djorgovski and Santiago that the number of globular clusters is a power-law function of the luminosity with an exponent greater than 1. We relate this higher specific frequency of globular clusters in more luminous galaxies to other observations which indicate that the physical conditions within elliptical galaxies at the time of their formation were dependent on galaxy mass.
The conditions under which a hybrid of clustering and canonical analysis for image classification produce optimum results were analyzed. The approach involves generation of classes by clustering for input to canonical analysis. The importance of the number of clusters input and the effect of other parameters of the clustering algorithm (ISOCLS) were examined. The approach derives its final result by clustering the canonically transformed data. Therefore the importance of number of clusters requested in this final stage was also examined. The effect of these variables were studied in terms of the average separability (as measured by transformed divergence) of the final clusters, the transformation matrices resulting from different numbers of input classes, and the accuracy of the final classifications. The research was performed with LANDSAT MSS data over the Hazleton/Berwick Pennsylvania area. Final classifications were compared pixel by pixel with an existing geographic information system to provide an indication of their accuracy.
Unsupervised clustering is a fundamental tool in numerous image processing and remote sensing applications. For example, unsupervised clustering is often used to obtain vegetation maps of an area of interest. This approach is useful when reliable training data are either scarce or expensive, and when relatively little a priori information about the data is available. Unsupervised clustering methods play a significant role in the pursuit of unsupervised classification. One of the most popular and widely used clustering schemes for remote sensing applications is the ISOCLUS algorithm, which is based on the ISODATA method. The algorithm is given a set of n data points (or samples) in d-dimensional space, an integer k indicating the initial number of clusters, and a number of additional parameters. The general goal is to compute a set of cluster centers in d-space. Although there is no specific optimization criterion, the algorithm is similar in spirit to the well known k-means clustering method in which the objective is to minimize the average squared distance of each point to its nearest center, called the average distortion. One significant feature of ISOCLUS over k-means is that clusters may be merged or split, and so the final number of clusters may be different from the number k supplied as part of the input. This algorithm will be described in later in this paper. The ISOCLUS algorithm can run very slowly, particularly on large data sets. Given its wide use in remote sensing, its efficient computation is an important goal. We have developed a fast implementation of the ISOCLUS algorithm. Our improvement is based on a recent acceleration to the k-means algorithm, the filtering algorithm, by Kanungo et al.. They showed that, by storing the data in a kd-tree, it was possible to significantly reduce the running time of k-means. We have adapted this method for the ISOCLUS algorithm. For technical reasons, which are explained later, it is necessary to make a minor modification to the ISOCLUS specification. We provide empirical evidence, on both synthetic and Landsat image data sets, that our algorithm's performance is essentially the same as that of ISOCLUS, but with significantly lower running times. We show that our algorithm runs from 3 to 30 times faster than a straightforward implementation of ISOCLUS. Our adaptation of the filtering algorithm involves the efficient computation of a number of cluster statistics that are needed for ISOCLUS, but not for k-means.