In the framework of the European VO-Tech project, we are implementing new machine learning methods specifically tailored to match the needs of astronomical data mining. In this paper, we shortly present the methods and discuss an application to the Sloan Digital Sky Survey public data set. In particular, we discuss some preliminary results on the 3-D taxonomy of the nearby (z < 0.5) universe. Using neural networks trained on the available spectroscopic base of knowledge we derived distance estimates for ca. 30 million galaxies distributed over 8,000 sq. deg. We also use unsupervised clustering tools to investigate whether it is possible to characterize in broad morphological bins the nature of each object and produce a reliable list of candidate AGNs and QSOs

ANTONINO STAIANO

ELISABETTA DE FILIPPIS

GIUSEPPE LONGO

M. PAOLILLO

MASSIMO BRESCIA

RAFFAELE D'ABRUSCO

ROBERTO TAGLIAFERRI

English

Archivio della ricerca - Università degli studi di Napoli Federico II

arXiv:astro-ph/0701137v1  5 Jan 2007The use of neural networks to probe the structure ofthe nearby universeRaffaele d’AbruscoDept. of Physical Sciences, University Federico II in Napoli, via Cinthia 6, 80126-Napoli, Italy &Institute of Astronomy, Cambridge, UKGiuseppe Longo & Maurizio PaolilloDept. of Physical Sciences, University Federico II in Napoli, via Cinthia 6, 80126-Napoli, Italy &INFN, Napoli Unit, via Cinthia 6, 80126-Napoli, ItalyMassimo Brescia, Elisabetta De Filippis & Antonino StaianoINAF, Napoli Unit, via Moiariello 16, 80131-Napoli, ItalyRoberto TagliaferriDept. of Mathematics and Informatics, University of Salerno, Fisciano, ItalyAbstractIn the framework of the European VO-Tech project, we are implementing new machine learningmethods specifically tailored to match the needs of astronomical data mining. In this paper,we shortly present the methods and discuss an application to the Sloan Digital Sky Surveypublic data set. In particular, we discuss some preliminary results on the 3-D taxonomy of thenearby (z < 0.5) universe. Using neural networks trained on the available spectroscopic base ofknowledge we derived distance estimates for ∼ 30 million galaxies distributed over ∼ 8, 000 sq.deg. We also use unsupervised clustering tools to investigate whether it is possible to characterizein broad morphological bins the nature of each object and produce a reliable list of candidateAGNs and QSOs.Key words: elsart, document class, instructions for usePACS: 01.30.−yEmail address: dabrusco@na.infn.it (Roberto Tagliaferri).Preprint submitted to Elsevier 15 April 20191. IntroductionModern photometric digital surveys performed with dedicated instruments are in factproducing high quality multiband data at a huge rate, and a new generation of softwaretools is being developed to merge databases and to extract patterns, trends, etc. Fur-thermore, due the ongoing efforts to build a Virtual Observatory (VO) infrastructure(cf. the European VO-Tech, [1]), everyone will have at his fingertips the possibility toextract and manipulate such multi-wavelength, multi-epoch, multi-instrument datasetsof huge dimensionality. The extraction of useful information from such datasets imposesto abandon old conceptual schemes largely based on the 3-D visualization capability ofhuman minds, and to introduce in the astronomical practice fine tuned, machine learn-ing methods for statistical pattern recognition, classification and visualization [2]. In thiscontext, however, astronomical data pose non trivial problems due to the fact that theyare usually highly degenerate, present strong non linear correlations among parametersand are plagued by large fraction of missing data and upper limits.In what follows we shall shortly outline the first results of our ongoing effort to im-plement and apply a new generation of tools to the construction of a 3-D taxonomy ofthe nearby universe with a characterization of galaxy types in a few, broadly defined,catagories: normal (early and late type) galaxies, AGN, QSO, etc.For all the experiments described below, we used the Sloan Digital Sky Survey Data Re-lease 4 and/or 5 (hereafter SDSS4/5; [3]) which provides photometric data in 5 bands forseveral hundred million galaxies distributed over ∼ 8, 000 square degrees, together withadditional spectroscopic information for a subsample (hereafter SpS) of ∼ 106 galaxies.Such spectroscopic information include, among other things, redshifts and a spectral clas-sification index derived from an eigenvector analysis which allows to disentangle amongnormal galaxies (SP2), stars (SP1), nearby AGN (SP3), distant AGN (SP4), late typestars (SP6). This spectroscopic base of knowledge can be used as a training ground forinterpolative and supervised methods as well as a validation ground for unsupervisedones.2. The photometric redshiftOne of the main sources of degeneracy in the photometric properties of galaxies isthe lack of information on their distances and therefore any attempt to partition theparameter space requires first the derivation of distance estimates. In order to evaluatephotometric redshifts we made use of an improved version [4] of the Neural Networks(NNs) method presented in [5]. Both steps are accomplished using NNs architectureknown as MLP (Multi Layer Perceptron).We adopted a two steps approach based on the use of MLP [6,7]. First we trained aMLP to recognize nearby (id est with redshift z < 0.25) and distant (z > 0.25) objects,then we trained two separate MLPs to work in the two different redshift regimes. Suchapproach finds support in the fact that in the SDSS-5 catalogue, the distribution ofgalaxies inside the two different redshift intervals is dominated by two different galaxypopulations: the Main Galaxy (MG) sample in the nearby region, and the Luminous RedGalaxies (LRG) in the distant one. The use of two separate networks ensures that theNNs achieve a good generalization capabilities in the nearby sample, leaving the biases2Fig. 1. Left: distribution of spectroscopic versus photometric redshifts in the test set. Lighter grey pointsmark LRG galaxies. Notice the larger scatter of non-LRGs in the distant (z > 0.25) sample. Right:distribution of the errors computed over redshift bins for the distant sample.mainly in the distant one.To perform the separation between MG and LRG objects, we extracted from the SDSS-4 SpS training, validation and test sets weighting, respectively, 60%, 20% and 20% of thetotal number of objects (449,370 galaxies). The resulting test set, therefore, consisted of89,874 randomly extracted objects.After the first network has classified the objects in nearby and distant ones, the deriva-tion of the photometric redshifts is performed separately in the two regimes. Since NNsare excellent at interpolating data but very poor in extrapolating them, in order to min-imize the systematic errors at the extremes of the training redshift ranges we adoptedthe following procedure. For the nearby sample we trained the network using objectswith spectroscopic redshift in the range [0.0, 0.27] and then considered the results to bereliable in the range [0.01, 0.25]. In the distant sample, instead, we trained the networkover the range [0.23, 0.50] and then considered the results to be reliable in the range[0.25, 0.48].The error for our redshift estimates has been evaluated on the test set by measuringthe dispersion of points in the zspec vs zphot plane, i. e. the variance of the zspec − zphotvariable, after performing an interpolative correction in order to correct for systematics.The results can be summarized as follows:– For the MG sample, in the best experiment, the robust variance turned out to beσ3 = 0.0208 over the whole redshift range and 0.0197 and 0.0245 for the nearby anddistant objects, respectively.– For LRG sample we obtained σ3 ≃ 0.0163 over the whole range, and σ3 ≃ 0.0154 andσ3 ≃ 0.0189 for the nearby and distant samples, respectively.3. The clusteringThe implementation of a reliable classification scheme based on photometric informa-tion only, is an old problem in astronomy [8]. In few words, it consists in partitioningthe observed parameter space in clusters of objects sharing some underlying commonphysical property. Obviously, since there is no a priori reason to assume that photo-3t1t2t3y(x;w)x1x2Fig. 2. Scheme of a latent variable model mapping a 3D data space onto a 2D latent space with latentvariables uniformly arranged. Each latent variable is mapped, through the nonlinear function y, to thecenter of a Gaussian distribution.metric classification must reflect strictly any morphological classification (actually thedegeneracy observed in colours strongly argues against it), any effective classificationmethod must be unsupervised, id est it must partition the photometric parameter spaceusing only the statistical properties of the data themselves. We therefore implemented ahierarchical approach which starts from a preliminary clustering performed using as un-supervised clustering algorithm the so called ”Probabilistic Principal Surfaces” or PPS,and then makes use of the Negative Entropy concept and of a dendrogram structure toagglomerate the clusters found in the first phase.3.1. PPS - Principal Probabilistic SurfacesProbabilistic Principal Surfaces (PPS) [9,10] are a nonlinear extension of principalcomponents, in that each node on the PPS is the average of all data points that projectsnear/onto it. PPS define a non-linear, parametric mapping y(x;W) from aQ-dimensionallatent space (x ∈ RQ) to aD-dimensional data space (t ∈ RD), where normallyQ << D.The function y(x;W) (defined continuous and differentiable) maps every point in thelatent space to a point into the data space.Since the latent space isQ-dimensional, these points will be confined to aQ-dimensionalmanifold non-linearly embedded into the D-dimensional data space. In our method,the points belonging to the parameter space will be projected on the surface of a 2-dimensional sphere. The visualization capabilities of the PPS can prove very useful inseveral aspects of the data interpretation phase such as, for instance, the localization ofdata points lying far away from the more dense areas (outlayers), or of those lying in theoverlapping regions between clusters, or to identify data points for which a specific latentvariable is responsible. A visualization of the data clustering is achieved by PPS algo-rithm through the projection of objects from the multi-parametric D-dimesional space toa two dimensional surface. These clusters, representing groups of objects correspondingto the same latent variable, are then decimated using NEC: a hyerarchical non supervisedagglomeration algorithm.4Fig. 3. Probability density function of the SDSS data plotted on the latent sphere surface.3.2. NEC - Negentropy ClusteringMost unsupervised methods require the number of clusters to be provided a priori.This circumstance represents a serious problem when exploring large and complex datasets where the number of clusters can be very high or, in any case, unpredictable. Asimple treshold criterium is not satisfactory in most astronomical applications due to thehigh degeneracy and the noisiness of the data which lead to the erroneous agglomerationof data.We introduce the Fisher’s linear discriminant which is a classification method thatfirst projects high-dimensional data onto a line, and then performs a classification in theprojected one-dimensional space [6]. The projection is performed in order to maximizethe distance between the means of the two classes while minimizing the variance withineach class. On the other hand, we define the differential entropy H of a random vectory = (y1, . . . , yn)T with density f(y) as H(y) =∫f(y) log f(y)dy so that negentropyJ can be defined as J(y) = J(yGauss) − H(y), where yGauss is a Gaussian randomvector of the same covariance matrix as y. Negentropy can be interpreted as a measureof non-Gaussianity and, since it is invariant for invertible linear transformations, it isobvious that finding an invertible transformation that minimizes the mutual informationis roughly equivalent to finding directions in which the Negentropy is maximized.Our implementation of the method makes use of an approximation to Negentropy whichprovides a good compromise between the properties of the two classic non-Gaussianitymeasures given by kurtosis and skeweness. Negentropy can be used to agglomerate theclusters (regions) found by the PPS and the only a priori information is a dissimilaritythreshold T . Let us suppose to have c multi-dimensional regions Xi with i = 1, . . . , c thathave been defined by the PPS. The NEC algorithm, in practice, measures whether twoclusters can or cannot be modeled by one single Gaussian or, in other words, if the tworegions can be considered to be just aligned or rather as a part of a larger data set. Byincreasing the value of the dissimilarity treshold, the number of clusters decreases androbust partitions produce plateaus in the ”treshold vs number of clusters” diagram5Fig. 4. Exemplification of how the NEC algorithm works on two Gaussian distribution examples. Left:Negentropy = 2.6261 using the G1 function with a1 = 0.1; Right: Negentropy = 0.005 using the G1function with a1 = 0.13.3. Preliminary results on clusteringWe first applied the PPS algorithm to the sample of spectroscopically selected SDSSDR-4 objects using as parameters for the clustering the 4 colours obtained from modelmagnitudes (u-g,g-r,r-i,i-z) of SDSS archive. We fixed the number of latent variables andlatent bases of the PPS to 614 and 51 respectively, so obtaining at the end of this step614 clusters, each formed by objects which only respond to a certain latent variables. Wechoose a large number of latent variables in order to obtain an accurate separation ofobjects and to avoid that any group of distinct but near points in the parameter spacecould be projected onto the same cluster by chance. The clusters found by the PPS aregraphically represented by groups of points with the same colour (a different colour foreach cluster) on the surface of a 2-d sphere embedded in the 3-dimensional latent space.These first order clusters were then fed to the NEC algorithm which determined the finalnumber of clusters. The plateau analysis of the agglomerative process and the inspectionof the dendrogram allowed to set the treshold to a value corresponding to 31 clusters.We present in table (3.3) the ten most populous clusters together with the distributionof the specClass index within each cluster. The additional 21 clusters represent less than10 % of the objects and are still under investigation. It needs to be emphasized that theclustering makes use of the photometric data only and the spectroscopic information isused only to validate them. As it can be seen, galaxies (SP2) clearly dominate clusters1, 2 and 6. Whether this separation reflects or not some deeper differences among thethree groups (such as, for instance, different morphologies), cannot be assessed on thegrounds of presently available data. AGNs (SP3) dominate clusters 5 and 9 even thoughsome contamination from galaxies exists.Late type stars (SP6) populate mainly clusters 7 and 8 and are strong contaminantsof cluster 10 which also is dominated by galaxies.4. ConclusionsWe have produced a catalogue of redshifts for all galaxies in the SDSS database match-ing the following selection criteria and falling in the redshift range 0.005 − 0.5. These6Cl. n SP0 SP1 SP2 SP3 SP4 SP61 69 145 9362 48 0 122 25 133 13370 10 0 123 149 132 63 64 0 54 44 3396 1530 189 67 15 202 85 447 2428 6 106 26 125 13728 12 0 127 0 0 0 0 0 4848 1 1 1 0 0 3299 541 1507 127 4750 18 110 89 474 2117 19 4 529Table 1The table shows the distribution of objects in the most significant clusters found by our procedure. Thecolumns correspond to different values of the specClass index.redshifts have been used to produce a first 3-D map of the nearby universe which iscurrently under study to identify structures and to define accurate selection functions.Much work has instead still to be done in obtaining a reliable partition in hogeneoustypes. This is partly due to the lack of a reliable data sets for labeling and partly to astill poor understanding of how to deal in an effective way with upper limits and missingdata.Acknowledgements: this work was sponsored by the Italian MIUR through a PRIN grantand by the European Union through the VO-Tech project. The authors wish to thankG. Miele, G. Raiconi and N. Walton for useful discussions.References[1] WWW site: http://eurovotech.org.[2] Staiano A., De Vico L., Ciaramella A., Donalek C., Longo G., Raiconi G., Tagliaferri R., Amato R.,Del Mondo C., Mangano G. & Miele G., 2005, IEEE, in print.[3] Adelman-McCarthy J., et al., 2007, ApJS, submitted.[4] D’Abrusco R., Staiano A., Longo G., Brescia M., De Filippis E., Paolillo & Tagliaferri R. 2006, ApJ,submitted.[5] Tagliaferri R., Longo G., Andreon S., Capozziello S., Donalek C., & Giordano ,G., 2003, LecturesNotes in Computer Science, 2859, 226.[6] Bishop C. M., 1995, Neural Networks for Pattern Recognition, Oxford University Press.[7] Duda R.O., Hart P.E., Stork D.G., 2001, Pattern Classification, John Wiley & Sons Inc., SecondEdition.[8] de Vaucouleurs G, de Vaucouleurs A., Corwin H.C. jr, 1976, Second Reference Catalogue of BrightGalaxies, University of Texas Press.[9] Chang K., 2000, Nonlinear Dimensionality Reduction Using Probabilistic Principal Surfaces, PhDThesis, The University of Texas at Austin (USA).[10] Staiano A. 2003, Unsupervised Neural Networks for the Extraction of Scientific Information fromAstronomical Data, PhD thesis, University of Salerno (Italy).7

The use of neural networks to probe the structure of the nearby universe

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The use of neural networks to probe the structure of the nearby universe

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