IRAS and MSX satellites, and from the 2MASS survey. We applied a classification method based .... modes of the unfolded 1D pdf. figure (c). â¢Computation of the.
1
Asymmetric Planetary Nebulae 5 (Ebrary, Cambridge, USA), A.A Zijlstra, I. McDonald, eds.
A new tool for Post-AGB SED classification Philippe Bendjoya Laboratoire Fizeau, Campus Valrose, Cedex 2, 06108 Nice, France O. Suarez, L. Gallucio, O. Michel We present the results of an unsupervised classification method applied on a set of 344 spectral energy distributions (SED) of post-AGB stars extracted from the Toru´n catalogue of Galactic post-AGB stars. This method aims to find a new unbiased method for post-AGB star classification based on the information contained in the IR region of the SED (fluxes, IR excess, colours). We used the data from IRAS and MSX satellites, and from the 2MASS survey. We applied a classification method based on the construction of the dataset of a minimal spanning tree (MST) with the Prim’s algorithm. In order to build this tree, different metrics have been tested on both flux and color indices. Our method is able to classify the set of 344 post-AGB stars in 9 distinct groups according to their SEDs.
GIPSA-Lab Institut Polytechnique de Grenoble CNRS UMR 5216
4
Data
The classification of the SEDs of post-AGB stars has been performed up to now following the classes established by van der Veen et al. 1989. This classification is based on the aspect of the SED and is rather subjective. An ideal SED classification method would establish classes able to separate the objects into different groups according to characteristics as their evolutionary moment, mass, spectral type. Since theoretical studies as those in van Hoof et al. 1997 showed that at least the far IR part of the SED is not univocally related to a precise evolutionary moment, (or at least, the position of a postAGB star in the IRAS two colour diagram is not), a SED classification might not be able to fulfill these ideal conditions. The alternative approach to the SED classification is the study of the two colour diagrams. Especially, the IRAS [12]-[25] vs [25]-[60] diagram has been one of the most useful, mainly to identify postAGB stars. However up to now there is not a diagram capable to discriminate between the different post-AGB categories. Our approach consists in finding an unsupervised and automatic method of classification by means of «Prim Min Span Tree» clustering method performed with the use of a suited metric.
Objectives
vendredi 25 juin 2010
Our sample for the classification was extracted from the Toruń catalogue of post-AGB stars (Szczerba et al. 2007, A&A, 469, 799). We used the 344 objects classified as “Very likely post-AGB stars” in the version of January 2010. We used only the IR region of the data, to avoid the bias of choosing only stars with a counterpart in the visible range. We used data from the 2MASS survey (J, H and K bands), the MSX satellite (8.28 µm,12.13 µm, 14.65 µm, and 21.3 µm) and the IRAS satellite (12 µm, 25 µm and 60 µm).Treatment of the set of used data : initially 344 spectra were considered. From the set of 344 spectra, some values were missing at certain wavelengths. In order to have complete spectra (no values missing), we interpolate linearly the data when possible leading to a set 237 complete spectra. Figure below shows the consistency between a complete spectrum of a Post-AGB star and its interpolation when removing some values and doing an interpolation
•
•
•
•
Abstract
We present the results of an unsupervised classification method applied on a set of 344 spectral energy distributions (SED) of postAGB stars extracted from the Toruń catalogue of galactic post-AGB stars. This method aims to find a new unbiased method for postAGB star classification based on the information contained in the IR region of the SED (fluxes, IR excess, colours). We used the data from IRAS and MSX satellites, and from the 2MASS survey. We applied a classification method based on the construction of the dataset of a minimal spanning tree (MST) with the Prim's algorithm. In order to build this tree and to characterize dissimilarity between SEDs, a Kullback-Leibler divergence measure is used. Our method is able to classify the set of 344 post-AGB stars in 9 distinct groups according to their SEDs. c
Clustering Method
0.3
30
40
50
60
60
60
0
0.7
0
0.1
0.2
0.3
0.4
0.5
0.6
−40
0
0
0
−20
10
10
10
20
20
20
0
30
30
30
20
40
40
40
50
50
50
40
60
60
60
0.3
0
0.05
0.1
0.15
0.2
0.25
0
0
0
0.35
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0
0.05
0.1
0.15
0.2
0.25
60
10
10
10
80
20
20
20
30
30
30
Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9
40
40
40
50
50
50
60
60
60
some
van Hoof P.A.M., Oudmaijer R.D., Waters L.B.F.M, 1997, MNRAS, 289, 371-387
van der Veen, W.E.C.J., Habing H.J., Geballe T.R., 1989, A&A, 226, 108-136
Szczerba et al. 2007, A&A, 469, 799
Galluccio, L., Michel, O., Comon, P., Slezak, E., Hero, A. O., 2009. Initialization free graph based clustering. Tech. Rep. I3S/RR-2009-08-FR, Laboratoire I3S, CNRS and Nice-Sophia Antipolis University, 06903 Sophia Antipolis, France.
Prim, L. Shortest connection networks and generalizations. Bell Syst.Tech. J.36, 6 :1389-1401.
Basseville, M. Distance measures for signal processing and pattern recognition. Signal Processing, 18(4) :349-369, December 1989.
References
We propose for the first time an unsupervised classification approach to automatically determine clusters in Post-AGB population from the SEDs. We used a metric well apdapted to evaluate the distance between two spectra I.e the Kullback-Leibler divergence. The clustering method based on a Primʼs MST computation and thresholding in the trajectory leads to the identification of 9 groups.Some of these 9 groups show marked differences in their spectral types and their old classifications. They occupy roughly different regions in the two colour diagrams that allow the classification of a source in one of these groups just by its position in the diagram.
Conclusion
We have plotted the classification of van der Veen (blue) and the spectral type taken from the Torun catalogue (orange) for all the stars in each of the 9 groups. Some of our groups have marked characteristics: -groups 1 and 4 contain mainly old class 0 SEDs and intermediate spectral type stars (F, G) - group 2 contains mainly old class II and III SEDs and most of the sources are not classified spectroscopically, thus they might be obscured in the visible range - the same characteristics can be found in group 5, but wih a higher number of sources of the old SED class IV - group 8 contains mainly old class IV SEDs and early-intermediate spectral type stars (B-F) - group 9 contains mainly old class 0 and I SEDs and a mixture of spectral types However, not all the groups have homogeneous characteristics, since as we discused in the introduction, we can find two objects with different spectral types and evolutionary moments that show the same SED in the IR, thus they will be in the same group
Analysis
A projection of the cluster in the two principal color plane computed by a principal component analysis
−40 −80
−30
−20
−10
0
10
−60
0
20
0.2
0.1
0
0.15
0.1
0.3
0.2
0.05
0.5
0.4
0.25
0.7
30
50
50
0.6
20
40
40
0.3
10
30
30
0.35
0
20
20
0.8
10
10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.4
0
0.05
0.1
0.15
0.2
0.25
0
0
0.35
0.4
0
0.05
0.1
0.15
0.2
0.25
of the mean spectrum of each group of spectra detected by the Prim's trajectory. •Each spectrum is labelized at the closest mean spectrum group. •The mean spectrum of each group are recomputed. •The previous steps are realized until the mean spectrum converge.
•Computation
Construction of a MST •Prim's trajectory: g(i)=|ei| figure (a) •Threshold: = std {ei} figure (b) •Determination of the number of clusters detected and their positions: valleys in G can be associated to modes of the unfolded 1D pdf. figure (c)
Nine groups were found by the previous method. The fluxes corresponding to the Post-AG stars classified in each group are illustrated on the figure on the right. The different groups are represented in the usual color-color space and on the plane corresponding to the two first principal components. Each group occupy roughly different regions in the diagrams, showing that they represent different characteristics for the circumstellar envelopes of the objects they contain. We have also plotted (upper figures) the same colour-colour diagram for the classification of van der Veen et al. 1989. We see that our method has improved the separation of the objects in the two-colour diagrams, facilitating the inclusion of a source in one group just with the position in one of these diagrams.
Results
b
a
Primʼs MST (a) + Trajectory (b) illustrated on a toy model (c)
Assumption: each spectrum (a 10-dimensional vector) is considered as a vertex on an undirected graph G=(V,E) V={vi} is a set of vertices and E={eij} is a set of undirected edges beween vertices of V. The weights eij (the edge lengths of the segment between vertices vi and vj) measure the dissimilarity between two vertices. Algorithm: construction of a Prim's Minimal Spanning Tree (MST), which connects iteratively the closest non connected vertex to the graph partially connected until no vertex remains unconnected. Properties of the MST: acyclic, unique, fully connected and of minimal length Metric used to build the MST: symmetrized Kullback-Leibler divergence : L ! v˜il v dKL (vi , vj ) = (˜ vil − v˜jl ) log( ) where v˜i = !L i v˜jl l=1 z=1 viz Use of the MST's construction to detect groups of points Unfold the probability density function (pdf) of points in L dimensions into a 1-dimensional function
Proposed Approach
Method
Laboratoire H. Fizeau Université Nice Sophia Antipolis, Observatoire de la Côte dʼAzur, CNRS UMR 6525 de Astrofisica de Andalucia (IAA, CSIC), Granada, Spain I3S Université Nice Sophia Antipolis, CNRS UMR 6070
3 Laboratoire
2 Instituto
1
Ph Bendjoya1, O. Suarez1,2, L. Galluccio1,3, O. Michel4
A New Tool For Post-AGB SED Classification
Philippe Bendjoya 2
