Understanding how the physical properties of galaxies (e.g. their spectral
type or age) evolve as a function of redshift relies on having an accurate
representation of galaxy spectral energy distributions. While it has been known
for some time that galaxy spectra can be reconstructed from a handful of
orthogonal basis templates, the underlying basis is poorly constrained. The
limiting factor has been the lack of large samples of galaxies (covering a wide
range in spectral type) with high signal-to-noise spectrophotometric
observations. To alleviate this problem we introduce here a new technique for
reconstructing galaxy spectral energy distributions directly from samples of
galaxies with broadband photometric data and spectroscopic redshifts.
Exploiting the statistical approach of the Karhunen-Loeve expansion, our
iterative training procedure increasingly improves the eigenbasis, so that it
provides better agreement with the photometry. We demonstrate the utility of
this approach by applying these improved spectral energy distributions to the
estimation of photometric redshifts for the HDF sample of galaxies. We find
that in a small number of iterations the dispersion in the photometric
redshifts estimator (a comparison between predicted and measured redshifts) can
decrease by up to a factor of 2.Comment: 25 pages, 9 figures, LaTeX AASTeX, accepted for publication in A

Alexander S. Szalay

Andrew J. Connolly

Baum W. A.

Everson R.

Hogg D. W.

István Csabai

Karhunen H.

Koo D. C.

Mark Dickinson

Tamás Budavári

English

arXiv

Crossref

Creating Spectral Templates from Multicolor Redshift Surveys

Understanding how the physical properties of galaxies (e.g., their spectral type or age) evolve as a function of redshift relies on having an accurate representation of galaxy spectral energy distributions. While it has been known for some time that galaxy spectra can be reconstructed from a handful of orthogonal basis templates, the underlying basis is poorly constrained. The limiting factor has been the lack of large samples of galaxies (covering a wide range in spectral type) with high signal-to-noise spectrophotometric observations. To alleviate this problem we introduce here a new technique for reconstructing galaxy spectral energy distributions directly from samples of galaxies with broadband photometric data and spectroscopic redshifts. Exploiting the statistical approach of the Karhunen-Loeve expansion, our iterative training procedure increasingly improves the eigenbasis, so that it provides better agreement with the photometry. We demonstrate the utility of this approach by applying these improved spectral energy distributions to the estimation of photometric redshifts for the HDF sample of galaxies. We find that in a small number of iterations the dispersion in the photometric redshifts estimator (a comparison between predicted and measured redshifts) can decrease by up to a factor of 2

Budavari, T

Szalay, A S

Connolly, A J

Csabai, I

Dickinson, M

ELTE Digital Institutional Repository (EDIT)

arXiv:astro-ph/9908008v1  2 Aug 1999Creating Spectral Templates from Multicolor RedshiftSurveysT. Budavári1, A.S. Szalay, A.J. Connolly2, I. Csabai1, M.E. Dickinson3Department of Physics and Astronomy, The Johns Hopkins University,Baltimore, MD 21218and the HDF/NICMOS TeamAbstract. We present a novel method capable of creating optimal eigen-spectra from multicolor redshift surveys for photometric redshift estima-tion. Our iterative training algorithm modifies the templates to representthe photometric measurements better. We present a short description ofour algorithm here. We show that the corrected templates give more pre-cise photometric redshifts, essentially a “free” feature, since we were notfitting for the redshifts themselves.1. IntroductionIn the last couple of years photometric redshift estimation has become a verypowerful tool in getting statistical information about our universe and also fre-quently used for spectroscopic target selection to measure spectra of faint galax-ies. Different techniques have been developed since Baum (‘62) but all of themmay be classified into two groups based on the approach they use.The first are the so called empirical methods (Connolly et al. ‘95a). Havinga set of objects with spectroscopic and photometric data, one can easily obtainan analytic, usually polynomial fitting function for the redshift over the color“hyperplane”. Lower order piecewise fits can be also applied efficiently (Brunneret al. ‘99). The function can be very quickly evaluated at any photometricdata point. However, this method has some disadvantages. We need to havea very good training set to get the fitting formula and we still will not be ableto extrapolate far from the training set. The fitting formula only works fora specific set of passbands and limited redshift ranges, thus if we want to getphotometric redshifts for different catalogs, we need training sets for all thephotometric standards, so it is very hard to get consistent redshift estimationsfor them.In the template or SED (spectral energy distribution) fitting methods a setof restframe spectra are used to work out what spectrum and redshift give better1Department of Physics, Eötvös Loránd University, Budapest, H-10882Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA 152603Space Telescope Science Institute, Baltimore, MD 212181representation of the color of a galaxy. Using this approach gives us not justthe redshift of the galaxy but also an approximate spectral type information.The technique works very well on separate photometric catalogs with differentfilter sets but it is obvious that the basis spectra are crucial. The currentlyused template spectra come from both synthetic (Bruzual & Charlot ‘93) andmeasured (Coleman, Wu & Weedman ‘80) spectral libraries but generally theyare applied as they come with the hope that they are good enough. An advancedversion of the template fitting algorithm has been developed to use “continuumnumber” of spectra using eigenspectra instead of a large number of actual spectrain a very efficient way in terms of CPU time and memory. Using eigenspectra(Connolly et al. ‘95b) means we consider all spectra to be approximately a linearcombination of a small number of spectra, the eigenspectra. It also turns outthere is an optimal subspace filtering method (see AJC’s talk; Budavári et al.‘99) which gives more reliable restframe spectra than direct coefficient fitting.In this paper we describe a new method which has been developed to bringtogether the advantages of both techniques. We show how to train eigenspectrain an iterative procedure to represent the photometry better. In the trainingprocedure different catalogs and also measured spectra can be incorporated. Theresulting eigentemplates can be used for redshift estimation on any photometrycatalog and the redshift prediction becomes more accurate.2. Template ReconstructionOur goal is to find out the underlying basis template spectra that can be usedlater for photometric redshift estimation. Since we observe galaxies at differ-ent redshifts, their restframe spectra are sampled at different wavelengths bythe blueshifted filters. If we have a deep enough multicolor redshift survey (ormore) then the rough SEDs defined by the blueshifted passbands are overlappingwith each other. This means the eigenspectra are oversampled and this is thefact which allows us to derive applications for extracting high resolution eigen-templates from broadband photometry (Csabai et al. ‘99). Our new approach isdifferent mainly in spectrum repairation that we introduce in the next section.The iterative training procedure follows a fully statistical approach. Itsrobustness is provided by the Karhunen-Loève (KL) expansion (Karhunen ‘47;Loève ‘48). This transformation has been used to derive eigentemplates fromspectra and this is just what we do in our iterative training. Starting froma set of eigenspectra we can compute a best fitting type for each and everygalaxy in our photometric catalog based on the known redshift. This gives usan approximation to the restframe spectra of the galaxies. Now we can checkhow good the spectrum is at least at such wavelengths where photometric dataare available. We modify — repair — “smoothly” the spectra to represent themeasured values better. This spectrum correction is the only tricky step thatwe will describe later on. Having all the modified spectra we can invoke the KLtransformation and obtain a new set of eigentemplates. The iteration can becontinued until the correlation between simulated and measured flux values isstrong enough.2Figure 1. Correlation between spectroscopic and photometric red-shifts. On the right - using 3 CWW eigenspectra, left - using 3 KLeigenspectra after 30 iterations. See Figure 3. for corresponding eigen-spectra.Repairing Spectra. The way we modify the spectrum of a galaxy is the keystep of our training algorithm. The best fitting linear combination of the eigen-spectra is the current approximation of the galaxy that we are about to correctdue to the photometric values. This is a multidimensional minimization problemfor the spectrum itself, where the dimension of the problem is determined by thespectral resolution. The cost function is built up by two terms. The first partcorresponds to the deviation of the spectrum based on simulated fluxes from themeasured photometric data and the second part is the deviation of the spectrumfrom the template based linear combination.χ2(~s) =∑n1∆2n{fn − f̂n(~s)}2 +∑k1σ2k{sk − stk}2 (1)~s is the discrete representation of the spectrum and sk is the kth element of ~s,where k refers to the wavelength, λk. stkstands for the template based spec-trum and σk describes the ability of the spectrum to be changed at a givenwavelength. The photometric error in the nth passband is represented by ∆n.This minimization problem can be reduced to a system of linear equations, sincef̂n(~s) is linear in its variables, {sk}; so it is easy to solve. The new KL basisbuilt from the modified spectra will span a different subspace which representsthe photometry better. The details of this method will be described elsewhere(Budavári et al. ‘99).3. ApplicationOur training algorithm has been applied to the HDF/NICMOS catalog (Williamset al. ‘96; Dickinson et al. ‘99) which has unique photometric data quality andthere is a reasonable number of objects with spectroscopic redshift (Cohen, ‘98).3Figure 2. Redshift limited scatter plot showing a close-up of Figure 1.There are photometric data available in seven passbands: 4 HDF filters (F300W,F450W, F606W, F814W), 2 NICMOS filters (F110W, F160W) and a ground-based K’. Our initial eigentemplates were derived from the Coleman, Wu &Weedman (CWW) spectra that usually provide better redshift estimation thanothers (Fernandez-Soto et al., ‘99; Hogg et al., ‘98).After a few iterations the redshift scatter becomes tighter while the eigen-templates just slightly change. The comparison between the original and re-sulting eigenspectra after 30 iterations (KL-30 ) can be seen on Figure 3. Thistiny difference between them was enough to improve the redshift scatter plotby a factor of two as it can be seen on Figure 1. and 2. The correspondingstatistical errors were calculated for two redshift ranges. The overall statisticsand a redshift limited sample were evaluated. Beyond the standard root meansquare error (∆rms) the relative deviation of (1 + z) was also computed (∆rel)for comparison with estimates found by other authors. Table 1. contains thecalculated error estimates for both the CWW and KL-30 cases.Table 1. Comparison of the errors in the fits based on the Coleman,Wu & Weedman and the KL (after 30 iterations) eigenspectraError Range CWW KL-30∆rms z < 6 0.23 0.12∆rel z < 6 0.079 0.042∆rms z < .8 0.087 0.063∆rel z < .8 0.050 0.035In order to test the robustness of the method we started the iteration fromarbitrary constant functions, as well. After just a couple of iterations the 4000Åbreak was visible and other important features emerged soon from scratch.4Figure 3. These figures show the initial eigenspectra derived fromthe Coleman, Wu & Weedman spectra (dotted line) and the correctedKL spectra after 30 iterations (solid line). The horizontal dashed linesignals the zero level. See Figure 1. and 2. for related redshift scatterplots.The limitation on our current application is the relatively small number ofgalaxies with accurate multicolor photometry and redshifts in the Hubble DeepField. Consequently the third eigenspectrum appears to be being over fit (werun out of degrees of freedom). As our technique is constructed to be applied toany set of multicolor observations we can incorporate many multicolor datasetsfrom different sources (i.e. we do not need to train the relation for a given set offilters or magnitude limits). With the new generation of multicolor photometricand spectroscopic surveys nearing completion we, therefore, expect the accuracyof the derived spectral energy distributions to improve dramatically in the nearfuture.4. ConclusionsWe have shown that the current templates used in SED fitting photometric red-shift estimation can be modified to give better correlation between the actualphotometry and the template based simulated fluxes. We presented a robustmethod that creates better eigenspectra in an iterative way and converges veryquickly. The training procedure is very generic. Different catalogs can be incor-porated at the same time even with different filtersets. It is also easy to extendthe method to involve measured spectra if needed.We showed that the new modified eigentemplates give us a better basis forphotometric redshift estimation, even though the training procedure is not fittingfor the redshift, in fact the photometric redshift estimation is only performedfor monitoring purposes.5Acknowledgments. IC and TB acknowledges partial support from theMTA-NSF grant no. 124 and the Hungarian National Scientific Research Foun-dation (OTKA) grant no. T030836, AS from NASA LTSA (NAG53503), HSTGrant (GO-07817-04-96A), MED from HST (GO-07817-01-96A) and AJC fromHST (GO-07817-02-96A) and LTSA (NRA-98-03-LTSA-039).ReferencesBaum, W.A., 1962, IAU Symposium No. 15, 390Budavári, T., Szalay, A.S., Connolly, A.J., Csabai, I. & Dickinson, M.E., 1999,in preparation, to be submitted to AJBrunner, R.J., Connolly, A.J., Szalay, A.S., 1999, ApJ, 516:563-581Bruzual, A.G. & Charlot, S., 1993, ApJ, 405, 538Cohen, J.G., 1998, HDF Symposium (STScI symp. ser. 11), p. 52Coleman, G.D., Wu., C.-C. & Weedman, D.W., 1980, ApJS, 43, 393Connolly, A.J., Csabai, I., Szalay, A.S., Koo, D.C., Kron, R.G. & Munn, J.A.,1995a, AJ 110, 2655Connolly A.J., Szalay A.S., Bershady M.A., Kinney A.L. & Calzetti D., 1995b,AJ, 110, 1071Csabai, I., Connolly, A.J., Szalay, A.S. & Budavári, T., 1999, in press, submittedto AJDickinson, M., 1999, in “After the Dark Ages: When Galaxies Were Young,”eds. S. Holt and E. Smith, (AIP: Woodbury NY), p. 122Dickinson et al., 1999, in preparationFernández-Soto, A., Lanzetta, K.M. & Yahil, A., 1999, ApJ, 513:34-50Hogg et al., 1998, ApJ, 499, 555Karhunen, H., 1947, Ann. Acad. Science Fenn, Ser. A.I. 37Loève, M., 1948, Processus Stochastiques et Mouvement Brownien, Hermann,Paris, FranceWilliams, R.E. et al., 1996, AJ, 112, 13356

Creating spectral templates from multicolor redshift surveys

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Spectral Templates from Multicolor Redshift Surveys

Spectral Templates from Multicolor Redshift Surveys

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