We present a prototype-based machine learning analysis of labeled galaxy catalogue data containing parameters from the Galaxy and Mass Assembly (GAMA) survey. Using both an unsupervised and supervised method, the Self-Organizing Map and Generalized Relevance Matrix Learning Vec- tor Quantization, we find that the data does not fully support the popular visual-inspection-based galaxy classification scheme employed to categorize the galaxies. In particular, only one class, the Little Blue Spheroids, is consistently separable from the other classes. In a proof-of-concept experiment, we present the galaxy parameters that are most discriminative for this class

Biehl, M

Nolte, A.

Wang, L.

English

ARTS repository - University of Groningen

   University of GroningenPrototype-based analysis of GAMA galaxy catalogue dataNolte, A.; Wang, L.; Biehl, MPublished in:ESANN, European Symposium on Artificial Neural Networks, Computational Intelligence and Machine LearningIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionEarly version, also known as pre-printPublication date:2018Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Nolte, A., Wang, L., & Biehl, M. (2018). Prototype-based analysis of GAMA galaxy catalogue data. In M.Verleysen (Ed.), ESANN, European Symposium on Artificial Neural Networks, Computational Intelligenceand Machine Learning (Vol. 26, pp. 339-344). Ciaco - i6doc.com.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 30-10-2023Prototype-based Analysis ofGAMA Galaxy Catalogue DataAleke Nolte1∗, Lingyu Wang2, and Michael Biehl11 - Univ. of Groningen - Johann Bernoulli Inst. for Mathematics and Computer ScienceP.O. Box 407, 9700 AK Groningen - The Netherlands2- SRON Netherlands Institute for Space Researchand University of Groningen - Katpeyn Astronomical InstituteLandleven 12 - 9747 AD Groningen - The NetherlandsAbstract. We present a prototype-based machine learning analysis of labeledgalaxy catalogue data containing parameters from the Galaxy and Mass As-sembly (GAMA) survey. Using both an unsupervised and supervised method,the Self-Organizing Map and Generalized Relevance Matrix Learning Vec-tor Quantization, we find that the data does not fully support the popularvisual-inspection-based galaxy classification scheme employed to categorize thegalaxies. In particular, only one class, the Little Blue Spheroids, is consistentlyseparable from the other classes. In a proof-of-concept experiment, we presentthe galaxy parameters that are most discriminative for this class.1 IntroductionTelescope images of galaxies reveal a multitude of appearances, ranging from disk-like galaxies over galaxies with spiral arms to more irregular shapes. The study ofgalaxy morphological classification plays an important role in astronomy: The spatialdistribution of galaxy types provides valuable information for the understanding ofgalaxy formation and evolution.The assignment of morphological classes to observed galaxies is a task which iscommonly handled by astronomers. As manual labeling of galaxies is time consumingand expert-devised classification schemes may be subject to cognitive biases, machinelearning techniques have great potential to advance astronomy by: 1) investigatingautomatic classification strategies, and 2) by evaluating to which extent existingclassification schemes are supported by the observed data.In this work, we want to make a contribution along both lines by analyzing a galaxycatalogue which has been annotated using a popular classification scheme proposed byKelvin [1]. To this end, we apply both an unsupervised and a supervised prototype-based method. In our analysis, we first assess whether Kelvin’s scheme is consistentwith a clustering of the data generated by the unsupervised Self Organizing Map(SOM) [2]. We then investigate if the morphological classification can be reproducedby Generalized Relevance Matrix Learning Vector Quantization (GMLVQ) [3], apowerful supervised prototype-based method. In addition to providing an evaluationof the classification scheme via the proxy of classification performance (high accu-racies would indicate a good separation of classes), GMVLQ also allows to identify∗We acknowledge financial support by the EU’s Horizon 2020 research and innovation programmeunder Marie Sk lodowska-Curie grant agreement No 721463 to the SUNDIAL ITN network.the feature dimensions that are of particular relevance for a classification problem.As we find the employed classification scheme to be not fully supported by thedata, we present the parameters that are relevant for distinguishing the most clearlypronounced class, the Little Blue Spheroids (LBS), from the remaining classes.2 DataIn this work we analyze a labeled galaxy catalogue [4] containing 42 parameters whichhave been derived from spectroscopic and photometric observations (measurementsof flux intensities in different wavelength bands) from the Galaxy and Mass Assembly(GAMA) survey [5] for 7941 astronomical objects. For each galaxy, a class label hasbeen determined by astronomers following a visual inspection based classificationscheme described by Kelvin et al. [1]. The scheme assigns galaxies to 9 classes:1-Ellipticals (11%), 2-Little blue spheriods (11%), 3-Early-type spirals (10%), 4-Early-type barred spirals (1%), 5-Intermediate-type spirals (15%), 6-Intermediate-type barredspirals (2%), 7-Late-type spirals & Irregulars(45%), 8-Artefacts (0.4%) and 9-Stars(0.005%). Numbers in parantheses represent the class-wise prevalences in the availabledata. We will refer to the classes by their class index (1-9). We exclude the LOG-SurfaceDensityErr parameter due to numerous missing measurements, and removesamples with missing measurements in any of the remaining features, resulting ina final dataset of 7356 astronomical objects of dimensionality n=41.3 SOM AnalysisThe self-organizing map (SOM) [2, 6] is an unsupervised prototype-based methodwhich allows to generate topology-preserving low-dimensional representations of high-dimensional data. In the default formulation, a SOM is composed of map units thatare arranged in a two-dim. lattice. Each unit has a defined set of neighbors whichare influenced when the unit undergoes changes. A prototype of input dimensionalityis associated with each unit of the map and the training process follows a competitivelearning paradigm: For each data point ξ, the best matching unit (BMU) mc(ξ) isdetermined, and subsequently only mc and its neighbors are modified by the trainingprocedure. The BMU mc(ξ) is given by the map unit with the prototype closest toξ and therefore fulfills∀i,‖ξ−mc‖≤‖ξ−mi‖.In the SOM batch learning rule ([7]), the map units mi are updated followingmi(t+1)=(∑mj=1hj,i sj)/(∑mj=1nVjhj,i) ,with si=∑ξk∈Viξk ,where m is the number of map units, Vi is the set of data points with BMU mi(t),nVi is the cardinality of this set and hi,j is the neighborhood function determiningthe strength of the influence of unit i to j.We perform an analysis of the galaxy catalogue data using the SOM implementation“SOM toolbox for Matlab5” [7]. Due to the SOM’s sensitivity to outliers we normalizethe data using a centered logistic normalization:ξ′=1/(1+e−ξ̂)−0.5 with ξ̂=(ξ−ξ̄)/σξ ,112 377 511= empty = others1 2 3 75Fig. 1: SOM visualization displaying an unsupervised clustering of galaxy data fromGAMA. The SOM is post-labeled using galaxy-class information, where the piechart for each unit encodes a histogram of class-wise hits. A unit’s backgroundcolor indicates the class with the majority of hits. A color version of this plot canbe found at http://doi.org/10.5281/zenodo.1092761.where ξ̄ and σξ are the empirical mean and standard deviation of the feature.We initialize the SOM by spacing the prototypes regularly in the coordinate systemgiven by the two largest eigenvectors of the data matrix [7]. Based on preliminaryexperiments we choose a SOM of size 20×40 units with a rectangular neighborhoodfunction and toroidal topology and train using the option long of the default toolboxsettings. After training, the galaxy label information is used to display the specificityof a map unit via unit-wise histograms of hits (pie charts in Figure 1), i.e. thedistribution of the class labels of the data points that have a particular map unit asBMU. Additionally, the class with the majority of hits is indicated by the backgroundcolor of each pie plot.In the pie charts of Fig. 1 it can be seen that there is no pronounced separation ofclasses and that many units respond to samples from more than one class. Classes1 and 3 are particularly intermixed. However, class 2 (LBS) forms a fairly compactcluster. We note that the basic structure remains similar for larger or smaller maps,or when the over-represented class 7 is under-sampled.4 GMLVQ analysisGeneralized Relevance Matrix LVQ (GMLVQ) [3] is an extension of Learning VectorQuantization (LVQ) [6]. LVQ is a supervised prototype-based method, in whichprototypes are annotated with a class label. The prototypes are adapted based onthe label information of the training data: If the BMU is of the same class as agiven data point, the prototype is moved towards the data point, while in the caseof a BMU with incorrect class label, the prototype is repelled. While both SOMand LVQ assess similarities between prototypes and data points using the Euclideandistance, GMLVQ learns a distance measure that is tailored to the data, allowingit to suppress noisy feature dimensions or to emphasize distinctive features and theirpair-wise combinations. GMLVQ therefore considers a generalized distancedΛ(w,ξ)=(ξ−w)TΛ(ξ−w) with Λ=ΩTΩ and∑iΛii=1,where Λ is an n×n a positive semi-definite distance matrix, and w is a prototype.The parameters {wi} and Λ are optimized based on a heuristic cost function, see [3].EGMLVQ =∑Pi=1µΛi , with µΛi =(dΛJ (ξi)−dΛK(ξi))/(dΛJ (ξi)+dΛK(ξi)),where dΛJ (ξ)=dΛJ (wJ ,ξ) denotes the distance to the closest correctly labeled proto-type, and dΛK(ξ)=dΛK(wK,ξ) denotes the distance to the closest incorrect prototype.If the closest prototype has an incorrect label, dΛK(ξi) will be smaller than dΛJ (ξi),hence, the corresponding µΛi is positive. Minimization of EGMLVQ will therefore favorthe correctness of nearest prototype classification. In a stochastic gradient descentprocedure based on single examples the update readswJ,K←wJ,K−ηw∂µi/∂wJ,K and Ω←Ω−ηΩ∂µi/∂Ω . (1)Derivations and full update rules can be found in [3].To assess relevances of features and discriminability between classes we train andevaluate GMLVQ on the galaxy catalogue data making use of a publicly available imple-mentation which employs batch gradient descent [8]. To maintain consistency, we nor-malize the data using logistic normalization. As the GMLVQ cost function is implicitlybiased toward classes with larger numbers of samples, we disregard the classes whichcontain only few samples (classes 4,6,8,9) and train and evaluate the classifier on size-balanced random subsets from the remaining five larger classes. For our experiments,we specify one prototype per class and run the algorithm for 100 epochs using the imple-mentation’s default settings. We validate the algorithm by performing a class-balancedrepeated random sub-sampling validation by randomly selecting 743 datapoints (the car-dinality of the smallest class, class 3) per class for each validation run, for a total of 50runs. For each sub-sampled data set, both the training (3343 samples, 90%) and valida-tion set (371 samples (10%)) are class-balanced. The resulting relevances and confusionmatrix (both averaged over all validation runs) are displayed in Figure 2 and Table 4.The confusion matrix corroborates the findings from the SOM analysis: The classifierhas difficulty distinguishing classes 1 and 3, placing about 19% of samples from class 1into class 3 and vice versa. Also classes 5 and 7 can only be classified with limited accu-racies of 73% and 71%. In line with the compactness of its representation by the SOM,only class 2 (little blue spheroids, LBS), are classified with a high accuracy (90.7%).Both the lack of pronounced clusters in the SOM and the moderate classificationperformance of GMLVQ may be an indication for the presence of label noise that isRADECZ_TONRYLUMDIST_TONRYTHETA_IMAGETHETA_J2000ERRTHETA_J2000MAG_AUTO_rMAGERR_AUTO_rR_MODELR_MODEL_ERRlogmstardellogmstarlogmoverl_idellogmoverl_iabsmag_rGALMAG_rGALMAGERR_rGALELLIP_rGALELLIPERR_rgminusidelgminusiuminusrdeluminusrLOGKRON_ALOGKRON_BLOGKRON_A_ERRLOGKRON_B_ERRLOGGALINDEX_rLOGGALINDEXERR_rLOGFLUX_AUTO_rLOGFLUXERR_AUTO_rLOGGALRE_rLOGGALREERR_rLOGKRON_A_KPCLOGKRON_B_KPCLOGKRON_A_ERRKPCLOGKRON_B_ERRKPCLOGGALRE_r_KPCLOGGALREERR_KPCLOGSurfaceDensity00.050.10.150.20.250.35 classesLBS vs othersrelevancegalaxy parameter103927 8145 6105786129 43Fig. 2: Average GMVLQ feature relevances for the task of distinguishing the 5largest galaxy classes (dark) and the little blue spheroid class from the other 4classes (light). The 10 most relevant features are indicated by digits (dark/light)corresponding to the rank of the feature. Arrows indicate features selected in [4].Explanations of the galaxy parameters and additional references can be found athttp://doi.org/10.5281/zenodo.1092761.1 2 3 5 71 63.8 10.8 19.3 5.5 0.72 3.1 90.7 0.1 1.3 4.83 18.4 1.7 66.1 13.5 0.35 2.0 6.8 9.3 73.7 8.27 1.1 13.1 0.1 14.6 71.1Table 1: Averaged GMVLQ confusion ma-trix displaying the percentage of predictedclass labels (columns) for samples from eachclass (rows) for the 5 largest galaxy classes.rooted in the subjective nature of galaxy classification [4, 9]. It may also indicate thatthe classification scheme is not fully supported by the data. As the relevance profilereflects the discriminability of parameters only within this possibly ill-defined setting,we additionally consider the relevance profile for the less ambiguous sub-problemof separating LBS from other galaxies. Following the procedure described above(with 1512 training and 168 validation samples per validation run), we find that LBScan be well distinguished (AUC(ROC)=0.96) from the other classes. Furthermore,in the two-class setting 8 of the 10 most relevant features are also among the 10most relevant ones found for the full five-class setting. The coinciding parametersare GALELLIP r, LOGGALINDEXERR r, LOGGALINDEX r, LOGGALRE r,R MODEL, R MODEL ERR, deluminusr, and uminusr.5 DiscussionThe results presented above suggest that there may be some inconsistencies in theinvestigated morphological classification scheme: The clustering produced by theunsupervised SOM is only moderately consistent with the galaxy classes and it hasproven difficult to distinguish galaxy types using supervised GMLVQ. In our analysis,class 1 (Ellipticals) and 3 (Early-type spirals) are particularly difficult to differentiate,while class 2 (LBS) seems to be well separable.The difficulty of training a successful classifier was also observed in [4], where class-wiseaveraged accuracies are also around 75%. Possible explanations for poor classificationperformance may be lack of discriminative power of the employed classifiers, mis-labelings of certain galaxies (a possibility already discussed in [4]), or that essentialparameters are not contained in the data set. In the first case, employing evenmore flexible classifiers, e.g. GMLVQ with local relevance matrices [3], may improvethe classification performance. In the second case, if mis-labelings are restricted to“neighboring” classes in an assumed underlying class ordering, ordinal classificationmay provide further insights [10]. Yet, our results do not rule out the possibility thatthe true, underlying grouping of galaxies is considerably different and less clear-cutthan the investigated one. Further data-driven analyses of galaxy parameters andimages with advanced clustering methods might reveal alternative groupings.In [4], 10 of the 42 parameters were selected manually. Out of GMLVQ’s 8 most rele-vant features, 4 overlap or highly correlate with the 10 features selected in [4], namelyGALELLIP r, LOGGALINDEX r, uminus r, and LOGGALRE r. These parametersare related to the ellipticity, light distribution, size and color of the galaxy. Note thatthe non-overlapping features comprise model fitting errors (LOGGALINDEXERR r,R MODEL ERR, deluminusr), indicating that a galaxy’s model consistency variesover morphological classes.Acknowledgements GAMA is a joint European-Australasian project based around a spectroscopiccampaign using the Anglo- Australian Telescope. The GAMA input catalogue is based on datataken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementaryimaging of the GAMA regions is being obtained by a number of in- dependent survey programmesincluding GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAPproviding UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), theAAO, and the participating institutions. The GAMA website is http://www.gama-survey.org/.We thank Sreevarsha Sreejith and Lee Kelvin for helpful feedback and discussions.References[1] L.S. Kelvin, S.P. Driver, A. SG Robotham, et al. Galaxy and mass assembly (gama): ugrizyjhksérsic luminosity functions and the cosmic spectral energy distribution by hubble type. MonthlyNotices of the Royal Astronomical Society, 439(2):1245–1269, 2014.[2] T. Kohonen. The self-organizing map. Neurocomputing, 21(1):1–6, 1998.[3] P. Schneider, M. Biehl, and B. Hammer. Adaptive relevance matrices in Learning VectorQuantization. Neural Computation, 21(12):3532–3561, 2009.[4] S. Sreejith, S. Pereverzyev Jr., L. S. Kelvin, et al. Galaxy and mass assembly: Automaticmorphological classification of galaxies using statistical learning. in preparation.[5] J. Liske, IK Baldry, S.P. Driver, et al. Galaxy and mass assembly (gama): end of survey reportand data release 2. Monthly Notices of the Royal Astronomical Society, 452(2):2087–2126, 2015.[6] T. Kohonen. Self-organizing maps. Springer, 1997.[7] J. Vesanto, J. Himberg, E. Alhoniemi, and Juha Parhankangas. Som toolbox for matlab 5.Helsinki University of Technology, Finland, 2000.[8] M. Biehl. Gmlvq demo code. http://www.cs.rug.nl/˜biehl/gmlvq. Last accessed: 2017-11-21.[9] A Dressier, JP Huchra, S van den Bergh, and S Raychaudhury. Galaxies, human eyes, andartificial neural networks. Science, 267:859, 1995.[10] S. Fouad and P. Tino. Adaptive metric learning vector quantization for ordinal classification.Neural Computation, 24(11):2825–2851, 2012.

Prototype-based analysis of GAMA galaxy catalogue data

Supplementary Material for Prototype-based Analysis of GAMA Galaxy Catalogue Data (Nolte, Wang, Biehl)

Nolte, Aleke

Wang, Lingyu

Biehl, Michael

ZENODO

Supplementary Material: Prototype-based Analysis of GAMA Galaxy Catalogue Data

University of Groningen

   University of GroningenPrototype-based analysis of GAMA galaxy catalogue dataNolte, A.; Wang, L.; Biehl, MPublished in:ESANN, European Symposium on Artificial Neural Networks, Computational Intelligence and Machine LearningIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionEarly version, also known as pre-printPublication date:2018Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Nolte, A., Wang, L., & Biehl, M. (2018). Prototype-based analysis of GAMA galaxy catalogue data. In M.Verleysen (Ed.), ESANN, European Symposium on Artificial Neural Networks, Computational Intelligenceand Machine Learning (Vol. 26, pp. 339-344). Louvain-La-Neuve: Ciaco - i6doc.com.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019Prototype-based Analysis ofGAMA Galaxy Catalogue DataAleke Nolte1∗, Lingyu Wang2, and Michael Biehl11 - Univ. of Groningen - Johann Bernoulli Inst. for Mathematics and Computer ScienceP.O. Box 407, 9700 AK Groningen - The Netherlands2- SRON Netherlands Institute for Space Researchand University of Groningen - Katpeyn Astronomical InstituteLandleven 12 - 9747 AD Groningen - The NetherlandsAbstract. We present a prototype-based machine learning analysis of labeledgalaxy catalogue data containing parameters from the Galaxy and Mass As-sembly (GAMA) survey. Using both an unsupervised and supervised method,the Self-Organizing Map and Generalized Relevance Matrix Learning Vec-tor Quantization, we find that the data does not fully support the popularvisual-inspection-based galaxy classification scheme employed to categorize thegalaxies. In particular, only one class, the Little Blue Spheroids, is consistentlyseparable from the other classes. In a proof-of-concept experiment, we presentthe galaxy parameters that are most discriminative for this class.1 IntroductionTelescope images of galaxies reveal a multitude of appearances, ranging from disk-like galaxies over galaxies with spiral arms to more irregular shapes. The study ofgalaxy morphological classification plays an important role in astronomy: The spatialdistribution of galaxy types provides valuable information for the understanding ofgalaxy formation and evolution.The assignment of morphological classes to observed galaxies is a task which iscommonly handled by astronomers. As manual labeling of galaxies is time consumingand expert-devised classification schemes may be subject to cognitive biases, machinelearning techniques have great potential to advance astronomy by: 1) investigatingautomatic classification strategies, and 2) by evaluating to which extent existingclassification schemes are supported by the observed data.In this work, we want to make a contribution along both lines by analyzing a galaxycatalogue which has been annotated using a popular classification scheme proposed byKelvin [1]. To this end, we apply both an unsupervised and a supervised prototype-based method. In our analysis, we first assess whether Kelvin’s scheme is consistentwith a clustering of the data generated by the unsupervised Self Organizing Map(SOM) [2]. We then investigate if the morphological classification can be reproducedby Generalized Relevance Matrix Learning Vector Quantization (GMLVQ) [3], apowerful supervised prototype-based method. In addition to providing an evaluationof the classification scheme via the proxy of classification performance (high accu-racies would indicate a good separation of classes), GMVLQ also allows to identify∗We acknowledge financial support by the EU’s Horizon 2020 research and innovation programmeunder Marie Sk lodowska-Curie grant agreement No 721463 to the SUNDIAL ITN network.the feature dimensions that are of particular relevance for a classification problem.As we find the employed classification scheme to be not fully supported by thedata, we present the parameters that are relevant for distinguishing the most clearlypronounced class, the Little Blue Spheroids (LBS), from the remaining classes.2 DataIn this work we analyze a labeled galaxy catalogue [4] containing 42 parameters whichhave been derived from spectroscopic and photometric observations (measurementsof flux intensities in different wavelength bands) from the Galaxy and Mass Assembly(GAMA) survey [5] for 7941 astronomical objects. For each galaxy, a class label hasbeen determined by astronomers following a visual inspection based classificationscheme described by Kelvin et al. [1]. The scheme assigns galaxies to 9 classes:1-Ellipticals (11%), 2-Little blue spheriods (11%), 3-Early-type spirals (10%), 4-Early-type barred spirals (1%), 5-Intermediate-type spirals (15%), 6-Intermediate-type barredspirals (2%), 7-Late-type spirals & Irregulars(45%), 8-Artefacts (0.4%) and 9-Stars(0.005%). Numbers in parantheses represent the class-wise prevalences in the availabledata. We will refer to the classes by their class index (1-9). We exclude the LOG-SurfaceDensityErr parameter due to numerous missing measurements, and removesamples with missing measurements in any of the remaining features, resulting ina final dataset of 7356 astronomical objects of dimensionality n=41.3 SOM AnalysisThe self-organizing map (SOM) [2, 6] is an unsupervised prototype-based methodwhich allows to generate topology-preserving low-dimensional representations of high-dimensional data. In the default formulation, a SOM is composed of map units thatare arranged in a two-dim. lattice. Each unit has a defined set of neighbors whichare influenced when the unit undergoes changes. A prototype of input dimensionalityis associated with each unit of the map and the training process follows a competitivelearning paradigm: For each data point ξ, the best matching unit (BMU) mc(ξ) isdetermined, and subsequently only mc and its neighbors are modified by the trainingprocedure. The BMU mc(ξ) is given by the map unit with the prototype closest toξ and therefore fulfills∀i,‖ξ−mc‖≤‖ξ−mi‖.In the SOM batch learning rule ([7]), the map units mi are updated followingmi(t+1)=(∑mj=1hj,i sj)/(∑mj=1nVjhj,i) ,with si=∑ξk∈Viξk ,where m is the number of map units, Vi is the set of data points with BMU mi(t),nVi is the cardinality of this set and hi,j is the neighborhood function determiningthe strength of the influence of unit i to j.We perform an analysis of the galaxy catalogue data using the SOM implementation“SOM toolbox for Matlab5” [7]. Due to the SOM’s sensitivity to outliers we normalizethe data using a centered logistic normalization:ξ′=1/(1+e−ξˆ)−0.5 with ξˆ=(ξ−ξ¯)/σξ ,112 377 511= empty = others1 2 3 75Fig. 1: SOM visualization displaying an unsupervised clustering of galaxy data fromGAMA. The SOM is post-labeled using galaxy-class information, where the piechart for each unit encodes a histogram of class-wise hits. A unit’s backgroundcolor indicates the class with the majority of hits. A color version of this plot canbe found at http://doi.org/10.5281/zenodo.1092761.where ξ¯ and σξ are the empirical mean and standard deviation of the feature.We initialize the SOM by spacing the prototypes regularly in the coordinate systemgiven by the two largest eigenvectors of the data matrix [7]. Based on preliminaryexperiments we choose a SOM of size 20×40 units with a rectangular neighborhoodfunction and toroidal topology and train using the option long of the default toolboxsettings. After training, the galaxy label information is used to display the specificityof a map unit via unit-wise histograms of hits (pie charts in Figure 1), i.e. thedistribution of the class labels of the data points that have a particular map unit asBMU. Additionally, the class with the majority of hits is indicated by the backgroundcolor of each pie plot.In the pie charts of Fig. 1 it can be seen that there is no pronounced separation ofclasses and that many units respond to samples from more than one class. Classes1 and 3 are particularly intermixed. However, class 2 (LBS) forms a fairly compactcluster. We note that the basic structure remains similar for larger or smaller maps,or when the over-represented class 7 is under-sampled.4 GMLVQ analysisGeneralized Relevance Matrix LVQ (GMLVQ) [3] is an extension of Learning VectorQuantization (LVQ) [6]. LVQ is a supervised prototype-based method, in whichprototypes are annotated with a class label. The prototypes are adapted based onthe label information of the training data: If the BMU is of the same class as agiven data point, the prototype is moved towards the data point, while in the caseof a BMU with incorrect class label, the prototype is repelled. While both SOMand LVQ assess similarities between prototypes and data points using the Euclideandistance, GMLVQ learns a distance measure that is tailored to the data, allowingit to suppress noisy feature dimensions or to emphasize distinctive features and theirpair-wise combinations. GMLVQ therefore considers a generalized distancedΛ(w,ξ)=(ξ−w)TΛ(ξ−w) with Λ=ΩTΩ and ∑iΛii=1,where Λ is an n×n a positive semi-definite distance matrix, and w is a prototype.The parameters {wi} and Λ are optimized based on a heuristic cost function, see [3].EGMLVQ =∑Pi=1µΛi , with µΛi =(dΛJ (ξi)−dΛK(ξi))/(dΛJ (ξi)+dΛK(ξi)),where dΛJ (ξ)=dΛJ (wJ ,ξ) denotes the distance to the closest correctly labeled proto-type, and dΛK(ξ)=dΛK(wK,ξ) denotes the distance to the closest incorrect prototype.If the closest prototype has an incorrect label, dΛK(ξi) will be smaller than dΛJ (ξi),hence, the corresponding µΛi is positive. Minimization of EGMLVQ will therefore favorthe correctness of nearest prototype classification. In a stochastic gradient descentprocedure based on single examples the update readswJ,K←wJ,K−ηw∂µi/∂wJ,K and Ω←Ω−ηΩ∂µi/∂Ω . (1)Derivations and full update rules can be found in [3].To assess relevances of features and discriminability between classes we train andevaluate GMLVQ on the galaxy catalogue data making use of a publicly available imple-mentation which employs batch gradient descent [8]. To maintain consistency, we nor-malize the data using logistic normalization. As the GMLVQ cost function is implicitlybiased toward classes with larger numbers of samples, we disregard the classes whichcontain only few samples (classes 4,6,8,9) and train and evaluate the classifier on size-balanced random subsets from the remaining five larger classes. For our experiments,we specify one prototype per class and run the algorithm for 100 epochs using the imple-mentation’s default settings. We validate the algorithm by performing a class-balancedrepeated random sub-sampling validation by randomly selecting 743 datapoints (the car-dinality of the smallest class, class 3) per class for each validation run, for a total of 50runs. For each sub-sampled data set, both the training (3343 samples, 90%) and valida-tion set (371 samples (10%)) are class-balanced. The resulting relevances and confusionmatrix (both averaged over all validation runs) are displayed in Figure 2 and Table 4.The confusion matrix corroborates the findings from the SOM analysis: The classifierhas difficulty distinguishing classes 1 and 3, placing about 19% of samples from class 1into class 3 and vice versa. Also classes 5 and 7 can only be classified with limited accu-racies of 73% and 71%. In line with the compactness of its representation by the SOM,only class 2 (little blue spheroids, LBS), are classified with a high accuracy (90.7%).Both the lack of pronounced clusters in the SOM and the moderate classificationperformance of GMLVQ may be an indication for the presence of label noise that isRA DECZ_TONRYLUMDIST_TONRYTHETA_IMAGETHETA_J2000ERRTHETA_J2000MAG_AUTO_rMAGERR_AUTO_rR_MODELR_MODEL_ERRlogmstardellogmstarlogmoverl_idellogmoverl_iabsmag_rGALMAG_rGALMAGERR_rGALELLIP_rGALELLIPERR_rgminusidelgminusiuminusrdeluminusrLOGKRON_ALOGKRON_BLOGKRON_A_ERRLOGKRON_B_ERRLOGGALINDEX_rLOGGALINDEXERR_rLOGFLUX_AUTO_rLOGFLUXERR_AUTO_rLOGGALRE_rLOGGALREERR_rLOGKRON_A_KPCLOGKRON_B_KPCLOGKRON_A_ERRKPCLOGKRON_B_ERRKPCLOGGALRE_r_KPCLOGGALREERR_KPCLOGSurfaceDensity00.050.10.150.20.250.35 classesLBS vs othersrelevancegalaxy parameter103927 8145 61057 86129 43Fig. 2: Average GMVLQ feature relevances for the task of distinguishing the 5largest galaxy classes (dark) and the little blue spheroid class from the other 4classes (light). The 10 most relevant features are indicated by digits (dark/light)corresponding to the rank of the feature. Arrows indicate features selected in [4].Explanations of the galaxy parameters and additional references can be found athttp://doi.org/10.5281/zenodo.1092761.1 2 3 5 71 63.8 10.8 19.3 5.5 0.72 3.1 90.7 0.1 1.3 4.83 18.4 1.7 66.1 13.5 0.35 2.0 6.8 9.3 73.7 8.27 1.1 13.1 0.1 14.6 71.1Table 1: Averaged GMVLQ confusion ma-trix displaying the percentage of predictedclass labels (columns) for samples from eachclass (rows) for the 5 largest galaxy classes.rooted in the subjective nature of galaxy classification [4, 9]. It may also indicate thatthe classification scheme is not fully supported by the data. As the relevance profilereflects the discriminability of parameters only within this possibly ill-defined setting,we additionally consider the relevance profile for the less ambiguous sub-problemof separating LBS from other galaxies. Following the procedure described above(with 1512 training and 168 validation samples per validation run), we find that LBScan be well distinguished (AUC(ROC)=0.96) from the other classes. Furthermore,in the two-class setting 8 of the 10 most relevant features are also among the 10most relevant ones found for the full five-class setting. The coinciding parametersare GALELLIP r, LOGGALINDEXERR r, LOGGALINDEX r, LOGGALRE r,R MODEL, R MODEL ERR, deluminusr, and uminusr.5 DiscussionThe results presented above suggest that there may be some inconsistencies in theinvestigated morphological classification scheme: The clustering produced by theunsupervised SOM is only moderately consistent with the galaxy classes and it hasproven difficult to distinguish galaxy types using supervised GMLVQ. In our analysis,class 1 (Ellipticals) and 3 (Early-type spirals) are particularly difficult to differentiate,while class 2 (LBS) seems to be well separable.The difficulty of training a successful classifier was also observed in [4], where class-wiseaveraged accuracies are also around 75%. Possible explanations for poor classificationperformance may be lack of discriminative power of the employed classifiers, mis-labelings of certain galaxies (a possibility already discussed in [4]), or that essentialparameters are not contained in the data set. In the first case, employing evenmore flexible classifiers, e.g. GMLVQ with local relevance matrices [3], may improvethe classification performance. In the second case, if mis-labelings are restricted to“neighboring” classes in an assumed underlying class ordering, ordinal classificationmay provide further insights [10]. Yet, our results do not rule out the possibility thatthe true, underlying grouping of galaxies is considerably different and less clear-cutthan the investigated one. Further data-driven analyses of galaxy parameters andimages with advanced clustering methods might reveal alternative groupings.In [4], 10 of the 42 parameters were selected manually. Out of GMLVQ’s 8 most rele-vant features, 4 overlap or highly correlate with the 10 features selected in [4], namelyGALELLIP r, LOGGALINDEX r, uminus r, and LOGGALRE r. These parametersare related to the ellipticity, light distribution, size and color of the galaxy. Note thatthe non-overlapping features comprise model fitting errors (LOGGALINDEXERR r,R MODEL ERR, deluminusr), indicating that a galaxy’s model consistency variesover morphological classes.Acknowledgements GAMA is a joint European-Australasian project based around a spectroscopiccampaign using the Anglo- Australian Telescope. The GAMA input catalogue is based on datataken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementaryimaging of the GAMA regions is being obtained by a number of in- dependent survey programmesincluding GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAPproviding UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), theAAO, and the participating institutions. The GAMA website is http://www.gama-survey.org/.We thank Sreevarsha Sreejith and Lee Kelvin for helpful feedback and discussions.References[1] L.S. Kelvin, S.P. Driver, A. SG Robotham, et al. Galaxy and mass assembly (gama): ugrizyjhkse´rsic luminosity functions and the cosmic spectral energy distribution by hubble type. MonthlyNotices of the Royal Astronomical Society, 439(2):1245–1269, 2014.[2] T. Kohonen. The self-organizing map. Neurocomputing, 21(1):1–6, 1998.[3] P. Schneider, M. Biehl, and B. Hammer. Adaptive relevance matrices in Learning VectorQuantization. Neural Computation, 21(12):3532–3561, 2009.[4] S. Sreejith, S. Pereverzyev Jr., L. S. Kelvin, et al. Galaxy and mass assembly: Automaticmorphological classification of galaxies using statistical learning. in preparation.[5] J. Liske, IK Baldry, S.P. Driver, et al. Galaxy and mass assembly (gama): end of survey reportand data release 2. Monthly Notices of the Royal Astronomical Society, 452(2):2087–2126, 2015.[6] T. Kohonen. Self-organizing maps. Springer, 1997.[7] J. Vesanto, J. Himberg, E. Alhoniemi, and Juha Parhankangas. Som toolbox for matlab 5.Helsinki University of Technology, Finland, 2000.[8] M. Biehl. Gmlvq demo code. http://www.cs.rug.nl/˜biehl/gmlvq. Last accessed: 2017-11-21.[9] A Dressier, JP Huchra, S van den Bergh, and S Raychaudhury. Galaxies, human eyes, andartificial neural networks. Science, 267:859, 1995.[10] S. Fouad and P. Tino. Adaptive metric learning vector quantization for ordinal classification.Neural Computation, 24(11):2825–2851, 2012.

   University of GroningenPrototype-based analysis of GAMA galaxy catalogue dataNolte, A.; Wang, L.; Biehl, MPublished in:ESANN, European Symposium on Artificial Neural Networks, Computational Intelligence and Machine LearningIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionEarly version, also known as pre-printPublication date:2018Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Nolte, A., Wang, L., & Biehl, M. (2018). Prototype-based analysis of GAMA galaxy catalogue data. In M.Verleysen (Ed.), ESANN, European Symposium on Artificial Neural Networks, Computational Intelligenceand Machine Learning (Vol. 26, pp. 339-344). Ciaco - i6doc.com.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 10-09-2023Prototype-based Analysis ofGAMA Galaxy Catalogue DataAleke Nolte1∗, Lingyu Wang2, and Michael Biehl11 - Univ. of Groningen - Johann Bernoulli Inst. for Mathematics and Computer ScienceP.O. Box 407, 9700 AK Groningen - The Netherlands2- SRON Netherlands Institute for Space Researchand University of Groningen - Katpeyn Astronomical InstituteLandleven 12 - 9747 AD Groningen - The NetherlandsAbstract. We present a prototype-based machine learning analysis of labeledgalaxy catalogue data containing parameters from the Galaxy and Mass As-sembly (GAMA) survey. Using both an unsupervised and supervised method,the Self-Organizing Map and Generalized Relevance Matrix Learning Vec-tor Quantization, we find that the data does not fully support the popularvisual-inspection-based galaxy classification scheme employed to categorize thegalaxies. In particular, only one class, the Little Blue Spheroids, is consistentlyseparable from the other classes. In a proof-of-concept experiment, we presentthe galaxy parameters that are most discriminative for this class.1 IntroductionTelescope images of galaxies reveal a multitude of appearances, ranging from disk-like galaxies over galaxies with spiral arms to more irregular shapes. The study ofgalaxy morphological classification plays an important role in astronomy: The spatialdistribution of galaxy types provides valuable information for the understanding ofgalaxy formation and evolution.The assignment of morphological classes to observed galaxies is a task which iscommonly handled by astronomers. As manual labeling of galaxies is time consumingand expert-devised classification schemes may be subject to cognitive biases, machinelearning techniques have great potential to advance astronomy by: 1) investigatingautomatic classification strategies, and 2) by evaluating to which extent existingclassification schemes are supported by the observed data.In this work, we want to make a contribution along both lines by analyzing a galaxycatalogue which has been annotated using a popular classification scheme proposed byKelvin [1]. To this end, we apply both an unsupervised and a supervised prototype-based method. In our analysis, we first assess whether Kelvin’s scheme is consistentwith a clustering of the data generated by the unsupervised Self Organizing Map(SOM) [2]. We then investigate if the morphological classification can be reproducedby Generalized Relevance Matrix Learning Vector Quantization (GMLVQ) [3], apowerful supervised prototype-based method. In addition to providing an evaluationof the classification scheme via the proxy of classification performance (high accu-racies would indicate a good separation of classes), GMVLQ also allows to identify∗We acknowledge financial support by the EU’s Horizon 2020 research and innovation programmeunder Marie Sk lodowska-Curie grant agreement No 721463 to the SUNDIAL ITN network.the feature dimensions that are of particular relevance for a classification problem.As we find the employed classification scheme to be not fully supported by thedata, we present the parameters that are relevant for distinguishing the most clearlypronounced class, the Little Blue Spheroids (LBS), from the remaining classes.2 DataIn this work we analyze a labeled galaxy catalogue [4] containing 42 parameters whichhave been derived from spectroscopic and photometric observations (measurementsof flux intensities in different wavelength bands) from the Galaxy and Mass Assembly(GAMA) survey [5] for 7941 astronomical objects. For each galaxy, a class label hasbeen determined by astronomers following a visual inspection based classificationscheme described by Kelvin et al. [1]. The scheme assigns galaxies to 9 classes:1-Ellipticals (11%), 2-Little blue spheriods (11%), 3-Early-type spirals (10%), 4-Early-type barred spirals (1%), 5-Intermediate-type spirals (15%), 6-Intermediate-type barredspirals (2%), 7-Late-type spirals & Irregulars(45%), 8-Artefacts (0.4%) and 9-Stars(0.005%). Numbers in parantheses represent the class-wise prevalences in the availabledata. We will refer to the classes by their class index (1-9). We exclude the LOG-SurfaceDensityErr parameter due to numerous missing measurements, and removesamples with missing measurements in any of the remaining features, resulting ina final dataset of 7356 astronomical objects of dimensionality n=41.3 SOM AnalysisThe self-organizing map (SOM) [2, 6] is an unsupervised prototype-based methodwhich allows to generate topology-preserving low-dimensional representations of high-dimensional data. In the default formulation, a SOM is composed of map units thatare arranged in a two-dim. lattice. Each unit has a defined set of neighbors whichare influenced when the unit undergoes changes. A prototype of input dimensionalityis associated with each unit of the map and the training process follows a competitivelearning paradigm: For each data point ξ, the best matching unit (BMU) mc(ξ) isdetermined, and subsequently only mc and its neighbors are modified by the trainingprocedure. The BMU mc(ξ) is given by the map unit with the prototype closest toξ and therefore fulfills∀i,‖ξ−mc‖≤‖ξ−mi‖.In the SOM batch learning rule ([7]), the map units mi are updated followingmi(t+1)=(∑mj=1hj,i sj)/(∑mj=1nVjhj,i) ,with si=∑ξk∈Viξk ,where m is the number of map units, Vi is the set of data points with BMU mi(t),nVi is the cardinality of this set and hi,j is the neighborhood function determiningthe strength of the influence of unit i to j.We perform an analysis of the galaxy catalogue data using the SOM implementation“SOM toolbox for Matlab5” [7]. Due to the SOM’s sensitivity to outliers we normalizethe data using a centered logistic normalization:ξ′=1/(1+e−ξ̂)−0.5 with ξ̂=(ξ−ξ̄)/σξ ,112 377 511= empty = others1 2 3 75Fig. 1: SOM visualization displaying an unsupervised clustering of galaxy data fromGAMA. The SOM is post-labeled using galaxy-class information, where the piechart for each unit encodes a histogram of class-wise hits. A unit’s backgroundcolor indicates the class with the majority of hits. A color version of this plot canbe found at http://doi.org/10.5281/zenodo.1092761.where ξ̄ and σξ are the empirical mean and standard deviation of the feature.We initialize the SOM by spacing the prototypes regularly in the coordinate systemgiven by the two largest eigenvectors of the data matrix [7]. Based on preliminaryexperiments we choose a SOM of size 20×40 units with a rectangular neighborhoodfunction and toroidal topology and train using the option long of the default toolboxsettings. After training, the galaxy label information is used to display the specificityof a map unit via unit-wise histograms of hits (pie charts in Figure 1), i.e. thedistribution of the class labels of the data points that have a particular map unit asBMU. Additionally, the class with the majority of hits is indicated by the backgroundcolor of each pie plot.In the pie charts of Fig. 1 it can be seen that there is no pronounced separation ofclasses and that many units respond to samples from more than one class. Classes1 and 3 are particularly intermixed. However, class 2 (LBS) forms a fairly compactcluster. We note that the basic structure remains similar for larger or smaller maps,or when the over-represented class 7 is under-sampled.4 GMLVQ analysisGeneralized Relevance Matrix LVQ (GMLVQ) [3] is an extension of Learning VectorQuantization (LVQ) [6]. LVQ is a supervised prototype-based method, in whichprototypes are annotated with a class label. The prototypes are adapted based onthe label information of the training data: If the BMU is of the same class as agiven data point, the prototype is moved towards the data point, while in the caseof a BMU with incorrect class label, the prototype is repelled. While both SOMand LVQ assess similarities between prototypes and data points using the Euclideandistance, GMLVQ learns a distance measure that is tailored to the data, allowingit to suppress noisy feature dimensions or to emphasize distinctive features and theirpair-wise combinations. GMLVQ therefore considers a generalized distancedΛ(w,ξ)=(ξ−w)TΛ(ξ−w) with Λ=ΩTΩ and∑iΛii=1,where Λ is an n×n a positive semi-definite distance matrix, and w is a prototype.The parameters {wi} and Λ are optimized based on a heuristic cost function, see [3].EGMLVQ =∑Pi=1µΛi , with µΛi =(dΛJ (ξi)−dΛK(ξi))/(dΛJ (ξi)+dΛK(ξi)),where dΛJ (ξ)=dΛJ (wJ ,ξ) denotes the distance to the closest correctly labeled proto-type, and dΛK(ξ)=dΛK(wK,ξ) denotes the distance to the closest incorrect prototype.If the closest prototype has an incorrect label, dΛK(ξi) will be smaller than dΛJ (ξi),hence, the corresponding µΛi is positive. Minimization of EGMLVQ will therefore favorthe correctness of nearest prototype classification. In a stochastic gradient descentprocedure based on single examples the update readswJ,K←wJ,K−ηw∂µi/∂wJ,K and Ω←Ω−ηΩ∂µi/∂Ω . (1)Derivations and full update rules can be found in [3].To assess relevances of features and discriminability between classes we train andevaluate GMLVQ on the galaxy catalogue data making use of a publicly available imple-mentation which employs batch gradient descent [8]. To maintain consistency, we nor-malize the data using logistic normalization. As the GMLVQ cost function is implicitlybiased toward classes with larger numbers of samples, we disregard the classes whichcontain only few samples (classes 4,6,8,9) and train and evaluate the classifier on size-balanced random subsets from the remaining five larger classes. For our experiments,we specify one prototype per class and run the algorithm for 100 epochs using the imple-mentation’s default settings. We validate the algorithm by performing a class-balancedrepeated random sub-sampling validation by randomly selecting 743 datapoints (the car-dinality of the smallest class, class 3) per class for each validation run, for a total of 50runs. For each sub-sampled data set, both the training (3343 samples, 90%) and valida-tion set (371 samples (10%)) are class-balanced. The resulting relevances and confusionmatrix (both averaged over all validation runs) are displayed in Figure 2 and Table 4.The confusion matrix corroborates the findings from the SOM analysis: The classifierhas difficulty distinguishing classes 1 and 3, placing about 19% of samples from class 1into class 3 and vice versa. Also classes 5 and 7 can only be classified with limited accu-racies of 73% and 71%. In line with the compactness of its representation by the SOM,only class 2 (little blue spheroids, LBS), are classified with a high accuracy (90.7%).Both the lack of pronounced clusters in the SOM and the moderate classificationperformance of GMLVQ may be an indication for the presence of label noise that isRADECZ_TONRYLUMDIST_TONRYTHETA_IMAGETHETA_J2000ERRTHETA_J2000MAG_AUTO_rMAGERR_AUTO_rR_MODELR_MODEL_ERRlogmstardellogmstarlogmoverl_idellogmoverl_iabsmag_rGALMAG_rGALMAGERR_rGALELLIP_rGALELLIPERR_rgminusidelgminusiuminusrdeluminusrLOGKRON_ALOGKRON_BLOGKRON_A_ERRLOGKRON_B_ERRLOGGALINDEX_rLOGGALINDEXERR_rLOGFLUX_AUTO_rLOGFLUXERR_AUTO_rLOGGALRE_rLOGGALREERR_rLOGKRON_A_KPCLOGKRON_B_KPCLOGKRON_A_ERRKPCLOGKRON_B_ERRKPCLOGGALRE_r_KPCLOGGALREERR_KPCLOGSurfaceDensity00.050.10.150.20.250.35 classesLBS vs othersrelevancegalaxy parameter103927 8145 6105786129 43Fig. 2: Average GMVLQ feature relevances for the task of distinguishing the 5largest galaxy classes (dark) and the little blue spheroid class from the other 4classes (light). The 10 most relevant features are indicated by digits (dark/light)corresponding to the rank of the feature. Arrows indicate features selected in [4].Explanations of the galaxy parameters and additional references can be found athttp://doi.org/10.5281/zenodo.1092761.1 2 3 5 71 63.8 10.8 19.3 5.5 0.72 3.1 90.7 0.1 1.3 4.83 18.4 1.7 66.1 13.5 0.35 2.0 6.8 9.3 73.7 8.27 1.1 13.1 0.1 14.6 71.1Table 1: Averaged GMVLQ confusion ma-trix displaying the percentage of predictedclass labels (columns) for samples from eachclass (rows) for the 5 largest galaxy classes.rooted in the subjective nature of galaxy classification [4, 9]. It may also indicate thatthe classification scheme is not fully supported by the data. As the relevance profilereflects the discriminability of parameters only within this possibly ill-defined setting,we additionally consider the relevance profile for the less ambiguous sub-problemof separating LBS from other galaxies. Following the procedure described above(with 1512 training and 168 validation samples per validation run), we find that LBScan be well distinguished (AUC(ROC)=0.96) from the other classes. Furthermore,in the two-class setting 8 of the 10 most relevant features are also among the 10most relevant ones found for the full five-class setting. The coinciding parametersare GALELLIP r, LOGGALINDEXERR r, LOGGALINDEX r, LOGGALRE r,R MODEL, R MODEL ERR, deluminusr, and uminusr.5 DiscussionThe results presented above suggest that there may be some inconsistencies in theinvestigated morphological classification scheme: The clustering produced by theunsupervised SOM is only moderately consistent with the galaxy classes and it hasproven difficult to distinguish galaxy types using supervised GMLVQ. In our analysis,class 1 (Ellipticals) and 3 (Early-type spirals) are particularly difficult to differentiate,while class 2 (LBS) seems to be well separable.The difficulty of training a successful classifier was also observed in [4], where class-wiseaveraged accuracies are also around 75%. Possible explanations for poor classificationperformance may be lack of discriminative power of the employed classifiers, mis-labelings of certain galaxies (a possibility already discussed in [4]), or that essentialparameters are not contained in the data set. In the first case, employing evenmore flexible classifiers, e.g. GMLVQ with local relevance matrices [3], may improvethe classification performance. In the second case, if mis-labelings are restricted to“neighboring” classes in an assumed underlying class ordering, ordinal classificationmay provide further insights [10]. Yet, our results do not rule out the possibility thatthe true, underlying grouping of galaxies is considerably different and less clear-cutthan the investigated one. Further data-driven analyses of galaxy parameters andimages with advanced clustering methods might reveal alternative groupings.In [4], 10 of the 42 parameters were selected manually. Out of GMLVQ’s 8 most rele-vant features, 4 overlap or highly correlate with the 10 features selected in [4], namelyGALELLIP r, LOGGALINDEX r, uminus r, and LOGGALRE r. These parametersare related to the ellipticity, light distribution, size and color of the galaxy. Note thatthe non-overlapping features comprise model fitting errors (LOGGALINDEXERR r,R MODEL ERR, deluminusr), indicating that a galaxy’s model consistency variesover morphological classes.Acknowledgements GAMA is a joint European-Australasian project based around a spectroscopiccampaign using the Anglo- Australian Telescope. The GAMA input catalogue is based on datataken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementaryimaging of the GAMA regions is being obtained by a number of in- dependent survey programmesincluding GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAPproviding UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), theAAO, and the participating institutions. The GAMA website is http://www.gama-survey.org/.We thank Sreevarsha Sreejith and Lee Kelvin for helpful feedback and discussions.References[1] L.S. Kelvin, S.P. Driver, A. SG Robotham, et al. Galaxy and mass assembly (gama): ugrizyjhksérsic luminosity functions and the cosmic spectral energy distribution by hubble type. MonthlyNotices of the Royal Astronomical Society, 439(2):1245–1269, 2014.[2] T. Kohonen. The self-organizing map. Neurocomputing, 21(1):1–6, 1998.[3] P. Schneider, M. Biehl, and B. Hammer. Adaptive relevance matrices in Learning VectorQuantization. Neural Computation, 21(12):3532–3561, 2009.[4] S. Sreejith, S. Pereverzyev Jr., L. S. Kelvin, et al. Galaxy and mass assembly: Automaticmorphological classification of galaxies using statistical learning. in preparation.[5] J. Liske, IK Baldry, S.P. Driver, et al. Galaxy and mass assembly (gama): end of survey reportand data release 2. Monthly Notices of the Royal Astronomical Society, 452(2):2087–2126, 2015.[6] T. Kohonen. Self-organizing maps. Springer, 1997.[7] J. Vesanto, J. Himberg, E. Alhoniemi, and Juha Parhankangas. Som toolbox for matlab 5.Helsinki University of Technology, Finland, 2000.[8] M. Biehl. Gmlvq demo code. http://www.cs.rug.nl/˜biehl/gmlvq. Last accessed: 2017-11-21.[9] A Dressier, JP Huchra, S van den Bergh, and S Raychaudhury. Galaxies, human eyes, andartificial neural networks. Science, 267:859, 1995.[10] S. Fouad and P. Tino. Adaptive metric learning vector quantization for ordinal classification.Neural Computation, 24(11):2825–2851, 2012.

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