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In the modern galaxy surveys photometric redshifts play a central role in a broad range of studies, from gravitational lensing and dark matter distribution to galaxy evolution. Using a dataset of about 25,000 galaxies from the second data release of the Kilo Degree Survey (KiDS) we obtain photometric redshifts with five different methods: (i) Random forest, (ii) Multi Layer Perceptron with Quasi Newton Algorithm, (iii) Multi Layer Perceptron with an optimization network based on the Levenberg-Marquardt learning rule, (iv) the Bayesian Photometric Redshift model (or BPZ) and (v) a classical SED template fitting procedure (Le Phare). We show how SED fitting techniques could provide useful information on the galaxy spectral type which can be used to improve the capability of machine learning methods constraining systematic errors and reduce the occurrence of catastrophic outliers. We use such classification to train specialized regression estimators, by demonstrating that such hybrid approach, involving SED fitting and machine learning in a single collaborative framework, is capable to improve the overall prediction accuracy of photometric redshifts.
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Accurate photometric redshifts are a lynchpin for many future experiments to pin down the cosmological model and for studies of galaxy evolution. In this study, a novel sparse regression framework for photometric redshift estimation is presented. Simulated and real data from SDSS DR12 were used to train and test the proposed models. We show that approaches which include careful data preparation and model design offer a significant improvement in comparison with several competing machine learning algorithms. Standard implementations of most regression algorithms have as the objective the minimization of the sum of squared errors. For redshift inference, however, this induces a bias in the posterior mean of the output distribution, which can be problematic. In this paper we directly target minimizing $Delta z = (z_textrm{s} - z_textrm{p})/(1+z_textrm{s})$ and address the bias problem via a distribution-based weighting scheme, incorporated as part of the optimization objective. The results are compared with other machine learning algorithms in the field such as Artificial Neural Networks (ANN), Gaussian Processes (GPs) and sparse GPs. The proposed framework reaches a mean absolute $Delta z = 0.0026(1+z_textrm{s})$, over the redshift range of $0 le z_textrm{s} le 2$ on the simulated data, and $Delta z = 0.0178(1+z_textrm{s})$ over the entire redshift range on the SDSS DR12 survey, outperforming the standard ANNz used in the literature. We also investigate how the relative size of the training set affects the photometric redshift accuracy. We find that a training set of textgreater 30 per cent of total sample size, provides little additional constraint on the photometric redshifts, and note that our GP formalism strongly outperforms ANNz in the sparse data regime for the simulated data set.
This work emphasizes that heterogeneity, diversity, discontinuity, and discreteness in data is to be exploited in classification and regression problems. A global a priori model may not be desirable. For data analytics in cosmology, this is motivated by the variety of cosmological objects such as elliptical, spiral, active, and merging galaxies at a wide range of redshifts. Our aim is matching and similarity-based analytics that takes account of discrete relationships in the data. The information structure of the data is represented by a hierarchy or tree where the branch structure, rather than just the proximity, is important. The representation is related to p-adic number theory. The clustering or binning of the data values, related to the precision of the measurements, has a central role in this methodology. If used for regression, our approach is a method of cluster-wise regression, generalizing nearest neighbour regression. Both to exemplify this analytics approach, and to demonstrate computational benefits, we address the well-known photometric redshift or `photo-z problem, seeking to match Sloan Digital Sky Survey (SDSS) spectroscopic and photometric redshifts.
Photometric redshifts (photo-zs) are fundamental in galaxy surveys to address different topics, from gravitational lensing and dark matter distribution to galaxy evolution. The Kilo Degree Survey (KiDS), i.e. the ESO public survey on the VLT Survey Telescope (VST), provides the unprecedented opportunity to exploit a large galaxy dataset with an exceptional image quality and depth in the optical wavebands. Using a KiDS subset of about 25,000 galaxies with measured spectroscopic redshifts, we have derived photo-zs using i) three different empirical methods based on supervised machine learning, ii) the Bayesian Photometric Redshift model (or BPZ), and iii) a classical SED template fitting procedure (Le Phare). We confirm that, in the regions of the photometric parameter space properly sampled by the spectroscopic templates, machine learning methods provide better redshift estimates, with a lower scatter and a smaller fraction of outliers. SED fitting techniques, however, provide useful information on the galaxy spectral type which can be effectively used to constrain systematic errors and to better characterize potential catastrophic outliers. Such classification is then used to specialize the training of regression machine learning models, by demonstrating that a hybrid approach, involving SED fitting and machine learning in a single collaborative framework, can be effectively used to improve the accuracy of photo-z estimates.
Obtaining accurate photometric redshift estimations is an important aspect of cosmology, remaining a prerequisite of many analyses. In creating novel methods to produce redshift estimations, there has been a shift towards using machine learning techniques. However, there has not been as much of a focus on how well different machine learning methods scale or perform with the ever-increasing amounts of data being produced. Here, we introduce a benchmark designed to analyse the performance and scalability of different supervised machine learning methods for photometric redshift estimation. Making use of the Sloan Digital Sky Survey (SDSS - DR12) dataset, we analysed a variety of the most used machine learning algorithms. By scaling the number of galaxies used to train and test the algorithms up to one million, we obtained several metrics demonstrating the algorithms performance and scalability for this task. Furthermore, by introducing a new optimisation method, time-considered optimisation, we were able to demonstrate how a small concession of error can allow for a great improvement in efficiency. From the algorithms tested we found that the Random Forest performed best in terms of error with a mean squared error, MSE = 0.0042; however, as other algorithms such as Boosted Decision Trees and k-Nearest Neighbours performed incredibly similarly, we used our benchmarks to demonstrate how different algorithms could be superior in different scenarios. We believe benchmarks such as this will become even more vital with upcoming surveys, such as LSST, which will capture billions of galaxies requiring photometric redshifts.
The development of fast and accurate methods of photometric redshift estimation is a vital step towards being able to fully utilize the data of next-generation surveys within precision cosmology. In this paper we apply a specific approach to spectral connectivity analysis (SCA; Lee & Wasserman 2009) called diffusion map. SCA is a class of non-linear techniques for transforming observed data (e.g., photometric colours for each galaxy, where the data lie on a complex subset of p-dimensional space) to a simpler, more natural coordinate system wherein we apply regression to make redshift predictions. As SCA relies upon eigen-decomposition, our training set size is limited to ~ 10,000 galaxies; we use the Nystrom extension to quickly estimate diffusion coordinates for objects not in the training set. We apply our method to 350,738 SDSS main sample galaxies, 29,816 SDSS luminous red galaxies, and 5,223 galaxies from DEEP2 with CFHTLS ugriz photometry. For all three datasets, we achieve prediction accuracies on par with previous analyses, and find that use of the Nystrom extension leads to a negligible loss of prediction accuracy relative to that achieved with the training sets. As in some previous analyses (e.g., Collister & Lahav 2004, Ball et al. 2008), we observe that our predictions are generally too high (low) in the low (high) redshift regimes. We demonstrate that this is a manifestation of attenuation bias, wherein measurement error (i.e., uncertainty in diffusion coordinates due to uncertainty in the measured fluxes/magnitudes) reduces the slope of the best-fit regression line. Mitigation of this bias is necessary if we are to use photometric redshift estimates produced by computationally efficient empirical methods in precision cosmology.
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