METAPHOR: a machine-learning-based method for the probability density estimation of photometric redshifts

A variety of fundamental astrophysical science topics require the determination of very accurate photometric redshifts (photo-z). A wide plethora of methods have been developed, based either on template models fitting or on empirical explorations of the photometric parameter space. Machine-learning-based techniques are not explicitly dependent on the physical priors and able to produce accurate photo-z estimations within the photometric ranges derived from the spectroscopic training set. These estimates, however, are not easy to characterize in terms of a photo-z probability density function (PDF), due to the fact that the analytical relation mapping the photometric parameters on to the redshift space is virtually unknown. We present METAPHOR (Machine-learning Estimation Tool for Accurate PHOtometric Redshifts), a method designed to provide a reliable PDF of the error distribution for empirical techniques. The method is implemented as a modular workflow, whose internal engine for photo-z estimation makes use of the MLPQNA neural network (Multi Layer Perceptron with Quasi Newton learning rule), with the possibility to easily replace the specificmachine-learningmodel chosen to predict photo-z. We present a summary of results on SDSS-DR9 galaxy data, used also to perform a direct comparison with PDFs obtained by the LE PHARE spectral energy distribution template fitting. We show that METAPHOR is capable to estimate the precision and reliability of photometric redshifts obtained with three different self-adaptive techniques, i.e. MLPQNA, Random Forest and the standard K-Nearest Neighbors models.