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We introduce an ordinal classification algorithm for photometric redshift estimation, which significantly improves the reconstruction of photometric redshift probability density functions (PDFs) for individual galaxies and galaxy samples. As a use case we apply our method to CFHTLS galaxies. The ordinal classification algorithm treats distinct redshift bins as ordered values, which improves the quality of photometric redshift PDFs, compared with non-ordinal classification architectures. We also propose a new single value point estimate of the galaxy redshift, which can be used to estimate the full redshift PDF of a galaxy sample. This method is competitive in terms of accuracy with contemporary algorithms, which stack the full redshift PDFs of all galaxies in the sample, but requires orders of magnitude less storage space. The methods described in this paper greatly improve the log-likelihood of individual object redshift PDFs, when compared with a popular neural network code (ANNZ). In our use case, this improvement reaches 50 percent for high-redshift objects (z ≥ 0.75). We show that using these more accurate photometric redshift PDFs will lead to a reduction in the systematic biases by up to a factor of 4, when compared with less accurate PDFs obtained from commonly used methods. The cosmological analyses we examine and find improvement upon are the following: gravitational lensing cluster mass estimates, modelling of angular correlation functions and modelling of cosmic shear correlation functions. [ABSTRACT FROM AUTHOR]

Photometric redshift uncertainties are a major source of systematic error for ongoing and future photometric surveys. We study different sources of redshift error caused by choosing a suboptimal redshift histogram bin width and propose methods to resolve them. The selection of a too large bin width is shown to oversmooth small-scale structure of the radial distribution of galaxies. This systematic error can significantly shift cosmological parameter constraints by up to 6σ for the dark energy equation-of-state parameter w. Careful selection of bin width can reduce this systematic by a factor of up to 6 as compared with commonly used current binning approaches. We further discuss a generalized resampling method that can correct systematic and statistical errors in cosmological parameter constraints caused by uncertainties in the redshift distribution. This can be achieved without any prior assumptions about the shape of the distribution or the form of the redshift error. Our methodology allows photometric surveys to obtain unbiased cosmological parameter constraints using a minimum number of spectroscopic calibration data. For a DES-like galaxy clustering forecast, we obtain unbiased results with respect to errors caused by suboptimal histogram bin width selection, using only 5k representative spectroscopic calibration objects per tomographic redshift bin. [ABSTRACT FROM AUTHOR]

Photometric redshift uncertainties are a major source of systematic error for ongoing and future photometric surveys. We study different sources of redshift error caused by choosing a suboptimal redshift histogram bin width and propose methods to resolve them. The selection of a too large bin width is shown to oversmooth small-scale structure of the radial distribution of galaxies. This systematic error can significantly shift cosmological parameter constraints by up to 6σ for the dark energy equation-of-state parameter w. Careful selection of bin width can reduce this systematic by a factor of up to 6 as compared with commonly used current binning approaches. We further discuss a generalized resampling method that can correct systematic and statistical errors in cosmological parameter constraints caused by uncertainties in the redshift distribution. This can be achieved without any prior assumptions about the shape of the distribution or the form of the redshift error. Our methodology allows photometric surveys to obtain unbiased cosmological parameter constraints using a minimum number of spectroscopic calibration data. For a DES-like galaxy clustering forecast, we obtain unbiased results with respect to errors caused by suboptimal histogram bin width selection, using only 5k representative spectroscopic calibration objects per tomographic redshift bin. [ABSTRACT FROM AUTHOR]

Photometric redshift uncertainties are a major source of systematic error for ongoing and future photometric surveys. We study different sources of redshift error caused by choosing a suboptimal redshift histogram bin width and propose methods to resolve them. The selection of a too large bin width is shown to oversmooth small-scale structure of the radial distribution of galaxies. This systematic error can significantly shift cosmological parameter constraints by up to 6σ for the dark energy equation-of-state parameter w. Careful selection of bin width can reduce this systematic by a factor of up to 6 as compared with commonly used current binning approaches. We further discuss a generalized resampling method that can correct systematic and statistical errors in cosmological parameter constraints caused by uncertainties in the redshift distribution. This can be achieved without any prior assumptions about the shape of the distribution or the form of the redshift error. Our methodology allows photometric surveys to obtain unbiased cosmological parameter constraints using a minimum number of spectroscopic calibration data. For a DES-like galaxy clustering forecast, we obtain unbiased results with respect to errors caused by suboptimal histogram bin width selection, using only 5k representative spectroscopic calibration objects per tomographic redshift bin. [ABSTRACT FROM AUTHOR]

We present an analysis of a general machine learning technique called 'stacking' for the estimation of photometric redshifts. Stacking techniques can feed the photometric redshift estimate, as output by a base algorithm, back into the same algorithm as an additional input feature in a subsequent learning round. We show how all tested base algorithms benefit from at least one additional stacking round (or layer). To demonstrate the benefit of stacking, we apply the method to both unsupervised machine learning techniques based on self-organizing maps (SOMs), and supervised machine learning methods based on decision trees. We explore a range of stacking architectures, such as the number of layers and the number of base learners per layer. Finally we explore the effectiveness of stacking even when using a successful algorithm such as AdaBoost. We observe a significant improvement of between 1.9 per cent and 21 per cent on all computed metrics when stacking is applied to weak learners (such as SOMs and decision trees). When applied to strong learning algorithms (such as AdaBoost) the ratio of improvement shrinks, but still remains positive and is between 0.4 per cent and 2.5 per cent for the explored metrics and comes at almost no additional computational cost. [ABSTRACT FROM AUTHOR]

We showcase machine learning (ML) inspired target selection algorithms to determine which of all potential targets should be selected first for spectroscopic follow-up. Efficient target selection can improve the ML redshift uncertainties as calculated on an independent sample, while requiring less targets to be observed. We compare seven different ML targeting algorithms with the Sloan Digital Sky Survey (SDSS) target order, and with a random targeting algorithm. The ML inspired algorithms are constructed iteratively by estimating which of the remaining target galaxies will be most difficult for the ML methods to accurately estimate redshifts using the previously observed data. This is performed by predicting the expected redshift error and redshift offset (or bias) of all of the remaining target galaxies. We find that the predicted values of bias and error are accurate to better than 10-30 per cent of the true values, even with only limited training sample sizes. We construct a hypothetical follow-up survey and find that some of the ML targeting algorithms are able to obtain the same redshift predictive power with 2-3 times less observing time, as compared to that of the SDSS, or random, target selection algorithms. The reduction in the required follow-up resources could allow for a change to the follow-up strategy, for example by obtaining deeper spectroscopy, which could improve ML redshift estimates for deeper test data. [ABSTRACT FROM AUTHOR]

We present an analysis of anomaly detection for machine learning redshift estimation. Anomaly detection allows the removal of poor training examples, which can adversely influence redshift estimates. Anomalous training examples may be photometric galaxies with incorrect spectroscopic redshifts, or galaxies with one or more poorly measured photometric quantity. We select 2.5 million 'clean' SDSS DR12 galaxies with reliable spectroscopic redshifts, and 6730 'anomalous' galaxies with spectroscopic redshift measurements which are flagged as unreliable. We contaminate the clean base galaxy sample with galaxies with unreliable redshifts and attempt to recover the contaminating galaxies using the Elliptical Envelope technique. We then train four machine learning architectures for redshift analysis on both the contaminated sample and on the preprocessed 'anomaly-removed' sample and measure redshift statistics on a clean validation sample generated without any preprocessing. We find an improvement on all measured statistics of up to 80 percent when training on the anomaly removed sample as compared with training on the contaminated sample for each of the machine learning routines explored. We further describe a method to estimate the contamination fraction of a base data sample. [ABSTRACT FROM AUTHOR]

We present analyses of data augmentation for machine learning redshift estimation. Data augmentation makes a training sample more closely resemble a test sample, if the two base samples differ, in order to improve measured statistics of the test sample.We perform two sets of analyses by selecting 800 000 (1.7 million) Sloan Digital Sky Survey Data Release 8 (Data Release 10) galaxies with spectroscopic redshifts.We construct a base training set by imposing an artificial r-band apparent magnitude cut to select only bright galaxies and then augment this base training set by using simulations and by applying the K-CORRECT package to artificially place training set galaxies at a higher redshift. We obtain redshift estimates for the remaining faint galaxy sample, which are not used during training.We find that data augmentation reduces the error on the recovered redshifts by 40 per cent in both sets of analyses, when compared to the difference in error between the ideal case and the non-augmented case. The outlier fraction is also reduced by at least 10 per cent and up to 80 per cent using data augmentation.We finally quantify how the recovered redshifts degrade as one probes to deeper magnitudes past the artificial magnitude limit of the bright training sample.We find that at all apparent magnitudes explored, the use of data augmentation with tree-based methods provide an estimate of the galaxy redshift with a low value of bias, although the error on the recovered redshifts increases as we probe to deeper magnitudes. These results have applications for surveys which have a spectroscopic training set which forms a biased sample of all photometric galaxies, for example if the spectroscopic detection magnitude limit is shallower than the photometric limit. [ABSTRACT FROM AUTHOR]

We present analyses of data augmentation for machine learning redshift estimation. Data augmentation makes a training sample more closely resemble a test sample, if the two base samples differ, in order to improve measured statistics of the test sample. We perform two sets of analyses by selecting 800 000 (1.7 million) Sloan Digital Sky Survey Data Release 8 (Data Release 10) galaxies with spectroscopic redshifts. We construct a base training set by imposing an artificial r-band apparent magnitude cut to select only bright galaxies and then augment this base training set by using simulations and by applying the K-CORRECT package to artificially place training set galaxies at a higher redshift. We obtain redshift estimates for the remaining faint galaxy sample, which are not used during training. We find that data augmentation reduces the error on the recovered redshifts by 40 per cent in both sets of analyses, when compared to the difference in error between the ideal case and the non-augmented case. The outlier fraction is also reduced by at least 10 per cent and up to 80 per cent using data augmentation. We finally quantify how the recovered redshifts degrade as one probes to deeper magnitudes past the artificial magnitude limit of the bright training sample. We find that at all apparent magnitudes explored, the use of data augmentation with tree-based methods provide an estimate of the galaxy redshift with a low value of bias, although the error on the recovered redshifts increases as we probe to deeper magnitudes. These results have applications for surveys which have a spectroscopic training set which forms a biased sample of all photometric galaxies, for example if the spectroscopic detection magnitude limit is shallower than the photometric limit. [ABSTRACT FROM AUTHOR]

We present an analysis of importance feature selection applied to photometric redshift estimation using the machine learning architecture Decision Trees with the ensemble learning routine ADABOOST (hereafter RDF). We select a list of 85 easily measured (or derived) photometric quantities (or 'features') and spectroscopic redshifts for almost two million galaxies from the Sloan Digital Sky Survey Data Release 10. After identifying which features have the most predictive power, we use standard artificial Neural Networks (aNNs) to show that the addition of these features, in combination with the standard magnitudes and colours, improves the machine learning redshift estimate by 18 percent and decreases the catastrophic outlier rate by 32 percent. We further compare the redshift estimate using RDF with those from two different aNNs, and with photometric redshifts available from the Sloan Digital Sky Survey (SDSS). We find that the RDF requires orders of magnitude less computation time than the aNNs to obtain a machine learning redshift while reducing both the catastrophic outlier rate by up to 43 percent, and the redshift error by up to 25 percent. When compared to the SDSS photometric redshifts, the RDF machine learning redshifts both decreases the standard deviation of residuals scaled by 1/(1+z) by 36?per?cent from 0.066 to 0.041, and decreases the fraction of catastrophic outliers by 57?per?cent from 2.32 to 0.99?per?cent. [ABSTRACT FROM AUTHOR]