GPz: non-stationary sparse Gaussian processes for heteroscedastic uncertainty estimation in photometric redshifts

Abstract

The next generation of cosmology experiments will be required to use photometric redshifts rather than spectroscopic redshifts. Obtaining accurate and well-characterized photometric redshift distributions is therefore critical for Euclid, the Large Synoptic Survey Telescope and the Square Kilometre Array. However, determining accurate variance predictions alongside single point estimates is crucial, as they can be used to optimize the sample of galaxies for the specific experiment (e.g. weak lensing, baryon acoustic oscillations, supernovae), trading off between completeness and reliability in the galaxy sample. The various sources of uncertainty in measurements of the photometry and redshifts put a lower bound on the accuracy that any model can hope to achieve. The intrinsic uncertainty associated with estimates is often non-uniform and input-dependent, commonly known in statistics as heteroscedastic noise. However, existing approaches are susceptible to outliers and do not take into account variance induced by non-uniform data density and in most cases require manual tuning of many parameters. In this paper, we present a Bayesian machine learning approach that jointly optimizes the model with respect to both the predictive mean and variance we refer to as Gaussian processes for photometric redshifts (GPz). The predictive variance of the model takes into account both the variance due to data density and photometric noise. Using the SDSS DR12 data, we show that our approach substantially outperforms other machine learning methods for photo-z estimation and their associated variance, such as tpz and annz2. We provide a matlab and python implementations that are available to download at https://github.com/OxfordML/GPz.

Using Sparse Gaussian Processes for Predicting Robust Inertial Confinement Fusion Implosion Yields

“…A higher m corresponds to higher model complexity and longer training times. Figure 2 shows algorithm performance as a function of m; best performance is achieved for m ≈ 10 − 100, in line with findings in [6], [7].…”

“…GPz is a sparse Gaussian process based code, a fast and a scalable approximation of a full Gaussian Process [22], with the added feature of being able to produce input-dependent variance estimations (heteroscedastic noise). For the full details of the algorithm see [6], [7], [23], but we summarise the main details here. The model assumes that the probability of the observing a target variable y given the vector input x is p(y|x) = N (µ(x), σ(x) 2 ).…”

“…Unless otherwise stated, we use the settings in Table I (see [6], [7] for precise definitions and interpretations). GPz requires very little fine-tuning.…”

“…In the ICF case, we can also set particular parts of parameter space of interest e.g. areas around a cliff, parts of parameter [7] for the different options) normalize True Pre-process the input by subtracting the means and dividing by the standard deviations joint True Jointly learn a prior linear mean-function (learn the function means and variances jointly) heteroscedastic True…”

Estimating redshift distributions using hierarchical logistic Gaussian processes

“…Gaussian processes are a popular machine learning method in cosmology that was used for example for machine learning-based photometric redshift estimation (e.g. Almosallam et al 2016) or to interpolate, and smooth, redshift histograms obtained using cross-correlation measurements (Johnson et al 2017). Our method differs from these previous applications of Gaussian processes, as we use the logistic Gaussian process as a prior to the shape of the logarithm of the redshift distribution and not as a regression model.…”