Modeling Quasar Photo-$z$ Distribution and Uncertainty. A Study Based on the Kilo-Degree Survey

Abstract

We aim to determine the most effective approach for estimating uncertainties in quasar photo-$z$ and to evaluate the ability of different models to reconstruct the true redshift distribution under varying data quality. We use photometric magnitudes from the Kilo-Degree Survey Data Release 5 and spectroscopically confirmed quasars from the Dark Energy Spectroscopic Instrument Data Release 1. We compare artificial neural networks (ANNs), Mixture Density Networks (MDNs), and Bayesian Neural Networks (BNNs), both latter combined with Gaussian Mixture Model (GMM) outputs. To assess robustness to observational limitations, we construct four test sets covering all combinations of sources fainter than those in the training sample and missing photometric bands. ANNs show substantial deviations in reconstructing the redshift distribution. MDNs require at least two Gaussian components to achieve accurate reconstruction, with the three-component MDN providing the best performance in this class. BNNs improve results for sources fainter than the training range, yielding a negative log-likelihood (NLL) gain of $0.11$, but reduce performance for brighter data by $0.07$ NLL. Reconstruction remains feasible for either fainter data or missing magnitudes individually; however, their combination leads to pronounced deviations. Unsupervised clustering identifies two dominant degenerate solutions at redshift pairs of $(1.2, 2.3)$ and $(1.6, 2.5)$. Accurate uncertainty modeling is essential for reliable reconstruction of the redshift distribution directly from photo-$z$. BNNs are particularly beneficial for out-of-distribution inference, although at the expense of reduced accuracy for brighter sources. Our methodology enables the identification and removal of degenerate photo-$z$ estimates unsuitable for tomographic analyses.

Figures (4)

• Figure 1: Distributions of test sets over $r$ magnitude. The faint-extrapolation test set includes the top 10% of $r$ values, starting at $r > 22.9$. The random test contains a randomly selected 10% of the objects with $r < 22.9$. Both test sets are comparable in size. The faint-extrapolation set appears more prominent only because its objects are concentrated in a much narrower $r$ range, whereas the random set is spread widely. We further split the both test sets into subsets with complete features and at least one missing feature.
• Figure 2: Spectroscopic (red) and photo-$z$ (black) redshift distributions for ANN (column 1), MDN with 1 mixture component (column 2), MDN with 3 mixture components (column 3), and BNN with 3 mixture components (column 4). Histograms correspond to four subsets: random test data with complete features (row 1), random test data with at least one missing feature (row 2), faint-extrapolation test data with complete features (row 3), and faint-extrapolation test data with at least one missing feature (row 4). We compute the mean squared error multiplied by a factor of $10^3$ ($\mathrm{MSE}^\ast$) for each model with respect to the spectroscopic distributions.
• Figure 3: t-SNE projection of photo-$z$ estimates in the random test data with complete features. Top-left: mean photo-$z$, top-right: photo-$z$ standard deviation, bottom-left: $r$ magnitude, bottom-right: clusters. For each object, we use an array of equally spaced 100 probability density function (PDF) values calculated from model output.
• Figure 4: Spectroscopic and photometric redshift distributions for all clusters.