Z-Sequence: Photometric redshift predictions for galaxy clusters with sequential random k-nearest neighbours

We introduce Z-Sequence, a novel empirical model that utilises photometric measurements of observed galaxies within a specified search radius to estimate the photometric redshift of galaxy clusters. Z-Sequence itself is composed of a machine learning ensemble based on the k-nearest neighbours algorithm. We implement an automated feature selection strategy that iteratively determines appropriate combinations of filters and colours to minimise photometric redshift prediction error. We intend for Z-Sequence to be a standalone technique but it can be combined with cluster finders that do not intrinsically predict redshift, such as our own DEEP-CEE. In this proof-of-concept study we train, fine-tune and test Z-Sequence on publicly available cluster catalogues derived from the Sloan Digital Sky Survey. We determine the photometric redshift prediction error of Z-Sequence via the median value of $|\Delta z|/(1+z)$ (across a photometric redshift range of $0.05 \le \textit{z} \le 0.6$) to be $\sim0.01$ when applying a small search radius. The photometric redshift prediction error for test samples increases by 30-50 per cent when the search radius is enlarged, likely due to line-of-sight interloping galaxies. Eventually, we aim to apply Z-Sequence to upcoming imaging surveys such as the Legacy Survey of Space and Time to provide photometric redshift estimates for large samples of as yet undiscovered and distant clusters.