To probe the late evolution history of the Universe, we adopt two kinds of optimal basis systems. One of them is constructed by performing the principle component analysis (PCA) and the other is build by taking the multidimensional scaling (MDS) approach. Cosmological observables such as the luminosity distance can be decomposed into these basis systems. These basis are optimized for different kinds of cosmological models that based on different physical assumptions, even for a mixture model of them. Therefore, the so-called feature space that projected from the basis systems is cosmological model independent, and it provide a parameterization for studying and reconstructing the Hubble expansion rate from the supernova luminosity distance and even gamma-ray bursts (GRBs) data with self-calibration. The circular problem when using GRBs as cosmological candles is naturally eliminated in this procedure. By using the Levenberg-Marquardt (LM) technique and the Markov Chain Monte Carlo (MCMC) method, we perform an observational constraint on this kind of parameterization. The data we used include the "joint light-curve analysis" (JLA) data set that consists of $740$ Type Ia supernovae (SNIa) as well as $109$ long gamma-ray bursts with the well-known Amati relation.
