The ability to compute models that correctly predict the trajectories of a nonlinear system can become a significant challenge in systems and control. The introduction of Koopman operator theory helped to deal with this challenge. The Koopman operator is a composition operator that globally describes a nonlinear system in an infinite-dimensional linear framework. To implement this theory, the usual approach is to approximate the Koopman operator through data-driven methods. These algorithms use measurements of the nonlinear system to compute the approximated operator. Generally, noise can be present in real-world scenarios. Noisy measurements can have a considerable deteriorating effect on the data-driven approximation of Koopman operators. The approximation of this operator in presence of noisy training data is a necessary step for its implementation to a wider spectrum of real-world applications. Many robust numerical methods were designed to solve this issue. Koopman subspace identification (KSI) is a promising approach. As the name suggests, this algorithm employs subspace identification modeling to compute the matrix approximation of the Koopman operator. In this work, we test KSI against other state-of-the-art techniques. Additionally, we improve its performance in predicting the state trajectories of the nonlinear system in presence of noisy measurements. To this end, we propose a reducing-order routine that computes the most robust model against measurement noise. Furthermore, a randomized singular value decomposition is adopted to reduce computational times. The improved KSI is then compared against the other state-of-the-art algorithms in the presence of noisy data sets. We will show that the upgraded KSI outperforms most of the other techniques.