The visualization of objects within or beyond a turbulent medium is hampered by the aberrations the medium induces in the wavefront. The sharpness of an image is maximal when the incoming wavefront is flat, with aberrated wavefronts yielding distorted images of limited utility. In the particular case of astronomy, the flat wavefront of the light from a distant target is aberrated as it moves through the atmosphere, before reaching our telescopes. The field of Adaptive Optics (AO) is dedicated to the correction of these aberrations, with the goal of enabling the imaging of objects through turbulent media with as much detail as possible. Due to a delay inherent to the AO control loop, each control input is used to correct an aberration slightly ahead in time, which means that, for the correction to be effective, the controller must be equipped with predictive capabilities. Prediction demands a model, and fortunately, the dynamics of atmospheric turbulence can be effectively modelled using a linear system. This makes the Kalman filter, the statistically optimal state estimator for linear systems, a natural choice for prediction. However, the steady-state Kalman filter requires a solution to the computationally intensive Discrete-time Algebraic Riccati Equation (DARE), which precludes its application to large-scale systems. Furthermore, the Kalman filter assumes a precise model with particular properties is available, which often is not the case in practice. The advent of extremely large telescopes demands prediction algorithms that can handle likewise extremely large system dimensions, which means that application of the steady-state Kalman filter, in its conventional form, is unfeasible. This thesis proposes a data-driven approach to Kalman filtering that exploits the intuitive sparsity pattern of the matrices involved in the modelling of an AO system. The developed algorithm replaces the DARE and, with knowledge of the system matrices, estimates the Kalman gain from measurement data, with the exploitation of sparsity providing a substantial drop in complexity when compared to the DARE, and with its data-driven nature inherently compensating, to a certain extent, for modelling errors. This thesis further proposes a two-stage approach to reduce the burden of the online prediction operation. While the measurement and state-transition matrices of the AO system are sparse, the Kalman gain is generally dense, consequentially slowing down prediction, which should accompany the sampling period of the loop. Our proposal splits prediction into a sparse stage for prediction of local structures in the wavefront phase, and a low-dimensional dense stage for prediction of the remaining low-frequency aberrations. The two-stage predictor is inherently suboptimal, but allows for quicker prediction and identification for extremely large systems. Via tuning, the user strikes a balance between performance and execution time.