A Koopman Operator-Based Prediction Algorithm and its Application to COVID-19 Pandemic

The problem of prediction of behavior of dynamical systems has undergone a paradigm shift in the second half of the 20th century with the discovery of the possibility of chaotic dynamics in simple, physical, dynamical systems for which the laws of evolution do not change in time. The essence of the paradigm is the long term exponential divergence of trajectories. However, that paradigm does not account for another type of unpredictability: the ``Black Swan" event. It also does not account for the fact that short-term prediction is often possible even in systems with exponential divergence. In our framework, the Black Swan type dynamics occurs when an underlying dynamical system suddenly shifts between dynamics of different types. A learning and prediction system should be capable of recognizing the shift in behavior, exemplified by ``confidence loss". In this paradigm, the predictive power is assessed dynamically and confidence level is used to switch between long term prediction and local-in-time prediction. Here we explore the problem of prediction in systems that exhibit such behavior. The mathematical underpinnings of our theory and algorithms are based on an operator-theoretic approach in which the dynamics of the system are embedded into an infinite-dimensional space. We apply the algorithm to a number of case studies including prediction of influenza cases and the COVID-19 pandemic. The results show that the predictive algorithm is robust to perturbations of the available data, induced for example by delays in reporting or sudden increase in cases due to increase in testing capability. This is achieved in an entirely data-driven fashion, with no underlying mathematical model of the disease

On Positive Semidefinite Matrices With Known Null Space

. We show how the zero structure of a basis of the null space of a positive semidenite matrix can be exploited to very accurately compute its Cholesky factorization. We discuss consequences of this result for the solution of (constrained) linear systems and eigenvalue problems. The results are of particular interest if A and the null space basis are sparse. Key words. Positive semidenite matrices, Cholesky factorization, null space basis. AMS subject classications. 65F05, 65F50 1. Introduction. The Cholesky factorization A = R T R, R upper-triangular, exists for any symmetric positive semidenite matrix A. In fact, R is the upper triangular factor of the QR factorization of A 1=2 [11, x10.3]. R can be computed with the well-known algorithm for positive denite matrices. However, zero pivots may appear. As zero pivots come with a zero row/column in the reduced A, a zero pivot implies a zero row in R. To actually compute a numerically stable Cholesky factorization of a positive ..