This course will provide a self-contained introduction to some of the actively-researched areas in data-driven analysis of dynamical data. The focus will be on the extraction from data of modes of variability relevant for dimensionality reduction, data-driven modeling and prediction. These lectures will cover guiding theoretical principles as well as address practical aspects and challenges. Along the way, we will introduce and use tools from probability, statistical learning, and nonlinear dynamics, as well as teach the relevant numerical methods. A variety of concrete examples coming from nonlinear dynamics, fluid dynamics, and geophysical applications will serve as illustrations. The course will cover the following topics:
- Basic problems of data representation of time-evolving datasets issued from nonlinear systems
- Principal component analysis and extensions
- Markov matrices techniques (Transfer operator methods, Diffusion Map)
- Dynamic mode decomposition, Koopman modes and other data-driven spectral methods (e.g. multivariate singular spectral analysis, data-adaptive harmonic analysis)