Reducing the Dimensionality of Spatio-Temporal Data in Rayleigh- B\'{e}nard Convection using Homology {\&} Karhunen- Lo\`{e}ve Decomposition

We present two different approaches to obtain a reduced dynamical description of spatio-temporal chaos in Rayleigh B\'{e}nard convection experiments. Computational Homology, a topological characterization technique, and Karhunen- Lo\`{e}ve (KL) decomposition are applied to time series of shadowgraph images. Homology computations for each image produce a set of non-negative integers called Betti numbers defining different characteristic topological properties of convective flows. Quantitative information is obtained from the probability distributions constructed from a time series of Betti numbers to identify different spatio-temporal states at different control parameters in experiments. For comparison, analogous information content at the same parameter values is captured by normalized eigenvalue spectra obtained by KL decomposition of shadowgraph images. We discuss strengths and weaknesses of these methods for characterizing spatio-temporal dynamics.