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By utilizing diffusion maps embedding and transition matrix analysis weinvestigate sparse temperature measurement time-series data fromRayleigh--B\'enard convection experiments in a cylindrical container of aspectratio $\Gamma=D/L=0.5$ between its diameter ($D$) and height ($L$). We considerthe two cases of a cylinder at rest and rotating around its cylinder axis. Wefind that the relative amplitude of the large-scale circulation (LSC) and itsorientation inside the container at different points in time are associated toprominent geometric features in the embedding space spanned by the two dominantdiffusion-maps eigenvectors. From this two-dimensional embedding we can measureazimuthal drift and diffusion rates, as well as coherence times of the LSC. Inaddition, we can distinguish from the data clearly the single roll state (SRS),when a single roll extends through the whole cell, from the double roll state(DRS), when two counter-rotating rolls are on top of each other. Based on thisembedding we also build a transition matrix (a discrete transfer operator),whose eigenvectors and eigenvalues reveal typical time scales for the stabilityof the SRS and DRS as well as for the azimuthal drift velocity of the flowstructures inside the cylinder. Thus, the combination of nonlinear dimensionreduction and dynamical systems tools enables to gain insight into turbulentflows without relying on model assumptions.

Diffusion maps embedding and transition matrix analysis of the large-scale flow structure in turbulent Rayleigh--B\'enard convection