Fig. 2: Illustration of phase space reconstruction (PSR).

A Illustration of how downwards projection of system states can result in False Nearest Neighbours (FNN). In the (top) three-dimensional space the red and green states (points) are closest. When projected down to a two- and then a one-dimensional space, the mustard yellow and blue states ‘falsely’ appear to be nearer to the red and green states. The inverse process, projection from 1 to 2 dimensions and then from 2 to 3 dimensions reveals, which neighbouring points are ‘true’ and which ones are ‘false’ (adapted from Boker, 1996). B Illustration of Phase Space Reconstruction and how time-delayed copies of a measured scalar sequence or times series can be employed as surrogate data series to reconstruct an n-dimensional phase space that is isomorphic to the behavioural system’s true phase space. Here a three-dimensional phase space is reconstructed for illustration purposes, although a six-dimensional space was employed to analyse the movement data collected for the current study. The coloured states in the bottom surrogate time-series, x(t + 2τ), and in the two- and three-dimensional phase spaces are included to illustrate how the neighbours a state has can change as the dimension of a reconstructed phase space is increased. The dark and light blue points, for example, do not constitute neighbours in the one-dimensional time-series, but do constitute neighbours in both two and three-dimensional space. Conversely, the dark and light magenta points appear to be neighbours in two-dimensional space, but not in three-dimensional space. In phase space reconstruction, surrogate dimensions are added until %FNN = 0. See text for more details.