Local false nearest neighbors and dynamical dimensions from observed chaotic data

The time delay reconstruction of the state space of a system from observed scalar data requires a time lag and an integer embedding dimension. The minimum necessary global embedding dimension d₄ may still be larger than the actual dimension of the underlying dynamics d₋. The embedding theorem only guarantees that the attractor of the system is fully unfolded using d₄ greater than 2d₀, with d₀ the fractal attractor dimension. Using the idea of local false nearest neighbors, we discuss methods for determining the integer-valued d₋.