Geometrical techniques for finding the embedding dimension of time series

We present a technique to determine the embedding dimension of deterministic time series. The technique is based mainly on the assumption that if a continuous deterministic time series, with first derivative and underlying model f_S(x_t-1, ..., x _t-m)=x_t is represented on its convenient embedding dimension ℜ^m, then any two neighbouring input observations x_mⁱ and x_m^j, should correspond to similar output observations x_tⁱ and x_t^j. Thus the ratio between the input and output distances of neighbouring observations is used as a criterion to determine the time series embedding dimension. The performance of this technique is illustrated and compared using synthetic time series (Logistic map, Henon map and Mackey Glass function), as well as the Laser data from the Santa Fe competition. Our technique applied to the Laser data, in combination with a Limited Resource Allocating Neural Network (LRAN), has proved to be as successful as the techniques proposed by the winners of the competition on the 100-step prediction of this chaotic time series