Efficient computation of the correlation dimension from a time series on a LIW computer

In the last years analysis of nonlinear systems using chaos theory has widely increased. Many methods have been proposed for the computation of parameters able to give in a synthetic way informations about the considered system. One of the most used of such parameters is the dimension of the chaotic attractor, called fractal dimension. For practical purposes the correlation dimension ( D2) is often used which is strictly related to the fractal dimension, but much easier to compute using the algorithm proposed by Grassberger and Procaccia. This parameter can be obtained for any real time series, but its computation is very time consuming, then the use of vector or parallel computers can be very convenient. In this work, we propose two versions of this algorithm: the first one for the computation of a single correlation integral (C); the second one optimized to compute in a recursive way several C's in order to evaluate D2. An analysis of the computational kernels of the algorithm is presented and several different approaches are compared. An implementation of the algorithm is shown on the FPS M64/60 LIW computer (38 MFLOPS peak performance). The performance depends on the embedding dimension: we obtain a maximum asymptotic speed of 28 MFLOPS for the basic version and 16 MFLOPS for the recursive one; both versions run at about 12 MFLOPS for the dimensions used in practice. Nevertheless the recursive computation allows a reduction in the time spent for determining D2 of a factor ranging between 3 and 6 for practical applications.