Parallel kd-Tree Based Approach for Computing the Prediction Horizon Using Wolf’s Method

Algorithm 1

Wolf’s method by computing .

  Program  fet1(, , , )

  Input:

    : data record of scalar quantities;

    : embedding delay;

    : minimal embedding dimension;

    : fixed evolution time;

  Output:

  For each replacement step : (1) : maximal Lyapunov exponent estimation; (2) : evolving step;

  (3) : initial separation between points; (4) : final separation between points.

(1) begin

   /  Initialization.                                    /

(2) Set as the useful size of ;

(3) Set as the number of replacement steps;

(4) Build using (2) as the set of delay vectors;

(5) Compute as the standard deviation of ;

(6) Set scalmn = 0.0125σ as the noise scale;

(7) Set scalmx = 0.05σ as an estimation of the useful length scale;

(8) Set ;

   /  Computing maximal Lyapunox exponent.                         /

(9) Select as the fiducial point;

(10) Search such as ;

(11)  for    to    do

(12)   Set as the initial separation;

(13)   Set as the final separation;

(14)   Set ;

(15)   Print ;

(16)   repeat

(17)    Search such as ;

(18)   until    was a minimum;

(19)   if  a replacement point    was found  then    else  ;

(20)   Set ;

(21) end