Using Artificial Neural Networks and Support Vector Regression to Model the Lyapunov Exponent Adam Maus

Nonlinear Prediction Largest Lyapunov Exponent (LE) Measure of long term predictability Models used to reconstruct LE Artificial Neural Network Support Vector Regression Henon Map (HM) Henon Map’s LE = .42093 Strange Attractor of Hénon Map

Artificial Neural Network Single Layer Feed-Forward Architecture 8 neurons and 3 dimensions Trained using next step prediction Weights updated using a variant of simulated annealing 400 training points 1 million training epochs

Support Vector Regression Global Solution to a Convex Programming Problem Uses only a Subset of Points Points outside of ɛ-tube thrown out User-Defined Parameters C - control flatness of the output function ɛ - controls size of the tube kernel function and its parameters Training dataset size Many toolboxes available LibSVM Toolbox s = length of w vector K = kernel function Toolbox Available At: http://tinyurl.com/sxnse Picture: http://tinyurl.com/6pscdr

Dynamic and Attractor Reconstruction Results Artificial Neural Network LE = .38431 Mean Square Error = 4.65 x 10 Support Vector Regression LE = .41288 Weighted Mean Square Error = 4.69 x 10 -5 -5 Strange Attractor of Neural Network Mean Square Error Weighted Mean Square Error c = length of time series Strange Attractor of Support Vector Regression

Conclusions Hypothesis Dynamic and Attractor Reconstruction are correlated Neural Networks Perform proper reconstruction given adequate training Support Vector Regression Performs proper reconstruction given that we weight the model so that it replicates the first points more due to the chaotic nature of the data. Strange Attractor of the Logistic Map (LM) Actual NN SVR LM .69526 .63744 .68923 DHM .37038 .35030 .35550 HM .42093 .38431 .41288 Strange Attractor of the delayed Hénon map (DHM)