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

2. 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 =
Strange Attractor of Hénon Map

3. 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

4. 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: Picture:

5. Dynamic and Attractor Reconstruction
Results
Artificial Neural Network
LE =
Mean Square Error = 4.65 x 10
Support Vector Regression
LE =
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

6. 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)