Lecture 7 Nonlinear Recursions Time series prediction HillValley

Lecture 7 Nonlinear Recursions ØTime series prediction ØHill-Valley classification ØLorenz series 數值方法 2008, Applied Mathematics NDHU 1

Nonlinear recursion • Post-nonlinear combination of predecessors • z[t]=tanh(a 1 z[t-1]+ a 2 z[t-2]+…+ a z[t- ])+ e[t], t= , …, N 數值方法 2008, Applied Mathematics NDHU 2

Post-nonlinear Projection tanh 數值方法 2008, Applied Mathematics NDHU y 3

Data generation • z[t]=tanh(a 1 z[t-1]+ a 2 z[t-2]+…+ a z[t- ])+ e[t], t= , …, N tanh 數值方法 2008, Applied Mathematics NDHU 4

Nonlinear Recursion IEEE Trans. on Neural Network Wu JM(2008), MLPotts learning 數值方法 2008, Applied Mathematics NDHU 5

Prediction • Use initial -1 instances to generate the full time series (red) based on estimated postnonlinear filter Post-nonlinear filter 數值方法 2008, Applied Mathematics NDHU 6

Nonlinear recursions: hill-valley • Classify hill and valley UCI Machine Learning Repository 數值方法 2008, Applied Mathematics NDHU 7

Noise-free Hill Valley 數值方法 2008, Applied Mathematics NDHU 8

Noise-free Hill Valley 數值方法 2008, Applied Mathematics NDHU 9

Hill-Valley with noise 數值方法 2008, Applied Mathematics NDHU 10

Hill-Valley with noise 數值方法 2008, Applied Mathematics NDHU 11

Hill-Valley with noise 數值方法 2008, Applied Mathematics NDHU 12

Nonlinear recursion • MLPotts: z[t]=F(z[t-1], z[t-2], …, z[t- ])+ e[t], t= , …, N MLPotts 1 數值方法 2008, Applied Mathematics NDHU 13

MLPotts 1 MLPotts 2 數值方法 2008, Applied Mathematics NDHU 14

Nonlinear Recursion IEEE Trans. on Neural Network Wu JM(2008), MLPotts learning MLPotts 1 (H) 數值方法 2008, Applied Mathematics NDHU 15

Nonlinear Recursion IEEE Trans. on Neural Network Wu JM(2008), MLPotts learning MLPotts 2 (V) 數值方法 2008, Applied Mathematics NDHU 16

Approximating error Green: Original Hill Blue: Approximated Hill Red: Approximating error MLPotts 1 (H) 數值方法 2008, Applied Mathematics NDHU 17

Separable 2 D diagram of Hill-Valley MLPotts 1 (H) Noise-free data set 1 MLPotts 2 (V) Mean Approximating Errors of Two Models 數值方法 2008, Applied Mathematics NDHU 18

Separable 2 D diagram of Hill-Valley MLPotts 1 (H) Noise-free data set 2 MLPotts 2 (V) Mean Approximating Errors of Two Models 數值方法 2008, Applied Mathematics NDHU 19

Separable 2 D diagram of Hill-Valley MLPotts 1 (H) Dataset 1 (Noise) MLPotts 2 (V) Mean Approximating Errors of Two Models 數值方法 2008, Applied Mathematics NDHU 20

Separable 2 D diagram of Hill-Valley MLPotts 1 (H) Dataset 2 (Noise) MLPotts 2 (V) Mean Approximating Errors of Two Models 數值方法 2008, Applied Mathematics NDHU 21

Chaos Time Series • Eric's Home Page 數值方法 2008, Applied Mathematics NDHU 22

Lorenz Attractor 數值方法 2008, Applied Mathematics NDHU 23

Lorenz >> NN=20000; >> [x, y, z] = lorenz(NN); >> plot 3(x, y, z, '. '); 數值方法 2008, Applied Mathematics NDHU 24

數值方法 2008, Applied Mathematics NDHU 25

Nonlinear Recursion IEEE Trans. on Neural Network Wu JM(2008), MLPotts learning MLPotts 數值方法 2008, Applied Mathematics NDHU 26

Prediction • Use initial -1 instances to generate the full time series (red) based on derived MLPotts network MLPotts 數值方法 2008, Applied Mathematics NDHU 27

Rayleigh number ρ=14, σ=10, β=8/3 ρ=15, σ=10, β=8/3 ρ=13, σ=10, β=8/3 ρ=28, σ=10, β=8/3 數值方法 2008, Applied Mathematics NDHU 28