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  • 2008, Applied Mathematics NDHU*Nonlinear Recursive relationsNonlinear recursive relationsTime series predictionHill-Valley classificationLorenz series

    Neural Recursions

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Nonlinear recursionPost-nonlinear combination of predecessorsz[t]=tanh(a1z[t-1]+ a2z[t-2]++ az[t-])+ e[t], t= ,,N

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Post-nonlinear Projectiontanhy

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Data generation

    z[t]=tanh(a1z[t-1]+ a2z[t-2]++ az[t-])+ e[t], t= ,,Ntanh

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Nonlinear RecursionLM learning

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*PredictionUse initial -1 instances to generate the full time series (red) based on estimated post-nonlinear filterPost-nonlinear filter

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*UCI Machine Learning RepositoryNonlinear recursions: hill-valley Classify hill and valley

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Noise-free Hill Valley

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Noise-free Hill Valley

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Hill-Valley with noise

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Hill-Valley with noise

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Hill-Valley with noise

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Nonlinear recursion

    LM_RBF: z[t]=F(z[t-1], z[t-2], ,z[t-])+ e[t], t= ,,NLM_RBF 1

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*LM_RBF 1LM_RBF 2

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Nonlinear RecursionLM learning MLPotts 1 (H)

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Nonlinear RecursionLM learning MLPotts 2 (V)

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Approximating errorLM_RBF 1 (H)Green: Original HillBlue: Approximated HillRed: Approximating error

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Approximating errorLM_RBF 2 (V)Green: Original HillBlue: Approximated HillRed: Approximating error

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Separable 2D diagram of Hill-ValleyNoise-free data set 1LM_RBF 2 (V)LM_RBF 1 (H)Mean Approximating Errors ofTwo Models

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Separable 2D diagram of Hill-ValleyNoise-free data set 2Mean Approximating Errors ofTwo ModelsLM_RBF 2 (V)LM_RBF 1 (H)

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Separable 2D diagram of Hill-ValleyDataset 1 (Noise)Mean Approximating Errors ofTwo ModelsLM_RBF 2 (V)LM_RBF 1 (H)

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Separable 2D diagram of Hill-ValleyDataset 2 (Noise)Mean Approximating Errors ofTwo ModelsLM_RBF 2 (V)LM_RBF 1 (H)

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Chaos Time SeriesEric's Home Page

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Lorenz Attractor

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Lorenz>> NN=20000;>> [x,y,z] = lorenz(NN);>> plot3(x,y,z,'.');

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Nonlinear RecursionNeural nonlinear recursive relations

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*PredictionUse initial -1 instances to generate the full time series (red) based on derived neural recursions

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*=14, =10, =8/3=13, =10, =8/3 =15, =10, =8/3 =28, =10, =8/3 Rayleigh number

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Sunspot time seriessunpot dataload sunspot_train.mat;n=length(Y);plot(1712:1712+n-1,Y);

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Sunspot time seriessunpot data (testing)load sunspot_test_1.mat;n=length(Y);plot(1952:1952+n-1,Y);

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*S[1]S[2]S[3]..S[-1]S[]..S[t-]S[t- +1]..S[t-1]S[t]S[t+1]...x[t]y[t]x[-1]y[-1]Time-series 2 paired data

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*aMa2a1xNetworktanhtanhtanh.1b2b1bM1r1r2rMrM+1y

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Paired data : tau=2x=[];len=3;for i=1:n-len p=Y(i:i+len-1)'; x=[x p]; y(i)=Y(i+len);endfigure;plot3(x(2,:),x(1,:),y,'.');hold on;yx

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Learning and predictionM=input('keyin the number of hidden units:');[a,b,r]=learn_MLP(x',y',M);y=eval_MLP2(x,r,a,b,M);hold on;plot(1712+2:1712+2+length(y)-1,y,'r')

    MSE for training data 0.005029ME for training data 0.052191K

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Prediction

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*SunspotSource codes

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Paired data: tau=5x=[];len=5;for i=1:n-len p=Y(i:i+len-1)'; x=[x p]; y(i)=Y(i+len);end

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*Learning and predictionM=input('keyin the number of hidden units:');[a,b,r]=learn_MLP(x',y',M);y=eval_MLP2(x,r,a,b,M);hold on;plot(1712+2:1712+2+length(y)-1,y,'r')

    ???

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*ExampleRepeat numerical experiments in slide #34 with

    Summarize your numerical results in a table

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*RBFy

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU*addpath('..\..\..\Methods\FunAppr\LM_RBF');% x : Nxd% y : 1xNN=200;d=2;x=rand(N,d)*2-1;y=tanh(-x(:,1).^2-x(:,2).^2);eval(['load data\sunspot_train.mat']); %([1 2 4 6 7 10],:)tx = X'; ty = Y;Net=LM_RBF(tx,ty);[yhat,D]=evaRBF(tx,Net);mean((ty-yhat).^2)/2

    2008, Applied Mathematics NDHU

  • 2008, Applied Mathematics NDHU* eval(['load data\sunspot_test_1.mat']); tex1 = X'; tey1 = Y; [yhat,D]=evaRBF(tex1,Net); mean((tey1-yhat).^2)/2One_step_look_ahead prediction

    2008, Applied Mathematics NDHU