ketsonroberto / diffusion-forecast Goto Github PK

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This code implements the data-driven forecasting algorithm developed by Tyrus Berry, Dimitrios Giannakis, and John Harlim in \'93Nonparametric forecasting of low-dimensional dynamical systems\'94. \'a0This implementation uses the variable bandwidth diffusion maps (VBDM) algorithm developed by Tyrus Berry and John Harlim in "Variable Bandwidth Diffusion Kernels".\
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The main folder contains two script files, "torusExample.m" and "L63Example.m" which reproduce the first two examples of \'93Nonparametric forecasting of low-dimensional dynamical systems\'94. \'a0The folder "ExampleDynamics" contains functions which integrate the models in the examples. \'a0The folder "DiffusionForecast" contains the functions which build the data driven model.\
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DiffusionForecast/BuildModelVBDM.m - Given a time series, this functions builds a data-driven forecast model, including a basis of smooth functions and a forecasting operator projected into the basis. \'a0\
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DiffusionForecast/VBDM.m - Estimates the sampling/invariant measure of the time series and builds a basis of smooth functions on the manifold defined by the data set.\
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BASIC USAGE\
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Given an N-x-n time series "trainingData" which consists of N data points in R^n build the nonparametric model by calling:\
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[basis,ForwardOp,peq,qest,normalizer,delayEmbedding] = BuildModelVBDM(trainingData,M,k,k2,delays);
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Were "M" is the number of basis elements desired (0<M<N), "k" is the number of nearest neighbors used in VBDM (larger may improve results but will require more memory), "k2" is the number of neighbors used in the initial density estimate (larger "k2" gives a smoother basis), and "delays" is an optional vector containing lag lengths to use in a time delay embedding. \
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Given an initial density p(x,0) construct an N-x-1 vector p_i = p(trainingData(i,:),0) then project the density into the basis by the Monte-Carlo integral:\
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c = basis'*(p./qest)/N;\
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To forecast, multiply the coefficients "c" by the matrix "ForwardOp".\
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c = ForwardOp*c;
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This can be repeated "s" times in order to forecast "s" time steps into the future. The "ForwardOp" may not numerically maintain the density normalization, so renormalize the coefficients by:\
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c = c/(normalizer*c);\
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Finally, reconstruct the forecast density by:
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recon = peq.*(basis*c);\

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The vector "recon" will be N-x-1 and recon_i = p(trainingData(i,:),s*tau) represents the density at the future time "s*tau" where "tau" is the time step of the training time series.}