Abstract
We present a method for both cross-estimation and iterated time series prediction of spatio-temporal dynamics based on local modelling and dimension reduction techniques. Assuming homogeneity of the underlying dynamics, we construct delay coordinates of local states and then further reduce their dimensionality through Principle Component Analysis. The prediction uses nearest neighbour methods in the space of dimension reduced states to either cross-estimate or iteratively predict the future of a given frame. The effectiveness of this approach is shown for (noisy) data from a (cubic) Barkley model, the Bueno-Orovio–Cherry–Fenton model, and the Kuramoto–Sivashinsky model.
Highlights
In many experiments, some variables of the system are more observable than others
We shall use local modelling by selecting for each local delay coordinate vector similar vectors from a training data set whose relations to other observables and/or future temporal evolutions are known and can be exploited for cross-estimation or time series prediction
The KS model has previously been used by Pathak et al (2018) for evaluating the prediction performance of some reservoir computing methods. These authors reported for L = 22 and L = 200 prediction horizons of ≈ 3 t (Fig.2 in Pathak et al (2018)) when using a reservoir network and ≈ 4 t (Fig.6a in Pathak et al (2018) for RMSE threshold values between 0.08 and 0.09, which corresponds to our criterium of Normalized Mean Squared Error (NRMSE) = 0.1) for 64 reservoirs running in parallel
Summary
Some variables of the system are more observable than others. If the underlying dynamics is deterministic, in general, the observable of interest is nonlinearly related to other variables of the system which might be more accessible. In such cases, one may try to estimate any observable which is difficult to measure.
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