Spatiotemporal Timeseries Prediction

An application and extension of local modeling to spatiotemporal timeseries.

Previously we have only considered relatively small systems. If $B$ -dim. multivariate timeseries were given, complete states would be embedded into a reconstruction of dimension $D×B$ . Spatiotemporal timeseries could in principle be considered simply be considered in this way but it is far from being ideal.

One of the main reasons is that it does not utilize all available information. All spatially extended physical systems posses a finite speed at which information travels. Therefore the future value of any of the variables depends solely on its past and its immediate spatial neighbors. Instead of trying to reconstruct the state of the whole system into one vector, we limit ourselves to reconstructing small neighborhoods of all points that carry enough information to predict one point one timestep into the future.

Examples

Several example scripts can be found in TimeseriesPrediction/examples.

localmodel_stts
crosspred_stts
STReconstruction

Showcasing Results

Chaotic Barkley Model cross-prediction

Here we cross-predicting the $V$ field from $U$ in the chaotic Barkley model. It is defined by:

$\begin{align} \frac{\partial u }{\partial t} =& \frac{1}{\epsilon} u (1-u)\left(u-\frac{v+b}{a}\right) + \nabla^2 u \\ \frac{\partial v }{\partial t} =& u - v \end{align}$

Embedding parameters are D=30, τ=1, B=0 and training length 2000. The following figure shows the cross-prediction 11 frames into the "future" (i.e. after the training set)

Crossprediction U->V in Barkley .

You can find the script that produced this figure in DynamicalSystems/coolanimations/barkley_crosspred.jl.

Periodic Barkley Model timeseries prediction

Using different parameters in the Barkley model can produce periodic behavior. Here is an example of timeseries prediction with embedding parameters D=2, τ=1, B=2, k=1 and c=200.

Plotting the real evolution, prediction, and error side by side with Ttrain = 1000, p = 200 produces:

Barkley prediction

You can find the script that produced this animation in DynamicalSystems/coolanimations/barkley_stts.jl.