@article{TyFangGonzalezetal.2019,
  author    = {Ty, Alexander J. A. and Fang, Zheng and Gonzalez, Rivver A. and Rozdeba, Paul J. and Abarbanel, Henry D.},
  title     = {Machine learning of time series using time-delay embedding and precision annealing},
  series = {Neural Computation},
  volume    = {31},
  journal   = {Neural Computation},
  number    = {10},
  publisher = {MIT Press},
  address   = {Cambridge},
  issn      = {0899-7667},
  doi       = {10.1162/neco_a_01224},
  pages     = {2004 -- 2024},
  year      = {2019},
  abstract  = {Tasking machine learning to predict segments of a time series requires estimating the parameters of a ML model with input/output pairs from the time series. We borrow two techniques used in statistical data assimilation in order to accomplish this task: time-delay embedding to prepare our input data and precision annealing as a training method. The precision annealing approach identifies the global minimum of the action (-log[P]). In this way, we are able to identify the number of training pairs required to produce good generalizations (predictions) for the time series. We proceed from a scalar time series s(tn);tn=t0+n Delta t and, using methods of nonlinear time series analysis, show how to produce a DE>1-dimensional time-delay embedding space in which the time series has no false neighbors as does the observed s(tn) time series. In that DE-dimensional space, we explore the use of feedforward multilayer perceptrons as network models operating on DE-dimensional input and producing DE-dimensional outputs.},
  language  = {en}
}