Machine Learning of Time Series Using Time-Delay Embedding and Precision Annealing

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 ($$). In this way, we are able to identify the number of training pairs required to produce good generalizations (predictions) for the time series. We proceed from a scalar time series $$ and, using methods of nonlinear time series analysis, show how to produce a $$-dimensional time-delay embedding space in which the time series has no false neighbors as does the observed $$ time series. In that $$-dimensional space, we explore the use of feedforward multilayer perceptrons as network models operating on $$-dimensional input and producing $$-dimensional outputs.