# Machine Learning of Time Series Using Time-delay Embedding and Precision Annealing

## TL;DR

This paper introduces a method combining time-delay embedding and precision annealing to improve machine learning predictions of time series, ensuring better generalization and identifying optimal training data requirements.

## Contribution

It presents a novel approach that integrates time-delay embedding with a precision annealing training method for enhanced time series prediction.

## Key findings

- Effective identification of training data needed for good predictions
- Use of nonlinear time series analysis to create embedding spaces
- Application of multi-layer perceptrons in high-dimensional embedding space

## 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. Using the equivalence between statistical data assimilation and supervised machine learning, we revisit this task. The training method for the machine utilizes a precision annealing approach to identifying 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 $s(t_n); t_n = t_0 + n \Delta t$ and using methods of nonlinear time series analysis show how to produce a $D_E > 1$ dimensional time delay embedding space in which the time series has no false neighbors as does the observed $s(t_n)$ time series. In that $D_E$-dimensional space we explore the use of feed forward multi-layer perceptrons as network models operating on $D_E$-dimensional input and producing $D_E$-dimensional outputs.