# Model-free prognostication of non-linear time series

## TL;DR

This paper introduces a model-free method for forecasting non-linear time series, such as infectious disease spread, using machine learning and time-lagged analysis.

## Contribution

The novel contribution is a model-independent approach for short-term forecasting of non-linear time series using machine learning and time-lagged features.

## Key findings

- Machine learning using correlation coefficients provided excellent predictions of progression using 80% of the time series as training data.
- Feature-space plots with dynamic calibration revealed sharp spikes in the maximum local Lyapunov exponent during infectious spread peaks.
- Average mutual information over time lags and wave lengths anticipated peaks in new infections.

## Abstract

The COVID-19 pandemic has highlighted the importance of studying the course of infectious progression. Similar needs exist for time series of other origins. While models are commonly devised and fitted to the observed data, we recently demonstrated the feasibility to directly evaluate the noisy non-linear time series that characterize the occurrence. However, for practical utility, analytics alone has limited value. The requirement of forecasting – at least in the short term – needs to be met.

We initially utilized normalized new infections per day (7-day moving average for cases per million inhabitants) from Our World in Data. We then validated our method in unrelated non-linear time series of stock markets and blowfly populations. We studied a novel model-independent time series approach, time lagged analyses, and feature-space plots incorporating the time-lagged data.

1) Machine learning on the basis of correlation coefficient, utilizing about 80% of the time series as training sets, was able to generate excellent predictions for progression. 2) Feature-space plots of normalized new cases versus autocorrelation and average mutual information required a form of dynamic calibration to correct for differences in scale among the axes. With that adjustment, the maximum local Lyapunov exponent displayed sharp spikes concomitantly with peaks of infectious spread. 3) The average mutual information over various time lags and wave lengths displayed divergence and sums of absolute values that were anticipatory to peaks in new infections.

The study of non-linear time series with available techniques for observed complex data can extract characteristics that enable short-range forecasting without the need for model-building. Time-lagged analysis provides one suitable foundation. Among various approaches, machine learning achieved the best prognosticative results.