Quantile NARMAX Model: Modelling Uncertainty for Data Based Space Weather Forecasts

Abstract:

Comprehensive forecast of space weather parameters should also include some information about the uncertainty associated with the forecasted value. Not only the prediction of forthcoming space weather but also knowledge about how reliable the forecast is able to facilitate the mitigation of adverse space weather effect. While for models built based on adequate first principles the estimate of the uncertainty is often a straightforward procedure, for data-based empirical forecasting tools this is usually a more complex task. This study is concerned with forecasting the Dst index and fluxes of energetic electrons at GEO, and attempts to show how the thorough systems science and engineering approach enables the modelling of the prediction uncertainty.

Quantile regression (QR) methodology provides an innovatively different perspective for understanding regression relationships from a new angle: It allows for analysing relationships between variables outside of the mean of the data, and enables revealing hidden information that cannot be obtained by means of traditional regression model analysis.

The focus of quantile regression is concerned with the distribution of the data. Taking the simple linear regression as an example, the traditional mean regression fits a model of the form y =a₀ + a₁x₁ + ... + a_mx_m + error, to the mean of the data, whilst quantile regression looks for a curve for a given quantile α that divides the data into two groups (above and below the curve).

The nonlinear autoregressive moving average with exogenous inputs (NARMAX) model, if appropriately trained by using either greedy algorithms (e.g. orthogonal least squares, orthogonal matching pursuit) or lasso types of algorithms, can provide a simple representation for a wide range of real-world data modelling problems.

This study presents a novel quantile NARMAX (Q-NARMAX) model for Dst index and fluxes of energetic electrons at GEO forecast by taking the advantages of both quantile regression and NARMAX models. Results show that Q-NARMAX model provides reliable prediction of the Dst index and GEO fluxes, and meanwhile the proposed method provides a measure of the uncertainty of the predicted values.