Exploring the usage of Probabilistic Neural Networks for Ionospheric electron density estimation

Abstract

A fundamental limitation of traditional Neural Networks (NN) in predictive modelling is their inability to quantify uncertainty in their outputs. In critical applications like positioning systems, understanding the reliability of predictions is critical for constructing confidence intervals, early warning systems, and effectively propagating results. For instance, Precise Point Positioning in satellite navigation heavily relies on accurate error models for ancillary data (orbits, clocks, ionosphere, and troposphere) to compute precise error estimates. In addition, these uncertainty estimates are needed to establish robust protection levels in safety critical applications. To address this challenge, the main objectives of this paper aims at exploring a potential framework capable of providing both point estimates and associated uncertainty measures of ionospheric Vertical Total Electron Content (VTEC). In this context, Probabilistic Neural Networks (PNNs) offer a promising approach to achieve this goal. However, constructing an effective PNN requires meticulous design of hidden and output layers, as well as careful definition of prior and posterior probability distributions for network weights and biases. A key finding of this study is that the uncertainty provided by the PNN model in VTEC estimates may be systematically underestimated. In low-latitude areas, the actual error was observed to be as much as twice the model's estimate. This underestimation is expected to be more pronounced during solar maximum, correlating with increased VTEC values.