Deriving confidence from artificial neural networks for navigation

Artificial Neural Networks (ANNs) are increasingly used in modeling and prediction due to advances in deep learning architecture and training. As ANNs become a viable option for making state or measurement predictions in navigation algorithms, special care must be taken in order to determine the confidence (covariance) of the predictions. A special kind of ANN called a Mixture Density Network (MDN) approximates the prediction distribution with a mixture of distributions such as Gaussian distributions. This allows the MDN to output a full Probability Distribution Function (PDF) rather than just a mean estimate. The prediction mixtures can also be converted into other forms for processing by a filter. In this paper, MDN are trained to model a measurement equation which converts measurements to state estimates represented as a Gaussian Mixture Model (GMM). The GMM distributions are simplified into a single Gaussian distribution for each measurement for inclusion into an EKF. An experiment was conducted using magnetic field measurements and Very Low Frequency (VLF) signals to estimate position using an EKF with measurements from a MDN. This method shows a performance improvement using the MDN covariance estimate to determine measurement noise when compared to using static estimates for measurement noise. Thus, the MDN output allows the modeling and prediction power of ANN to be used in standard navigation filters with minimal changes to the filter.