Beyond Normal: Learning Spatial Density Models of Node Mobility

TL;DR

This work addresses the question of applicability for off-the-shelf mixture density network models for the description of mobile node density over a disk and proposes the use of M\"obius distributions to retain symmetric spatial relations, yet be flexible enough to capture changes as one radially traverses the disk.

Abstract

Learning models of complex spatial density functions, representing the steady-state density of mobile nodes moving on a two-dimensional terrain, can assist in network design and optimization problems, e.g., by accelerating the computation of the density function during a parameter sweep. We address the question of applicability for off-the-shelf mixture density network models for the description of mobile node density over a disk. We propose the use of Möbius distributions to retain symmetric spatial relations, yet be flexible enough to capture changes as one radially traverses the disk. The mixture models for Möbius versus Gaussian distributions are compared and the benefits of choosing Möbius distributions become evident, yet we also observe that learning mixtures of Möbius distributions is a fragile process, when using current tools, compared to learning mixtures of Gaussians.

Figures (9)

• Figure 1: Example of the exact solution for the node densities approximated. The supporting disk, centered at (0,0), has a radius of 1.
• Figure 2: Overview of the deep learning models using mixtures of Möbius versus Gaussian distributions.
• Figure 3: Mixture of three Gaussians (i.e., $K$=3), plotted from a training set of five training points, with MSE=0.0054 and KL=0.1048.
• Figure 4: Mixture of three Möbiuses (i.e., $K$=3), plotted from a training set of five training points, with MSE=0.0022 and KL=0.0219.
• Figure 5: Training set (with five training points) performance over 10 runs, with error bars from minimum to maximum values.