On Using “Stochastic Learning on the Line” to Design Novel Distance Estimation Methods for Three-Dimensional Environments

Abstract

We consider the unsolved problem of Distance Estimation (DE) when the inputs are the x and y coordinates (i.e., the latitudinal and longitudinal positions) of the points under consideration, and the elevation/altitudes of the points specified, for example, in terms of their z coordinates (3DDE). The aim of the problem is to yield an accurate value for the real (road) distance between the points specified by all the three coordinates of the cities in question (This is a typical problem encountered in a GISs and GPSs.). In our setting, the distance between any pair of cities is assumed to be computed by merely having access to the coordinates and known inter-city distances of a small subset of the cities, where these are also specified in terms of their 3D coordinates. The 2D variant of the problem has, typically, been tackled by utilizing parametric functions called “Distance Estimation Functions” (DEFs). To solve the 3D problem, we resort to the Adaptive Tertiary Search (ATS) strategy, proposed by Oommen et al., to affect the learning. By utilizing the information provided in the 3D coordinates of the nodes and the true road distances from this subset, we propose a scheme to estimate the inter-nodal distances. In this regard, we use the ATS strategy to calculate the best parameters for the DEF. While “Goodness-of-Fit” (GoF) functions can be used to show that the results are competitive, we show that they are rather not necessary to compute the parameters. Our results demonstrate the power of the scheme, even though we completely move away from the traditional GoF-based paradigm that has been used for four decades. Our results conclude that the 3DDE yields results that are far superior to those obtained by the corresponding 2DDE.