Novel Distance Estimation Methods Using “Stochastic Learning on the Line” Strategies

Abstract

In this paper, we consider the problem of Distance Estimation (DE) when the inputs are the $x$ and $y$ coordinates (or equivalently, the latitudinal and longitudinal positions) of the points under consideration. The aim of the problem is to yield an accurate value for the real (road) distance between the points specified by the latter coordinates. 1 This problem has, typically, been tackled by utilizing parametric functions called the “Distance Estimation Functions” (DEFs). The parameters are learned from the training data (i.e., the true road distances) between a subset of the points under consideration. We propose to use Learning Automata (LA)-based strategies to solve the problem. In particular, 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 coordinates of the nodes and the true 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. Traditionally, the parameters of the DEF are determined by minimizing an appropriate “Goodness-of-Fit” (GoF) function. As opposed to this, the ATS uses the current estimate of the distances, the feedback from the Environment, and the set of known distances, to determine the unknown parameters of the DEF. While the goodness-of-fit functions can be used to show that the results are competitive, our research shows that they are rather not necessary to compute the parameters themselves. The results that we have obtained using artificial and real-life data sets demonstrate the power of the scheme, and also validate our hypothesis that we can completely move away from the GoF-based paradigm that has been used for four decades. Based on the latter results, the paper also suggests a completely novel method by which we can extend the traditionally-studied DE problem where the road distances were, typically, estimated using only the $x$ and $y$ coordinates (or equivalently, the latitudinal and longitudinal positions). In such a generalized model, we hypothesize that one can also provide to the system the additional $z$ coordinate, that represents the height or elevation of the subset of nodes and of the cities whose inter-city distance is to be estimated. The results for this generalized model are currently being compiled into a companion paper. 1 [1] This is a typical problem encountered in a Geographic Information System (GIS), or in a GPS. However, unlike the traditional systems where all the inter-city distances are assumed to be stored, in our setting, the distance between any pair of cities is assumed to be computed by merely having access to a small subset of known inter-city distances.