Approximate relations between Manhattan and Euclidean distance regarding Latin hypercube experimental design

Abstract

Among several existing distance measures, perhaps Euclidean distance is the most used tool to calculate distances and then Manhattan distance measure, though which distance measure is suitable depends on the problems. In the field of Design of Experiments (DoEs) especially in simulation domain Latin Hypercube design (LHD) is a well-known approach to find out design points for experiments/simulations. Generally, randomly generated LHDs show poor space-filling property. But space-filling is one of the most required impotent properties for DoE. So researchers search optimal LHD in the sense of space-filling. Some authors considered Euclidean distance measure on the other hand some other authors considered Manhattan distance measure to find out optimal LHD regarding space-filling. It is obvious that both optimal LHDs are not identical. So it is a crucial asked - which measure provided better space-filling? The main problem is that there is no any relation among the distance measures especially Euclidean distance measure and Manhattan distance measure. In this article we have established some relations and bounds between Euclidean distance measures and Manhattan distance measures in perspective of LHD. Finally some experimental results are compared namely maximin LHDs measured by both Euclidean distance measure and Manhattan distance measures by using some proposed transformation techniques.