3. Circular Scaling
Society for Industrial and Applied Mathematics eBooks · 2006
This chapter will discuss circular unidimensional scaling (CUS), where the objective is to place the n objects around a closed continuum such that the reconstructed distance between each pair of objects, defined by the minimum length over the two possible paths that join the objects, reflects the given proximities as well as possible. Explicitly, and in analogy with the loss function for linear unidimensional scaling (LUS) in (2.2), we wish to find a set of coordinates, x1,… ,xn, and an (n + 1 )st value, x 0 ≥ | xj − xi | for all 1 ≤ i ≠ j ≤ n, minimizing ∑ i<j ( pij +c−min {| xj − xi |, x0 − | xj − xi | } )2 3.1 or, equivalently, ∑ i