Smooth multidimensional scaling for time-indexed dissimilarities

Abstract: We introduce a new variant of multidimensional scaling (MDS) for dissimilarities among observations that are indexed by time. Instead of a set of points in k-dimensional space as in classical MDS, the proposed method produces a smooth trajectory in that space. The key to the proposed continuous-time variant of MDS is to replace the usual double-centered matrix, whose eigendecomposition determines the classical solution, with a smooth function of two variables, which can be estimated by tensor product B-splines. This function is the kernel of an integral operator whose eigendecomposition yields the required solution. We extend the key MDS theorems from the classical, discrete-observation setting to the continuous-time setting of our proposal. The method is illustrated with cursive handwriting data, a time series of components of gross domestic product in Sweden, and historical text data from State of the Union addresses in the United States. This is joint work with Biplab Paul and Karel Hron.