Functional Multidimensional Scaling

Abstract

Multidimensional scaling is an important component in analyzing proximity (similarity or dissimilarity) between objects and plays a key role in creating low-dimensional visualizations of objects. Regardless of the progress in this area, traditional solutions of multidimensional scaling problems are inapplicable to the proximity which change in time. In this dissertation, we focus on dissimilarity instead of similarity. Motivated by the studies of functional data analysis, we extend the current multidimensional scaling techniques and propose a functional method to obtain lower-dimensional smooth representations in terms of time-varying dissimilarities. This method incorporates the smoothness approach of functional data analysis by using cubic B-spline basis functions. The model is also designed to arrive at optimal representations such that dissimilarities evaluated by estimated representations are almost the same as original dissimilarities of objects in a low dimension which is easier for people to recognize. We verify the feasibility of the model by running simulations, as well as using the closing prices of the S&P 500 stocks as a real case to analyze their dissimilarities. This case study reconstructs the 500 stocks with this functional multidimensional scaling method and provides us a good visualization on a 2D map for the 500 stocks so that we can see how their dissimilarities change smoothly in each month of the year 2018. Following the analysis of all of the 500 stocks, the cluster analysis of the first 15 stocks is displayed based on some conditions so that it helps us see how the stocks move from month to month and offers a new tool to cluster the stocks in the future.