Analysis of stock market indices through multidimensional scaling J. Tenreiro Machado a , Fernando B. Duarte b,⇑ , Gonçalo Monteiro Duarte c a Dept. of Electrical Engineering, Institute of Engineering, Porto, Portugal b Dept. of Mathematics, School of Technology, Viseu, Portugal c Lusofona University, Lisboa, Portugal article info Article history: Available online 9 May 2011 Keywords: Multidimensional scaling Stock market daily values Time-varying correlation Econophysics abstract We propose a graphical method to visualize possible time-varying correlations between fif- teen stock market values. The method is useful for observing stable or emerging clusters of stock markets with similar behaviour. The graphs, originated from applying multidimen- sional scaling techniques (MDS), may also guide the construction of multivariate econo- metric models. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction Economical indexes measure the performance of segments of the stock market and are normally used to benchmark the performance of stock portfolios. This paper proposes a descriptive method which analyses possible correlations/similarities in international stock markets. Its results are expected to guide the design of statistical models aiming to test hypotheses of interest. Ultimately, the method can even lead to the postulation of new hypotheses. The study of the correlation of inter- national stock markets may have different motivations. Economic motivations to identify the main factors which affect the behaviour of stock markets across different exchanges and countries. Statistical motivations to visualize correlations in order to suggest some potentially plausible parameter relations and restrictions.The understanding of such correlations would be helpful to the design good portfolios [16,18]. Bearing these ideas in mind the outline of our paper is as follows. In Section 2 we give the fundamentals of the multidi- mensional scaling (MDS) technique, which is the core of our method, and we discuss the details that are relevant for our specific application. In Section 3 we apply our method for daily data on fifteen stock markets, including major American, Asian/Pacific, and European stock markets. In Section 4 we conclude the paper with some final remarks and potential topics for further research. 2. Multidimensional scaling Generally speaking MDS techniques develop spatial representations of psychological stimuli or other complex objects about which people make judgements (e.g., preference, relatedness), that is they represent each object as a point in a m-dimensional space. What distinguishes MDS from other similar techniques (e.g., factor analysis, cluster analysis) is that in MDS there are no preconceptions about which factors might drive each dimension. Therefore, the only data needed is a measure for the similarity between each possible pair of objects under study. The result is the transformation of the data into similarity measures which can be represented by Euclidean distances in a space of unknown dimensions [4]. The greater 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.04.027 ⇑ Corresponding author. E-mail addresses: jtm@isep.ipp.pt (J.T. Machado), fduarte@estv.ipv.pt (F.B. Duarte), monteiro.duarte@gmail.com (G.M. Duarte). Commun Nonlinear Sci Numer Simulat 16 (2011) 4610–4618 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns