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Procrustes Analysis and Stock Markets

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Procrustes Analysis and Stock Markets CS-BIGS 4(2): 93-100 http://legacy.bentley.edu/csbigs/documents/camiz.pdf © 2011 CS-BIGS Procrustes Analysis and Stock Markets Sergio Camiz Sapienza Università di Roma, Italy Jean Jacques Denimal Université des Sciences et Technologies de Lille, France In this paper, Procrustes Analyses are introduced with their various facets and set in an economics framework with an application to stock markets. The time series of the daily indexes of eight major world stock markets are considered for a four-year period (1988-91) and five two-year mobile windows are built to find a common graphical representation in which the evolution of the relations among the stock markets can be appreciated. Both Procrustes and Generalized Procrustes Analyses are applied in order to illuminate this evolution. Introduction In ancient Greek mythology, the bandit Procrustes (“the adjustment is searched among several sets of points. The Stretcher”) was a son of Poseidon with a stronghold on method is largely used both in pattern recognition and in Mount Korydallos, situated on the sacred way between the so-called shape analysis (Dryden and Mardia, 1998) as Athens and Eleusis. There, he had an iron bed in which a first adjustment of more complex transformations, but it he invited every passer-by to spend the night. According may be applied to all situations in which direct to the guest's size with respect to the bed, he would either comparisons among configurations of the same objects stretch him with his smith’s hammer to fit the bed, or under different representations are requested. amputate the excess length. Indeed, nobody ever fit the bed exactly because Procrustes would take advantage of The exploratory analysis of time-series through classical two beds which differed in size. Procrustes continued his methods may be performed by considering the reign of terror until Theseus, travelling to Athens along observations as units and the occasions of observation as the sacred way, “fitted” him to his own bed (Plutarch). variables: its drawback is the very large number of variables in respect of the usually smaller number of units. In data analysis, the Procrustes Analysis (PA) is a method Bry (1995) proposes to transpose the data table, so that that provides the best adjustment of a set of points, called the series represent the variables and the occasions the test cloud, to a given set, called target cloud, according to units. In this way, classical scaling methods, such as transformations that do not change, up to a scale factor, Principal Component Analysis (PCA, Bry, 1995) and its the reciprocal distances among the points of the test variations, may be adopted. This allows to investigate the cloud. Originally proposed by Mosier (1939), its name is overall relations among time-series, based on their due to Hurley and Cattel (1962), and it later underwent correlation matrix, and to detect the influence of the further developments, in particular the Generalized occasions as displayed by their position on the PCA Procrustes Analysis (GPA, Gower, 1975), in which a best principal axes and planes. If some further information is - 94 - Procrustes Analysis and Stock Markets / Camiz & Denimal searched, in particular to identify specific time-periods depending on the metric M, a symmetric definite during which the pattern of correlations may be different M positive matrix. from the overall one, mobile windows on which to apply the same analyses may be adopted. Camiz et al. (2010) Search for the translation a applied this method to a set of yearly time-series of tree- ring width of Notofagus trees of Tierra del Fuego n (Argentina), resulting in a time-series of PCA, whose If we introduce both clouds centroids G p .x and synthesis seemed difficult. The application of Procustes i i i1 Analysis to this case arose as a possible solution. n H pi .zi , it is easy to see that the transformation of In this paper, we aim at describing both PA and GPA i1 principles and we show their application to time-series H, U P(H) s.T.H a minimizes PA when G U , through an example: the evolution of eight international which implies: a G s.T.H . Indeed, from the objective stock-exchange markets, as represented by five tables of function it follows that: daily observations for two years. For this task a pre- treatment is needed, in order to get a set of points in the 2 2 2 space corresponding to the series; for our purposes, we pi . xi ui pi . xi G pi . ui U iI iI iI shall use first a Non-Metric Multidimensional Scaling n (NMMDS, Borg and Groenen, 2005), based on a distance 2 2. pi . xi G,ui U G U matrix derived from the correlation between series, to i1 obtain a graphical display representing the relations between time-series. This will provide us with the clouds 2 where the only member depending on a is G U , of points on which PA may be applied. hence its minimum for G U . The Procrustes Transformation Search for the rescaling s The aim of a Procrustes Analysis (PA) is to best adjust two clouds of points in a geometrical space, that is to adjust a If we replace the found translation a into the objective test cloud to a target cloud as best as possible through a function, this becomes rigid transformation. As rigid transformations we consider n n only translations, rotations, and rescaling, or a 2. 2 composition of these. s . pi . T.zi H 2.s. pi . xi G,T.zi H i1 i1 p Let there be in R two clouds of points, the target cloud whose minimum is reached when its derivative is zero, N x of n points xi and the test cloud Nz of n points zi that is indexed by the same set I (with n card(I) ), and given n the same weights pi assumed to be strictly positive and 2 2s. p . T.z H to sum up to 1. In addition, we assume that R p is given a i i i1 metric represented by a positive definite matrix M. This is n the most general framework: normally, all points have the 2 pi . xi G,T.zi H 0 same weight and the metric M is given by the identity i1 matrix I. hence The PA consists in finding the so-called Procrustes Transformation (PT) P(a,T,s), composed of a translation n p . x G,T.z H a , a R p , a rotation T, T R pp , and a rescaling s, i i i s i1 s R , such that the images of the points of the test n cloud N under P, that is u P(z ) s.T.z a , i I 2 z i i i pi . T.zi H are as close as possible to the points of the target cloud i1 N x in the least-squares sense. The objective function to 2 Since any rotation is a norm-preserving isometry, T does minimize is thus: PA p . x u , with the norm i i i M not influence the denominator and may be removed, iI giving - 95 - Procrustes Analysis and Stock Markets / Camiz & Denimal t V.M.e p n M-orthonormal vectors f R , 1,..,r , pi . xi G,T.zi H s i1 that may be completed to an M-orthonormal basis n 2 p f1, f2,..., f p of R .Indeed, pi . zi H i1 t f,.. f f M f Search for the rotation T tt e...... M V M V M e 0 if Once the found rescaling s is plugged into the objective . 1 else function, it becomes Bourgois (1978) showed that a possible T that gives the 22 pi.. x i u i p i x i G searched optimum is defined by T. f e , 1, r. i I i I 2 n Remark 1 pi. x i G , T .( z i H ) i1 n 2 Let X and Z be two tables with dimensions n, p whose pii. z H i1 rows correspond to the two clouds N x and Nz 2 n respectively. When the metric M of R p is the classical p. x G , T .( z H ) i i i Euclidean one and the weights p / i I are all equal, 2.i1 , i n then the three steps of the transformation, say the search p. z H 2 ii for the rotation T, the translation a, and the rescaling s i1 may be described as (see Borg and Groenen, 2005): simplifying to ~ ~ 1) Centering the tables X and Z , ~ ~ 22 2) Compute the tables product1 Ct X.Z , pi.. x i u i p i x i G i I i I 3) Compute the Singular Value Decomposition of 2 t n C P.. Q , pi. x i G , T .( z i H ) i1 n . 2 with the following results: pii. z H i1 t the rotation T P. Q t ~ ~ t If we center both vectors xi and zi to their respective trace( X.Z. T) the rescaling s ~ ~ t ~ ~ centroids, say xi xi G and zi zi H , the search trace( Z.Z ) for the rotation T is reduced to the maximization of 1 t t the translation a . (X s.Y. T).1n , n n ~ ~ pi . xi ,T.zi . i1 with 1n a vector with all components equal to 1. n Remark 2 ~ t ~ Let us now define two matrices V pi .xi . zi and 2 i1 The objective function PA p .
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