In this paper we propose a new approach to measure stresses in a capital stored into a certain economic system. Our propositions appeal to statistical theory of shape (STS) and data depth concept. We use among other a pair of thin-plane splines deformation of (proposed by Bookstein) multivariate medians and measures of a shape variability.

We start our considerations with a concept according to which a capital stored in an economic system is connected with an ability of this system to perturb a certain economic space of values (prices, utilities etc.). A flow of the capital between systems is associated with activities of internal and external forces and could be described by an equation similar to the famous Navier–Stockes equation. Stresses in the capital represent the activity of the inner forces and are proportional to the stresses in the space of values.

In this paper we treat companies belonging to sector stock indices as particles of the capital. Using robust methods (functional data depth procedures) we choose a representative particle for each sector index on base of time series of a relative price change to relative volume change ratio analysis. We treat such the representative particle as a landmark – a basic notion of the STS. A set of representative particles considered with respect to change of price and change of volume we treat as a configuration. We apply tools of the STS to analyze configurations of representative companies belonging to the branch indices. In particular we consider changes of a median size-and-shape before and after financial crisis of 2007 year using deformation proposed by Bookstein. We analyze size-and-shape variability before and after the crisis.

We show also certain aspects of the probability distribution of the proposed stress measures for particles behaviors ruled by well known and commonly used in the empirical finance time series models.

References:

Bookstein, F. L. (1989) Principal warps: thin-plate splines and the decompositions of deformations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, 567 – 585.

Kosiorowski, D. (2007). Nonparametric Equity of Two Shapes Test Based on Multivariate Quantile Functional. Bulletin of the International Statistical Institute, 56th Session of the ISI – Lisbon.

Lopez–Pintado, S., Romo J. (2009). On the Concept of Depth for Functional Data. Journal of the American Statistical Association 104(486), 718 – 734.

Mizera, I., Muller, C. H. (2004). Location – Scale Depth (with Discussion and Rejoinder).Journal of the American Statistical Association 99(4), 981 – 989

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