A Topological Approach to Scaling in Financial Data

Date: October 2017
Authors: Jean de Carufel, Martin Brooks, Michael Stieber, Paul Britton
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There is a large body of work, built on tools developed in mathematics and physics, demonstrating that financial market prices exhibit self-similarity at different scales. In this paper, we explore the use of analytical topology to characterize financial price series. While wavelet and Fourier transforms decompose a signal into sets of wavelets and power spectrum respectively, the approach presented herein decomposes a time series into components of its total variation. This property is naturally suited for the analysis of scaling characteristics in fractals.


Varilets: Additive Decomposition, Topological Total Variation, and Filtering of Scalar Fields

Date: April 2016
Authors: Martin Brooks
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Continuous interpolation of real-valued data is characterized by piecewise monotone functions on a compact metric space. Topological total variation of piecewise monotone function f:X->R is a homeomorphism-invariant generalization of 1D total variation. A varlet basis is a collection of piecewise monotone functions { gi |i = 1…n}, called varlets, such that every linear combination ∑aigi has topological total variation ∑|ai|. A varlet transform for f is a varlet basis for which f=∑αigi. Filtered versions of f result from altering the coefficients αi.