From phenomenological modelling of anomalous diffusion through continuoustime random walks and fractional calculus to correlation analysis of complex systems
This document contains more than one topic, but they are all connected in ei ther physical analogy, analytic/numerical resemblance or because one is a building block of another. The topics are anomalous diffusion, modelling of stylised facts based on an empirical random walker diffusion model...
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Format:  Dissertation 
Language:  English 
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PhilippsUniversität Marburg
2009

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Summary:  This document contains more than one topic, but they are all connected in ei
ther physical analogy, analytic/numerical resemblance or because one is a building
block of another. The topics are anomalous diffusion, modelling of stylised facts
based on an empirical random walker diffusion model and nullhypothesis tests in
time series dataanalysis reusing the same diffusion model. Inbetween these topics
are interrupted by an introduction of new methods for fast production of random
numbers and matrices of certain types. This interruption constitutes the entire
chapter on random numbers that is purely algorithmic and was inspired by the
need of fast random numbers of special types. The sequence of chapters is chrono
logically meaningful in the sense that fast random numbers are needed in the first
topic dealing with continuoustime random walks (CTRWs) and their connection
to fractional diffusion. The contents of the last four chapters were indeed produced
in this sequence, but with some temporal overlap.
While the fast Monte Carlo solution of the time and space fractional diffusion
equation is a nice application that spedup hugely with our new method we were
also interested in CTRWs as a model for certain stylised facts. Without knowing
economists [80] reinvented what physicists had subconsciously used for decades
already. It is the so called stylised fact for which another word can be empirical
truth. A simple example: The diffusion equation gives a probability at a certain
time to find a certain diffusive particle in some position or indicates concentration
of a dye. It is debatable if probability is physical reality. Most importantly, it
does not describe the physical system completely. Instead, the equation describes
only a certain expectation value of interest, where it does not matter if it is of
grains, prices or people which diffuse away. Reality is coded and “averaged” in the
diffusion constant.
Interpreting a CTRW as an abstract microscopic particle motion model it
can solve the time and space fractional diffusion equation. This type of diffusion
equation mimics some types of anomalous diffusion, a name usually given to effects
that cannot be explained by classic stochastic models. In particular not by the
classic diffusion equation. It was recognised only recently, ca. in the mid 1990s, that
the random walk model used here is the abstract particle based counterpart for the
macroscopic time and spacefractional diffusion equation, just like the “classic”
random walk with regular jumps ±∆x solves the classic diffusion equation. Both
equations can be solved in a Monte Carlo fashion with many realisations of walks.
Interpreting the CTRW as a time series model it can serve as a possible null
hypothesis scenario in applications with measurements that behave similarly. It
may be necessary to simulate many nullhypothesis realisations of the system to
give a (probabilistic) answer to what the “outcome” is under the assumption that
the particles, stocks, etc. are not correlated.
Another topic is (random) correlation matrices. These are partly built on the
previously introduced continuoustime random walks and are important in null
hypothesis testing, data analysis and filtering. The main ob jects encountered in
dealing with these matrices are eigenvalues and eigenvectors. The latter are car
ried over to the following topic of mode analysis and application in clustering. The
presented properties of correlation matrices of correlated measurements seem to
be wasted in contemporary methods of clustering with (dis)similarity measures
from time series. Most applications of spectral clustering ignores information and
is not able to distinguish between certain cases. The suggested procedure is sup
posed to identify and separate out clusters by using additional information coded
in the eigenvectors. In addition, random matrix theory can also serve to analyse
microarray data for the extraction of functional genetic groups and it also suggests
an error model. Finally, the last topic on synchronisation analysis of electroen
cephalogram (EEG) data resurrects the eigenvalues and eigenvectors as well as the
mode analysis, but this time of matrices made of synchronisation coefficients of
neurological activity. 

DOI:  https://doi.org/10.17192/z2009.0105 