Incollection,
A sum rule approach to detect complex correlation in time series
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Abstract
A basic problem in the analysis of time series consists in unveiling and
characterizing correlations among the variables at different times.
In practice inmost cases this consists in considering the two point
correlations over a long time series. Often complex properties are related
to the long time behavior of these correlations.
However, in many systems, like for example financial time series, simple
correlations are intrinsically excluded by the arbitrage hypothesis.
This leaves space for subtle complex correlations which are clearly
difficult to detect.
The usual approach is to focus on the pair correlations for
grouped variables like in the problem of volatility clustering.
Also in this case the availability of long time series is fundamental.
This poses another problem because the stationarity hypothesis is not
always appropriate.
Inspired by these problems we introduce a new method to detect complex
correlations in time series of finite size.
The method comes from the Spitzerกวs identity which controls
the extremal values for sums of random variables.
The basic idea is that a deviation from this identity is a sign of
correlations in the variables and it corresponds to a sort of sum
rule for correlations of any extension also in non stationary processes.
We have tested the method which has only four point correlations.
The application to real financial data shows that the method is a practical
tool to detect correlations of any type even in finite time series.
This is usually not possible with the standard statistical tools.
