Patrice Abry

Title :
Fractal Connectivity in Multivariate Long Memory?

Abstract :
It is common nowadays that complex or large systems are 
monitored via large size array of sensors, hence yielding 
multi-component (or multivariate) time series for the analysis.
Such data, for a large number of applications of possibly 
very different natures producing such data, are often 
characterized by a scaling (or scale invariance) property, 
where scaling means that the dynamics underlying the time course 
of the data series develops with no preferred scale of time, or frequency, or period.

When measured on each sensor, the scaling parameters that 
quantify scale invariance can be found close one to another 
or very different, hence yielding natural questions such as~: are all 
measured scaling parameters equal,  up to statistical fluctuations 
or are they really different~? And even, more generally, is the scaling 
seen on all sensors caused by a unique and same phenomenon in the systems 
or are they different mechanisms at work producing different form of scaling 
properties that can be mixed up on the sensors~?

To investigate such issues,  "fractal connectivity" (cf. Achard et al.)  is 
a particular model, within the framework of long memory multivariate stochastic 
processes, that has been proposed to account for the "simplest" situation where 
all the scaling properties seen on each components are related one to the others.
It assumes that the low frequencies (coarse scales) of the interspectrum of each 
pair of process components are determined by the autospectra of the components. The 
underlying intuition is that long memories in each components are likely to arise 
from a same and single mechanism.

The present contribution aims at defining and characterizing a statistical procedure for 
testing actual fractal connectivity amongst data.
The test is based on Fisher's Z transform and Pearson correlation coefficient, and 
anchored in a wavelet framework. Its performance are analyzed theoretically and validated on 
synthetic data. Its usefulness is illustrated on the analysis of Internet traffic 
Packet and Byte count time series.

Further research directions will be discussed:
- Non Gaussian data,
- Long range dependence un-mixing,
- Sensor resolution heterogeneity (in time or space)