Multiscale Statistics for Evolving Complex Systems

Research projects


The student projects under my supervision form integrated parts of
on-going and developing research projects. Spanning disciplines,
departments and scientific institutions, these projects have
received substantial funding from various agencies. My involvement
ranges from contributing as a consultant to active member of the
founding team to initiating and leading the effort.

Ad hoc networking (2003)

With the prevalence of digital information, the dependence of
today's industrialized societies on networking infrastructure is
growing to a critical degree. For one, it aggravates the digital
divide by limiting digital communication to regions where this
infrastructure is technically and economically feasible. Equally
important, it renders economies and societies vulnerable to
disasters and attacks.

The vision of the Safari project is to provide network
connectivity and basic network services in a self-organizing
fashion, using available infrastructure only when available.
At the core of Safari stands a highly scalable architecture which
combines an adaptive, proximity-based self-organization with
hierarchical routing schemes and a distributed algorithm to
provide locality awareness. Current research interests include
information diffusion and data delivery, inference of mobility and
topology from limited local observation. Modeling approaches draw
from branching processes, percolation theory and random geometric
graphs. Several students and senior personnel are involved in this
larger project.

Mapping and simulating the Internet (2005)

Synthetic internet topologies and graphs are indispensable in
order to test and design certain internet applications. To be
realistic such simulations need to capture crucial spatial
properties such as a hierarchy of clusters and inhomogeneous
spatial distributions on a large range of scales. Extremely large
data sets require novel hierarchical versions of classical
methodologies such as model-based clustering and clustered point
processes. Again, several students and senior personnel work on
this project.

Statistical Inference of secret keys (2005)

Secret keys lie at the basis of providing privacy in digital
communication. As is appears, processing times of guesses of the
key can leak a part of the secret, if measured accurately enough.
The challenge of this project is to perform such measurements
across the internet. To deal with noise of multiple magnitude as
compared to the measured signal, robust quantile estimation
procedures are developed.

Traffic modeling (2000)

In networking, the importance of multiscale properties becomes
apparent in various ways. Intuitively, hierarchies are omnipresent
in networks which may be one of many reasons for the strong
scaling properties of traffic dynamics. More practically, network
control entities and protocols operate on different characteristic
time scales. Queues for instance fill up typically over a
characteristic duration which depends on service speed, buffer
size but also traffic variability. Ongoing research concerns
network weather tools and algorithms for high bandwidth data
transfer, both based on inference from opportunistic limited
observation and designed, optimal sampling or probing, involving
students and postdocs.

Multiscale Stochastic Processes (2002)

Stochastic processes with scaling properties have for long been
recognized as parsimonious models for complex observations such as
financial data, fully developed turbulence or internet traffic.
Seeking models with rich scaling structure, yet desirable
statistical properties such as stationary increments let to the
conception of Infinitely Divisible Cascades and the Products of
Stationary Processes. Current interests concern extensions to
multivariate and spatial-temporal settings, in particular
theoretical aspects such as convergence in regularity spaces and
estimation procedures.

Default dependence (since summer 2005)

In risk management the standard methodology for modeling default
dependence employs factor models and copulas. In a nutshell,
default can be modeled within a structural framework as the first
time the value of a firm crosses some barrier. For pricing
multi-name products, such as synthetic collateralized debt
obligations (CDO), a copula approach is usually employed. The
industry standard is the normal copula. The marginal distributions
are calibrated to match the term structure of individual credit
default swap prices and the correlation matrix is usually
estimated using a single factor equity index model. The Wall
Street Journal online on September 12, 2005, spells out a clear
warning for the pitfalls of a shortsighted application of this
standard pricing technologies for prediction, elaborating on the
GM and Ford case of this year.

The development of this area of research is ongoing.

Further Reading on Multifractals on this web-site