Conferencias

In this talk I will consider critical percolation on the triangular lattice, which is simply the random black or white coloring of a regular hexagonal tiling of the plane. Scaling the diameter of the hexagons to zero and focusing on the boundaries between black and white clusters reveals a complex geometric structure, with the appearance of conformally invariant, random, fractal curves. In recent years, substantial progress has been made in understanding this scaling limit in terms of the Stochastic Loewner Evolution with parameter 6.
I will describe some of this progress, including work done in collaboration with C. M. Newman.

We will present a model of language evolution based on a dynamics of probability measures on finite sets of grammars.
Such models show through bifurcations, transitions between different situations of pure grammars reminiscent of the transitions historically observed in several languages.

A discriminant rule allows allocating an observation to a certain group, depending on the characteristics of the observation. An example from medicine is where one wishes to allocate a patient in the group of persons having a certain disease, or not having the disease. The outcome of the classification rule will depend here on the patients' age, sex, results on several medical tests... Classification rules are constructed from a so-called training sample, being a set of observations for which the group-membership is known. Aim is to make predictions for new observations, with an unobserved group membership.
If the training sample comes from a mixture of two normal distributions, then it is well known that linear or quadratic discriminant analysis is optimal. But, we can expect that for many data configurations such a normal model hypothesis will not be valid. In particular, there will be the risk of presence of outliers in the training sample. When using a non robust classification rule, outliers may lead to a corrupted classification rule, leading to poor prediction rules for new observations with unknown group membership.
The performance of a classification rule is typically measured by the error rate, being the percentage of incorrectly predicted observations. Aim is to study the sensitivity of the classification rule with respect to the observations in the training sample. Such a robustness study has already been performed for linear and quadratic discrimination (Croux and Dehon 2001, Croux and Joossens 2005). Robustness is measured here by computing the influence that observations in the training sample have on the error rate of the classification rule, merely than by measuring parameter estimates' sensitivity, making this approach different from previous studies.
Besides a classification rule being robust, we would also like it to be efficient, where efficiency is measured by the closeness of the classifiers' error rate to the lowest possible error rate one could get, the Bayes error rate. For parameter estimates there is a close link between the influence function of an estimator and its efficiency. Preliminary calculations indicate, surprisingly, that in the setting of discriminant anallysis the second order influence function of the error rate needs to be used to compute the classification efficiency. The standard first order influence function cancels out here. The classification efficiency of robust logistic discriminantion (e.g. Croux and Haesbroek 2003) will be studied in more detail.
References:
Croux, C., and Joossens, K. (2005), "Influence of Observations on the Misclassification Probability in Quadratic Discriminant Analysis", Journal of Multivariate Analysis, 96, 384-403.
Croux, C., and Dehon, C. (2001), "Robust Linear Discriminant Analysis using S-estimators", The Canadian Journal of Statistics, 29, 473-492.
Croux, C., and Haesbroeck, G. (2003), "Implementing the Bianco and Yohai estimator for Logistic Regression", Computational Statistics and Data Analysis, 44, 273-295.

We consider the discrete Hammersley-Aldous-Diaconis process (HAD) and the totally asymmetric simple exclusion process (TASEP) in Z.
The basic coupling induces a multiclass process which is useful in discussing shock measures and other important properties of the processes.
The invariant measures of the multiclass systems are the same for both processes, and can be constructed as the law of the output process of a system of multiclass queues in tandem; the arrival and service processes of the queueing system are a collection of independent Bernoulli product measures.
The proof of invariance involves a new coupling between stationary versions of the processes called a multi-line process; this process has a collection of independent Bernoulli product measures as an invariant measure.
We emphasize a new approach via dual points: when the graphical construction is used to construct a trajectory of the TASEP or HAD process as a function of a Poisson process in ZxR, the dual points are those which govern the time-reversal of the trajectory.
Each line of the multi-line process is governed by the dual points of the line below. We note an extension of Burke's theorem to multiclass queues which follows from the results.

Trimming techniques are without a doubt the first designed tool to prevent the excessive deviations that the representatives of the data can present.
They suppose therefore a referring one forced in the analysis of the robustness of the statistical procedures, but in addition they have evolved adapting to problems of a great diversity and opening new methodologic perspective in all the scopes of statistics.
The conference will begin with the introduction of impartial trimming in problems of location and their later extension to problems of cluster analysis.
We will also present the most recent applications of the trimming procedures in classification of functional data, for the construction of adaptive estimators in parametric models and mixture, as well as to goodness of fit tests and comparison of samples.

Understanding the performance trade-offs in multi-hop wireless networks is essential to build good protocols for ad hoc networks.
In this talk I will present results on the scaling laws for capacity/delay tradeoffs for ad hoc networks with random mobility.
The results will be asymptotic in nature as the density of the ad hoc nodes grows. In particular we provide explicit results for both random walk and random waypoint mobility models that are canonical models of mobility in the plane.
The key ideas are of hitting time distributions for random walks and estimating queueing delays in queues with non-iid inputs.
I will conclude with a discussion of more general random mobility models and showing that there is the notion of *critical delay* associated with various models.
(Joint work with G. Sharma, X. Lin, and N. Shroff of Purdue University)

Abstract not avariable