Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densities by generalizing the Aitchison geometry for compositions in the simplex into the set probability densities

Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Central notations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform. In this way very elaborated aspects of mathematical statistics can be understood easily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating, combination of likelihood and robust M-estimation functions are simple additions/ perturbations in A2(Pprior). Weighting observations corresponds to a weighted addition of the corresponding evidence. Likelihood based statistics for general exponential families turns out to have a particularly easy interpretation in terms of A2(P). Regular exponential families form finite dimensional linear subspaces of A2(P) and they correspond to finite dimensional subspaces formed by their posterior in the dual information space A2(Pprior). The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P. The discussion of A2(P) valued random variables, such as estimation functions or likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

Bayesian tools for zero counts in compositional data

The log-ratio methodology makes available powerful tools for analyzing compositional data. Nevertheless, the use of this methodology is only possible for those data sets without null values. Consequently, in those data sets where the zeros are present, a previous treatment becomes necessary. Last advances in the treatment of compositional zeros have been centered especially in the zeros of structural nature and in the rounded zeros. These tools do not contemplate the particular case of count compositional data sets with null values. In this work we deal with \count zeros" and we introduce a treatment based on a mixed Bayesian-multiplicative estimation. We use the Dirichlet probability distribution as a prior and we estimate the posterior probabilities. Then we apply a multiplicative modi¯cation for the non-zero values. We present a case study where this new methodology is applied. Key words: count data, multiplicative replacement, composition, log-ratio analysis

Evaluating the probability and the impact of a failure in GMPLS based networks

In this paper, we define a new scheme to develop and evaluate protection strategies for building reliable GMPLS networks. This is based on what we have called the network protection degree (NPD). The NPD consists of an a priori evaluation, the failure sensibility degree (FSD), which provides the failure probability, and an a posteriori evaluation, the failure impact degree (FID), which determines the impact on the network in case of failure, in terms of packet loss and recovery time. Having mathematical formulated these components, experimental results demonstrate the benefits of the utilization of the NPD, when used to enhance some current QoS routing algorithms in order to offer a certain degree of protection

Estimation of the probability of congestion using Monte Carlo method in OPS networks

In networks with small buffers, such as optical packet switching based networks, the convolution approach is presented as one of the most accurate method used for the connection admission control. Admission control and resource management have been addressed in other works oriented to bursty traffic and ATM. This paper focuses on heterogeneous traffic in OPS based networks. Using heterogeneous traffic and bufferless networks the enhanced convolution approach is a good solution. However, both methods (CA and ECA) present a high computational cost for high number of connections. Two new mechanisms (UMCA and ISCA) based on Monte Carlo method are proposed to overcome this drawback. Simulation results show that our proposals achieve lower computational cost compared to enhanced convolution approach with an small stochastic error in the probability estimation

Partial nitritation of landfill leachate in a SBR prior to an anammox reactor : operation and modelling

Els lixiviats d'abocadors urbans són aigües residuals altament contaminades, que es caracteritzen per les elevades concentracions d'amoni i el baix contingut de matèria orgànica biodegradable. El tractament dels lixiviats a través dels processos de nitrificació-desnitrificació convencionals és costós a causa de la seva elevada demanda d'oxigen i la necessitat d'addició d'una font de carboni externa. En els darrers anys, la viabilitat del tractament d'aquest tipus d'afluents per un procés combinat de nitritació parcial-anammox ha estat demostrada. Aquesta tesi es centra en el tractament de lixiviats d'abocador a través d'un procés de nitritació parcial en SBR, com un pas preparatori per a un reactor anammox. Els resultats de l'estudi han demostrat la viabilitat d'aquesta tecnologia per al tractament de lixiviats d'abocador. El treball va evolucionar des d'una escala inicial de laboratori, on el procés va ser testat inicialment, a uns exitosos experiments d'operació a llarg termini a escala pilot. Finalment, la tesi també inclou el desenvolupament, calibració i validació d'un model matemàtic del procés, que té com a objectiu augmentar el coneixement del procés.

Congestion probability routing in virtual path ATM network

This dissertation focuses on the problem of providing mechanisms for routing point to point and multipoint connections in ATM networks. In general the notion of multipoint connection refers to connections that involve a group of users with more than two members. The main objective of this dissertation is to contribute to design efficient routing protocols with alterative routes in fully connected VP-based ATM Networks for call establishment of point to point and multipoint VC connections. An efficient route should be computed during this connection establishment phase.