Hana Baili

Statistical characterization of a measurement approach using MCMC

A measurement is any quantity to be observed within a system; we talk about indirect measurement when this quantity cannot be directly given by some sensors. This paper proposes a probabilistic approach to characterize a dynamic continuous measurement by a knowledge-based uncertain model, using a Monte-Carlo technique with Markov chains (MCMC). The method is far simpler than the Monte-Carlos one or the numerical resolution of the Fokker-Planck equation; looking at the precision, it is also quite satisfactory.

Indirect measurement within dynamical context: probabilistic approach to deal with uncertainty

This paper deals with the general question of indirect measurement within dynamical continuous context. The proposed answer is of probabilistic nature in the sense that: the modeling, which is the first element of the answer, consists in transforming the initial model into a stochastic differential equation (SDE) such that, estimating the probability density function (pdf) of its process achieves the measurement, which is indeed the second element of the answer.

Probabilistic approach for indirect measurement within dynamical uncertain context application from electrical engineering

A “measurement” is any quantity to be observed within a system. We talk about indirect measurement when the observation cannot be directly given by some sensors. “Indirect measurement” raises in a variety of applications and leads to the paramount estimation theory. The starting point in the resolution of such an estimation problem is the modela mathematical description of the system. When the model comes from physics, it is said knowledge-based model, as opposed to black-box model. A model consists of a set of relations between some quantities among them appears the measurement. The term “dynamical” in the title refers to the evolution of the measurement in time in one hand, and means that the model comprises at least one dynamical relation a time differentiation or integration, in the other band. The term “continuous” in the abstract means that the quantities depending on time are homogenous to functions continuous on t, or random processes with sample functions almost surely continuous on t . The quantities that when fixed cause the others to be determined uniquely are called model’s data such as initial conditions, observations, controls, parameters, etc. Often some of them are unknown, this is what expresses the term “uncertain” in the title. A prior information about some unknown can be acquired. It will consist of its average and dispersion if i t is random, or of some set that specifies its values if it is deterministic.

Optimal scheduling and power allocation in wireless networks with heavy traffic: the infinite time horizon case

ABSTRACTThis paper addresses the problem of joint transmit power allocation and time-slot scheduling in a wireless communication system with time-varying traffic. The system is handled by a single base station transmitting over time-varying channels. The operating time horizon is divided into time slots; a fixed amount of power is available at each time slot. The mobile users share each time slot and the power available at this time slot with the objective of minimising their expected total delay. This is reformulated, via a heavy traffic approximation, as the optimal control of a reflected diffusion in the positive orthant. We address this optimisation problem using the dynamic programming principle and establish Bellmans equation; here the main feature consists in how to deal with the reflection of the diffusion process on the boundary of its state space. A closed-form solution for the optimal control problem is provided. The proposed solution relies on, among other issues, the knowledge of the intensi...

Optimal Scheduling and Power Allocation in Wireless Networks with Heavy Traffic

This paper addresses the problem of joint transmit power allocation and time slot scheduling in a wireless communication system with time varying traffic. The system is handled by a single base station transmitting over time varying channels. This may be the case in practice of a hybrid TDMA-CDMA (Time Division Multiple Access-Code Division Multiple Access) system. The operating time horizon is divided into time slots; a fixed amount of power is available at each time slot. The users share each time slot and the power available at this time slot with the objective of minimizing the expected total queue length. The problem is reformulated, via a heavy traffic approximation, as the optimal control of a reflected diffusion in the positive orthant. We establish a closed form solution for the obtained control problem. The main feature that makes it possible is an astute choice of some auxiliary weighting matrices in the cost rate. The proposed solution relies also on the knowledge of the covariance matrix of the non-standard multi-dimensional Wiener process which is the driving process in the reflected diffusion. We then compute this covariance matrix given the stationary distribution of the multi-dimensional channel process. Further stochastic analysis is provided: the cost variance, and the Fokker–Planck equation for the distribution density of the queue length.

Stochastic Analysis and Particle Filtering of the Volatility

Let \(S = {({S}_{t})}_{t\in {\rm{IR}}_{+}}\) be an \({\rm{IR}}_{+}\)-valued semimartingale based on a filtered probability space \((\Omega,\mathcal{F},{({\mathcal{F}}_{t})}_{t\in {\rm{IR}}_{+}}, \rm{IP})\) which is assumed to be continuous. The process S is interpreted to model the price of a stock. A basic problem arising in Mathematical Finance is to estimate the price volatility, i.e., the square of the parameter σ in the following stochastic differential equation

Ergodic optimization of stochastic differential systems in wireless networks

d{S}_{t} = \mu {S}_{t}\,dt + \sigma {S}_{t}\,d{W}_{t}

Optimal filtering of piecewise deterministic processes for source detection and separation in electric load monitoring

where \(W = {({W}_{t})}_{t\in {\rm{IR}}_{+}}\) is a Wiener process. It turns out that the assumption of a constant volatility does not hold in practice. Even to the most casual observer of the market, it should be clear that volatility is a random function of time which we denote σ t 2. Ito’s formula for the return \({y}_{t} =\log ({S}_{t}/{S}_{0})\) yields