Probability density function

About: Probability density function is a research topic. Over the lifetime, 22321 publications have been published within this topic receiving 422885 citations. The topic is also known as: probability function & PDF.

Uncertainty propagation for nonlinear dynamic systems using gaussian mixture models

Abstract: A Gaussian-mixture-model approach is proposed for accurate uncertainty propagation through a general nonlinear system. The transition probability density function is approximated by a finite sum of Gaussian density functions for which the parameters (mean and covariance) are propagated using linear propagation theory. Two different approaches are introduced to update the weights of different components of a Gaussian-mixture model for uncertainty propagation through nonlinear system. The first method updates the weights such that they minimize the integral square difference between the true forecast probability density function and its Gaussian-sum approximation. The second method uses the Fokker-Planck-Kohnogorov equation error as feedback to adapt for the amplitude of different Gaussian components while solving a quadratic programming problem. The proposed methods are applied to a variety of problems in the open literature and are argued to be an excellent candidate for higher-dimensional uncertainty-propagation problems.

Angle and time of arrival statistics for the Gaussian scatter density model

Abstract: Starting from a Gaussian distribution of scatterers around a mobile station, expressions are provided for the probability density function (PDF) in the angle of arrival, the power azimuth spectrum, the PDF in the time of arrival, and the time delay spectrum, all as seen from a base station. Expressions are also provided for some of the quantities of practical interest such as the root-mean-square (RMS) angular spread, the RMS delay spread, and the spatial cross-correlation function. Results for the Gaussian scatter density model are compared with those for the circular scattering model and the elliptical scattering model as well as with experimental results available for outdoor and indoor environments. Comparison is shown for the PDFs as well as for the power spectra in angle and delay. It is shown that the present model, in contrast to the previous models, produces results that closely agree with experimental results. With an appropriate choice of the standard deviation of the scattering region, the Gaussian density model can be made suitable both for environments with very small angular spreads as well as those with very large angular spreads. Consequently, the results provided in the paper are applicable to both macrocellular as well as picocellular environments.

Independent component analysis based on nonparametric density estimation

Abstract: In this paper, we introduce a novel independent component analysis (ICA) algorithm, which is truly blind to the particular underlying distribution of the mixed signals. Using a nonparametric kernel density estimation technique, the algorithm performs simultaneously the estimation of the unknown probability density functions of the source signals and the estimation of the unmixing matrix. Following the proposed approach, the blind signal separation framework can be posed as a nonlinear optimization problem, where a closed form expression of the cost function is available, and only the elements of the unmixing matrix appear as unknowns. We conducted a series of Monte Carlo simulations, involving linear mixtures of various source signals with different statistical characteristics and sample sizes. The new algorithm not only consistently outperformed all state-of-the-art ICA methods, but also demonstrated the following properties: 1) Only a flexible model, capable of learning the source statistics, can consistently achieve an accurate separation of all the mixed signals. 2) Adopting a suitably designed optimization framework, it is possible to derive a flexible ICA algorithm that matches the stability and convergence properties of conventional algorithms. 3) A nonparametric approach does not necessarily require large sample sizes in order to outperform methods with fixed or partially adaptive contrast functions.

Least-Cost Design of Water Distribution Networks under Demand Uncertainty

Abstract: Due to inherent variability in instantaneous water consumption levels, values of demands at nodes in a water distribution system remain one of the major sources of uncertainty in the network design process. Uncertainty in demand leads to uncertainty in head at the nodes, which, in turn, affects the system performance and has to be taken into account when designing new water distribution systems or extending/rehabilitating existing ones. One approach to dealing with this difficulty is to formulate and solve the stochastic optimization problem providing a robust, cost-effective solution. However, stochastic formulation usually requires Monte Carlo simulation, which involves calculation of a large number of state estimates, even for relatively simple networks. This renders the approach intractable when combined with heuristic adaptive search techniques, such as genetic algorithms (GAs) or simulated annealing. These methodologies require the fitness function to be evaluated for thousands of possible network configurations in the course of the search process. In this paper a new approach to quantifying the influence of demand uncertainty on nodal heads is proposed. The original stochastic model is reformulated as a deterministic one, which uses standard deviation as a natural measure of variability. Such an approach allows the use of effective numerical methods to quantify the influence of uncertainty on the robustness of water distribution system solutions. The deterministic equivalent is then coupled with an efficient GA solver to find robust and economic solutions. The proposed methodology was tested on the New York tunnels and Anytown problems. A number of low cost network solutions were found for different levels of reliability and different forms of probability distribution function for demands. The robustness of the solutions found was compared to known solutions for deterministic formulations, whose results were postprocessed using full Monte Carlo simulation.

Outage mutual information of space-time MIMO channels

Abstract: We derive analytical expressions for the probability density function (pdf) of the random mutual information between transmitted and received vector signals of a random space-time independent and identically distributed (i.i.d.) multiple-input multiple-output (MIMO) channel, assuming that the transmitted signals from the multiple antennas are Gaussian i.i.d.. We show that this pdf can be well approximated by a Gaussian distribution, and such a Gaussian approximation is based on expressions for the given pdfs mean and variance that we derive. We prove that at high signal-to-noise ratio (SNR), every factor of 2 increase in SNR leads to an increase in outage rate in the amount of min(M,N) bits, where M and N denote the number of transmit and receive antennas, respectively. A simple expression for the moment generating function (MGF) of the mutual information pdf is also provided, based on which we establish normality of the pdf, when both M and N are large, and the SNR is large.