Densities Of Random Variables Research Articles

Estimating the Integral of the Square of Derivatives of Symmetric Probability Densities of One-Dimensional Random Variables

In justifying a method for rapid choice of the spread coefficient of kernel probability density estimates, a constant has been found which is a functional of the second derivative of the density. This result is generalized to the derivatives of symmetric probability densities of different orders in this paper. Functional dependences of these constants on the coefficient of antikurtosis of a random variable are found. The features of these constants are studied. A method for estimating functionals of derivatives of the probability densities that assumes fulfillment of the following steps is developed on the basis of these results. The mean square deviation of the one-dimensional random variables and the antikurtosis coefficient are estimated from the original sample. The recovered functional dependences of the antikurtosis are used to estimate the values of constants which are functionals of the derivatives of the probability density. For the known estimates of the mean square deviation of random variable being studied and the constant examined here, the values of the functional of the derivative of the probability distribution of the chosen order are calculated. The results are confirmed by analyzing data from a numerical simulation. It is found that with increasing order of the derivative, the estimates of these functionals increase. This fact is explained by the increasing complexity of the integrand in these functionals. The proposed method provides objective results for the first three derivatives of the probability density of the random variable. The results of this study are confirmed by a confidence estimate for the functionals studied here.

ЭФФЕКТИВНЫЙ АЛГОРИТМ ГЕНЕРАЦИИ СЛУЧАЙНЫХ ВЕЛИЧИН С КУСОЧНО-ЛИНЕЙНЫМИ ПЛОТНОСТЯМИ РАСПРЕДЕЛЕНИЯ

The PURPOSE of the article is to improve the efficiency of simulation modeling in various subject domains through the use of piecewise linear functions that allow high accuracy in the determination of arbitrary random variables. METHODS. It is proposed to use piecewise linear functions to specify the density of random variables under the transformation of a standard random variable to build simulation models. RESULTS. Based on the type of the random variable density specified by a piecewise linear function three algorithms of action sequence have been proposed for generating the mixture of the components that must have a defined discrete probability distribution. CONCLUSION. Due to their versatility the piecewise linear functions when applied provide sufficiently high generation accuracy of arbitrary random variables. A high simulation efficiency of piecewise-linear functions can improve the accuracy and adequacy of simulation models in different subject domains.

Evaluation of Variable Density and Data-Driven K-Space Undersampling for Compressed Sensing Magnetic Resonance Imaging.

The aim of this study was to investigate the influence of variable density and data-driven k-space undersampling patterns on reconstruction quality for compressed sensing (CS) magnetic resonance imaging to provide recommendations on how to avoid suboptimal CS reconstructions. First, we investigated the influence of randomness and sampling density on the reconstruction quality when using random variable density and variable density Poisson disk undersampling. Compressed sensing reconstructions on 1 knee and 2 brain data sets were compared with fully sampled data sets and reconstruction errors were measured. Sampling coherence was evaluated on the undersampling patterns to investigate whether there was a relation between this coherence measure and reconstruction error.Second, we investigated whether data-driven undersampling methods could improve reconstruction quality when 1 or more fully sampled scans are available as a training set. We implemented 3 different data-driven undersampling methods: (1) Monte Carlo optimization of variable density and variable density Poisson disk undersampling, (2) calculating sampling probabilities directly from the k-space power spectra of the training data, and (3) iterative design of undersampling patterns based on CS reconstruction errors in k-space.Two cross-validation experiments were set up using retrospective undersampling to evaluate the 3 data-driven methods and the influence of the size of the training set. Furthermore, in an experiment that included prospective under sampling, we show the practical applicability of 2 of the data-driven methods. Compressed sensing reconstruction quality was measured with both the normalized root-mean-square error metric and the mean structural similarity index measure. Different optimal variable sampling densities were found for each of the data sets, showing that the optimal sampling density is data dependent. Choosing a sampling density other than the optimal density decreased reconstruction quality. These results suggest that choosing a sampling density without having any reference scans is likely suboptimal. Furthermore, no meaningful correlation was found between sampling coherence and reconstruction error.For the data-driven methods, the iterative method yielded statistically significantly higher reconstruction quality in both retrospective and prospective experiments. In retrospective experiments, the power spectrum method yielded a reconstruction quality that was comparable with the data-driven variable density method. The size of the training set had only a minor influence on the reconstruction quality. Data-driven undersampling methods can be used to avoid suboptimal reconstruction quality in CS magnetic resonance imaging, provided that at least 1 fully sampled scan is available to train the data-driven method. The iterative design method resulted in the highest reconstruction quality.

Imitating modelling of data processing in information system

The information system with using distributed databases, where inquiry processing is performed in the order of the queue at each workstation is described. It is found that predictive methods are focused on the narrow use and not suitable for widespread use in information systems. The operation logic of the imitating model in terms of events, related to issuing commands from the main system to the control object is described. The steps to ensure that the data on the duration of executing the commands, received from control objects, turned into the probability density of random variables are defined. The relative frequency of events in the system is determined. Using predictive methods and knowledge about previous states of information system, progressive function, which can provide information about its future states with a certain probability, is formed. As well as all other imitating models with discrete events, this model describes the situation with the queue, in which data flows arrive from control objects before constructing the information system model. The considered system consists of one queue and N workstations (N control objects in information system). The queue is set by sequence of processing control objects in the initial priorities, calculated based on the workstation parameters. Empirical and theoretical functions of the system states are created. During the imitation, it was found that the average deviation of the theoretical distribution from empirical does not exceed 10% that indicates the feasibility of using the exponential function as predictive for the given system with the queue. Time of staying of the simulated system in the steady state is estimated and is equal to 2.7 seconds. It is concluded that this time is enough to predict the next state of information system that will allow timely and uniform load distribution among the workstations.

Fractional calculus approach to the statistical characterization of random variables and vectors

Fractional moments have been investigated by many authors to represent the density of univariate and bivariate random variables in different contexts. Fractional moments are indeed important when the density of the random variable has inverse power-law tails and, consequently, it lacks integer order moments. In this paper, starting from the Mellin transform of the characteristic function and by fractional calculus method we present a new perspective on the statistics of random variables. Introducing the class of complex moments, that include both integer and fractional moments, we show that every random variable can be represented within this approach, even if its integer moments diverge. Applications to the statistical characterization of raw data and in the representation of both random variables and vectors are provided, showing that the good numerical convergence makes the proposed approach a good and reliable tool also for practical data analysis.

Calcul des variations stochastique pour la mesure de densité uniforme

A gradient operator is defined for the functionals of a non-Markovian jump process Y whose jump times are given by uniform probability laws. The adjoint of this gradient extends the compensated stochastic integral with respect to Y. An explicit representation of the functionals of Y as stochastic integrals is obtained via a Clark formula in two different approaches. The associated Dirichlet forms is studied in order to obtain criteria for the existence and regularity of densities of random variables in infinite dimension.