Probability Density Function Of Data Research Articles

Data-Driven Sensor Scheduling for Remote Estimation in Wireless Networks

Sensor scheduling is a well-studied problem in signal processing and control with numerous applications. Despite its successful history, most of the related literature assumes the knowledge of the underlying probabilistic model of the sensor measurements such as the correlation structure or the entire joint probability density function. Herein, a framework for sensor scheduling for remote estimation is introduced in which the system design and the scheduling decisions are based solely on observed data. Unicast and broadcast networks and corresponding receivers are considered. In both cases, the empirical risk minimization can be posed as a difference-of-convex optimization problem, and locally optimal solutions are obtained efficiently by applying the convex-concave procedure. Our results are independent of the data's probability density function, correlation structure, and the number of sensors.

Parameter Estimation via Fokker–Planck Type Residual: Application to Linear Stationary Random Vibration

In this study, we propose a new method that is useful for estimating unknown parameter values of stochastic differential equation (SDE) models, based on probability density function (PDF) data measured from random dynamical systems. As our method does not require explicit description of PDF, it can be applied to the SDE models even when their PDFs are hardly derived in explicit forms due to multiplicative-noise terms, nonlinear terms, and so on. Therefore, our method is expected to provide a versatile tool to dynamically parameterize measured PDF data. In our proposed method, it is assumed that a measured PDF is obtained from a random dynamical system whose structure is described by a known SDE model with unknown parameter values. With the help of It\^o calculus, the Fokker-Planck equation (FPE) is derived from the SDE model. The measured PDF and a candidate of parameter values are substituted into the FPE to calculate a FPE residual. Our method is applied to two random vibration systems. Their FPE residuals tend to zero as the parameter values tend to exact values, showing that our proposed FPE residual can be utilized for unknown parameter estimation of SDE models.

Flow Structure of Lifted Swirling Jet Flames

Swirling combustion is widely applied in various applications such as gas turbines, utility boilers or waste incinerators. This article contributes to the ongoing research by providing experimental data that are gathered in the mixing zone of a lifted swirling premixed natural gas flame. The objective of this paper is fivefold: (1) to introduce the lifted swirling flame featuring low NOx emissions (2) to provide experimental data such as major species distributions, temperature and streamlines of the flow pattern, (3) to report on velocity bias in probability density function (PDF) distributions and to present PDF sequences of velocities in medium scale swirling flows, (4) to make an assessment on the local small-scale turbulence that is present in the swirling mixing layer and (5) to provide new experimental data for model verification and development. The PDFs are corrected in order to compensate for the velocity bias phenomenon, which is typical for randomly sampled LDA data. Sequences of axial PDF data are presented and measure- ment locations of interest are selected to look at the PDF characteristics of the internal and external recirculation zones, the mixing layer and the onset of the reacting flow into detail. The mixing layer PDFs covered a wide velocity range and revealed bimodality; even the concept of multi-modality is suggested and explored. Analysis showed that a sum of two Gaussian distributions can accurately envelop the experimental PDFs. The reason for this broadband turbulence behavior is to be found in combination of precessing and flapping motion of the flow structures, and also in combustion generated instabilities of the lifted flame. As a result, the flame brush is wide (large scale motion) and the mixing (small-scale turbulence) flattens any high temperatures in the combustion process. The multi-scale turbulence concept is subsequently used to make an assessment of the local turbulence characteristics in the mixing layer. The idea is that the PDFs capture both contributions of the flow-inherent fine grain turbulence (u � ) which is superposed on slow large scale fluctuating structures. It is this uthat will be of interest in continued research on the classification of the lifted flame into a combustion regime diagram (e.g. Borghi diagram). Finally, the bimodality character in reacting flows and the prediction of large-scale structures may be a challenge for LES researchers.

Comparison of closure schemes used to specify the velocity PDF in Lagrangian stochastic dispersion models for convective conditions

Lagrangian stochastic dispersion models make use of the probability density function (PDF) of the Eulerian vertical turbulent velocities. For convective conditions, the PDF is often assumed to have a bi-Gaussian form. Using new laboratory measurements of velocity PDFs in the convective boundary layer (CBL), we propose a new closure for constructing this bi-Gaussian PDF and compare results with three other closure schemes in current use. Of the three existing closures, two utilize the second and third moments of the vertical velocity as inputs, while the third one also incorporates the fourth moment. The new closure is defined with the desirable property that it collapses to a simple Gaussian in the limit of zero skewness. The value of an adjustable parameter in this closure scheme is selected using laboratory data for the third and fourth velocity moments. We determine the parameters in the PDF expression obtained using the four closures, and compare them with those derived by fitting velocity PDF data from the convection tank experiments. Significant differences are found between the values of the PDF parameters from the various closures and the water tank data. The performance of the closure schemes is compared by using a Lagrangian stochastic model to compute ground-level crosswind-integrated concentrations from particles released at four source heights. It is shown that the differences between the concentration estimates obtained using various closures increase as the source height increases. Using, as the benchmark, the dispersion results calculated from the Lagrangian stochastic model incorporating the laboratory velocity data without any closure, we recommend our new closure scheme. The results highlight the importance of turbulence observations in the CBL for accurate dispersion modelling.

Deriving parameter probability density functions

The long-term performance of a nuclear fuel waste disposal system is typically studied by modelling potential releases of contaminants to the environment and the consequent health risk to humans. To deal with uncertainty in the estimated consequences of the releases, the behaviour of the disposal system may be repeatedly simulated under the direction of a probabilistic assessment code. In each simulation, parameters in the system model take different values to reflect the uncertainty about their values in the real system. The selection of the possible values of the parameters is governed by a probability density function (PDF) for each parameter. This paper examines the question of how to derive parameter PDFs for such assessments. The paper includes: 1.1. A definition of the concept of probability, and the mathematical properties of PDFs and the related cumulative distribution functions (CDFs).2.2. A discussion of the influence of the context in which PDFs are derived on their form and interpretation. Key elements of the context include what is known about the disposal system, the structure and extent of the model description of the system, and how the results of the assessment will be used.3.3. A description of approaches to deriving parameter PDFs. In cases where data are unavailable or incomplete, reliance must be placed on the judgement of experts to generate subjective probability distributions. 1. A definition of the concept of probability, and the mathematical properties of PDFs and the related cumulative distribution functions (CDFs). 2. A discussion of the influence of the context in which PDFs are derived on their form and interpretation. Key elements of the context include what is known about the disposal system, the structure and extent of the model description of the system, and how the results of the assessment will be used. 3. A description of approaches to deriving parameter PDFs. In cases where data are unavailable or incomplete, reliance must be placed on the judgement of experts to generate subjective probability distributions. Examples are given of PDFs defined for recent Canadian assessments. The paper concludes with remarks on managing reliably the large body of PDF data for an assessment, how values are sampled from PDFs, and desirable future developments.

Volume scattering: Echo peak PDF

The statistical properties of the backscattered acoustic echo from clouds of scatterers are examined. These clouds could be plankton, nekton, bubbles, etc., or a combination. The clouds of interest are sufficiently dense so that echoes from the individual scatterers overlap. The scatterers are assumed to be randomly distributed in space so that, at high enough acoustic frequencies, the individual echoes are not correlated (random phase). The probability density function (PDF) of the maximum echo value achieved in a depth gate has been analytically derived, computer simulated, and measured in the field. It is shown to be quite different than the Rayleigh PDF of the envelope, even for gate durations smaller than one transmission ping duration. An approximate analytical expression for the echo peak PDF is derived by first modeling the echo as a discrete ‘‘stepped’’ waveform where the ‘‘steps’’ are statistically independent of each other. The problem is then solved by straightforward extremal statistics. There is excellent agreement between the theoretical PDF, computer simulations, and field data. While we are concerned with gate durations several times one ping duration, the analytical expression for the PDF is shown to accurately describe the PDF over ranges of the gate duration less than the ping duration to much greater than the ping duration. From the echo peak PDF one can (1) estimate the abundance of occasional large scatterers that are within the clouds (for example large fish feeding on plankton) since their occurrence is differentiable from the rest of the cloud in the PDF, and (2) clearly differentiate between conditions where the echoes overlap and do not overlap. Once it is determined whether or not they overlap, lower or upper bounds in scatterer density (number per unit volume) may be calculated. Both of these applications can be performed in situ.

The neural path probabilistic delay model

The neural path probabilistic delay (NPPD) algorithm models the neural network in a discrete-time manner analogous to the computer network. The four stages of the basic neuron cell component, i.e. neuron cell body, axon branches, synapse region, and dendrite trees are represented by the mathematical structures of cascaded Markov chains. A state probability transition matrix for the neuron cell is generated by a queueing model of signal interarrival and service rate probability density function (PDF) data. To obtain the delay value for the soma, the PDF functions respectively describing the rate of action potential generation at each cell body and neuron signal arrival at synaptic dendrite terminals, are subtracted and quantized horizontally. This quantization of service minus interarrival PDFs is analogous to the human nervous system encoding of action potential pulses. The specific state transition probability matrices associated with the axon, synapse, and dendrite stages are computed from experimental link structure data. Both PDF and link structure data are measured from neurons located in the specific section of the nervous system considered. An example of the NPPD measure is derived for a neural path in a segment of the cerebellar folium in the cerebellar cortex of the human brain, because a path in this area is initiated by a pain sensation stimulus to the cerebral cortex. Therefore, the motor response path delay parameters for the pain stimulus are of interest in the implementation of the robotic system.

The interrelation between void fraction fluctuations and flow patterns in two-phase flow

A fast response, linearized X-ray void measurement system has been used to obtain statistical measurements in normally fluctuating air-water flow in a rectangular channel. It is demonstrated that the probability density function (PDF) of the fluctuations in void fraction may be used as an objective and quantitative flow pattern discriminator for the three dominant patterns of bubbly, slug, and annular flow. This concept is applied to data over the range of 0.0 to 37 m/sec mixture velocities to show that slug flow is simply a transitional, periodic time combination of bubbly and annular flows. Film thicknesses calculated from the PDF data are similar in magnitude in both slug and annular flows. Calculation of slug length and residence time ratios along with bubble lengths in slug flow are also readily obtainable from the statistical measurements. Spectral density measurements showed bubbly flow to be stochastic while slug and annular flows showed periodicities correlatable in terms of the liquid volume flux.