Respective Probability Density Research Articles

Experimental investigations on an FSO-fiber converged communication system under fog and turbulence

This paper presents experimental results for a free-space optical-fiber converged (FSO-FC) communication system under varying turbulence and foggy conditions. Based on the experimental measurements, we statistically characterize the FSO channel under different levels of turbulence and fog in terms of their respective probability density functions (PDFs). Our experimental PDFs fit well with the theoretical PDFs proposed in the literature. We experimentally evaluate the average bit error rate (ABER) of the considered FSO-FC communication system under different levels of turbulence and fog. Further, to compensate for the effects of turbulence and fog, we use multimode fiber as the receiver and compare the results with single-mode fiber as the receiver. Finally, we show the improvement in ABER under varying turbulence and fog by adding cyclic redundancy check bits to data bits.

User Behavior Clustering Based Method for EV Charging Forecast

The increasing adoption of electric vehicles poses new problems for the electrical distribution network. For this reason, proper electric vehicle forecasting will be of fundamental importance for a predictive energy management system, which could greatly help the operation of the grid. This paper proposes a comprehensive novel methodology to forecast single charging sessions of electric vehicle and the resulting cumulative energy forecast of the charging infrastructure. Historical charging sessions are first clustered on the basis of similar user characteristics and their respective probability density functions are defined. From this, every charging session is predicted with a triplet of parameters, namely the arrival time, the charging duration and the average power expected during the process. The proposed method has been evaluated by considering a real case study. The results showed the ability to greatly improve the accuracy with respect to the chosen benchmark, both in terms of energy required by the station and the predicted number of charging sessions. The overall performance measured by Skill Score is 0.37 for the year 2019.

Fractional derivatives as weighted average of historical values: an application to COVID-19 in Brazil

The memory effect is an interesting tool that can be seen in fractional differential equations. To show this clearly, in this paper we prove that the Caputo derivative of a function 𝑓, as well as the Riemann-Liouville integral and derivative, are proportional to a weighted average of the historical values of 𝑓 or 𝑓 ′. For this, we use the statistical expectation of functions, whose random variable follows a beta distribution. Moreover, through the respective probability density functions, for each operator we specify the weight of the historical values of the function to determine its current value, according to the values of the fractional order of the derivative. Furthermore, to prove the effectiveness of the memory effect to describe real phenomena, we compared a classic model with its fractional version to model COVID-19 in Brazil.

A Flexible Top-Down Data-Driven Stochastic Model for Synthetic Load Profiles Generation

The study of the behavior of smart distribution systems under increasingly dynamic operating conditions requires realistic and time-varying load profiles to run comprehensive and accurate simulations of power flow analysis, system state estimation and optimal control strategies. However, due to the limited availability of experimental data, synthetic load profiles with flexible duration and time resolution are often needed to this purpose. In this paper, a top-down stochastic model is proposed to generate an arbitrary amount of synthetic load profiles associated with different kinds of users exhibiting a common average daily pattern. The groups of users are identified through a preliminary Ward’s hierarchical clustering. For each cluster and each season of the year, a time-inhomogeneous Markov chain is built, and its parameters are estimated by using the available data. The states of the chain correspond to equiprobable intervals, which are then mapped to a time-varying power consumption range, depending on the statistical distribution of the load profiles at different times of the day. Such distributions are regarded as Gaussian Mixture Models (GMM). Compared with other top-down approaches reported in the scientific literature, the joint use of GMM models and time-inhomogeneous Markov chains is rather novel. Furthermore, it is flexible enough to be used in different contexts and with different temporal resolution, while keeping the number of states and the computational burden reasonable. The good agreement between synthetic and original load profiles in terms of both time series similarity and consistency of the respective probability density functions was validated by using three different data sets with different characteristics. In most cases, the median values of synthetic profiles’ mean and standard deviation differ from those of the original reference distributions by no more than ±10% both within a typical day of each season and within the population of a given cluster, although with some significant outliers.

Spatiotemporal just noticeable difference modeling with heterogeneous temporal visual features

Developing accurate Just-noticeable difference (JND) models are challenged by complicated HVS characteristics and nonstationary features of video sequence. Great efforts have been devoted to JND modeling, and inspiring performance improvements are witnessed in the literature, especially spatial JND models. However, there are not only urgent requirement but also technical potentiality for improving temporal JND models fully accounting for the temporal perception characteristics. In terms of temporal JND modeling, there are two challenges, one is how to extract perceptual feature parameters of source video, and the other is how to quantitatively characterize the interaction relationship between feature parameters and HVS characteristics? Firstly, this work extracts content-aware temporal feature parameters having predominate impacts on vision perception, including motion (foreground/background), pixel-correspondence duration and inter-frame residue fluctuation intensity along temporal trajectory, and investigates the HVS responses to these four heterogeneous feature parameters. Secondly, this work proposes respective probability density functions (PDF) in the perception sense to quantitatively depict the attention and suppression perception responses of feature parameters, accounting for the temporal perception characteristics. Using these PDF models, we fuse the heterogeneous feature parameters from the viewpoint of uniform dimension,i.e. self-information measured visual attention and information entropy measured masking uncertainty, achieving heterogeneous parameter homogenization. Thirdly, with self-information and entropy results, this work then proposes a temporal weight model, by striking the balance between visual attention and masking suppression, to adjust the spatial JND threshold, and then develops the improved spatiotemporal JND model. Intensive simulation results verity the effectiveness of the proposed spatiotemporal JND profile, with competitive model accuracy compared with the-state-of-the-art candidate models.

On the mean value of the force operator for 1D particles in the step potential

In the one-dimensional Klein-Fock-Gordon theory, the probability density is a discontinuous function at the point where the step potential is discontinuous. Thus, the mean value of the external classical force operator cannot be calculated from the corresponding formula of the mean value. To resolve this issue, we obtain this quantity directly from the Klein-Fock-Gordon equation in Hamiltonian form, or the Feshbach-Villars wave equation. Not without surprise, the result obtained is not proportional to the average of the discontinuity of the probability density but to the size of the discontinuity. In contrast, in the one-dimensional Schr\"odinger and Dirac theories this quantity is proportional to the value that the respective probability density takes at the point where the step potential is discontinuous. We examine these issues in detail in this paper. The presentation is suitable for the advanced undergraduate level.

The statistical finite element method (statFEM) for coherent synthesis of observation data and model predictions

The increased availability of observation data from engineering systems in operation poses the question of how to incorporate this data into finite element models. To this end, we propose a novel statistical construction of the finite element method that provides the means of synthesising measurement data and finite element models. The Bayesian statistical framework is adopted to treat all the uncertainties present in the data, the mathematical model and its finite element discretisation. From the outset, we postulate a statistical generating model which additively decomposes data into a finite element, a model misspecification and a noise component. Each of the components may be uncertain and is considered as a random variable with a respective prior probability density. The prior of the finite element component is given by a conventional stochastic forward problem. The prior probabilities of the model misspecification and measurement noise, without loss of generality, are assumed to have a zero-mean and a known covariance structure. Our proposed statistical model is hierarchical in the sense that each of the three random components may depend on one or more non-observable random hyperparameters with their own corresponding probability densities. We use Bayes rule to infer the posterior densities of the three random components and the hyperparameters from their known prior densities and a data dependent likelihood function. Because of the hierarchical structure of our statistical model, Bayes rule is applied on three different levels in turn. On level one, we determine the posterior densities of the finite element component and the true system response using the prior finite element density given by the forward problem and the data likelihood. In this step, approximating the prior finite element density with a multivariate Gaussian distribution allows us to obtain a closed-form expression for the posterior. On the next level, we infer the hyperparameter posterior densities from their respective priors and the marginal likelihood of the first inference problem. These posteriors are sampled numerically using the Markov chain Monte Carlo (MCMC) method. Finally, on level three we use Bayes rule to choose the most suitable finite element model in light of the observed data by computing the respective model posteriors. We demonstrate the application and versatility of statFEM with one and two-dimensional examples.

Observational evidence of non-extensive behavior of seismic coda waves

The last decades a lot of progress has been made in studying Earth and planetary Physics based on the science of complexity and the application of generalized formalisms of statistical physics and entropy. Towards this direction, investigation of the dynamical features of seismic waves is of fundamental importance since they could provide information on the statistical properties of the lithosphere.The non extensive statistical mechanics (NESM) seems to be a promising framework for studying complex earthquake systems that exhibit long-range interactions and memory effects. In this work, coda waves are studied from the point of view of non extensive statistical mechanics. Our aim is to demonstrate the applicability of Non extensive statistical mechanics to the fluctuations of coda amplitudes and to define the entropic q parameters obtained in the coda waves recorded by seismological stations in the South Aegean, and generated by three strong earthquake events in the front of the Hellenic Arc.Coda waves are interpreted as scattered seismic waves by heterogeneities in the earth (cracks, faults, velocity and/or density anomalies, etc.). Due to the nature of those scatterers distribution we follow the NESM approach and study the probability distribution functions of the increments of seismic coda waves generated by seismic events recorded in South Aegean and generated by three strong events.We defined the increment function X(t)=V(t+1)−Vt, where V(t) is the measured horizontal ground velocity. We proceed by analyzing the normalized increments X(t) constructing the probability density function PDFp(x), where x=X−XσX, σX being the standard deviation of X(t) and <X> the observed mean value on the same data set. The results of the present work suggests that the seismic coda waves increments deviate from the Gaussian shape and their respective probability density function px could be adequately described by a q-Gaussian with entropic q-values in the range of1.35<q<2.02 with an error of δq=±0.05.This observation implies that seismic coda wave complexity originating by multiscattering effects, can be described by Tsallis entropy terms. The entropic q-values estimated vary along each seismic path for each of the three strong earthquake events analyzed. This observation implies that the value of q is determined by the particular nature of path between the source and the recorded station.

Probability Navigation Function for Stochastic Static Environments

The navigation function (NF) is widely used for motion planning of autonomous vehicles. Such a function is bounded, analytic, and guarantees convergence due to its Morse nature, while having a single minimum value at the target point. This results in a safe path to the target. Originally, the NF was developed for deterministic scenarios where the positions of the robot and the obstacles are known. Here we extend the concept of NF for static stochastic scenarios. We assume the robot, the obstacles and the workspace geometries are known, while their positions are random variables. We define a new Probability NF that we call PNF by introducing an additional permitted collision probability, which limits the risks (to a set value) during the robot’s motion. The Minkowski sum is generalized for the geometries of the robot and the obstacles with their respective Probability Density Functions (PDF), that represent their locations’ uncertainties. The probability for a collision is therefore the convolution of the robot’s geometry, the obstacles’ geometries and the PDFs of their locations The novelty of the proposed algorithm is in its ability to provide a converging trajectory in stochastic environment without inflating the ambient space dimension. We demonstrate our algorithm performances using a simulator, and compare its results with the conventional NF algorithm and with a version of the well known RRT* and Voronoi Uncertainty Fields methods for uncertain scenarios. Finally, we show simulation results of the PNF in disc-shaped world, as well as in star-shaped world.

A priori study of the subgrid energy transfers for small-scale dynamo in kinematic and saturation regimes

The statistical properties of the subgrid energy transfers of homogeneous small-scale dynamo are investigated during the kinematic, nonlinear, and statistically saturated stages. We carry out an a priori analysis of data obtained from an ensemble of direct numerical simulations on 5123 grid points and at unity magnetic Prandtl number. In order to provide guidance for subgrid-scale (SGS) modelling of different types of energy transfer that occur in magnetohydrodynamic dynamos, we consider the SGS stress tensors originating from inertial dynamics, Lorentz force, and the magnetic induction separately. We find that all SGS energy transfers display some degree of intermittency as quantified by the scale-dependence of their respective probability density functions. Concerning the inertial dynamics, a depletion of intermittency occurs in the presence of a saturated dynamo.

Random encounters in probabilistic time geography

ABSTRACTProbabilistic time geography considers the encounter of moving agents to be random; therefore, a quantitative time geography analysis must consider the actual encounter probability. The existing algorithm of encounter probability is oriented over a discrete space and is sensitive to the unit definition of a virtual grid; thus, it is not suitable for continuous space. For this reason, a new method is presented in this paper for the encounter of two moving agents in continuous space. When the encounters are less than a specified distance threshold apart, an encounter event occurs based on the probability of the product, which is calculated by their respective probability density functions over their respective potential location areas. This probability provides a quantitative basis for predicting the likelihood of two agents meeting, as well as the location of this meeting point. Finally, the validity of the proposed model is verified by an experiment, which uses tracking data to calculate the encounter probabilities of three zebras and analyse the distribution characteristics of these probabilities over time and space.

Performance analysis of DF-relayed cognitive underlay networks over EGK fading channels

In this paper, we analyse the performance of a dual hop cognitive radio underlay network employing a decode and forward relay. The fading channels have been modeled using extended generalized-K fading statistics. The interference temperature is taken as a constraint so that the transmit power of the secondary source and the relay do not exceed this value. This limits the transmit powers of the secondary network to ensure no interference with the primary communication which is imperative in cognitive underlay systems. First the signal to noise ratio at the relay and the destination of the system under consideration are analyzed and then the respective cumulative distribution function and probability density function are evaluated. This is followed by the derivation of statistical expression for end-to-end signal to noise ratio of the relay network, it’s cumulative distribution function and probability density function. These expressions are further utilized to derive the analytical expressions for the outage probability and the ergodic capacity of the underlay dual-hop cognitive relay network. Our results demonstrate the dependence of the performance of the system on various parameters under the constraints of transmit power and interference temperature already defined. The mathematical analysis is followed by various numerical examples to illustrate how the various parameters of the system affect it’s performance.

Thermodynamics of a real fluid near the critical point in numerical simulations of isotropic turbulence

We investigate the behavior of a fluid near the critical point by using numerical simulations of weakly compressible three-dimensional isotropic turbulence. Much has been done for a turbulent flow with an ideal gas. The primary focus of this work is to analyze fluctuations of thermodynamic variables (pressure, density, and temperature) when a non-ideal Equation Of State (EOS) is considered. In order to do so, a hybrid lattice Boltzmann scheme is applied to solve the momentum and energy equations. Previously unreported phenomena are revealed as the temperature approaches the critical point. Fluctuations in pressure, density, and temperature increase, followed by changes in their respective probability density functions. Due to the non-linearity of the EOS, it is seen that variances of density and temperature and their respective covariance are equally important close to the critical point. Unlike the ideal EOS case, significant differences in the thermodynamic properties are also observed when the Reynolds number is increased. We also address issues related to the spectral behavior and scaling of density, pressure, temperature, and kinetic energy.

Applied problems of the identification and regression analysis in stochastic structures with multimodal properties

We have been elaborating an approach founded on the identification of multimodal laws of the complex structure distribution in medicine, biology, chemistry of ultrapure materials and membrane technology as well as in technical applications. The method is based on the formulation and solution of inverse problems in mathematical physics for the respective probability density functions. The verification of the used algorithmic tools is carried out on model limited-scope samples. For stochastic structures and systems under study the method is supplemented with an original option of a regression analysis taking into account the identified stochastic laws displaying numerical parameters into the binary space. The proposed approach has been tested on clinical material in practical medicine.

Forecasting seeing and parameters of long-exposure images by means of ARIMA

Atmospheric turbulence is the one of the major limiting factors for ground-based astronomical observations. In this paper, the problem of short-term forecasting seeing is discussed. The real data that were obtained by atmospheric optical turbulence (OT) measurements above Mount Shatdzhatmaz in 2007--2013 have been analysed. Linear auto-regressive integrated moving average (ARIMA) models are used for the forecasting. A new procedure for forecasting the image characteristics of direct astronomical observations (central image intensity, full width at half maximum, radius encircling 80% of the energy) has been proposed. Probability density functions of the forecast of these quantities are 1.5--2 times thinner than the respective unconditional probability density functions. Overall, this study found that the described technique could adequately describe temporal stochastic variations of the OT power.

Probabilistic-Based Available Transfer Capability Assessment Considering Existing and Future Wind Generation Resources

This paper presents a probabilistic-based approach for available transfer capability (ATC) assessment. A composite algorithm is developed to generate ensembles of future wind generation scenarios for the existing and planned wind sites using both measured and model-produced wind data. Then, the ensembles of wind and load are used to calculate their respective probability density functions (pdfs), which are subsequently used to calculate the probabilistic-based ATC for a selected transmission corridor. The method has been tested and validated using historical and operational data provided by the Idaho Power Co. The results show that the method can effectively quantify the uncertainties in the ATC assessment introduced by variable generation resources and load variations. As a result, the grid planners will inform the likelihood for the transmission corridor to exceed its transfer capacity in any targeted future years as well as the duration of such events.

Simplified probabilistic voltage stability evaluation considering variable renewable distributed generation in distribution systems

This study investigates the evaluation of static voltage stability using a two-point estimation method (2PEM) and continuation power flow (CPF) in distribution systems. The stochastic distribution generation (DG) units have been considered with their respective probability density function (PDF). The proposed static voltage stability evaluation method first chooses several sample points to replace the PDFs using 2PEM. Then, based on each sample point, the critical static voltage stability value is calculated by CPF. On the basis of the critical values corresponding to all selected sample points, Cornish–Fisher series are used to evaluate the PDF of the critical static voltage stability, and then the static voltage stability state can be obtained. The proposed method has been tested on IEEE 33-bus, PG&E 69-bus and a real case with two stochastic DG units. Comparisons have been made with the Monte Carlo simulation and the first-order second-moment method. The results show that the proposed method has better efficiency and accuracy.

Adaptive Covariance Matrix Estimation for Multi-Baseline InSAR Data Stacks

For many multidimensional applications of synthetic aperture radar (SAR) imaging, the estimation of the covariance matrix for each resolution cell is a critical processing step. The context of this work is the application of covariance matrix estimation for multi-baseline interferometric SAR data sets. In order to ensure local stationarity, which is needed for an unbiased estimation, adaptive techniques are necessary. In this paper, a new approach for adaptive covariance matrix estimation is proposed and evaluated based on measures known from the field of image processing. The procedure is centered around the idea of checking whether the neighboring pixels belong to the same statistical distribution as the currently investigated pixel by applying a threshold to the respective probability density function. All inlier pixels are then used to estimate the complex covariance matrix of the reference pixel. From this covariance matrix, both amplitude and interferometric phase values are extracted, which are then combined for all pixels in the stack in order to employ techniques for the evaluation of filtering efficiency that are typically used in image denoising research. It is found that the proposed algorithm provides high filtering efficiency and good detail preservation at the same time. Apart from that, it is found to be particularly suitable for small-sized stacks of coregistered SAR imagery.

A Data-based Method for Harmonising Heterogeneous Data Modelling Techniques Across Data Mining Applications

We propose an iterative graphical data visualisation algorithm for optimal model selection. The algorithm is implemented on three domain-partitioning techniques - decision trees, neural networks and support vector machines. Each model is trained and tested on the Pima Indians and Bupa Liver Disorders datasets with the performance being assessed in a multi-step process. Firstly, the conventional ROC curves and the Youden Index are applied to determine the optimal model then sequential moving differences involving the fitted parameters - true and false positives – are extracted and their respective probability density estimations are used to track their variability using the proposed algorithm. The algorithm allows the use of data-dependent density bandwidths as tuning parameters in determining class separation across applications. Our results suggest that this novel approach yields robust predictions and minimizes data obscurity and over-fitting. The algorithm’s simple mechanics which derive from the standard confusion matrix and built-in graphical data visualisation and adaptive bandwidth features make it multidisciplinary compliant and easily comprehensible to non-specialists. The paper’s main outcomes are two-fold. Firstly, it combines the power of domain partitioning techniques on Bayesian foundations with graphical data visualisation to provide a dynamic, discernible and comprehensible information representation. Secondly, it demonstrates that by converting mathematical formulation into visual objects, multi-disciplinary teams can jointly enhance the knowledge of concepts and positively contribute towards global consistency in the data-based characterisation of various phenomena across disciplines.

Toward a quantitative account of pitch distribution in spontaneous narrative: Method and validation

Pitch is well-known both to animate human discourse and to convey meaning in communication. The study of the statistical population distributions of pitch in discourse will undoubtedly benefit from methodological improvements. The current investigation examines a method that parameterizes pitch in discourse as musical pitch interval H measured in units of cents and that disaggregates the sequence of peak word-pitches using tools employed in time-series analysis and digital signal processing. The investigators test the proposed methodology by its application to distributions in pitch interval of the peak word-pitch (collectively called the discourse gamut) that occur in simulated and actual spontaneous emotive narratives obtained from 17 middle-aged African-American adults. The analysis, in rigorous tests, not only faithfully reproduced simulated distributions imbedded in realistic time series that drift and include pitch breaks, but the protocol also reveals that the empirical distributions exhibit a common hidden structure when normalized to a slowly varying mode (called the gamut root) of their respective probability density functions. Quantitative differences between narratives reveal the speakers' relative propensity for the use of pitch levels corresponding to elevated degrees of a discourse gamut (the "e-la") superimposed upon a continuum that conforms systematically to an asymmetric Laplace distribution.