Estimation Of Probability Density Function Research Articles

A maximum entropy model with fractional moments for probability density function estimation of wind pressures on low-rise building

Due to the fickle distribution characteristics, good approximation of the wind pressure probability density function for an entire roof is challenging, especially for tail region. Maximum entropy model (MEM) can theoretically generate the least biased distribution. However, classical MEM may lead to unreliable estimates of the corresponding integer moments. This study developed a new maximum entropy model with fractional moments. There are several important features: A new translation function is introduced, thus allowing negative data to be modeled. For a good bias-variance trade-off, the constraint number M was fixed as four. After estimating the initial value of the Lagrange multiplier λ by a computationally simple linear equation system, a simple search was carried out to find a better solution. The generalized pattern search was adopted for determining the global optimal solution for the fractional moment orders α. The performance was benchmarked through typical field measurement data of wind pressures on the roof of a low-rise building under typhoon conditions. Compared with common models, the proposed method had better performance and more stable results for the dataset examined for the whole roof. This method is also beneficial for peak value evaluation and the simulation of wind pressures.

Stochastic Energy Management of Electric Bus Charging Stations With Renewable Energy Integration and B2G Capabilities

In this paper, the stochastic energy management of electric bus charging stations (EBCSs) is investigated, where the photovoltaic (PV) with integrated battery energy storage systems (BESS) and bus-to-grid (B2G) capabilities of electric buses (EBs) are included for cost-effective charging of EBs. Also, the day-ahead dynamic prices are derived to mitigate charging impacts on power distribution systems. This problem is formulated as a distributionally robust Markov decision process (DRMDP) with uncertain transition probabilities and costs to address the impacts of random bus loads with inaccurate probability density function estimation. An event-based ambiguity set with combined statistical distance and moment information is developed to achieve minimax-regret criteria for less-conservative and robust solutions. To facilitate practical applications with reduced computational complexity, a heuristic regret function is proposed, based on which the dynamic prices are derived. Case studies based on EB data from St. Albert Transit and IEEE test feeders indicate that the proposed method can minimize EB charging cost with mitigated impacts on power distribution systems.

Stationary distribution and probability density function of a stochastic SIRSI epidemic model with saturation incidence rate and logistic growth

Focusing on the results of Rajasekar (2020) and the continuous dynamics of stochastic differential equation (SDE) developed by Mao (1997), a stochastic SIRSI epidemic model with saturation incidence rate and logistic growth is investigated in this paper. First, we propose and prove that the unique solution of stochastic model is globally positive. By constructing some suitable Lyapunov functions, the sufficient condition R0h>1 is obtained for the unique stationary distribution which has ergodicity property. Next, by solving the corresponding Fokker-Planck equation, we derive the approximate probability density function around the quasi-endemic equilibrium of the stochastic system. The above stationary distribution and density function can reveal all statistical properties of the disease persistence. In addition, by comparison with other existing articles, our developed theoretical results and some numerical simulations are introduced at the end of this paper.

A novel mobile agent-based distributed evidential expectation maximization algorithm for uncertain sensor networks

In this paper, a novel mobile agent-based distributed evidential expectation maximization (MADEEM) algorithm is presented for sensor networks. The proposed algorithm is used for probability density function estimation and data clustering in the presence of uncertainties in sensor measurements. It is assumed that the sensor measurements are statistically modeled by a common Gaussian mixture model. The proposed algorithm maximizes a new generalized likelihood criterion in an iterative and distributed manner. For this purpose, mobile agents compute some local sufficient statistics by using local measurements of each sensor node. After the local computations, the global sufficient statistics are updated. At the end of iterations, the parameters of the probability density function are updated by using the global sufficient statistics. The mentioned process will be continued until the convergence criterion is satisfied. Convergence analysis of the proposed algorithm is also presented in this paper. After the convergence analysis, the simulation results show the promising performance of the proposed algorithm. Finally, the last part of the paper is devoted to the concluding remarks.

Application and Comprehensive Analysis of Neighbor Approximated Information Theoretic Configurational Entropy Methods to Protein-Ligand Binding Cases.

The binding entropy is an important thermodynamic quantity which has numerous applications in studies of the biophysical process, and configurational entropy is often one of the major contributors in it. Therefore, its accurate estimation is important, though it is challenging mostly due to sampling limitations, anharmonicity, and multimodality of atomic fluctuations. The present work reports a Neighbor Approximated Maximum Information Spanning Tree (A-MIST) method for conformational entropy and presents its performance and computational advantage over conventional Mutual Information Expansion (MIE) and Maximum Information Spanning Tree (MIST) for two protein-ligand binding cases: indirubin-5-sulfonate to Plasmodium falciparum Protein Kinase 5 (PfPK5) and P. falciparum RON2-peptide to P. falciparum Apical Membrane Antigen 1 (PfAMA1). Important structural regions considering binding configurational entropy are identified, and physical origins for such are discussed. A thorough performance evaluation is done of a set of four entropy estimators (Maximum Likelihood (ML), Miller-Madow (MM), Chao-Shen (CS), and James and Stein shrinkage (JS)) with known varying degrees of sensitivity of the entropy estimate on the extent of sampling, each with two schemes for discretization of fluctuation data of Degrees of Freedom (DFs) to estimate Probability Density Functions (PDFs). Our comprehensive evaluation of influences of variations of parameters shows Neighbor Approximated MIE (A-MIE) outperforms MIE in terms of convergence and computational efficiency. In the case of A-MIE/MIE, results are sensitive to the choice of root atoms, graph search algorithm used for the Bond-Angle-Torsion (BAT) conversion, and entropy estimator, while A-MIST/MIST are not. A-MIST yields binding entropy within 0.5 kcal/mol of MIST with only 20-30% computation. Moreover, all these methods have been implemented in an OpenMP/MPI hybrid parallel C++11 code, and also a python package for data preprocessing and entropy contribution analysis is developed and made available. A comparative analysis of features of current implementation and existing tools is also presented.

Prediction of Elastic Properties Within CO2 Plume at Sleipner Field Using AVS Inversion Modified for Thin‐Layer Reflections Guided by Uncertainty Estimation

AbstractExisting amplitude variation with offset (AVO) and slowness (AVS) theories fail to interpret observed amplitudes in terms of actual elastic properties of the media for a reflection event resulting from a stack of thin layers. We propose a method to replace the stack of thin layers with an equivalent medium of elastic properties using the Backus averaging theory. Numerical examples show considerable deviation in intercept calculated from the modified AVO and AVS theories from the conventional methods. AVS theory is preferred as it can be applied irrespective of impedance contrast in both precritical and postcritical reflections, and no spherical divergence correction is required. P and S wave velocities, density, and thickness of thin layers are determined using the inversion scheme of very fast simulated annealing (VFSA). Uncertainty in prediction is evaluated using an approximate marginal posterior probability density function and a parameter correlation matrix. We demonstrate our methodology on synthetic and real seismic data from the Sleipner field, which is a good example of enhanced seismic amplitudes due to interference among reflections from thin layers. We select 11 common depth point (CDP) gathers for amplitude analysis of four identified reflectors along a line of 3‐D seismic data acquired in 2008. Predicted mean models of CO2‐saturated sand layers at all locations show that the thickness, density, and P and S wave velocities vary from 5–7 m, 1.8–2.0 g/cm3, and 1,460–1,490 and 630–650 m/s, respectively. Our analysis suggests that the model parameters are well constrained and independent except at a few locations.

Estimating of Survival Function under Type One Censoring Sample for Mixture Distribution

In this article, it is interesting to estimate and derive the three parameters which contain two scales parameters and one shape parameter of a new mixture distribution for the singly type one censored data which is the branch of right censored sample. Then to define some special mathematical and statistical properties for this new mixture distribution which is considered one of the continuous distributions characterized by its flexibility. Next, using maximum likelihood estimator method for singly type one censored data based on the Newton-Raphson matrix procedure to find and estimate values of these three parameter by utilizing the real data taken from the National Center for Research and Treatment of Hematology/University of Mustansiriyah for leukemia diseases. After that we find and derive the estimate of probability density function, estimate survival function and finally estimate the hazard function.

Compositional splines for representation of density functions

In the context of functional data analysis, probability density functions as non-negative functions are characterized by specific properties of scale invariance and relative scale which enable to represent them with the unit integral constraint without loss of information. On the other hand, all these properties are a challenge when the densities need to be approximated with spline functions, including construction of the respective spline basis. The Bayes space methodology of density functions enables to express them as real functions in the standard $$L^2$$ space using the centered log-ratio transformation. The resulting functions satisfy the zero integral constraint. This is a key to propose a new spline basis, holding the same property, and consequently to build a new class of spline functions, called compositional splines, which can approximate probability density functions in a consistent way. The paper provides also construction of smoothing compositional splines and possible orthonormalization of the spline basis which might be useful in some applications. Finally, statistical processing of densities using the new approximation tool is demonstrated in case of simplicial functional principal component analysis with anthropometric data.

Generic evaluation of FSO system over Málaga turbulence channel with MPPM and non‐zero‐boresight pointing errors

Free space optical (FSO) communication channels are affected by fluctuations in irradiance due to atmospheric turbulence and pointing errors. Recently, a generalized statistical model knows as Malaga ( M ) was developed to describe irradiance fluctuations of the beam propagating through a turbulent medium. In this paper, an approximate finite-series probability density function (PDF) for composite M turbulence with pointing errors is verified. Considering multiple pulse-position-modulation (MPPM) with intensity modulation and direct detection, specific closed-form expressions for average symbol error rate (ASER) and outage probability are obtained. Moreover, asymptotic expressions for ASER and outage probability over the Malaga channel with pointing errors are presented. Finally, ASER and outage probability expressions are deduced in three turbulence regimes, strong, moderate, and weak. Therefore, simulation results are given to confirm the accuracy of the proposed expressions. The results show that the optimum beam divergence angle is 1 mrad for zero-boresight displacement and 2 mrad for nonzero-boresight at a distance of 1 km. By increasing the range to 5 km, atmospheric turbulence is found to have the dominant effect over boresight displacement. Moreover, the system performance at the optimized divergence angle is studied in terms of transmit power, normalized boresight, and link distance.

Bayesian selector of adaptive bandwidth for gamma kernel density estimator on simulations and applications

Bayesian bandwidth selections in continuous associated kernel estimation of probability density function are very good alternatives to classical methods like cross-validation techniques. In this paper, we examined the behavior of Bayesian variable bandwidths in gamma kernel estimation, developed theoretically in Wansouwé et al. [Ake: An R Package for Discrete and Continuous Associated Kernel Estimations, The R journal 8 (2016), pp. 259–276], and appropriated to smooth densities of support Simulations studies point out remarkable performance of the proposed approach, comparing to the global cross-validation bandwidth selection, and under integrated squared errors. Two applications related to CO 2 emissions and medical bills of bodily injury claims, respectively, are finally made.

Damage diagnosis and prognosis in composite double cantilever beam coupons by particle filtering and surrogate modelling

Delamination is a failure mechanism which is intrinsic of laminated fibre-reinforced plastics and possibly one of the major concerns of laminated composite structures, since, under certain conditions, delaminations can grow up to an hazardous extent without visible traces. In order to keep pace with recent condition-based maintenance requirements, proper validated diagnostic and prognostic methods which should be capable of operating on-line and in real time are required. In this respect, particle filters provide a consistent Bayesian framework, where the posterior distribution of the system degradation status is recursively approximated based on a time-growing stream of observations measuring the system response. However, the real-time operation capability of such methods is hindered by their requirements in terms of analysis time, which is mainly due to the complexity of the models they rely upon. Within this work, a particle filter framework, able to deal with the inherent stochasticity of fatigue delamination growth – while simultaneously relieving the computational burden associated with the evaluation of the trajectory likelihoods – is provided, leveraging on surrogate modelling strategies. Simultaneous diagnosis and prognosis of a simulated carbon fibre-reinforced plastics double cantilever beam specimen subject to fatigue delamination growth are performed, based on the observation of the strain field pattern acquired at some specific locations. The posterior probability density function of the delamination extent during propagation is updated at each inspection time as well as the probability density function of the remaining useful life. Ultimately, the adoption of the augmented state formulation allows for the estimation and updating of the joint probability density function of the parameters driving the stochastic delamination propagation model. Results demonstrate the feasibility and potential of the proposed approach as a tool able to monitor the progressing delamination while simultaneously providing estimates about the remaining useful life of composite structures.

Nakagami's Approximation for Distribution of Signal Envelope in TWDP Channel

Problem statement. With the development of radio communication technologies, especially mobile, the choice of radio channel models is playing an increasingly important role. In this paper, one of the currently popular models is considered, namely, a Two-Wave radio propagation with Diffuse Power scattering, which is abbreviated in the literature as TWDP. For this model, both exact and approximate expressions for the Probability Density Function (PDF) and the Cumulative Distribution Function (CDF) of the signal envelope are known. However, their common disadvantage is the high complexity in terms of practical application. In this regard, this paper considers the possibility of a much simpler approximation of the PDF and CDF of a signal envelope based on the Nakagami’s distribution. Method development. The approximation of distributions by the Nakagami method comes down to the calculation of the second and fourth moments of a signal envelope at a channel output. Such calculations are performed in the paper, which allows to represent the specified moments as functions of two parameters of the TWDP channel: the K-factor and Δ-parameter depending from the ratio of beams amplitudes.Analysis of the approximation errors. Errors were estimated by comparing the calculations for the PDF and CDF as per exact formulas and calculations using the Nakagami approximation. In parallel, calculations were also carried out as per the approximation formulas known in the literature. The analysis showed that the Nakagami method gives errors in the calculation of the CDF that are acceptable for practical needs in all ranges of parameters values K and Δ. At the same time, the accuracy of the PDF approximation in comparison with other known methods was not very satisfactory.Conclusions. Studies carried out in this paper have confirmed the possibility of approximating the CDF of the signal envelope in the TWDP channel by the Nakagami distribution. However, the results of such approximation for the PDF were less satisfactory. The usefulness of the obtained results for the interpretation of the data of field experiments is also noted.

Uncertainty Analysis of Greenhouse Gas (GHG) Emissions Simulated by the Parametric Monte Carlo Simulation and Nonparametric Bootstrap Method

Uncertainty of greenhouse gas (GHG) emissions was analyzed using the parametric Monte Carlo simulation (MCS) method and the non-parametric bootstrap method. There was a certain number of observations required of a dataset before GHG emissions reached an asymptotic value. Treating a coefficient (i.e., GHG emission factor) as a random variable did not alter the mean; however, it yielded higher uncertainty of GHG emissions compared to the case when treating a coefficient constant. The non-parametric bootstrap method reduces the variance of GHG. A mathematical model for estimating GHG emissions should treat the GHG emission factor as a random variable. When the estimated probability density function (PDF) of the original dataset is incorrect, the nonparametric bootstrap method, not the parametric MCS method, should be the method of choice for the uncertainty analysis of GHG emissions.

Nonparametric maximum likelihood estimation using neural networks

Abstract Estimation of probability density functions is an essential component of various applications. Nonparametric techniques have been widely used for this task owing to the difficulty in parameterization of data. In particular, certain kernel density estimation methods have been developed. However, they are either incapable of maximum likelihood estimation or require the maintenance of a training set to process new patterns. In this study, a new approach, called the nonparametric maximum likelihood neural network (MLNN), is proposed. This is a nonparametric method, relying on maximum likelihood and neural network. It is compact in form and does not require the maintenance of training patterns. Theoretical and experimental analyses demonstrate the efficacy of the proposed approach.

A kernel discriminant analysis for spatially dependent data

We propose a novel supervised classification algorithm for spatially dependent data, built as an extension of kernel discriminant analysis, that we named Spatial Kernel Discriminant Analysis (SKDA). Our algorithm is based on a kernel estimate of the spatial probability density function, which integrates a second kernel to take into account spatial dependency of data. In fact, classical data mining algorithms assume that data samples are independent and identically distributed. However, this assumption is not verified when dealing with spatial data characterized by spatial autocorrelation phenomenon. To make an accurate analysis, it is necessary to exploit this rich source of information and to capture this property. We have applied our algorithm to a relevant domain, which consist of the classification of remotely sensed hyperspectral images. In order to assess the efficiency of our proposed method, we conducted experiments on two remotely sensed images datasets (Indian Pines and Pavia University) with different characteristics and scenarios. The experimental results show that our method is competitive and achieves higher classification accuracy compared to other contextual classification methods.

The Quijote Simulations

The Quijote simulations are a set of 44,100 full N-body simulations spanning more than 7,000 cosmological models in the $\{\Omega_{\rm m}, \Omega_{\rm b}, h, n_s, \sigma_8, M_\nu, w \}$ hyperplane. At a single redshift the simulations contain more than 8.5 trillions of particles over a combined volume of 44,100 $(h^{-1}{\rm Gpc})^3$; each simulation follow the evolution of $256^3$, $512^3$ or $1024^3$ particles in a box of $1~h^{-1}{\rm Gpc}$ length. Billions of dark matter halos and cosmic voids have been identified in the simulations, whose runs required more than 35 million core hours. The Quijote simulations have been designed for two main purposes: 1) to quantify the information content on cosmological observables, and 2) to provide enough data to train machine learning algorithms. In this paper we describe the simulations and show a few of their applications. We also release the Petabyte of data generated, comprising hundreds of thousands of simulation snapshots at multiple redshifts, halo and void catalogs, together with millions of summary statistics such as power spectra, bispectra, correlation functions, marked power spectra, and estimated probability density functions.

A study on performance improvement of IMDD-UFMC with modified K-means non-uniform quantization

In this paper, a modified K-means-based low complexity the nonuniform quantization scheme is proposed in the intensity-modulated direct detection universal filter multicarrier (IMDD-UFMC) system to improve system performance. Owing to the good cluster property of our scheme, the optimal quantization levels can be directly figured out, and the complexity induced by the probability density function (PDF) estimation can be reduced. By utilizing this scheme, the low quantization noise and peak-to-average power ratio (PAPR) can be achieved by assigning finer quantization steps for the low-level area signals. And, to manifest this scheme, the IMDD-UFMC-based numerical simulation and experimental platform with 10-Gb/s 16-QAM signal over 20 km fiber are constructed. The corresponding results show that our method always achieves the better performance, measuring in the form with the error-vector-magnitudes and bit error rate, compared to the common uniform quantization scheme even with the low-bit resolution case.

Dimension Reduction Based Non-Parametric Disaggregation for Dependence Modeling in Composite System Reliability Evaluation

The correlated variation of bus loads and wind powers has posed a challenge to power system security. How to construct an accurate dependence model becomes vital for power system reliability evaluation. Most currently used methods in estimating the multivariate probability density function (PDF) not only ignore the aggregation constraint but also encounter the curse of dimensionality. In this paper, a new non-parametric disaggregation (ND) technique is presented to disaggregate an aggregate variable (e.g. system load) into non-aggregate variables (e.g. bus loads) while meeting the aggregation constraint. An innovative hierarchical dimension reduction technique is proposed and integrated into the ND model to transform a high-dimensional conditional PDF estimation into some low-dimensional estimations, which effectively resolves the difficulty associated with high dimensionality. The above two techniques are incorporated into an effective three-stage conditional sampling method to form the proposed dimension reduction based ND model (DRND) for high-dimensional dependence modeling in composite system reliability evaluation. The validity of the proposed DRND is verified by three test systems, i.e. MRTS79, MRTS96, and the latest RTS-GMLC.

A distributed averaging-based evidential Expectation-Maximization algorithm for density estimation in unreliable sensor networks

Abstract In this paper, a novel distributed averaging-based evidential Expectation-Maximization algorithm is presented for data clustering and probability density function estimation in unreliable sensor networks in the presence of uncertain observations. All observations of the sensor network are statistically modelled by a Gaussian mixture model. A generalized likelihood function is maximized in a distributed manner for density estimation. For this purpose, the parameters of the mixture model such as weighting coefficients, mean vectors, and covariance matrices are updated by using a distributed averaging method. The proposed algorithm can be applied to any network with a mesh topology. The convergence of algorithm is also investigated and guaranteed by using a mathematical analysis. In the simulation result section, the proposed algorithm has been applied to a simulated uncertain sensor network with a mesh topology. The simulations confirm the favourable performance of the distributed averaging-based evidential EM algorithm.

Time-variant reliability analysis via approximation of the first-crossing PDF

Time-variant reliability analysis can effectively estimate the safe state of structures under dynamic uncertainties during their lifecycle. However, one of its key challenging issues is computational efficiency. To improve the efficiency, this paper develops an approximation of the first-crossing probability density function (PDF) method, termed AFC-PDFM, based on the first-crossing theory. The response surface of the first-crossing time point (FCTP) about input variables is first estimated by using the moving least squares method (MLSM) and uniform design. Furthermore, an iterative algorithm is developed to compute the origin moments of the FCTP based on the MLSM response surface. Finally, the PDF of the FCTP is obtained by combining the origin moments and the maximum entropy method to avoid direct calculations of the first-crossing rate, which is often intractable, and then the time-variant reliability is estimated. Three examples are implemented to testify and validate the effectiveness of the proposed method and potential applications to engineering problems.