Output Probability Density Function Research Articles

Emulators for stochastic simulation codes

Numerical simulation codes are very common tools to study complex phenomena, but they are often time-consuming and considered as black boxes. For some statistical studies (e.g. asset management, sensitivity analysis) or optimization problems (e.g. tuning of a molecular model), a high number of runs of such codes is needed. Therefore it is more convenient to build a fast-running approximation - or metamodel - of this code based on a design of experiments. The topic of this paper is the definition of metamodels for stochastic codes. Contrary to deterministic codes, stochastic codes can give different results when they are called several times with the same input. In this paper, two approaches are proposed to build a metamodel of the probability density function of a stochastic code output. The first one is based on kernel regression and the second one consists in decomposing the output density on a basis of well-chosen probability density functions, with a metamodel linking the coefficients and the input parameters. For the second approach, two types of decomposition are proposed, but no metamodel has been designed for the coefficients yet. This is a topic of future research. These methods are applied to two analytical models and three industrial cases.

Probabilistic Modeling of Partial Shading of Photovoltaic Arrays

The probabilistic modeling technique is used in this paper to ascertain the performance of energy conversion of a photovoltaic (PV) array under the influence of partial shading. Three cases of increasing complexity are studied in this paper. The first case is to find the probability density function (pdf) of the maximum power output (MPO) of a single PV module exposed to a random level of solar irradiance. Given that the probability distribution of solar irradiance is knowna priori, the pdf of the MPO of the PV module is derived analytically. A Monte Carlo simulation is then conducted to validate the analysis. This is followed in the second case by studying the MPO of an array composed of two series PV modules, each exposed to a random and independent level of irradiance. The third case further involves the effects of bypass diodes which are commonly installed to reduce partial shading losses.

Iterative Learning Double Closed-Loop Structure for Modeling and Controller Design of Output StochAstic Distribution Control Systems

Stochastic distribution control (SDC) systems are known to have the 2-D characteristics regarding time and probability space of a random variables, respectively. A double closed-loop structure, which includes iterative learning modeling (ILM) and iterative learning control (ILC), is proposed for non-Gaussian SDC systems. The ILM is arranged in the outer loop, which takes a longer period for each cycle termed as a BATCH. Each BATCH is divided into a modeling period and a number of control intervals, called batches, being arranged in the inner loop for ILC. The output probability density functions (PDFs) of the system are approximated by a radial basis function neural network (RBFNN) model, whose parameters are updated via ILM in each BATCH. Based on the RBFNN approximation of the output PDF, a state-space model is constructed by employing the subspace parameter estimation method. An IL optimal controller is then designed by decreasing the PDF tracking errors from batch to batch. Model simulations are carried out on a forth-order numerical example to examine the effectiveness of the proposed algorithm. To further assess its application feasibility, a flame shape distribution control simulation platform for a combustion process in a coal-fired gate boiler system is constructed by integrating WinCC interface, MATLAB simulation programs, and OPC communication together. The simulation study over this industrial simulation platform shows that the output PDF tracking performance can be efficiently achieved by this double closed-loop iterative learning strategy.

An Analytical Solution for Wind Farm Power Output

An ad-hoc probabilistic analytical solution for modeling wind farm power output is proposed in this letter. In accordance with the unit impulse function described in signals and systems engineering, and combined with the output power versus wind speed curve with wake effect consideration and the wind speed probability density function (PDF), the PDF of wind farm power output is deduced analytically. The comparisons with Monte Carlo method are given to demonstrate the validity of the proposed model.

Roughness and Discharge Uncertainty in 1D Water Level Calculations

In this study, we investigate the effect of two key uncertainty sources in one-dimensional (1D) water level calculations: the roughness coefficient and the upstream discharge. The work shows how these two uncertainties, separately and together, propagate through the hydraulic model and result in the uncertainty of water levels. The analysis is conducted for the case of uniform flow in rectangular channels and for steady gradually varied flow in real rivers. In the first case, the exact probability density functions (PDFs) of water levels are obtained analytically through the derived distribution method, while in the second case, the output PDFs are heuristically obtained via Monte Carlo simulations. The results show that (1) the water level PDFs have a lower coefficient of variation than the input PDFs due to the mathematical nature of the relationship between input and output; (2) the propagation of symmetric input distributions through the uniform and steady flow equations determines asymmetric output distributions, due to model nonlinearities. In particular, discharge uncertainty leads to left skewed water level PDFs while roughness uncertainty is responsible for output distributions with heavier right tails. Therefore, in the case of roughness uncertainty, the adoption of symmetrical PDFs would lead to underestimation of high quantiles; (3) water level calculations are more sensitive to uncertainty in the Strickler coefficient rather than in upstream discharge, when the two variables are characterised by the same level of uncertainty, and (4) roughness and discharge uncertainties have a significant effect on the predicted water levels, and they should not be neglected in the practical applications, such as flood forecasting, floodplain mapping and design of flood protection solutions.

A new robust ELM method based on a Bayesian framework with heavy-tailed distribution and weighted likelihood function

In actual industrial processes, the data used for modeling usually contains some outliers, which may lead to a corrupted approximation function. In this paper, we propose a new Robust Extreme Learning Machine method based on a Bayesian framework (RBELM). The main idea of RBELM is to replace the Gaussian distribution with a heavy-tailed distribution as the probability density function of the model output and give weights to the likelihood function based on the error of the model output, which makes the model more robust to outliers. Two different heavy-tailed distributions are used in this paper. One is the Laplace distribution, and the corresponding model is denoted as RBELM-l. In RBELM-l, the posterior distribution is replaced by an approximate surrogate function, and then the parameters learning problem can be solved by means of Maximum Likelihood-type II (ML-II). The other is the Student's t-distribution, and the corresponding model is denoted as RBELM-t. In order to solve RBELM-t, the Student's t-distribution is written as a scale-mixture of infinitely many Normal distributions and a Gamma distribution. Then, the variational inference method is used to obtain an approximate solution for the parameters. Also, we propose two methods for giving weights to the likelihood function based on the concepts of robust statistic. Finally, a robust error measure is proposed for evaluating the performance of models with outliers in the testing data. Results of numerical comparison based on one synthesis data set, four real benchmark regression problems, and two actual industrial modeling problems show the usefulness of the proposed RBELM-l and RBELM-t.

A new method for evaluating Borgonovo moment-independent importance measure with its application in an aircraft structure

The moment-independent importance measure proposed by Borgonovo, which is defined as the average shift between the unconditional and conditional probability density functions (PDFs) of model output, is widely used to evaluate the influence of input uncertainty on the entire output distribution. And how to exactly and efficiently estimate the PDFs remains a crucial and challenging problem. In this paper, a novel PDF estimation based method is proposed to efficiently evaluate the moment-independent index. Firstly, the PDF of the model output is obtained based on the concepts of maximum entropy, fractional moment and high dimensional model representation. Secondly, the Nataf transformation is utilized to estimate the joint PDF of the output and input variable. Finally, the index can be easily computed using the generated correlated standard normal samples. Thus the importance measure can be calculated with high efficiency and accuracy using this proposed composited method. Several examples are employed to demonstrate the advantages of the proposed method. Meanwhile, the importance analysis of a stiffening rib of the wing leading edge in a certain aircraft also verifies its good engineering applicability.

A Natural Gradient Algorithm for Stochastic Distribution Systems

In this paper, we propose a steepest descent algorithm based on the natural gradient to design the controller of an open-loop stochastic distribution control system (SDCS) of multi-input and single output with a stochastic noise. Since the control input vector decides the shape of the output probability density function (PDF), the purpose of the controller design is to select a proper control input vector, so that the output PDF of the SDCS can be as close as possible to the target PDF. In virtue of the statistical characterizations of the SDCS, a new framework based on a statistical manifold is proposed to formulate the control design of the input and output SDCSs. Here, the Kullback–Leibler divergence is presented as a cost function to measure the distance between the output PDF and the target PDF. Therefore, an iterative descent algorithm is provided, and the convergence of the algorithm is discussed, followed by an illustrative example of the effectiveness.

Robust tracking controller design for non-Gaussian singular uncertainty stochastic distribution systems

In this paper, the robust tracking control problem for uncertain singular stochastic distribution control (SDC) systems is considered. A new control target, where the distribution tracking error at each time instant satisfies a certain upper bound beyond a limited time, is proposed. This control target is different from the tracking control in the output SDC systems which makes the output probability density function (PDF) track a desired PDF as close as possible. Then an instant performance index instead of the infinite integration index is adopted, and the upper bound of this index is taken as the stability condition of a Lyapunov function to obtain a robust tracking controller via an augmentation control and linear matrix inequality (LMI). Simulations are also included to show the effectiveness of the proposed algorithm and encouraging results have been obtained.

Probability Density Function of Bubble Size Based Reagent Dosage Control for Flotation Process

AbstractAs an effective measurement of bubble stability, bubble size structure is believed to be closely related to flotation performance. Reagent dosage has a very important influence on bubble size. A novel probability density function (PDF) of bubble size based reagent dosage control method is proposed in the paper. First, froth images acquired by camera with big and tiny blobs are segmented by use of an improved two‐pass watershed algorithm. With non‐Gaussian features, the PDF of the bubble size is hence approximated by the estimator based on B‐spline such that the PDF of the bubble size is characterized by a weight vector of the B‐spline. Finally, a PDF‐based reagent dosage control algorithm is proposed to make the output PDF track the given PDF according to the linear matrix inequality technique, which is established to solve its stability condition by Lyapunov stability analysis. Experimental results show the effectiveness of the proposed method.

Blind equalization based on pdf fitting and convergence analysis

In this paper, we address M-QAM blind equalization by fitting the probability density functions (pdf) of the equalizer output with the constellation symbols. We propose two new cost functions, based on kernel pdf approximation, which force the pdf at the equalizer output to match the known constellation pdf. The kernel bandwidth of a Parzen estimator is updated during iterations to improve the convergence speed and to decrease the residual error of the algorithms. Unlike related existing techniques, the new algorithms measure the distance error between observed and assumed pdfs for the real and imaginary parts of the equalizer output separately. The advantage of proceeding this way is that the distributions show less modes, which facilitates equalizer convergence, while as for multi-modulus methods phase recovery keeps being preserved. The proposed approaches outperform CMA and classical pdf fitting methods in terms of convergence speed and residual error. We also analyse the convergence properties of the most efficient proposed equalizer via the ordinary differential equation (ODE) method.

Anti-disturbance fault diagnosis for non-Gaussian stochastic distribution systems with multiple disturbances

In this paper, an anti-disturbance fault diagnosis scheme is proposed for non-Gaussian stochastic distribution systems (SDSs) with multiple disturbances. The available driven information for fault diagnosis is probability density functions (PDFs) of output rather than output value. Using B-spline expansion technique, the output PDFs can be approximated in terms of dynamic weights of B-spline neural network by which a nonlinear model can be established between input and weights. Therefore, the concerned problem is transformed into fault diagnosis problem of the weighting system presented by an uncertain nonlinear system with multiple disturbances and time-varying fault. Different from most of the existing results, the multiple disturbances are supposed to include unknown disturbance modeled by an exo-system and norm bounded uncertain disturbances. In the proposed approach, a disturbance observer is designed to estimate and compensate the modeled disturbance, and H∞ optimization technology is applied to attenuate the norm bounded disturbance. Finally, simulation results are given to show the efficiency of the proposed approach.

Fast identification algorithms for Gaussian process model

A class of fast identification algorithms is introduced for Gaussian process (GP) models. The fundamental approach is to propose a new kernel function which leads to a covariance matrix with low rank, a property that is consequently exploited for computational efficiency for both model parameter estimation and model predictions. The objective of either maximizing the marginal likelihood or the Kullback–Leibler (K–L) divergence between the estimated output probability density function (pdf) and the true pdf has been used as respective cost functions. For each cost function, an efficient coordinate descent algorithm is proposed to estimate the kernel parameters using a one dimensional derivative free search, and noise variance using a fast gradient descent algorithm. Numerical examples are included to demonstrate the effectiveness of the new identification approaches.

Robust H<SUB align="right">∞; fault diagnosis for stochastic distribution systems with disturbance rejection performance

The task of robust fault diagnosis of stochastic distribution control (SDC) systems with unknown external disturbance is to use the measured input and the system output PDFs to still obtain possible faults information of the system. In this paper, an enhanced robust fault diagnosis scheme is presented for the non-Gaussian stochastic distribution systems (SDCs). The available driven information for fault diagnosis is the probability density functions (PDFs) rather than the real output value. Different from conventional FDD problems, the measured information for FDD is the output probability density functions and the stochastic variables involved are not conned to Gaussian ones. A B-spline neura l network model is established, where a static neural network is applied to model the output PDFs and a nonlinear dynamic model is used to describe the relationships between the input and the weight. The concerned problem is transformed into the fault diagnosis problem of the weighting system presented by a nonlinear system with unknown external disturbance. The composite observer for SDCs is constructed by combining a fault diagnosis observer with a disturbance observer, with which the fault can be diagnosed and the disturbance can be rejected simultaneously. At last, an illustrated example is given to demonstrate the effectiveness of the proposed algorithm, and satisfactory results have been obtained.

Wind Power Bidding Strategy Based on the Minimax Regret Criterion with Limited Distribution Information

In optimal wind bidding strategy related literatures, it is usually assumed that the full distribution information (for example, the cumulative distribution function or the probability density function) of wind power output is known. In real world applications, however, only very limited distribution information can be obtained. Therefore, the “optimal bidding strategy” obtained based on the hypothetical distribution may be far away from the true optimal one. In this paper, an optimal bidding strategy is obtained based on the minimax regret criterion. The salient feature of the new approach is that it requires only partial information of wind power distribution, for example, the expectation and the support set. Numerical test is then performed and the results suggest that the method established in this paper is effective.

A New PDF Modelling Algorithm and Predictive Controller Design

Output distribution control is required in many industrial processes mainly for the purpose of improving product qualities. Different from the traditional mean and variance control of stochastic processes, the probability density function (PDF) control provides a comprehensive solution to deal with outputs with general distributions. Various models based on B-splines have been developed to approximate the output PDF required for closed-loop control, among them the rational square-root (RSR) B-spline model can guarantee the nonnegativeness and the integration constraint of a PDF. In this paper, the relationship between the so-called actual weights and pseudo weights of the RSR B-spline PDF model is investigated so as explore a simplified modelling algorithm for the very complex nonlinear PDF modelling problem. Based on the proposed modelling algorithm, a predictive PDF control strategy has been established and applied to an exemplar system of closed-loop molecular weight distribution (MWD) control in a polymerisation process. The merit of predictive control over conventional PDF control is clearly demonstrated through the simulation study.

Fault diagnosis and fault tolerant control for the non-Gaussian time-delayed stochastic distribution control system

The purpose of fault diagnosis of stochastic distribution control (SDC) systems is to use the measured input and the system output probability density functions (PDFs) to obtain the fault information of the SDC system. When the target PDF is known, the purpose of fault tolerant control of stochastic distribution control system is to make the output PDF still track the given distribution using the fault tolerant controller. However, in practice, time delay may exist in the data (or image) processing, the modeling and transmission phases. When time delay is not considered, the effectiveness of the fault detection, diagnosis and fault tolerant control of stochastic distribution systems will be reduced. In this paper, the rational square-root B-spline is used to approach the output probability density function. In order to diagnose the fault in the dynamic part of such systems, it is then followed by the novel design of a nonlinear neural network observer-based fault diagnosis algorithm. The time delay term will be deleted in the stability proof of the observation error dynamic system. Based on the fault diagnosis information, a new fault tolerant controller based on PI tracking control is designed to make the post-fault probability density function still track the given distribution, which is dependent of the time delay term. Finally, simulations for the particle distribution control problem are given to show the effectiveness of the proposed approach.

A Guaranteed Cost Control for Stochastic Distribution System with Time-delay

This article investigates the guaranteed cost control problem for the stochastic distribution system with time-delay. Attention is focused on a memory state feedback control law based on Linear Matrix Inequality (LMI) with model transformation such that the closed-loop system is asymptotically stable and the guaranteed cost index is not more than a specied upper bound, which ensure that the system output Probability Density Function (PDF) follows the target probability density function. A comparative study of an example problem has show to illustrate the proposed method.

A 10-state model for an AMC scheme with repetition coding in mobile wireless networks

Abstract In modern broadband wireless access systems such as mobile worldwide interoperability for microwave access (WiMAX) and others, repetition coding is recommended for the lowest modulation level, in addition to the mandatory concatenated Reed-Solomon and convolutional code data coding, to protect vital control information from deep fades. This paper considers repetition coding as a time-diversity technique using maximum ratio combining (MRC) and proposes techniques to define and to calculate the repetition coding gain G r and its effect on bit error rate (BER) under the two fading conditions: correlated lognormal shadowing and composite Rayleigh-lognormal fading also known as Suzuki fading. A variable-rate, variable-power 10-state finite-state Markov channel (FSMC) model is proposed for the implementation of the adaptive modulation and coding (AMC) scheme in mobile WiMAX to maximize its spectral efficiency under constant power constraints in the two fading mechanisms. Apart from the proposed FSMC model, the paper also presents two other significant contributions: one is an innovative technique for accurate matching of moment generating functions, necessary for the estimation of the probability density function of the combiner's output signal-to-noise ratio, and the other is efficient and fast expressions using Gauss-Hermite quadrature approximation for the calculation of BER of QPSK signal using MRC diversity reception.

Natural gradient algorithm for stochastic distribution systems with output feedback

In this paper, we use natural gradient algorithm to control the shape of the conditional output probability density function for the stochastic distribution systems from the viewpoint of information geometry. The considered system here is of multi-input and single output with an output feedback and a stochastic noise. Based on the assumption that the probability density function of the stochastic noise is known, we obtain the conditional output probability density function whose shape is only determined by the control input vector under the condition that the output feedback is known at any sample time. The set of all the conditional output probability density functions forms a statistical manifold (M), and the control input vector and the output feedback are considered as the coordinate system. The Kullback divergence acts as the distance between the conditional output probability density function and the target probability density function. Thus, an iterative formula for the control input vector is proposed in the sense of information geometry. Meanwhile, we consider the convergence of the presented algorithm. At last, an illustrative example is utilized to demonstrate the effectiveness of the algorithm.