Reference Probability Distribution Research Articles

Risk-based distributionally robust optimal dispatch for multiple cascading failures in regional integrated energy system using surrogate modeling

Regional integrated energy system (RIES) with the close coupling of multiple energies faces various cascading failure risks in its operation, and formulating the risk-based optimal dispatch scheme is important to ensure the secure and economic operation of the RIES. Considering that the occurrence probability of each subsequent failure path caused by the initial failure in the cascading failure chain varies with the change of dispatch scheme, the cooling/heating and gas pipeline dynamics and the uncertainty of renewable energy, a risk-based distributionally robust optimal dispatch model for multiple cascading failures in a RIES is proposed. A kernel ambiguity set is constructed by using the maximum mean discrepancy between the real and reference probability distributions and the kernel mean embedding technology, which can obtain the more accurate worst-case probability distribution. Moreover, a surrogate modeling method based on piecewise convex quadratic functions is proposed to learn the mapping between the cascading failure risks and the decision variables, and the original large-scale mixed integer nonlinear programming model is transformed into a small-scale mixed integer quadratic constrained programming model for efficient solution. Finally, case study on an actual RIES demonstrates the effectiveness and advantages of the proposed method. Compared to optimal dispatch without considering the cascading failure risks, the proposed method can reduce the total failure risk cost by nearly 60%.

Fault detection strategy combining NARMAX model and Bhattacharyya distance for process monitoring

The complexity of modern chemical and petrochemical plants is becoming increasingly problematic in the recent years. At the same time, the demands to ensure safety and reliability of process operations rise. Early detection of abnormal event in complex real systems decrease maintenance cost and lead to guarantee the safety of human operators and environment. In the present work, a fault detection (FD) method which exploits the advantages of black-box modeling and statistical measure for fault detection in real chemical process as a distillation column is proposed. This technique is developed by applying the Nonlinear Auto-Regressive Moving Average with eXogenous input (NARMAX) model and Bhattacharyya distance (BD). In order to determine the NARMAX model, a real data set recorded during normal operations is used. Then, the BD is used to quantify on-line the dissimilarity between the current and reference probability distributions of the residual obtained from the NARMAX model for fault detection purposes. The ability of the proposed FD approach is demonstrated using real fault of separation unit. The obtained results indicate that the developed technique produces favorable performance compared to the conventional Cumulative Sum (CUSUM) test.

Efficient ab initio calculation of electronic stopping in disordered systems via geometry pre-sampling: Application to liquid water

Knowledge of the electronic stopping curve for swift ions, Se(v), particularly around the Bragg peak, is important for understanding radiation damage. Experimentally, however, the determination of such a feature for light ions is very challenging, especially in disordered systems such as liquid water and biological tissue. Recent developments in real-time time-dependent density functional theory (rt-TDDFT) have enabled the calculation of Se(v) along nm-sized trajectories. However, it is still a challenge to obtain a meaningful statistically averaged Se(v) that can be compared to observations. In this work, taking advantage of the correlation between the local electronic structure probed by the projectile and the distance from the projectile to the atoms in the target, we devise a trajectory pre-sampling scheme to select, geometrically, a small set of short trajectories to accelerate the convergence of the averaged Se(v) computed via rt-TDDFT. For protons in liquid water, we first calculate the reference probability distribution function (PDF) for the distance from the proton to the closest oxygen atom, ϕR(rp→O), for a trajectory of a length similar to those sampled experimentally. Then, short trajectories are sequentially selected so that the accumulated PDF reproduces ϕR(rp→O) to increasingly high accuracy. Using these pre-sampled trajectories, we demonstrate that the averaged Se(vp) converges in the whole velocity range with less than eight trajectories, while other averaging methods using randomly and uniformly distributed trajectories require approximately ten times the computational effort. This allows us to compare the Se(vp) curve to experimental data and assess widely used empirical tables based on Bragg’s rule.

Probability distributions control for multi-input multi-output stationary non-Gaussian random vibration test

An inverse system method for multi-input multi-output stationary non-Gaussian random vibration test is proposed in the paper to control the response characteristics both in time-domain and frequency-domain simultaneously. Hence, the control objectives are not only the traditional response power spectral densities but also probability distributions. The control of the probability distributions is more comprehensive and reasonable than the control of kurtoses in order to duplicate the time-domain characteristics of the measured data because kurtoses provide only partial behaviors of the probability distributions. The nonlinear transformation process is introduced to generate non-Gaussian random signals with specified probability distributions. To obtain the desired vibration environments for the test, the reference response signals are synthesized first by the reference spectra and probability distributions, and then the coupled drive signals are generated via the inverse system in the time domain. The close loop correction algorithms are implemented to update the drive signals, actually to update the reference response signals according to the deviations between the references and the measured responses. Finally, a numerical example by a cantilever beam and a biaxial vibration test are carried out and the results demonstrate the effectiveness and feasibility of the proposed method.

Improved nonlinear fault detection strategy based on the Hellinger distance metric: Plug flow reactor monitoring

Fault detection has a vital role in the process industry to enhance productivity, efficiency, and safety, and to avoid expensive maintenance. This paper proposes an innovative multivariate fault detection method that can be used for monitoring nonlinear processes. The proposed method merges advantages of nonlinear projection to latent structures (NLPLS) modeling and those of Hellinger distance (HD) metric to identify abnormal changes in highly correlated multivariate data. Specifically, the HD is used to quantify the dissimilarity between current NLPLS-based residual and reference probability distributions obtained using fault-free data. Furthermore, to enhance further the robustness of these methods to measurement noise, and reduce the false alarms due to modeling errors, wavelet-based multiscale filtering of residuals is used before the application of the HD-based monitoring scheme. The performances of the developed NLPLS-HD fault detection technique is illustrated using simulated plug flow reactor data. The results show that the proposed method provides favorable performance for detection of faults compared to the conventional NLPLS method.

Kullback-Leibler distance-based enhanced detection of incipient anomalies

Accurate and effective anomaly detection and diagnosis of modern engineering systems by monitoring processes ensure reliability and safety of a product while maintaining desired quality. In this paper, an innovative method based on Kullback-Leibler divergence for detecting incipient anomalies in highly correlated multivariate data is presented. We use a partial least square (PLS) method as a modeling framework and a symmetrized Kullback-Leibler distance (KLD) as an anomaly indicator, where it is used to quantify the dissimilarity between current PLS-based residual and reference probability distributions obtained using fault-free data. Furthermore, this paper reports the development of two monitoring charts based on the KLD. The first approach is a KLD-Shewhart chart, where the Shewhart monitoring chart with a three sigma rule is used to monitor the KLD of the response variables residuals from the PLS model. The second approach integrates the KLD statistic into the exponentially weighted moving average monitoring chart. The performance of the PLS-based KLD anomaly-detection methods is illustrated and compared to that of conventional PLS-based anomaly detection methods. Using synthetic data and simulated distillation column data, we demonstrate the greater sensitivity and effectiveness of the developed method over the conventional PLS-based methods, especially when data are highly correlated and small anomalies are of interest. Results indicate that the proposed chart is a very promising KLD-based method because KLD-based charts are, in practice, designed to detect small shifts in process parameters.

Information infrastructure risk prediction through platform vulnerability analysis

The protection of information infrastructures is important for the function of other infrastructure sectors. As vital parts for the information infrastructure operation, software-based platforms, face a series of vulnerabilities and threats. This paper aims to provide a complementary approach to existing vulnerability prediction solutions and launch the measurement of zero-day risk by introducing a risk prediction methodology for an information infrastructure. The proposed methodology consists of four steps and utilizes the outcomes of a proper analysis of security measurements provided by specifications from the Security Content Automation Protocol. First, we identify software platform assets that support an information infrastructure and second we measure the historical rate of vulnerability occurrences. Third, we use a distribution fitting procedure to estimate the statistical correlation between empirical and reference probability distributions and verify the statistical significance of the distribution fitting results with the Kolmogorov–-Smirnov test. Fourth, we develop conditional probability tables that constitute a Bayesian Belief Network topology as means to enable risk prediction and estimation on security properties. The practicality of the risk prediction methodology is demonstrated with an implementation example from the electronic banking sector. The contribution of the proposed methodology is to provide auditors with a proactive approach about zero-day risks.

Family of probability distributions derived from maximal entropy principle with scale invariant restrictions

Using statistical thermodynamics, we derive a general expression of the stationary probability distribution for thermodynamic systems driven out of equilibrium by several thermodynamic forces. The local equilibrium is defined by imposing the minimum entropy production and the maximum entropy principle under the scale invariance restrictions. The obtained probability distribution presents a singularity that has immediate physical interpretation in terms of the intermittency models. The derived reference probability distribution function is interpreted as time and ensemble average of the real physical one. A generic family of stochastic processes describing noise-driven intermittency, where the stationary density distribution coincides exactly with the one resulted from entropy maximization, is presented.

A Multicriteria Statistical Based Comparison Methodology for Evaluating Evolutionary Algorithms

This paper presents a statistical based comparison methodology for performing evolutionary algorithm comparison under multiple merit criteria. The analysis of each criterion is based on the progressive construction of a ranking of the algorithms under analysis, with the determination of significance levels for each ranking step. The multicriteria analysis is based on the aggregation of the different criteria rankings via a non-dominance analysis which indicates the algorithms which constitute the efficient set. In order to avoid correlation effects, a principal component analysis pre-processing is performed. Bootstrapping techniques allow the evaluation of merit criteria data with arbitrary probability distribution functions. The algorithm ranking in each criterion is built progressively, using either ANOVA or first order stochastic dominance. The resulting ranking is checked using a permutation test which detects possible inconsistencies in the ranking—leading to the execution of more algorithm runs which refine the ranking confidence. As a by-product, the permutation test also delivers <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$p$</tex></formula> -values for the ordering between each two algorithms which have adjacent rank positions. A comparison of the proposed method with other methodologies has been performed using reference probability distribution functions (PDFs). The proposed methodology has always reached the correct ranking with less samples and, in the case of non-Gaussian PDFs, the proposed methodology has worked well, while the other methods have not been able even to detect some PDF differences. The application of the proposed method is illustrated in benchmark problems.

Real and spurious contributions for the Shannon, Rényi and Tsallis entropies

A two-parameter probability distribution is constructed by dilatation (or contraction) of the escort probability distribution. This transformation involves a physical probability distribution P associated with the system under study and an almost arbitrary reference probability distribution P ′ . In contrast to the Shannon and Rényi entropies, the Tsallis entropy does not decompose as the sum of the physical contribution due to P and the reference or spurious part owing to P ′ . For solving this problem, a slight modification to the relation between Tsallis and Rényi entropies must be introduced. The procedure in this paper gives rise to a nonconventional one-parameter Shannon entropy and to two-parameter Rényi and Tsallis entropies associated with P . It also contributes to clarify the meaning and role of the escort probabilities set.

Computationally efficient algorithms for the two-dimensional Kolmogorov–Smirnov test

Goodness-of-fit statistics measure the compatibility of random samples against some theoretical or reference probability distribution function. The classical one-dimensional Kolmogorov-Smirnov test is a non-parametric statistic for comparing two empirical distributions which defines the largest absolute difference between the two cumulative distribution functions as a measure of disagreement. Adapting this test to more than one dimension is a challenge because there are 2d-1 independent ways of ordering a cumulative distribution function in d dimensions. We discuss Peacock's version of the Kolmogorov-Smirnov test for two-dimensional data sets which computes the differences between cumulative distribution functions in 4n2 quadrants. We also examine Fasano and Franceschini's variation of Peacock's test, Cooke's algorithm for Peacock's test, and ROOT's version of the two-dimensional Kolmogorov-Smirnov test. We establish a lower-bound limit on the work for computing Peacock's test of Ω(n2lgn), introducing optimal algorithms for both this and Fasano and Franceschini's test, and show that Cooke's algorithm is not a faithful implementation of Peacock's test. We also discuss and evaluate parallel algorithms for Peacock's test.

Data-driven parceling and entropic inference in MEG

In Amblard et al. [Amblard, C., Lapalme, E., Lina, J.M. 2004. Biomagnetic source detection by maximum entropy and graphical models. IEEE Trans. Biomed. Eng. 55 (3) 427–442], the authors introduced the maximum entropy on the mean (MEM) as a methodological framework for solving the magnetoencephalography (MEG) inverse problem. The main component of the MEM is a reference probability density that enables one to include all kind of prior information on the source intensity distribution to be estimated. This reference law also encompasses the definition of a model. We consider a distributed source model together with a clustering hypothesis that assumes functionally coherent dipoles. The reference probability distribution is defined as a prior parceling of the cortical surface. In this paper, we present a data-driven approach for parceling out the cortex into regions that are functionally coherent. Based on the recently developed multivariate source prelocalization (MSP) principle [Mattout, J., Pelegrini-Issac, M., Garnero, L., Benali, H. 2005. Multivariate source prelocalization (MSP): Use of functionally informed basis functions for better conditioning the MEG inverse problem. NeuroImage 26 (2) 356–373], the data-driven clustering (DDC) of the dipoles provides an efficient parceling of the sources as well as an estimate of parameters of the initial reference probability distribution. On MEG simulated data, the DDC is shown to further improve the MEM inverse approach, as evaluated considering two different iterative algorithms and using classical error metrics as well as ROC (receiver operating characteristic) curve analysis. The MEM solution is also compared to a LORETA-like inverse approach. The data-driven clustering allows to take most advantage of the MEM formalism. Its main trumps lie in the flexible probabilistic way of introducing priors and in the notion of spatial coherent regions of activation. The latter reduces the dimensionality of the problem. In so doing, it narrows down the gap between the two types of inverse methods, the popular dipolar approaches and the distributed ones.

Sensitivity of some biological reference points to shifts in exploitation patterns and inputs uncertainty for three North Sea demersal stocks

This study examines the sensitivity of six biological reference points (BRPs), both to variability and uncertainty of their input parameters, and to shifts in exploitation patterns. These shifts arise from accounting for the trends in catchability in extended survivors analysis (XSA) tuning fleets derived from the “hybrid method”. By using non-parametric resampling and Monte Carlo stochastic procedures, we test whether such shifts in exploitation patterns significantly affect reference point values and probability distributions, for various levels of variation of input parameters. This method is applied to North Sea plaice, sole and cod stocks. We show that accounting for catchability trend generally induces only slight changes in reference point mean values. In particular, it is mostly significant over the last decade, especially for plaice, but not when introducing medium levels of uncertainty for the natural mortality.

A Screening Level Probabilistic Risk Assessment of Mercury in Florida Everglades Food Webs

A screening level probabilistic assessment of risks was performed on three species of piscivorous wildlife at the top of Everglades aquatic food webs: the American alligator (Alligator mississippiensis), the great egret (Egretta alba), and the raccoon (Procyon lotor varius). Ranges of dietary exposure concentrations (and probability distribution functions) were derived for two general areas of the Everglades: Shark Slough and the south-central Everglades (highly contaminated with Hg), and the northern Everglades (a lower Hg contaminated area in and near Water Conservation Area 1). Ranges of toxicity reference values and probability distribution functions were derived from literature on the toxicity of dietary methyl Hg to birds and mammals. Probability distributions of risk estimates for each receptor were generated using Monte Carlo simulations and indicated that piscivorous wildlife feeding in the south-central region of the Everglades are at high risk from consumption of Hg-contaminated prey. Alligators had 100% exceedences of chronic risk thresholds, and great egrets had 99% exceedences. In the northern Everglades, exceedences of chronic risk thresholds were substantially lower but were still present (6–34% exceedences). Results support previous studies suggesting that top predators of the Everglades may be at risk from Hg contamination and indicate that Hg risks are location-dependent.

Prediction Of Uncertainty In Reservoir Performance Forecast

Abstract Geostatistical models are used to provide equally probable reservoir descriptions that honour available data of a given reservoir. The differences between these descriptions provide an indication of reservoir uncertainty due to lack of information. Transferring this uncertainty into the reservoir performance forecast would require flow simulation of a large number of these equally probable descriptions. The spread in the response derived from the flow simulation measures its uncertainty. However, this approach is not a viable option in most cases because it would require excessive amounts of computer time. A new approach for transferring geological uncertainty is presented. Each reservoir description is first ranked using a Fast Simulator (FS) rather than the Comprehensive flow Simulation (CS). A few selected descriptions are then processed through the CS to generate an approximate probability distribution of tile reservoir production response. This approach yields a considerable saving in the computer time over the use of the CS alone. This approach is tested using a waterflood example in a quarter of a five-spot. A standard black-oil type simulator is used as the CS and a tracer model is used as the FS. Approximate probability distribution of production parameters (water breakthrough rime, cumulative oil recovery and cumulative water-oil ratio) are generated by selecting just five reservoir descriptions among 100, and processing them through the CS. The approximation is tested by running the CS on all 100 cases to generate the reference probability distributions. The tracer concentration at the producer provided very good results with a 90% reduction in computer time over the use of the CS for all 100 descriptions. Introduction Many complex geological processes, such as sedimentation, erosion, migration, compaction and diagenesis produce complex spatial distributions of reservoir properties. Rock type, porosity, permeability, fault occurrence, degree of cementation, and hydrocarbon saturation are some of these properties. However, the exact conditions that result from these complex geological processes are never known e.xacrly. This spatia [distribution of reservoir properties generally presems some degree of Variability, often shared with a complex geological architecture. Moreover, in practice, samples obtained represem only a very small fraction of [he whole reservoir volume, resulting in incomplete knowledge about the reservoir structure at all scales. For all these reasons, it is not advisable to consider only one single reservoir description or image and describe the spatial distribution of reservoir properties in a purely deterministic manner. Rather, a statistic.a1 treatment of the variables involved is desirable one that recognizes the lack of knowledge or uncertainty associated with any description selected. The geological uncertainty can be determined through the differences between many equally probable reservoir descriptions generated using a geostatistical technique known as slochastic simulation(l,2). Each of the images reproduces a prior measure of spatial continuity (covariances) and may be conditioned to honour available data, such as, core and log data, well test, geophysical and geological interpretation, etc. The variables that should mostly be described using probablistic techniques are those that influence the amount, position, accessibility and flow of fluids through the reservoir (e.g. lithofacies or flow unit distributions, porosity, permeability, and fault occurrences).

Modified Regression Approach for Modeling Probability Distributions

ABSTRACT Amodified regression approach is presented for modeling probability distributions. The approach combines a fitted probability distribution with polynomial regression. The method is based on fitting a reference probability distribution to a set of data. Polynomial regression is then used to modify the fitted reference distribution for a closer fit to the data. The modified regression approach is a fast and statistically efficient method of modeling probability distributions for random variable generation and percentile estimation.