Conditional Probability Density Function Research Articles

Formulating the history matching problem with consistent error statistics

It is common to formulate the history-matching problem using Bayes’ theorem. From Bayes’, the conditional probability density function (pdf) of the uncertain model parameters is proportional to the prior pdf of the model parameters, multiplied by the likelihood of the measurements. The static model parameters are random variables characterizing the reservoir model while the observations include, e.g., historical rates of oil, gas, and water produced from the wells. The reservoir prediction model is assumed perfect, and there are no errors besides those in the static parameters. However, this formulation is flawed. The historical rate data only approximately represent the real production of the reservoir and contain errors. History-matching methods usually take these errors into account in the conditioning but neglect them when forcing the simulation model by the observed rates during the historical integration. Thus, the model prediction depends on some of the same data used in the conditioning. The paper presents a formulation of Bayes’ theorem that considers the data dependency of the simulation model. In the new formulation, one must update both the poorly known model parameters and the rate-data errors. The result is an improved posterior ensemble of prediction models that better cover the observations with more substantial and realistic uncertainty. The implementation accounts correctly for correlated measurement errors and demonstrates the critical role of these correlations in reducing the update’s magnitude. The paper also shows the consistency of the subspace inversion scheme by Evensen (Ocean Dyn. 54, 539–560 2004) in the case with correlated measurement errors and demonstrates its accuracy when using a “larger” ensemble of perturbations to represent the measurement error covariance matrix.

Gibbs Sampling for Damage Detection Using Complex Modal Data from Multiple Setups

This paper presents a novel Gibbs sampling approach for structural health monitoring (SHM) with detection of structural changes/damages using incomplete complex modal data measured with a limited number of sensors. The usual difficulty with the availability of sensors in SHM practices and enforcing data acquisition in multiple setups is thoroughly addressed. Structural modeling incorporated with damping is considered in this proposed inverse problem exercise to calibrate damping parameters along with the stiffness and mass parameters facilitating SHM. Both proportional and nonproportional viscous damping are adopted in structural modeling. Detailed formulations on the probabilistic detection of changes/damages are presented in detail. Moreover, a Gibbs sampling technique is introduced to quantify uncertainties of the various sets of uncertain parameters, where samples of the conditional probability density function of a parameter set are obtained iteratively. The proposed approach retains the typical advantage of the nonrequirement of mode-matching. A validation exercise is performed using a three-dimensional building structure (attached with supplementary viscous dampers) and a laboratory steel structure considering multiple damage cases and different sensor placements. The proposed methodology is observed to be efficient for SHM using incomplete complex modal data measured with a limited number of sensors.

The joint probability density function of mixture fraction, reaction progress variable, and total enthalpy in a stratified, swirl-stabilized turbulent flame

The three-condition version of the uniform conditional state combustion model makes use of the mixture fraction, progress variable, and normalized total enthalpy as conditioning variables to build a three-dimensional conditional manifold for chemistry. In order to map the solution in conditional space into the flow domain, the joint Probability Density Function (PDF) of the conditioning variables needs to be modeled. In simulations, presumed functions (i.e., β-PDF for the mixture fraction and progress variable and δ-PDF for total enthalpy) are often used for modeling the marginal PDFs. In this work, the measurements from the Cambridge/Sandia burner are employed to obtain the marginal PDFs for the conditioning variables at various points in the reacting domain. The measurements are then combined from all positions in space to form conditional PDFs of the normalized total enthalpy for various values of the other two variables. In the vicinity of the flame brush, the marginal PDF of the normalized total enthalpy resembles a bimodal Gaussian distribution; nonetheless, the conditional PDFs for this variable are nearly Gaussian distributions. The correlation coefficients between the conditioning variables are also investigated, and the assumption of their statistical independence is examined. To consider the association between the conditioning variables for modeling, the copula concept is introduced, and the performances of three different copulas are tested. Furthermore, the statistical moments of the conditioning variables are computed from the experimental data at different points and are utilized for modeling the joint PDF of the conditioning variables from two different approaches that are compared.

Uncertainty Assessment over any Volume without Simulation: Revisiting Multi-Gaussian Kriging

Assessing spatial uncertainty over an arbitrary volume is usually done by generating multiple simulations of the random function and averaging the property over each realization to build its uncertainty distribution. However, this is a cumbersome process for practitioners, as they need to compute and process a large number of realizations. Multi-Gaussian kriging provides a simpler alternative, by directly computing the conditional probability density functions of the random variables. In this work, we revisit the multi-Gaussian framework and present the implementation details to determine the conditional distribution at any support, by numerical integration of the conditional probabilities, using an importance sampling approach. We demonstrate the use of this approach and assess its accuracy in the lognormal and exponential cases with synthetic data. We also apply it to a real three-dimensional mining case, where the uncertainty over scheduled production volumes is determined. The ability to assess this uncertainty may prove valuable, as it enables schedule changes to be made in a mining setting in order to ensure the smooth running of downstream processes.

Forecasting Earnings with Predicted, Conditional Probability Density Functions

This study provides empirical evidence for the efficacy of deriving firms' earnings forecasts from predictions of the complete, conditional probability density function (pdf). Relative to cross-sectional earnings forecasts based on OLS regressions, improvements of accuracy, bias and measures for the validity as an expectation's proxy amount to approximately two fifths, when conditional pdfs are obtained via quantile regressions. In turn, another fifth is gained substituting quantile regressions by artificial neural networks. Cross-sectional analyses are consistent with improvements deriving from taking into consideration pdfs of firms which are particular peculiar. Furthermore, also recent point estimation methods fall behind the pdf-based approach.

Distributed Sensor Local Linear Fusion Detection of Weak Pulse Signal in Chaotic Background

This paper combines the distributed sensor fusion system with the signal detection under chaotic noise to realize the distributed sensor fusion detection from chaotic background. First, based on the short‐term predictability of the chaotic signal and its sensitivity to small interference, the phase space reconstruction of the observation signal of each sensor is carried out. Second, the distributed sensor local linear autoregressive (DS‐LLAR) model is constructed to obtain the one‐step prediction error of each sensor. Then, we construct a Bayesian risk model and also obtain the corresponding conditional probability density function under each sensor’s hypothesis test which firstly needs to fit the distribution of prediction errors according to the parameter estimation. Finally, the fusion optimization algorithm is designed based on the Bayesian fusion criterion, and the optimal decision rule of each sensor and the optimal fusion rule of the fusion center are jointly solved to effectively detect the weak pulse signal in the observation signal. Simulation experiments show that the proposed method which used a distributed sensor combined with a local linear model can effectively detect weak pulse signals from chaotic background.

Distribution of Selling Duration for Zero Ending Inventory Dynamic Pricing Model with Exponentially Distributed Purchases

Abstract We consider a perishable product with a fixed shelf life under fixed-order quantity policy. The demand is a compound Poisson process with a price sensitive intensity and an exponential batch size’s distribution. A model of retail price control is considered allowing us to sell a lot without leftovers almost surely during the product’s lifetime. The Laplace transforms of the conditional probability density function of the time to the end of the session and the conditional mean of this time are obtained. The simulation results of the session duration are also presented and compared with previously obtained approximate ones.

Modeling Time-Varying Variability and Reliability of Freeway Travel Time Using Functional Principal Component Analysis

This paper presents an aproach for modeling travel time variability and reliability that accounts for time-of-day effects on travel time variations. The conditional probability density functions of travel time in each specific period play a fundamental role in the modeling of time-varying variability and reliability. Nonparametric kernel density estimation is applied to travel time data, which enhances the flexibility and accuracy in accommodating travel time distributions under various complex traffic conditions. The proposed functional travel time density model (FTTDM) uses functional principal component analysis in kernel density function estimates for modeling and depicting time-varying variability through the dependence on the departure time of day. The resulting quantile estimates are used to obtain time-varying reliability, which considers the addition of extra time to the average or the median for an on-time arrival. This study uses linked travel time data of an electronic toll collection system to estimate route travel times in the Taiwan Freeway system. As illustrated in the data applications, the FTTDM effectively captures the time-varying feature of travel time variability and reliability, and the reliability indicators identify the unreliable departure time of day of a route.

Bayesian uncertainty quantification for guided-wave-based multidamage localization in plate-like structures using Gibbs sampling

In this article, a new Bayesian approach for guided-wave-based multidamage localization by employing Gibbs sampling is proposed. By using the information of time-of-flight (ToF) embedded in guided wave signals, the posterior probability distributions of three parameter groups, that is, the horizontal and vertical coordinates of the multidamage locations (x, y) and wave velocity v, are characterized using Gibbs sampling samples. To obtain the analytical form of the conditional posterior probability density function of each parameter group conditional on the other two and the available ToF data, a first-order Taylor expansion of the nonlinear ToF-based damage localization model with respect to each parameter group is performed. Two Gibbs sampling algorithms are proposed, which differ in their strategies to address the posterior uncertainty of the prediction error parameter; however, both algorithms iteratively sample from conditional posterior probability density functions of three parameter groups. Therefore, the effective number of dimensions for Gibbs sampling is always three, regardless of the number of defects. The final damage localization results are obtained by grouping all ToFs and then comparing the posterior uncertainty of localization results of each grouping scheme to obtain the most reliable sampling results among all candidates. The proposed method not only identifies the group velocity but also localizes multiple defects by sharing the same characteristics of damage localization. Furthermore, this method can quantify the uncertainty of multidamage localization to automatically find the most reliable damage locations. The effectiveness and robustness of the proposed algorithms are validated by both numerical and experimental examples.

Direct Multivariate Simulation - A stepwise conditional transformation for multivariate geostatistical simulation

Several applications in geoscience require the generation of multiple realizations of random fields of physical properties to mimic their spatial distribution and quantify the model uncertainty. Some modeling problems present complex multivariate distributions with heteroscedasticity and non-linear relations among the variables. We propose a new algorithm, namely Direct Multivariate Simulation, for the simulation of random fields of non-parametric multivariate joint distributions. The methodology is based on a generalization of the normal score and back transformation for multivariate distributions, also called Stepwise Conditional transformation. The sequential sampling of each variable is performed by decomposing the target joint distribution into a product of univariate marginal and conditional probability density functions. We propose numerical solutions to improve the algorithm efficiency for real case applications with a large number of variables. The method is demonstrated through an application where we sample a 6-variate joint distributions with strong nonlinear dependence among the variables. The results are validated by comparing the results to the Projection Pursuit Multivariate Transform and through the computation of the experimental semi-variograms, the marginal distributions, and the bi-histograms of the simulated variables.

Trajectory Generation by Chance-Constrained Nonlinear MPC With Probabilistic Prediction.

Continued great efforts have been dedicated toward high-quality trajectory generation based on optimization methods; however, most of them do not suitably and effectively consider the situation with moving obstacles; and more particularly, the future position of these moving obstacles in the presence of uncertainty within some possible prescribed prediction horizon. To cater to this rather major shortcoming, this work shows how a variational Bayesian Gaussian mixture model (vBGMM) framework can be employed to predict the future trajectory of moving obstacles; and then with this methodology, a trajectory generation framework is proposed which will efficiently and effectively address trajectory generation in the presence of moving obstacles, and incorporate the presence of uncertainty within a prediction horizon. In this work, the full predictive conditional probability density function (PDF) with mean and covariance is obtained and, thus, a future trajectory with uncertainty is formulated as a collision region represented by a confidence ellipsoid. To avoid the collision region, chance constraints are imposed to restrict the collision probability, and subsequently, a nonlinear model predictive control problem is constructed with these chance constraints. It is shown that the proposed approach is able to predict the future position of the moving obstacles effectively; and, thus, based on the environmental information of the probabilistic prediction, it is also shown that the timing of collision avoidance can be earlier than the method without prediction. The tracking error and distance to obstacles of the trajectory with prediction are smaller compared with the method without prediction.

Hybrid Discriminator With Correlative Autoencoder for Anomaly Detection

Advances in deep neural networks (DNNs) have led to impressive results and in recent years many works have exploited DNNs for anomaly detection. Among others, generative/reconstruction model-based methods have been frequently used for anomaly detection because they do not require any labels for training. The anomaly detection performance of these methods, however, varies a lot, due to the change of the intra-class variance and the difference in complexity of input samples. In addition, most previous state-of-the-art works on anomaly detection have empirically adjusted several hyperparameters to heighten their performance of anomaly detection. These sorts of procedures are known to be impractical and create obstacles in real world anomaly detection. To solve these problems, we propose a hybrid discriminator with a correlative autoencoder for anomaly detection. In the proposed framework, the discriminator implicitly estimates the conditional probability density function and the autoencoder has improved ability to control the reconstruction error. We provide theoretical foundation of our method and verify it through various experiments. We also confirm practical benefits of our interpretation of the conditional expectation and the proposed framework by comparing our results with other state-of-the-art methods.

Bivariate Gompertz generator of distributions: statistical properties and estimation with application to model football data

In this paper, the bivariate extension of the so called Gompertz-G family was introduced and studied in detail. Marshall and Olkin shock model was used to build the proposed bivariate family. The new family was constructed from three independent Gompertz-H families using a minimisation process. Some of its statistical properties such as joint probability density function, coefficient of median correlation, moments, product moment, covariance, conditional probability density function, joint reliability function, stress-strength reliability and joint reversed (hazard) rate function were derived. After introducing the general class, three special models of the new family were discussed. Maximum likelihood method was used to estimate the family parameters. A simulation study was carried out to examine the bias and mean square error of the maximum likelihood estimators. Finally, the importance of the proposed bivariate family was illustrated by means of real dataset, and it was found that the proposed model provides better fit than other well-known models in the statistical literature such as bivariate Gompertz, bivariate generalised Gompertz, bivariate Gumbel Gompertz, bivariate Burr X Gompertz and bivariate exponentiated Weibull-Gomperz.

High-cycle fatigue and fracture behaviours of SLM AlSi10Mg alloy

Abstract The high-cycle fatigue and fracture behaviours of the selective laser melting (SLM) AlSi10Mg alloy were investigated. Flat specimens were designed directly in the shape required for the fatigue tests under pulsating loading in tension (R=0, R is the dynamic factor). The fatigue–life (S–N) curves were modelled with a conditional Weibull's probability density function, where the real-valued genetic algorithm (GA) and the differential ant-stigmergy algorithm (DASA) were applied to estimating the needed Weibull's parameters. The fractography of the fatigue specimens showed that the fatigue cracks initiated around the surface defects produced by SLM and then propagated in an unstable manner. However, the presence of large SLM defects mainly influenced the crack initiation period and did not have a strong influence on the crack propagation. The obtained experimental results present a basis for further investigation of the fatigue behaviour of advanced materials and structures (e.g. cellular metamaterials) fabricated by additive manufacturing (AM). Especially, in the case of two-dimensional cellular structures, the cross-section of cellular struts is usually rectangular which corresponds to the specimen shape considered in this work.

Properties and methods of estimation for a bivariate exponentiated Fréchet distribution

Abstract In this paper, a bivariate extension of exponentiated Fréchet distribution is introduced, namely a bivariate exponentiated Fréchet (BvEF) distribution whose marginals are univariate exponentiated Fréchet distribution. Several properties of the proposed distribution are discussed, such as the joint survival function, joint probability density function, marginal probability density function, conditional probability density function, moments, marginal and bivariate moment generating functions. Moreover, the proposed distribution is obtained by the Marshall-Olkin survival copula. Estimation of the parameters is investigated by the maximum likelihood with the observed information matrix. In addition to the maximum likelihood estimation method, we consider the Bayesian inference and least square estimation and compare these three methodologies for the BvEF. A simulation study is carried out to compare the performance of the estimators by the presented estimation methods. The proposed bivariate distribution with other related bivariate distributions are fitted to a real-life paired data set. It is shown that, the BvEF distribution has a superior performance among the compared distributions using several tests of goodness–of–fit.

Multipoint reconstruction of wind speeds

Abstract. The most intermittent behaviour of atmospheric turbulence is found for very short timescales. Based on a concatenation of conditional probability density functions (cpdf's) of nested wind speed increments, inspired by a Markov process in scale, we derive a short-time predictor for wind speed fluctuations around a non-stationary mean value and with a corresponding non-stationary variance. As a new quality this short-time predictor enables a multipoint reconstruction of wind data. The used cpdf's are (1) directly estimated from historical data from the offshore research platform FINO1 and (2) obtained from numerical solutions of a family of Fokker–Planck equations in the scale domain. The explicit forms of the Fokker–Planck equations are estimated from the given wind data. A good agreement between the statistics of the generated and measured synthetic wind speed fluctuations is found even on timescales below 1 s. This shows that our approach captures the short-time dynamics of real wind speed fluctuations very well. Our method is extended by taking the non-stationarity of the mean wind speed and its non-stationary variance into account.

Data-driven sensitivity indices for models with dependent inputs using polynomial chaos expansions

Uncertainties exist in both physics-based and data-driven models. Variance-based sensitivity analysis characterizes how the variance of a model output is propagated from the model inputs. The Sobol index is one of the most widely used sensitivity indices for models with independent inputs. For models with dependent inputs, different approaches have been explored to obtain sensitivity indices in the literature. Typical approaches are based on procedures of transforming the dependent inputs into independent inputs. However, such transformation requires additional information about the inputs, such as the dependency structure or the conditional probability density functions. In this paper, data-driven sensitivity indices are proposed for models with dependent inputs. We first construct ordered partitions of linearly independent polynomials of the inputs. The modified Gram-Schmidt algorithm is then applied to the ordered partitions to generate orthogonal polynomials with respect to the empirical measure based on observed data of model inputs and outputs. Using the polynomial chaos expansion with the orthogonal polynomials, we obtain the proposed data-driven sensitivity indices. The sensitivity indices provide intuitive interpretations of how the dependent inputs affect the variance of the output without a priori knowledge on the dependence structure of the inputs. Four numerical examples are used to validate the proposed approach.

Service life design for carbonation-induced corrosion based on air-permeability requirements

Full probabilistic service life design is the most accurate approach to address the durability of reinforced concrete structures. The aim of this study was to develop a full probabilistic service life design method based on a durability indicator that can be assessed on site, thus allowing simple conformity control. Experimental data and a combination of existing models were applied to estimate the service life of reinforced concrete structures, subject to carbonation-induced corrosion. Furthermore, a conditional probability density function of carbonation rates, given an air-permeability coefficient, was established. Probability density functions of service life, for different scenarios, were obtained through the Monte Carlo simulation technique. The analysis of those curves enabled the definition of design charts, that allow practitioners to set cover depth and air-permeability specifications for an intended service life and also to find the design service life of a reinforced concrete structure from actual cover depth and air-permeability, assessed in the as-built structure. The results of the proposed approach show a fair agreement with other performance-based design methodologies.

BetheSF: Efficient computation of the exact tagged-particle propagator in single-file systems via the Bethe eigenspectrum

Single-file diffusion is a paradigm for strongly correlated classical stochastic many-body dynamics and has widespread applications in soft condensed matter and biophysics. However, exact results for single-file systems are sparse and limited to the simplest scenarios. We present an algorithm for computing the non-Markovian time-dependent conditional probability density function of a tagged-particle in a single-file of N particles diffusing in a confining external potential. The algorithm implements an eigenexpansion of the full interacting many-body problem obtained by means of the coordinate Bethe ansatz. While formally exact, the Bethe eigenspectrum involves the generation and evaluation of permutations, which becomes unfeasible for single-files with an increasing number of particles N. Here we exploit the underlying exchange symmetries between the particles to the left and to the right of the tagged-particle and show that it is possible to reduce the complexity of the algorithm from the worst case scenario O(N!) down to O(N). A C++ code to calculate the non-Markovian probability density function using this algorithm is provided. Solutions for simple model potentials are readily implemented including single-file diffusion in a flat and a ‘tilted’ box, as well as in a parabolic potential. Notably, the program allows for implementations of solutions in arbitrary external potentials under the condition that the user can supply solutions to the respective single-particle eigenspectra. Program summaryProgram Title: BetheSFCPC Library link to program files:http://dx.doi.org/10.17632/3bs74vf72n.1Licensing provisions: MITProgramming language: C++ (C++17 support required)Supplementary material: makefile, README, SingleFileBluePrint.hppNature of problem: Diffusive single-files are mathematical models of effectively one-dimensional strongly correlated many-body systems. While the dynamics of the full system is Markovian, the diffusion of a tracer-particle in a single-file is an example of non-Markovian and anomalous diffusion. The many-body Fokker–Planck equation governing the system’s dynamics can be solved using the coordinate Bethe ansatz. A naïve implementation of such a solution runs in non-polynomial time since it requires the generation of permutations of the elements of a multiset.Solution method: In this paper we show how, exploiting the exchange symmetries of the system, it is possible to reduce the complexity of the algorithm to evaluate the solution, using a permutation-generation algorithm, from O(N!) in the worst case scenario to O(N) in the best case scenario, which corresponds to tagging the first or the last particle, where N stands for the number of particles in the single-file.Additional comments including restrictions and unusual features: The code may overflow for large single-files N≥170. All the benchmarks ran on the following CPU: Intel Xeon E3-1270 v2 3.50 GHz 4 cores. The compiler used is g++ 7.3.1 (SUSE Linux) with the optimization-O3 turned on. The code to produce all the data in the figures is included in the files: figure2.cpp, figure3.cpp, figure4.cpp, figure5a.cpp and figure5b.cpp

Probabilistic estimation of specific surface area and cation exchange capacity: a global multivariate distribution

Specific surface area (SSA) and cation exchange capacity (CEC) are two fundamental clay properties. However, the determination of CEC and SSA is challenging due to inherent uncertainties and difficulty in experimental measurement. A popular approach is to employ transformation models for their estimation. However, most of the existing models were developed on limited sample sizes and quantification of uncertainty associated with the estimate is not possible. Therefore, this study proposes a multivariate probabilistic approach for estimation of CEC and SSA. First, a five-dimensional database (278 × 5) for the parameters liquid limit (LL), plasticity index (PI), clay fraction (CF), CEC, and SSA (labelled as CLAY/C-S/5/278) is developed. Thereafter, multivariate distribution for the five parameters in the database is constructed using the vine copula approach. Implementation of the proposed approach is demonstrated by updating the prior–unconditional probability density functions (PDFs) of CEC and SSA given single or multiple clay parameters using Bayes’ rule. The posterior or conditional PDFs of CEC and SSA are also summarized as practitioner-friendly analytical expressions. Two geotechnical application examples are shown as well. In the proposed approach, CEC and SSA are characterized by their complete joint distribution and therefore this approach is superior to the popular deterministic transformation approach in literature.