Probabilistic Formulation Research Articles

Streamer inception probability in air at atmospheric pressure – an experimental and theoretical analysis

Abstract We consider a self-consistent and probabilistic formulation of the classical Raether–Meek streamer inception model, and compare it with dedicated experiments. Our approach is based on the statistical evaluation of a first critical electron. This is done by evaluating the Raether–Meek integral for arbitrary starting positions of the first electron, and evaluating the size of the region in which the electrons can transition into a streamer. Then, using an ensemble-averaged model for the negative ion distribution which acts as a source of initial electrons, we compute the time-dependent probability of streamer inception. Unlike the standard Meek-based inception model, the experiments show that the generalized formulation is valid for both voltage polarities, but in return it requires knowledge about the initial ion distribution and electron detachment rate.

Statistical Metamodel of Liner Acoustic Impedance Based on Neural Network and Probabilistic Learning for Small Datasets

The main novelty of this paper consists of presenting a statistical artificial neural network (ANN)-based model for a robust prediction of the frequency-dependent aeroacoustic liner impedance using an aeroacoustic computational model (ACM) dataset of small size. The model, focusing on percentage of open area (POA) and sound pressure level (SPL) at a zero Mach number, takes into account uncertainties using a probabilistic formulation. The main difficulty in training an ANN-based model is the small size of the ACM dataset. The probabilistic learning carried out using the probabilistic learning on manifolds (PLoM) algorithm addresses this difficulty as it allows constructing a very large training dataset from learning the probabilistic model from a small dataset. A prior conditional probability model is presented for the PCA-based statistical reduced representation of the frequency-sampled vector of the log-resistance and reactance. It induces some statistical constraints that are not straightforwardly taken into account when training such an ANN-based model by classical optimizations methods under constraints. A second novelty of this paper consists of presenting an alternate solution that involves using conditional statistics estimated with learned realizations from PLoM. A numerical example is presented.

Measurement uncertainty evaluation of equivalent roughness in hydraulic pipes

This paper addresses the quantification of the dispersion of equivalent roughness values obtained from the experimental study of hydraulic pipes used, for example, in water supply networks. This quantity is mainly used in the determination of the friction factor related to pipe fluid flow. In this context, non-linear and complex mathematical models, such as the Colebrook-White equation, are applied to characterize the equivalent roughness of hydraulic pipes composed of different types of materials. However, knowledge about the measurement uncertainty of the obtained estimates is still reduced, having a direct impact in the conformity assessment of this type of hydraulic component and in the technical comparison between different types of pipes (materials and manufacturers). The paper describes the application of a Monte Carlo method (MCM) in the measurement uncertainty evaluation of equivalent roughness. In addition to presenting the theoretical and experimental background, the paper describes the measurement uncertainty propagation, from the probabilistic formulation of the input quantities up to the output quantity. A numerical example, based on experimental data retrieved from field testing of hydraulic pipes integrated in a large-scale agricultural irrigation network, is shown in the paper, illustrating the suitability, advantages, and limitations of the proposed approach.

Probabilistic PARAFAC2.

The Parallel Factor Analysis 2 (PARAFAC2) is a multimodal factor analysis model suitable for analyzing multi-way data when one of the modes has incomparable observation units, for example, because of differences in signal sampling or batch sizes. A fully probabilistic treatment of the PARAFAC2 is desirable to improve robustness to noise and provide a principled approach for determining the number of factors, but challenging because direct model fitting requires that factor loadings be decomposed into a shared matrix specifying how the components are consistently co-expressed across samples and sample-specific orthogonality-constrained component profiles. We develop two probabilistic formulations of the PARAFAC2 model along with variational Bayesian procedures for inference: In the first approach, the mean values of the factor loadings are orthogonal leading to closed form variational updates, and in the second, the factor loadings themselves are orthogonal using a matrix Von Mises-Fisher distribution. We contrast our probabilistic formulations to the conventional direct fitting algorithm based on maximum likelihood on synthetic data and real fluorescence spectroscopy and gas chromatography-mass spectrometry data showing that the probabilistic formulations are more robust to noise and model order misspecification. The probabilistic PARAFAC2, thus, forms a promising framework for modeling multi-way data accounting for uncertainty.

Application of a new method for probabilistic fatigue analysis from strain measurements on bridges

Increasing loads, ageing of materials as well as environmental change make the issue of fatigue damage critical on bridges, especially in the case of old steel or iron structures. The assessment of the fatigue loads on specific elements of these assets can be carried out from on-field strain measurements with the application of classical methods involving rainflow cycle counting and the use of Palmgren-Miner's rule. However, such a fatigue assessment suffers of significant uncertainties about the real fatigue damage ratio and the consecutive residual fatigue life. A new probabilistic method for the fatigue assessment is proposed, as the result of a joint research by OSMOS Group and the Ecole des Ponts. The theoretical background of the method is described, involving a probabilistic formulation of Miner's rule and the use of a Weibull-Basquin model to describe the health of structural elements. Real field case studies are presented, on two historical iron truss road bridges in France, with a dataset of more than 18 months of continuous strain measurements used for the fitting of the probabilistic model of fatigue loads. The incidence of the quantity of data and of the number of sensors used for the analysis is discussed, along with the conclusive results concerning the probabilistic description of the residual fatigue life.

Biological optimization for hybrid proton-photon radiotherapy

Objective. Hybrid proton-photon radiotherapy (RT) is a cancer treatment option to broaden access to proton RT. Additionally, with a refined treatment planning method, hybrid RT has the potential to offer superior plan quality compared to proton-only or photon-only RT, particularly in terms of target coverage and sparing organs-at-risk (OARs), when considering robustness to setup and range uncertainties. However, there is a concern regarding the underestimation of the biological effect of protons on OARs, especially those in close proximity to targets. This study seeks to develop a hybrid treatment planning method with biological dose optimization, suitable for clinical implementation on existing proton and photon machines, with each photon or proton treatment fraction delivering a uniform target dose. Approach. The proposed hybrid biological dose optimization method optimized proton and photon plan variables, along with the number of fractions for each modality, minimizing biological dose to the OARs and surrounding normal tissues. To mitigate underestimation of hot biological dose spots, proton biological dose was minimized within a ring structure surrounding the target. Hybrid plans were designed to be deliverable separately and robustly on existing proton and photon machines, with enforced uniform target dose constraints for the proton and photon fraction doses. A probabilistic formulation was utilized for robust optimization of setup and range uncertainties for protons and photons. The nonconvex optimization problem, arising from minimum monitor unit constraint and dose-volume histogram constraints, was solved using an iterative convex relaxation method. Main results. Hybrid planning with biological dose optimization effectively eliminated hot spots of biological dose, particularly in normal tissues surrounding the target, outperforming proton-only planning. It also provided superior overall plan quality and OAR sparing compared to proton-only or photon-only planning strategies. Significance. This study presents a novel hybrid biological treatment planning method capable of generating plans with reduced biological hot spots, superior plan quality to proton-only or photon-only plans, and clinical deliverability on existing proton and photon machines, separately and robustly.

Density-based in-orbit collision risk model valid for any impact geometry

Neglecting small fragments in space debris evolutionary models can lead to a significant underestimation of the collision risk for operational satellites. However, when scaling down to the millimetre range, the debris population grows to over a hundred million objects, making deterministic approaches too computationally expensive. On the contrary, probabilistic models provide a more efficient alternative, which however typically works under some simplifying assumptions on the dynamics, limiting their field of applicability. This work proposes an extension of the density-based collision risk models to any orbital dynamics and impact geometry. The impact rate with a target satellite is derived from a multi-dimensional phase space density function in orbital elements, which discretely varies over both phase space and time. The assumption of a bin-wise constant cloud density allows for the analytical transformation of the six-dimensional distribution in orbital elements into the three-dimensional spatial density function, guaranteeing an efficient and accurate evaluation of the fragments flux. The proposed method is applied to the assessment of the collision risk posed by occurred fragmentation events in different orbital regions on a high-risk target object. The effect on the impact rate of the modelling improvements, compared to previous probabilistic formulations, is discussed.

A robust probabilistic approach to stochastic subspace identification

Modal parameter estimation of operational structures is often a challenging task when confronted with unwanted distortions (outliers) in field measurements. Atypical observations present a problem to operational modal analysis (OMA) algorithms, such as stochastic subspace identification (SSI), severely biasing parameter estimates and resulting in misidentification of the system. Despite this predicament, no simple mechanism currently exists capable of dealing with such anomalies in SSI. Addressing this problem, this paper first introduces a novel probabilistic formulation of stochastic subspace identification (Prob-SSI), realised using probabilistic projections. Mathematically, the equivalence between this model and the classic algorithm is demonstrated. This fresh perspective, viewing SSI as a problem in probabilistic inference, lays the necessary mathematical foundation to enable a collection of new OMA approaches. To this end, a statistically robust SSI algorithm (robust Prob-SSI) is developed, capable of providing a principled and automatic way of handling outlying or anomalous data in the measured timeseries, such as may occur in field recordings, e.g. intermittent sensor dropout. Robust Prob-SSI is shown to outperform conventional SSI when confronted with ‘corrupted’ data, exhibiting improved identification performance and higher levels of confidence in the found poles when viewing stabilisation diagrams. Similar benefits are also demonstrated on the Z24 Bridge benchmark dataset, highlighting enhanced performance on measured systems.

Compactness property of the linearized Boltzmann collision operator for a multicomponent polyatomic gas

The linearized Boltzmann collision operator is fundamental in many studies of the Boltzmann equation and its main properties are of substantial importance. The decomposition into a sum of a positive multiplication operator, the collision frequency, and an integral operator is trivial. Compactness of the integral operator for monatomic single species is a classical result, while corresponding results for monatomic mixtures and polyatomic single species are more recently obtained. This work concerns the compactness of the operator for a multicomponent mixture of polyatomic species, where the polyatomicity is modeled by a discrete internal energy variable. With a probabilistic formulation of the collision operator as a starting point, compactness is obtained by proving that the integral operator is a sum of Hilbert-Schmidt integral operators and operators, which are uniform limits of Hilbert-Schmidt integral operators, under some assumptions on the collision kernel. The assumptions are essentially generalizations of the Grad's assumptions for monatomic single species. Self-adjointness of the linearized collision operator follows. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere like model. Then it follows that the linearized collision operator is a Fredholm operator, and its domain is also obtained.

Compactness Property of the Linearized Boltzmann Collision Operator for a Mixture of Monatomic and Polyatomic Species

The linearized Boltzmann collision operator has a central role in many important applications of the Boltzmann equation. Recently some important classical properties of the linearized collision operator for monatomic single species were extended to multicomponent monatomic gases and polyatomic single species. For multicomponent polyatomic gases, the case where the polyatomicity is modelled by a discrete internal energy variable was considered lately. Here we consider the corresponding case for a continuous internal energy variable. Compactness results, stating that the linearized operator can be decomposed into a sum of a positive multiplication operator, the collision frequency, and a compact operator, bringing e.g., self-adjointness, is extended from the classical result for monatomic single species, under reasonable assumptions on the collision kernel. With a probabilistic formulation of the collision operator as a starting point, the compactness property is shown by a decomposition, such that the terms are, or at least are uniform limits of, Hilbert–Schmidt integral operators and therefore are compact operators. Moreover, bounds on—including coercivity of—the collision frequency are obtained for hard sphere like, as well as hard potentials with cutoff like, models, from which Fredholmness of the linearized collision operator follows, as well as its domain.

Uncertainty in Distinct Element Modeling of Freestanding Structures Considering Stiffness Parameters

ABSTRACT Numerical studies on freestanding structures typically leverage the Distinct Element Method where the response is dictated by joints; however, the response carries significant uncertainty due to joint stiffness parameters. This study aims to quantify the influence of joint stiffness parameters on the global response of freestanding structures subjected to seismic excitation in a probabilistic formulation. By using a range of joint stiffness values with rectangular rigid blocks of different aspect ratios, the impact of joint stiffness parameters is evaluated against a large suite of historical ground motions. Results indicate that increasing contact stiffness results in increased stability against overturning.

Constructions of statistical estimates of digital measurement error in the case of small samples

The paper presents a computational-experimental method for the construction of statistical estimates of the error of digital measurements carried out on complex technical objects with metrological support. The algorithm of construction of the estimation of the distribution density function of random error of measurements on the basis of complex application of methods of the theory of characteristic functions, the theory of Lie operator series and methods of statistical modeling in the processing of a limited amount of statistical information in the conditions of small samples is described. The solution of the practical problem on the construction of the estimation of the measurement error distribution density and the problem on the zero mark of the measuring instrument in the probabilistic formulation is given. The presented computational and experimental method can be used to construct statistical estimates for the probability distribution density of the general type, including those given by implicit functions, characteristic functions and operator series.

Probabilistic Neural-Symbolic Models With Inductive Posterior Constraints.

Neural-symbolic models provide a powerful tool to tackle complex visual reasoning tasks by combining symbolic program execution for reasoning and deep representation learning for visual recognition. A probabilistic formulation of such models with stochastic latent variables can obtain an interpretable and legible reasoning system with less supervision. However, it is still nontrivial to generate reasonable symbolic structures without the guidance of domain knowledge, since it generally involves an optimization problem with both continuous and discrete variables. Despite the challenges, the interpretability of such symbolic structures provides an interface to regularize their generation by domain knowledge. In this article, we propose to incorporate the available domain knowledge into the learning process of probabilistic neural-symbolic (PNS) models via posterior constraints that directly regularize the structure posterior. In this way, our model is able to identify a middle point where the structure generation process mainly learns from data but also selectively borrows information from domain knowledge. We further present inductive reasoning where the posterior constraints can be automatically reweighted to handle noisy annotations. The experimental results show that our method achieves state-of-the-art performance on major abstract reasoning datasets and enjoys good generalization capability and data efficiency.

Reliability Analysis of Crack Growth Occurrence for a Secondary Hull Component due to Vibration Excitation

Abstract Ship hull vibration is a significant contributor to fatigue crack growth and the major sources of vibrations are found to be the main engine vibration excitation, the wave-induced springing and whipping loads, and the action of the propeller. In the midship region, wave-induced loads and the main engine are the major contributors, whereas propeller excitation dominates in the aft region of the ship hull. No general method exists to solve all kinds of vibration problems and they need to be evaluated through a cost-by-case approach. The complex and uncertain aspects of hull vibration and fatigue crack growth motivate the need for a reliability-based scheme for assessing the resulting fatigue crack propagation. In the present paper, a probabilistic formulation for the failure probability of the occurrence of crack propagation of a secondary hull component is outlined. A generic cargo hold model is analyzed with engine excitation and wave-induced loading as vibration sources, and a stochastic model for vibration response is outlined. The limit state is formulated as the possible occurrence of fatigue crack growth. The secondary hull component considered is a pipe stack support, which is a supporting component that attaches the cargo pipes to the wall inside a cargo tank. Different initial crack sizes are implemented to evaluate the adequacy of the applied stochastic model for vibration response and the accuracy of the estimated failure probability is assessed.

A Gibbs Posterior Framework for Fair Clustering.

The rise of machine learning-driven decision-making has sparked a growing emphasis on algorithmic fairness. Within the realm of clustering, the notion of balance is utilized as a criterion for attaining fairness, which characterizes a clustering mechanism as fair when the resulting clusters maintain a consistent proportion of observations representing individuals from distinct groups delineated by protected attributes. Building on this idea, the literature has rapidly incorporated a myriad of extensions, devising fair versions of the existing frequentist clustering algorithms, e.g., k-means, k-medioids, etc., that aim at minimizing specific loss functions. These approaches lack uncertainty quantification associated with the optimal clustering configuration and only provide clustering boundaries without quantifying the probabilities associated with each observation belonging to the different clusters. In this article, we intend to offer a novel probabilistic formulation of the fair clustering problem that facilitates valid uncertainty quantification even under mild model misspecifications, without incurring substantial computational overhead. Mixture model-based fair clustering frameworks facilitate automatic uncertainty quantification, but tend to showcase brittleness under model misspecification and involve significant computational challenges. To circumnavigate such issues, we propose a generalized Bayesian fair clustering framework that inherently enjoys decision-theoretic interpretation. Moreover, we devise efficient computational algorithms that crucially leverage techniques from the existing literature on optimal transport and clustering based on loss functions. The gain from the proposed technology is showcased via numerical experiments and real data examples.

Efficient optimization in stochastic production planning problems with product substitution

We consider the stochastic production planning problem with product substitution, which can be decomposed into several optimization subproblems with sequential decisions. The decision variables in each time period include (1) the product substitution decision and (2) the recipe input quantity decision. The goal is to minimize the total of production cost, holding cost, and shortage cost, while achieving a service level for demand satisfaction.Since this optimization model involves analytically intractable probabilistic formulation, traditional mathematical programming techniques cannot be readily applied. We develop the deterministic SPLINE and the stochastic R-SPLINE algorithms for different scenarios. The probability generating function is embedded into the deterministic algorithm to exactly calculate the desired performance measures, which is reasonable when dealing with independent data (with a small number of product classes as well). The stochastic R-SPLINE algorithm uses simulation to estimate the desired probabilistic measures, allowing correlations between different production recipes as well as between different demand classes. We also present a convergence analysis for the stochastic R-SPLINE algorithm. Experimental results demonstrate the efficiency of the developed algorithms compared to other existing approaches.

Effective Semi-Supervised Medical Image Segmentation with Probabilistic Representations and Prototype Learning.

Label scarcity, class imbalance and data uncertainty are three primary challenges that are commonly encountered in the semi-supervised medical image segmentation. In this work, we focus on the data uncertainty issue that is overlooked by previous literature. To address this issue, we propose a probabilistic prototype-based classifier that introduces uncertainty estimation into the entire pixel classification process, including probabilistic representation formulation, probabilistic pixel-prototype proximity matching, and distribution prototype update, leveraging principles from probability theory. By explicitly modeling data uncertainty at the pixel level, model robustness of our proposed framework to tricky pixels, such as ambiguous boundaries and noises, is greatly enhanced when compared to its deterministic counterpart and other uncertainty-aware strategy. Empirical evaluations on three publicly available datasets that exhibit severe boundary ambiguity show the superiority of our method over several competitors. Moreover, our method also demonstrates a stronger model robustness to simulated noisy data. Code is available at https://github.com/IsYuchenYuan/PPC.

Contextualized Trajectory Parsing with Spatio-Temporal Graph

This work investigates how to automatically parse object trajectories in surveillance videos, which aims at jointly solving three subproblems: 1) spatial segmentation, 2) temporal tracking, and 3) object categorization. We present a novel representation spatiotemporal graph (ST-Graph) in which: 1) Graph nodes express the motion primitives, each representing a short sequence of small-size patches over consecutive images, and 2) every two neighbor nodes are linked with either a positive edge or a negative edge to describe their collaborative or exclusive relationship of belonging to the same object trajectory. Phrasing the trajectory parsing as a graph multicoloring problem, we propose a unified probabilistic formulation to integrate various types of context knowledge as informative priors. An efficient composite cluster sampling algorithm is employed in search of the optimal solution by exploiting both the collaborative and the exclusive relationships between nodes. The proposed framework is evaluated over challenging videos from public datasets, and results show that it can achieve state-of-the-art tracking accuracy.

Robust and fast stochastic 4D Flow Vector-field signature technique for quantifying composite flow dynamics from 4D Flow MRI: Application to left atrial flow in atrial fibrillation

4D flow MRI is an emerging imaging modality that maps voxel-wise blood flow information as velocity vector fields that is acquired in 7-dimensional image volumes (3 spatial dimensions + 3 velocity directions + time). Blood flow in the cardiovascular system is often complex and composite involving multiple flow dynamics and patterns (e.g., vortex flow, jets, stagnating flow) that occur and interact simultaneously. The spectrum of such complex flow dynamics is embedded in the velocity vector field dynamics derived from 4D Flow MRI. However, current flow metrics cannot fully measure high-dimensional vector-field data and embedded complex composite flow data. Instead, these methods need to break down the vector-field data into secondary scalar fields of individual flow components using fluid dynamics operators. These methods are gradient-based and sensitive to data uncertainties, and only focus on individual flow components of the overall composite flow, therefore potentially underestimating the severity of overall flow changes associated with cardiovascular diseases. To address these limitations, in MICCAI 2021, we introduced a novel comprehensive stochastic 4D Flow vector-field signature technique that works directly on the entire spatiotemporal velocity vector field. This technique uses efficient stochastic gradient-free interrogation of multi-million flow vector-pairs per patient to derive the patient's unique flow profile of the complex composite flow alterations and in real-time processing. The signature technique's probabilistic gradient-free formulation should allow for highly robust quantification despite inherent errors in 4D flow MRI acquisitions. Here, we extend the application of the 4D flow vector-field signature technique to the left atrium to analyze complex composite flow changes in patients with atrial fibrillation. In 128 subjects, we performed extensive sensitivity testing and determined that the vector-field signature technique is highly robust to typical sources of data uncertainties in 4D flow MRI: degradation in spatiotemporal resolution, added gaussian noise, and segmentation errors. We demonstrate the excellent generalizability of the stochastic convergence from the aorta to the left atrium and between different 4D Flow MRI acquisition protocols. We compare the robustness of our technique to existing advanced flow quantification metrics of kinetic energy, vorticity, and energy loss demonstrating a superior performance of up-to 14-fold. Our results show the potential diagnostic and clinical utility of our signature technique in identifying distinctly altered composite flow signatures in atrial fibrillation patients independent of existing flow metrics.

Probabilistic assessment of the safety profile of the Fischer–Tropsch process with a supercritical solvent

AbstractInherent safety assessment during the design stage of chemical processes is typically conducted based on average values for design parameters. Under those conditions, the single‐point deterministic process performance assessment may be affected by the phenomenon known as the ‘flaw of averages’ in the presence of irreducible sources of uncertainty (performance evaluated at average conditions does not represent average performance). In this work, an inherent process safety assessment developed under a probabilistic formulation is presented. An evaluation of the proposed approach is performed in the case of a gas‐to‐liquid process system using a supercritical solvent for Fischer–Tropsch reactor systems. The pertinent uncertainty analysis is carried out using Monte Carlo simulation techniques to account for the propagation of uncertainty through the inherent process safety model and the derivation of probability distribution profiles for the associated metrics, thus statistically characterizing ranges of potential performance outcomes. The response variables were the autothermal reactor and the syngas flows. The results show that the input variables associated to the autothermal flow potentially generate the most hazardous conditions for the process. The results also show how the metrics are affected when uncertainty is explicitly taken into account at the design stage of the process, offering a more nuanced assessment and characterization of the inherent process safety profile.