Second-order Probability Research Articles

Thermal modelling of a small satellite data processing unit aided by sensitivity analysis and uncertainty quantification

Sensitivity analysis (SA) and uncertainty quantification (UQ) were used to verify the thermal design of the Intuition-1 nanosatellite data processing unit (DPU). For that purpose, a numerical model of the satellite was developed and validated against experimental data, which were obtained in thermal-vacuum chamber experiments. First, a simplified approach was applied to identify and eliminate less important parameters from further analysis. In the second step, a second-order probability approach was utilized to take into account the epistemic and aleatory uncertainties in the UQ study. The results identified crucial design parameters and the separate influence of aleatory and epistemic input uncertainty on the output variables. The results showed that the design of the nanosatellite DPU does not require modification and that minimizing the uncertainty of the input could considerably reduce the output uncertainty. The presented methodology can be applied to other systems to identify the importance of individual design parameters and predict the uncertainty of key responses under mixed aleatory and epistemic uncertainty. This process is crucial for the robustness of thermal design and provides valuable information supporting project decisions.

Variability of Single Pulse Electrical Stimulation Responses Recorded with Intracranial Electroencephalography in Epileptic Patients

Cohort studies of brain stimulations performed with stereo-electroencephalographic (SEEG) electrodes in epileptic patients allow to derive large scale functional connectivity. It is known, however, that brain responses to electrical or magnetic stimulation techniques are not always reproducible. Here, we study variability of responses to single pulse SEEG electrical stimulation. We introduce a second-order probability analysis, i.e. we extend estimation of connection probabilities, defined as the proportion of responses trespassing a statistical threshold (determined in terms of Z-score with respect to spontaneous neuronal activity before stimulation) over all responses and derived from a number of individual measurements, to an analysis of pairs of measurements.Data from 445 patients were processed. We found that variability between two equivalent measurements is substantial in particular conditions. For long ( > ~ 90 mm) distances between stimulating and recording sites, and threshold value Z = 3, correlation between measurements drops almost to zero. In general, it remains below 0.5 when the threshold is smaller than Z = 4 or the stimulating current intensity is 1 mA. It grows with an increase of either of these factors. Variability is independent of interictal spiking rates in the stimulating and recording sites.We conclude that responses to SEEG stimulation in the human brain are variable, i.e. in a subject at rest, two stimulation trains performed at the same electrode contacts and with the same protocol can give discrepant results. Our findings highlight an advantage of probabilistic interpretation of such results even in the context of a single individual.

Bayesian analysis for the Lomax model using noninformative priors

The Lomax distribution is an important member in the distribution family. In this paper, we systematically develop an objective Bayesian analysis of data from a Lomax distribution. Noninformative priors, including probability matching priors, the maximal data information (MDI) prior, Jeffreys prior and reference priors, are derived. The propriety of the posterior under each prior is subsequently validated. It is revealed that the MDI prior and one of the reference priors yield improper posteriors, and the other reference prior is a second-order probability matching prior. A simulation study is conducted to assess the frequentist performance of the proposed Bayesian approach. Finally, this approach along with the bootstrap method is applied to a real data set.

Application of objective priors for the multivariate Lomax distribution

For a model incorporating the effect of a common environment on several components of a system, a multivariate Lomax distribution (MLD) is generally considered by mixing exponential variables. Objective Bayesian has very good frequentist properties and provides a moderate solution for the prior elicitation which is one of important and difficult issues on Bayesian analysis. In this paper, we develop noninformative priors, such as the probability matching priors and reference priors, for the parameters of the MLD. We proved that a reference prior for the shape parameter is a first-order probability matching prior, but the reference priors for the scale parameters do not satisfy the first-order matching criterion. In addition, a second-order probability matching prior does not exist for all parameters. We also presented the conditions that make the posterior distributions for the general prior, including the probability matching prior and reference priors, to be proper. In particular, Jeffreys’ prior and probability matching priors for all parameters give proper posteriors, whereas reference priors for scale parameters give improper posteriors.

Quaternionic second-order freeness and the fluctuations of large symplectically invariant random matrices

We present a definition for second-order freeness in the quaternionic case. We demonstrate that this definition on a second-order probability space is asymptotically satisfied by independent symplectically invariant quaternionic matrices. This definition is different from the natural definition for complex and real second-order probability spaces, those motivated by the asymptotic behavior of unitarily invariant and orthogonally invariant random matrices respectively. Most notably, because the quaternionic trace does not have the cyclic property of a trace over a commutative field, the asymmetries which appear in the multi-matrix context result in an asymmetric contribution from the terms which appear symmetrically in the complex and real cases.

A probabilistic procedure for quantifying the relative importance of model inputs characterized by second-order probability models

This paper develops a new global sensitivity analysis (GSA) framework for computational models with input variables being characterized by second-order probability models due to epistemic uncertainties. Firstly, two graphical tools, called individual effect (IE) function and total effect (TE) function, are defined for identifying the influential and non-influential input variables. Secondly, two probabilistic GSA indices, called T-indices, are introduced for comparing the relative importance of pairwise influential input variables. Thirdly, the expected Sobol' indices are introduced for ranking the importance of the input variables. For efficiently estimating the proposed GSA indices, the extended Monte Carlo simulation (EMCS), whose computational cost is the same as the Monte Carlo simulation for estimating the Sobol' indices, is firstly introduced, and then a procedure combining Kriging surrogate model and EMCS procedure is introduced for further reducing the computational cost. Three numerical examples and a ten-bar structure are introduced for illustrating the significance of the proposed GSA framework and demonstrating the effectiveness of the computational methods.

Second-order matching prior family parametrized by sample size and matching probability

We propose a family of priors that satisfies the second-order probability matching property. The posterior quantile of a probability matching prior is exactly or approximately equal to the frequentist one. Most models lack an exact matching prior. If all quantiles of a prior’s posterior converge to the frequentist ones up to $$o(n^{-1/2})$$ or $$o(n^{-1})$$ as the sample size n increases, the prior is called a first-order probability matching prior and a second-order probability matching prior, respectively. Although a second-order matching prior does not necessarily exist, a first-order matching prior always exists. We introduce a class of priors that depend on the sample size and matching probability. We derive the condition under which the family satisfies the second-order probability matching property even when a second-order probability matching prior does not exist. The superiority of the proposed priors is illustrated in several numerical experiments.

Continuous-time smooth ambiguity preferences

This study extends the smooth ambiguity preferences model proposed by Klibanoff et al. (2005) to a continuous-time dynamic setting. It is known that the original smooth ambiguity preferences converge to the subjective expected utility as the time interval shortens so that decision makers do not exhibit any ambiguity-sensitive behavior in the continuous-time limit. Accordingly, this study proposes an alternative model of these preferences that interchanges the role of the second-order utility function with that of the second-order probability to prevent the smooth ambiguity attitude of decision maker from evaporating in the continuous-time limit. By utilizing the utility convergence results established by Kraft and Seifried (2014), our model is eventually represented by the stochastic differential utility with distorted beliefs so that most existing techniques in economics and financial studies can be made applicable together with these distorted beliefs. We give an asset-pricing example to demonstrate the applicability of our model.

Ambiguity preferences, risk taking and the banking firm

This paper examines the risk taking behavior of a banking firm facing ambiguity and possessing smooth ambiguity preferences. Ambiguity is modeled by a second-order probability distribution that captures the bank’s uncertainty about which of the subjective beliefs govern the return on its loans. Ambiguity aversion is modeled by a concave transformation of the (first-order) expected utility of profit conditional on each plausible subjective distribution of the random loan return. Within this framework, we show that the bank finds it less attractive to take risk in the presence than in the absence of ambiguity. This result extends to the case of greater ambiguity aversion. Given that the bank’s smooth ambiguity preferences exhibit non-increasing absolute ambiguity aversion, imposing a more stringent capital requirement to the bank has the desired effect that limits the bank’s incentive to take on excessive risk.

Extension of quasi-Newton approximation-based SORM for series system reliability analysis of geotechnical problems

The quasi-Newton approximation-based second-order reliability method is extended, through the incorporation of a linearization approach, to analyze the series system reliability of geotechnical problems. Two types of quasi-Newton approximations are applied to identify the design point and to compute the second-order probability of failure, respectively. The equivalent second-order reliability index is obtained using an approximation hyperplane for each limit state function. Then, the computed equivalent second order reliability indices can be employed, together with the corresponding unit direction vectors, to estimate the series system probability of failure using a linearization approach. Three geotechnical problems are employed to demonstrate the efficiency and accuracy of the suggested procedure, and advantages with respect to alternative computational tools are discussed.

A Continuous-Time Asset Pricing Model with Smooth Ambiguity Preferences

This study extends the smooth ambiguity preferences model proposed by Klibanoff et al. (2005, 2009) to a continuous-time dynamic setting. It is known that these preferences converge to the subjective expected utility as the time interval shortens so that decision makers do not exhibit any ambiguity-sensitive behavior in the continuous-time limit. Accordingly, this study proposes an alternative model of decision making that applies Yaari’s (1987) dual theory to the original preferences and interchanges the role of the second-order utility function with that of the second-order probability. We then formulate a recursive utility of smooth ambiguity-sensitive decision makers in continuous-time. Our model is represented by the stochastic differential utility with distorted beliefs so that most existing techniques in financial studies can be made applicable together with these distorted beliefs. We give an asset-pricing example to demonstrate the applicability of our model.

Optimal sequence of tests for the mediastinal staging of non-small cell lung cancer.

BackgroundNon-small cell lung cancer (NSCLC) is the most prevalent type of lung cancer and the most difficult to predict. When there are no distant metastases, the optimal therapy depends mainly on whether there are malignant lymph nodes in the mediastinum. Given the vigorous debate among specialists about which tests should be used, our goal was to determine the optimal sequence of tests for each patient.MethodsWe have built an influence diagram (ID) that represents the possible tests, their costs, and their outcomes. This model is equivalent to a decision tree containing millions of branches. In the first evaluation, we only took into account the clinical outcomes (effectiveness). In the second, we used a willingness-to-pay of € 30,000 per quality adjusted life year (QALY) to convert economic costs into effectiveness. We assigned a second-order probability distribution to each parameter in order to conduct several types of sensitivity analysis.ResultsTwo strategies were obtained using two different criteria. When considering only effectiveness, a positive computed tomography (CT) scan must be followed by a transbronchial needle aspiration (TBNA), an endobronchial ultrasound (EBUS), and an endoscopic ultrasound (EUS). When the CT scan is negative, a positron emission tomography (PET), EBUS, and EUS are performed. If the TBNA or the PET is positive, then a mediastinoscopy is performed only if the EBUS and EUS are negative. If the TBNA or the PET is negative, then a mediastinoscopy is performed only if the EBUS and the EUS give contradictory results. When taking into account economic costs, a positive CT scan is followed by a TBNA; an EBUS is done only when the CT scan or the TBNA is negative.This recommendation of performing a TBNA in certain cases should be discussed by the pneumology community because TBNA is a cheap technique that could avoid an EBUS, an expensive test, for many patients.ConclusionsWe have determined the optimal sequence of tests for the mediastinal staging of NSCLC by considering sensitivity, specificity, and the economic cost of each test. The main novelty of our study is the recommendation of performing TBNA whenever the CT scan is positive. Our model is publicly available so that different experts can populate it with their own parameters and re-examine its conclusions. It is therefore proposed as an evidence-based instrument for reaching a consensus.

Ambiguity and the multinational firm

This paper examines the optimal production and hedging decisions of the multinational firm under exchange rate uncertainty when the multinational firm possesses smooth ambiguity preferences. Ambiguity is modeled by a second-order probability distribution that captures the multinational firm's uncertainty about which of the subjective beliefs govern the exchange rate risk. Ambiguity preferences are modeled by the (second-order) expectation of a concave transformation of the (first-order) expected utility of home currency profit conditional on each plausible subjective distribution of the exchange rate risk. Within this framework, we show that the separation and full-hedging theorems are robust to the incorporation of ambiguity and ambiguity preferences. The presence of the currency hedging opportunity induces the multinational firm to produce more and sell less at home, and export more abroad. When the forward exchange rate is biased, we derive necessary and sufficient conditions under which the multinational firm optimally opts for a forward position that is closer to a full-hedge in response to either the introduction of ambiguity or greater ambiguity aversion.

The incentive to trade under ambiguity aversion

This paper examines the behavior of an exporting firm that sells in both the home country and a foreign country. The firm makes its optimal production and export decisions when facing ambiguous exchange rate risk. Ambiguity is modeled by a second-order probability distribution that captures the firm's uncertainty about which of the subjective beliefs govern the exchange rate risk. Ambiguity preferences are modeled by the (second-order) expectation of a concave transformation of the (first-order) expected utility of profit conditional on each plausible subjective distribution of the exchange rate risk. Within this framework, we derive necessary and sufficient conditions under which the ambiguity-averse firm optimally sells more in the home country and exports less to the foreign country in response either to the introduction of ambiguity or to greater ambiguity aversion when ambiguity prevails. We further show that ambiguity and ambiguity aversion have adverse effect on the firm's incentive to export to the foreign country.

On mechanics and material length scales of failure in heterogeneous interfaces using a finite strain high performance solver

Three-dimensional simulations capable of resolving the large range of spatial scales, from the failure-zone thickness up to the size of the representative unit cell, in damage mechanics problems of particle reinforced adhesives are presented. We show that resolving this wide range of scales in complex three-dimensional heterogeneous morphologies is essential in order to apprehend fracture characteristics, such as strength, fracture toughness and shape of the softening profile. Moreover, we show that computations that resolve essential physical length scales capture the particle size-effect in fracture toughness, for example. In the vein of image-based computational materials science, we construct statistically optimal unit cells containing hundreds to thousands of particles. We show that these statistically representative unit cells are capable of capturing the first- and second-order probability functions of a given data-source with better accuracy than traditional inclusion packing techniques. In order to accomplish these large computations, we use a parallel multiscale cohesive formulation and extend it to finite strains including damage mechanics. The high-performance parallel computational framework is executed on up to 1024 processing cores. A mesh convergence and a representative unit cell study are performed. Quantifying the complex damage patterns in simulations consisting of tens of millions of computational cells and millions of highly nonlinear equations requires data-mining the parallel simulations, and we propose two damage metrics to quantify the damage patterns. A detailed study of volume fraction and filler size on the macroscopic traction-separation response of heterogeneous adhesives is presented.

Quantification of Margins and Uncertainties for Integrated Spacecraft Systems Models

The objective of this study was to introduce an efficient and accurate approach to the quantification of margins and uncertainties for integrated spacecraft systems models. In this study, stochastic expansions, based on nonintrusive polynomial chaos, were used for efficient representation of uncertainty both in design metrics and associated performance limits of a system. Additionally, procedures were outlined for analyzing systems that contain different uncertainty types between the performance metrics and performance limits. These methodologies were demonstrated on two model problems, each possessing mixed (epistemic and aleatory) uncertainty, which was propagated through the models using second-order probability. The first was a complex system model of highly nonlinear analytical functions. The second was a coupled multisystem, physics-based model for spacecraft reentry. The performance metrics consisted of two systems used to determine the maximum load, the necessary bank angle correction, and maximum convective heat load along a reentry trajectory. Overall, the methodologies and examples of this work have detailed an efficient approach for measuring the reliability of complex spacecraft systems models, as well as the importance of quantifying margins and uncertainties for the design of reliable systems.

Through-space and through-bridge interactions in the correlation analysis of chemical bonds

The formalism of the through-space and through-bridge communications proposed by Nalewajski within the framework of the Orbital Communication Theory is used to evaluate the correlation effect between electron populations of two chemical bonds. Proposed in this paper purely-probabilistic formalism defines the correlation between two chemical bonds as determined by the difference between appropriately normalized fourth-order joint probabilities corresponding to direct (through-space) and indirect (through-bridge) communication within a four-state time-homogeneous Markov chain. Alternative bond-correlation coefficient introduced earlier by Yamasaki and Goddard is based on the concept of the hierarchy of statistical covariance operators. The proposed correlation coefficient requires computing only the second-order probability terms when working within the condensed-atomic resolution and thus they are far less computationally expensive making them suitable for correlation analysis of chemical interactions between large molecular fragments.

Uncertainty Levels of Second-Order Probability

Since second-order probability distributions assign probabilities to probabilities there is uncertainty on two levels. Although different types of uncertainty have been distinguished before and corresponding measures suggested, the distinction made here between first- and second-order levels of uncertainty has not been considered before. In this paper previously existing measures are considered from the perspective of first- and second-order uncertainty and new measures are introduced. We conclude that the concepts of uncertainty and informativeness needs to be qualified if used in a second-order probability context and suggest that from a certain point of view information can not be minimized, just shifted from one level to another

A qualified description of extended fuzzy logic

The sense of approximate reasoning within fuzzy logic has given rise to speculation as to whether precise reasoning within classical logic can even be considered a feasible way of handling most of the world’s problems. Now, another question can be raised; can fuzzy logic, by itself, justify/handle the sort of uncertainty found in problems that belong to the class of open world problems? This family of problems corresponds to the case of ambiguous mathematical theories mainly due to the fact that there is just imperfect/incomplete information about them. Recently, Zadeh introduced an extension of fuzzy logic, claimed to bridge the gaps in conventional approaches to the open world problems. In this paper, we present a basic background of extended fuzzy logic (FLe). We address several main concepts such as f-transformation and f-validity, enhance the definition of generalized constraints, and propose that FLe’s rules have two layers: one based on the implication model and another based on the conjunction model. In addition, we explore how FLe can handle those conditions described in antonym pairs as well as those occurred without considering rationality in the framework. Both cases challenge most decision-making theories. We also suggest FLe and second-order probability are f-complementary rather than f-competitive. The paper asserts that FLe is highly capable of constructing mathematical solutions to problems (including computational problems) that are stated in natural language.

Reliability Analysis for Rotor System with Oil Whip and Resonance Based on Frequency

Parameter uncertainty of general systems is inherent in most engineering problems. Based on the regularity of oil whip and resonance for rotor system, the reliability problem of rotor system with oil whip and resonance is studied by applying random perturbation technology and reliability theory considering the correlation of the multi-order natural frequency. The reliability mode and the failure probability of rotor systems are defined. The second-order joint failure probability is obtained by using the numerical integration method. The presented method provided the theoretic basis for the reliability design of the rotor system. A numerical example demonstrated that the proposed method is effective.