Bayesian Inference in Credibility Theory

In many practical periodic parameter estimation problems, the appropriate cost function is periodic with respect to the unknown parameter. In this paper a new class of cyclic Bayesian lower bounds on the mean cyclic error (MCE) is developed. The new class includes the cyclic version of the Bayesian Cramér-Rao bound (BCRB). The cyclic BCRB requires milder regularity conditions compared to the conventional BCRB. The tightest bound in the proposed class is derived and it is shown that under a certain condition it achieves the minimum MCE (MMCE). The new lower bounds are compared with the cyclic version of the Ziv-Zakai lower bound (ZZLB) and the MCE's of the MMCE and maximum aposteriori probability (MAP) estimators for frequency estimation with uniform a-priori probability density function (pdf) of the unknown parameter. In this common estimation problem, the conventional BCRB does not exist, while the proposed cyclic BCRB provides a valid lower bound for parameter estimation.

PoCET: a Polynomial Chaos Expansion Toolbox for Matlab

Dithering improves the linearity of A/D converters (ADCs). Wagdy and Goff (1994) have developed a criterion for assessing the deviation of ADC transfer characteristics from the unity-gain line. This criterion is based on a closed-form formula for the deviation factor D derived from the characteristic function of the ADC quantization error function. This approach was extended further by Wagdy (1996) to include nonideal ADCs. The results in that work pertain to additive dither, but a closed-form formula for subtractive dither may be very difficult to derive. In this paper, the ADC is modeled and dither is simulated using a software random number generator. First, additive dithering with continuous (analog) uniform (rectangular) probability density function (RPDF) is considered. The cases of an ADC with no errors, single-bit error, or multi-bit errors are investigated. Results are in excellent agreement with the previous ones, which validates the formulae therein. Second, the author presents new results using discrete (digital) rectangular dither (DRPDF). This dither type is generated via a D/A converter (DAC) whose number of bits, k, is greater by del than the ADC number of bits, m. For various values of del, subtractively-dithered ADCs outperform additively-dithered ones as far as D is concerned. Also, the effect of increasing del on D for an ADC is investigated for the subtractive dither case, thus providing some quantitative results which are useful for the designer.

We investigate subcritical resonant heat transfer in a heated periodic grooved channel by modulating the flow with an oscillation of random amplitude. This excitation effectively destabilizes the flow at relatively low Reynolds number and establishes strong communication between the grooved flow and the Tollmien–Schlichting channel waves, as revealed by various statistical quantities we analysed. Both single-frequency and multi-frequency responses are considered, and an optimal frequency value is obtained in agreement with previous deterministic studies. In particular, we employ a new approach, the multi-element generalized polynomial chaos (ME-gPC) method, to model the stochastic velocity and temperature fields for uniform and Beta probability density functions (PDFs) of the random amplitude. We present results for the heat transfer enhancement parameter $E$ for which we obtain mean values, lower and upper bounds as well as PDFs. We first study the dependence of the mean value of $E$ on the magnitude of the random amplitude for different Reynolds numbers, and we demonstrate that the deterministic results are embedded in the stochastic simulation results. Of particular interest are the PDFs of $E$, which are skewed with their peaks increasing towards larger values of $E$ as the Reynolds number increases. We then study the effect of multiple frequencies described by a periodically correlated random process. We find that the mean value of $E$ is increased slightly while the variance decreases substantially in this case, an indication of the robustness of this excitation approach. The stochastic modelling approach offers the possibility of designing ‘smart’ PDFs of the stochastic input that can result in improved heat transfer enhancement rates.

The classical intermittency theory developed forty years ago considers a uniform reinjection probability density function (RPD). However, in the last ten years, studies have found a more general RPD, which depends on the type of prereinjection points, extreme or infinite derivative points. Recently, we have introduced an analytical scheme to estimate the non-uniform RPD when the pre-reinjection points are extreme. Here, this theoretical scheme is extended to embrace pre-reinjection points with infinite derivative. The proposed theoretical method, to get the RPD, needs only the explicit expression of the map, and it is general and direct to use. For different intermittency types and different non-linearity, the theoretical predictions show a very high agreement with the numerical results.

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A node in a switch architecture is examined. The node receives messages, stores and processes them locally, and transmits them to other nodes. Both message transmission and processing times depend on the message length, which is a random variable. The node is modeled as a queue with a single server where the interarrival and service times are related, and different cases are studied where this relation ranges from direct dependence to moderate correlation. Exponential and uniform probability density functions (pdf's) are examined, and the results are compared to the independent model. A relationship is developed for the system time pdf of a message, which is used to find the system time pdf and the system state probabilities. These quantities are directly related to buffer sizes and transmission delays within the architecture. >

Combined heat power dispatch with improved operation remains a popular preference in microgrid having uncertain and non-convex nature of the system. This research work deploys an efficient energy management system driven by a novel stochastic transitions tuned unscented Gaussian distribution (ST-UnGD) for optimal energy transaction. The proposed scheme has been introduced with uncertainty modeling of available generations and load using probabilistic transition-based timed petri net. Echo algorithm has been used to influence prioritized transition for appropriate number of token distribution. The control over the process during transition has been supervised with UnGD. A uniform probability density function has been created using Gaussian distribution to characterize random variables for effective time series scenario generation. The mean and variance which helps in sorting ten competent scenarios from one thousand scenarios, has been computed with the proposed strategy and compared with other methods. The improvement has been noticed with standard deviation of 0.0162% after 40 trials run with average operational cost of 254.211€ when token distribution was updated by 100, 200, 300, 400, and 500 firings. A statistical analysis by parametric and non-parametric test confirmed the effectiveness of the proposed algorithm showing appropriate confidence level in the mentioned case study.

In recent decades, analysis of structures considering variability of some parameters for more reliable design has attracted the attention of researchers. In this paper, the stochastic analysis of a cantilever deep beam made of large deformable neo-hookean material is carried out. For this purpose, the meshless local Petrov-Galerkin (MLPG) method is developed to obtain the geometrically non-linear equilibrium equations. The radial point interpolation method is used for generating the shape functions. The incremental iterative Newton-Raphson method with suitable load steps is used to solve the non-linear governing equations. The results of deterministic analysis obtained with proposed method are compared with the finite element results and good agreement is achieved. The initial elasticity modulus of neo-hookean material is considered to be uncertain variable. To generate random field of uncertain variable with normal, lognormal and uniform probability density functions (PDFs), the Monte Carlo Simulation (MCS) technique was employed. The sufficient number of simulations for convergence the results was determined experimentally. The effect of elasticity modulus, PDF and coefficients of variation (COV) on maximum vertical displacement, PDF and COV of results are studied in details. Comparing the stochastic and deterministic results shows that the uncertainty in mechanical properties has significant effect on results.

The problem of statistical pattern recognition with noisy or imprecise feature measurements is considered. An exact analytical expression is found for the probability of misclassification under this condition, for multiclass multivariate systems. The probability of error exceeds that of the ideal case for the special case of two classes, the a priori conditional probability density functions are assumed to be normal, along with the two cases of feature measurement error, namely normal and uniform probability density functions. Monotonicity of the misclassification probability with measurement error variance is shown. Numerical results are presented for both cases over a workable range of parameters. The study is useful in practical pattern recognition problems.

In this study, we investigate the impact of pore distribution and size on the mechanical response of elastic porous materials. Two-dimensional porous media with convex porosity are considered (e.g., foams and sponges). These structures are numerically generated by subtracting randomly dispersed circular holes from a solid square domain. The in-plane position vector of pore centers is given by a uniform probability density function (PDF). The pore diameters are, instead, drawn from a non-uniform two-parameter continuous probability distribution. The influence of the void ratio, i.e., the volume fraction of voids, and the parameters of the diameter PDF are studied over a wide range of values, including the percolation threshold, where the effective elastic modulus of the material abruptly drops to zero. This transition takes place at a critical value of the void ratio, which turns out to be unaffected by the parameters of the pore size distribution. Our results demonstrate that the elastic response of these structures is primarily governed by the void ratio, with the correlation length playing a secondary role. While the pore size distribution may affect the correlation length, it has a minimal impact on the foam’s overall behavior.

In most studies of flood levee reliability the probability of failure is taken to be equal to the probability that the flood peak discharge will exceed the capacity of the channel‐levee system. Such an analysis assumes that levees fail only by overtopping, whereas in many cases, levees fail at discharges much below the channel capacity. In the present paper the discharge at which levees fail is assumed to be a random variable with a particular distribution. The results on the flood frequency curve are presented for two conditions: one for the failure discharge having a uniform probability density function and the other for failure discharges having a quadratic distribution. In a similar manner the effects on the net benefits from levee construction were also analyzed and results are given.

In windy conditions, the measurement of liquid and solid atmospheric precipitation is still a challenge even using the most advanced automatic instrumentation (Cauteruccio et al., 2021). The measurement accuracy is affected by various environmental sources of bias, including siting issues and exposure. These add to the instrumental bias, which can be minimized in case of accurate instrument calibration. Wind is however recognised as the most impactful source of environmental bias, outperforming by 3 to 50 times the total impact of all other environmental factors. Computational Fluid Dynamics simulation with embedded liquid (raindrops) and solid (snowflakes) particle tracking is here used to quantify the wind-induced bias of catching-type precipitation gauges. Starting from the numerically calculated catch ratios, six common commercial gauges having different outer geometry are compared in terms of their expected performance under various precipitation intensity and wind speed conditions. Preliminary wind tunnel experiments allowed full validation of the simulated aerodynamic behaviour and its effect on water drop trajectories. The overall collection efficiency is shown to depend on the precipitation intensity and its functional dependence is quantitatively derived as a measure of the instrument performance under a wind climatology characterised by a uniform probability density function. A less pronounced diversion of hydrometeor trajectories is shown – at any given size – by instruments with aerodynamic design than in case of more traditional geometry. Chimney-shaped instruments rank low in case of liquid precipitation measurements, while a high performance is shown by inverted conical and Nipher shielded instruments and the investigated quasi-cylindrical gauges have intermediate behaviour, which depends on their specific aerodynamic features. All instruments rank low at light to moderate precipitation intensity for the measurement of solid precipitation, except the Nipher shielded gauge. This work provides the basic information needed to apply adjustments to the measured data and supports manufacturers in upgrading instruments with an existing design by introducing on-board adjustments of the measured precipitation. These would only require contemporary measurement of the wind velocity (often included in typical meteorological stations). The full work and the numerically derived adjustments for the six investigated commercial gauges are published in Cauteruccio et al. (2024).

Credibility theory: a new view from the theory of second order optimal statistics

One approach to solve the Electric Impedance Tomography inverse problem is the Bayesian Inference. The use of Bayes’ Rule of conditional posterior probability density functions requires the definition of two probability density functions, a prior probability density function and a likelihood. The prior probability density function is often called Anatomical Atlas when the information is based on anatomy and physiology. Sufficient prior information of a pathology is necessary in order to have this pathology correctly represented in the tomographic image. A measure of how much a particular pathology is represented in an anatomical atlas is proposed.