Finding the parameters of a Bernoulli-Gaussian Model

<div><div><div><p>A transmission medium perturbed by an additive noise from which the estimated noise power is all information known, is better modeled as a Gaussian channel. Since the Gaussian channel is, according to Information Theory, the worst channel to transmit information through, this is the most pessimistic assumption. When noise samples are available though, choosing to model the transmission medium using a more sophisticated model pays off. The Bernoulli-Gaussian channel, would be one such a choice. Finding the three parameters that characterize the Bernoulli-Gaussian stochastic process which mathematically models the noise is a task of paramount importance. Many algorithms can be used to estimate the parameters of this model based on numerical methods. In the current work a closed form expression to estimate the model parameters is presented. All that is required besides the estimation of the power of Bernoulli-Gaussian process from the available noise samples is the estimation of two additional quantities: the expected value of the absolute value of the amplitude of the process—the first absolute moment—plus the third absolute moment, viz., the expected value of the third power of the absolute value of the process. An alternative option, often used for power line communication, is to model the transmission medium as a channel in which the noise is represented by a three parameter stochastic process called Middleton Class A. Other models (like generalized-Bernoulli-Gaussian, or Bernoulli- Gaussian with memory) might render a better medium model than the Bernoulli-Gaussian channel. Estimating the parameters of these processes is however a cumbersome task and, as we show in the current work, the rate harvested by using the simple, yet more sophisticated, Bernoulli-Gaussian channel is increased as compared to the, more pessimistic, Gaussian channel, allowing one thus to more closely approach the true capacity. The communication system design can be much improved if a well fit Bernoulli-Gaussian stochastic process is selected to model the true noise. The incorporation of the Bernoulli-Gaussian channel in the communication system model leads to a better design as corroborated by the computer simulation results presented.</p></div></div></div>

Finding the parameters of a Bernoulli-Gaussian Model

<div><div><div><p>A transmission medium perturbed by an additive noise from which the estimated noise power is all information known, is better modeled as a Gaussian channel. Since the Gaussian channel is, according to Information Theory, the worst channel to transmit information through, this is the most pessimistic assumption. When noise samples are available though, choosing to model the transmission medium using a more sophisticated model pays off. The Bernoulli-Gaussian channel, would be one such a choice. Finding the three parameters that characterize the Bernoulli-Gaussian stochastic process which mathematically models the noise is a task of paramount importance. Many algorithms can be used to estimate the parameters of this model based on numerical methods. In the current work a closed form expression to estimate the model parameters is presented. All that is required besides the estimation of the power of Bernoulli-Gaussian process from the available noise samples is the estimation of two additional quantities: the expected value of the absolute value of the amplitude of the process—the first absolute moment—plus the third absolute moment, viz., the expected value of the third power of the absolute value of the process. An alternative option, often used for power line communication, is to model the transmission medium as a channel in which the noise is represented by a three parameter stochastic process called Middleton Class A. Other models (like generalized-Bernoulli-Gaussian, or Bernoulli- Gaussian with memory) might render a better medium model than the Bernoulli-Gaussian channel. Estimating the parameters of these processes is however a cumbersome task and, as we show in the current work, the rate harvested by using the simple, yet more sophisticated, Bernoulli-Gaussian channel is increased as compared to the, more pessimistic, Gaussian channel, allowing one thus to more closely approach the true capacity. The communication system design can be much improved if a well fit Bernoulli-Gaussian stochastic process is selected to model the true noise. The incorporation of the Bernoulli-Gaussian channel in the communication system model leads to a better design as corroborated by the computer simulation results presented.</p></div></div></div>

Gaussian Stochastic Process Modeling of Blend Repaired Airfoil Modal Response Using Reduced Basis Mode Shape Approach

A machine learning (ML) approach is developed to predict the effect of blend repairs on airfoil frequency, modal assurance criterion (MAC), and modal displacement vectors. The method is demonstrated on a transonic research rig compressor rotor airfoil. A parametric definition of blend geometry is developed and shown to be capable of encompassing a large range of blend geometry. This blend repair geometry is used to modify the airfoil surface definition and a mesh morphing process transforms the nominal finite element model (FEM) to the repaired configuration. A multi-level full factorial sampling of the blend repair design space provides training data to a Guassian stochastic process (GSP) regressor. The frequency and MAC results create a vector of training data for GSP calibration, but the airfoil mode shapes require further mathematical manipulation to avoid creating GSP models for each nodal displacement. This paper develops a method to significantly reduce blended airfoil mode shape emulation cost by transforming the mode shape training data into a reduced basis space using principal component analysis (PCA). The coefficients of this reduced basis are used to train a GSP that can then predict the values for new blended airfoils. The emulated coefficients are used with the reduced basis vectors in a reconstruction of blended airfoil mode shape. Validation data is computed at a full-factorial design that maximizes the distance from training points. It is found that large variations in modal properties from large blend repairs can be accurately emulated with a reasonable number of training points. The reduced basis approach of mode shape variation is shown to more accurately predict MAC variation when compared to direct MAC emulation. The added benefit of having the full modal displacement field also allows determination of other influences such as tip-timing limits and modal force values.

Tabu efficient global optimization with applications in additive manufacturing

Methods based on Gaussian stochastic process (GSP) models and expected improvement (EI) functions have been promising for box-constrained expensive optimization problems. These include robust design problems with environmental variables having set-type constraints. However, the methods that combine GSP and EI sub-optimizations suffer from the following problem, which limits their computational performance. Efficient global optimization (EGO) methods often repeat the same or nearly the same experimental points. We present a novel EGO-type constraint-handling method that maintains a so-called tabu list to avoid past points. Our method includes two types of penalties for the key “infill” optimization, which selects the next test runs. We benchmark our tabu EGO algorithm with five alternative approaches, including DIRECT methods using nine test problems and two engineering examples. The engineering examples are based on additive manufacturing process parameter optimization informed using point-based thermal simulations and robust-type quality constraints. Our test problems span unconstrained, simply constrained, and robust constrained problems. The comparative results imply that tabu EGO offers very promising computational performance for all types of black-box optimization in terms of convergence speed and the quality of the final solution.

Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

We combine the stochastic perturbation method with the maximum entropy principle to construct approximations of the first probability density function of the steady-state solution of a class of nonlinear oscillators subject to small perturbations in the nonlinear term and driven by a stochastic excitation. The nonlinearity depends both upon position and velocity, and the excitation is given by a stationary Gaussian stochastic process with certain additional properties. Furthermore, we approximate higher-order moments, the variance, and the correlation functions of the solution. The theoretical findings are illustrated via some numerical experiments that confirm that our approximations are reliable.

Sequentially Estimating the Approximate Conditional Mean Using Extreme Learning Machines

This study examined the extreme learning machine (ELM) applied to the Wald test statistic for the model specification of the conditional mean, which we call the WELM testing procedure. The omnibus test statistics available in the literature weakly converge to a Gaussian stochastic process under the null that the model is correct, and this makes their application inconvenient. By contrast, the WELM testing procedure is straightforwardly applicable when detecting model misspecification. We applied the WELM testing procedure to the sequential testing procedure formed by a set of polynomial models and estimate an approximate conditional expectation. We then conducted extensive Monte Carlo experiments to evaluate the performance of the sequential WELM testing procedure and verify that it consistently estimates the most parsimonious conditional mean when the set of polynomial models contains a correctly specified model. Otherwise, it consistently rejects all the models in the set.

Stochastic modeling of extreme El-Niño and La Niña events by nonlinearly coupled oscillators

&lt;p&gt;The El-Ni&amp;#241;o index behaves as a nonlinear and non-Gaussian stochastic process. A well-known characteristic is its positive skewness coming from the occurrence of stronger episodes of El-Ni&amp;#241;o than of La Ni&amp;#241;a. Here, we use the period 1870-2018 of the standardized El-Ni&amp;#241;o index x(t), sampled in trimesters to analyze the spectral origin of the bicorrelation: sk(t1,t2)=E[x(t)x(t+t1)x(t+t2)] and skewness sk(0,0). For that, we estimate the two-dimensional Fourier transform of sk(t1,t2) or bispectrum B(f1,f2). Its sum over bi-frequencies (f1,f2) equals the skewness (0.45 in our case). Positive and negative bispectrum peaks are due to phase locking of frequency triplets: (f1,f2,f1+f2), contributing to extreme El-Ni&amp;#241;os and La Ni&amp;#241;as respectively. Moreover, the most significant positive and/or negative bispectrum regions are rather well localized in the bispectrum domain. Here, we propose a partition of the El Ni&amp;#241;o signal into a set of band-pass spectrally separated components whose self and cross interactions can explain the broad structure of bispectrum. In the simplest case where the signal is decomposed into a fast and a slow component (with a cutoff frequency of (1/2.56) cycles/yr.), we verifty that slow-slow interactions (or phase locking) explain most of La-Ni&amp;#241;as, particularly at the frequency triplet (1/4.9, 1/15 and 1/3.7 cycles/yr) whereas the fast-slow interactions explain most of El Ni&amp;#241;os, particularly at the frequency triplet (1/4.9, 1/4.9 and 1/2.5 cycles/yr). In order to simulate this stochastic behavior, we calibrate a set of nonlinearly coupled oscillators (Auto-regressive processes, forced by self and cross quadratic component terms), one for each component. In the case of weak cross-component interactions, and thus weak nonlinearity, the coupling coefficients between spectral-band components are proportional to the corresponding cross-skewnesses, which represent good first guesses in the calibration of the model parameters. The predictability of the model is then assessed, in particular for the anticipation of big El Ni&amp;#241;os and la Ni&amp;#241;as. The authors would like to acknowledge the financial support FCT through project&amp;#160;&lt;strong&gt;UIDB/50019/2020 &amp;#8211; IDL.&lt;/strong&gt;&lt;/p&gt;

The accuracy of modeling of Gaussian stochastic process in some Orlicz spaces

The main purpose of this study is the construction of a model of a Gaussian stochastic process with given reliability and accuracy in some Orlicz spaces. In the paper, a suitable model is presented, conditions for the model parameters are derived, and some examples of their calculations are given.