non-Gaussian Power Research Articles

Neural network-aided simulation of non-Gaussian stochastic processes

Simulation of non-Gaussian stochastic processes is of paramount importance to dynamic reliability assessment of structures driven by non-Gaussian loads. In this paper, a novel translation model based on neural network is proposed to convert the target non-Gaussian power spectrum to the underlying Gaussian power spectrum for simulating non-Gaussian stochastic processes. First, a valid neural network model, whose structure is similar to the transformation from a non-Gaussian power spectrum to its underlying Gaussian power spectrum, is established. Then, starting with an initial underlying Gaussian power spectrum, the non-Gaussian power spectrum corresponding to the target non-Gaussian distribution is obtained through the Wiener-Khintchine transform and translation process. After that, the discrete data of non-Gaussian power spectrum is employed as the input layer in the neural network above, while the discrete data of Gaussian power spectrum serves as the output layer. With the trained neural network, the required underlying Gaussian power spectrum can be directly obtained by inputting the target non-Gaussian power spectrum. Finally, Gaussian samples can be generated by using the spectral representation method, which are then converted into non-Gaussian samples via the translation process. The effectiveness, accuracy, and applicability of the proposed method are demonstrated through two numerical examples.

Dynamic response and reliability analyses of non-linear structures driven by non-stationary non-Gaussian stochastic ground motions

In this paper, stochastic dynamic response and reliability assessment of non-linear structures driven by non-stationary non-Gaussian ground motions are investigated. First, the non-stationary non-Gaussian ground motions are simulated by using the translation process theory. To circumvent the problem of a large number of iterations, an improved upgrading model is developed to obtain a power spectral density which is compatible with the target non-Gaussian probability distribution and is as close as possible to the target non-Gaussian power spectral density. To conduct stochastic dynamic response and reliability analyses of non-linear structures driven by non-stationary non-Gaussian ground motions, the probability density evolution method is employed, whereby a new high-dimensional points selection strategy based on the clustering algorithm is put forward. The new strategy of points selection is quite flexible and simple while taking into account both efficiency and accuracy. The efficacy and applicability of the proposed method are verified through a numerical example and a practical engineering example, in which the influence of non-Gaussianity on the probabilistic seismic response and reliability of the structure is discussed.

Simulation of stationary Gaussian/non-Gaussian stochastic processes based on stochastic harmonic functions

A new model is proposed to represent and simulate Gaussian/non-Gaussian stochastic processes. In the proposed model, stochastic harmonic function (SHF) is extended to represent multivariate Gaussian process firstly. Compared with the conventional spectral representation method (SRM), the SHF based model requires much fewer variables and Cholesky decompositions. Then, SHF based model is further extended to univariate/multivariate non-Gaussian stochastic process simulation. The target non-Gaussian process can be obtained from the corresponding underlying Gaussian processes by memoryless nonlinear transformation. For arbitrarily given marginal probability distribution function (PDF), the covariance function of the underlying multivariate Gaussian process can be determined easily by introducing the Mehler’s formula. And when the incompatibility between the target non-Gaussian power spectral density (PSD) or PSD matrix and marginal PDF exists, the calibration of the target non-Gaussian spectrum will be required. Hence, the proposed model can be regarded as SRM to efficiently generate Gaussian/non-Gaussian processes. Finally, several numerical examples are addressed to show the effectiveness of the proposed method.

Kernel Density Estimation Based Gaussian and Non-Gaussian Random Vibration Data Induction for High-Speed Train Equipment

Because general statistics tolerance is not applicable to the induction of non-Gaussian vibration data and the methods for converting non-Gaussian data into Gaussian data are not always effective and can increase the estimation error, a novel kernel density estimation method in which induction is carried out on power spectral density data for the measured vibration of high-speed trains is proposed in this paper. First, data belonging to the same population of power spectral density are merged into the same feature sample. Then, the probability density function of all power spectral density values at the first frequency point is calculated through the kernel density estimation method, and the upper-limit estimate of all power spectral density values under the set quantile is obtained. This process is repeated, and the upper limit values of the power spectral density values at all frequency points can be obtained to convert the measured acceleration data to the acceleration power spectral density spectrum of the vibration test. Engineering examples are used to verify the proposed method. For the same Gaussian power spectral density data, the relative error between the root mean squares of the power spectral density spectrum obtained from induction by the kernel density estimation method and the statistics tolerance is 0.155% ~1.55%; for the non-Gaussian power spectral density data, the acceleration power spectral density spectrum of the non-Gaussian vibration can be obtained with the induction by the kernel density estimation method. The proposed kernel density estimation method satisfies the induction requirements for the measured Gaussian and non-Gaussian vibration data of high-speed trains with two different distributions, and its induction results have very good universality and estimation accuracy.

Statistical anisotropies in temperature and polarization fluctuations from a scale-dependent trispectrum

We study statistical anisotropies generated in the observed two-point function of the cosmic microwave background (CMB) fluctuations if the primordial statistics are non-Gaussian. Focusing on the dipole modulations of the anisotropies, we find that the hemispherical power asymmetry observed in the CMB temperature fluctuations can be modeled by a local-type trispectrum with amplitude $\tau_{\rm NL}(k_p=0.05~{\rm Mpc}^{-1}) \approx 2 \times 10^4$ and a large red tilt $n\approx -0.68$. We numerically evaluate the non-Gaussian covariance of the modulation estimators for both temperature and E-mode polarization fluctuations and discuss the prospects of constraining the model using Planck satellite data. We then discuss other effects of the scale-dependent trispectrum that could be used to distinguish this scenario from other explanations of the power asymmetry: higher-order modulations of the two-point function and the non-Gaussian angular power spectrum covariance. As an important consequence of the non-Gaussian power spectrum covariance, we discuss how the CMB-inferred spectral index of primordial scalar fluctuations can be significantly biased in the presence of a scale-dependent local-type trispectrum.

Energy-Per-Bit Performance Analysis of Relay-Assisted Power Line Communication Systems

This paper provides a comprehensive analysis of the energy efficiency performance for different relaying schemes over the non-Gaussian power line communication (PLC) channel. Specifically, amplify-and-forward (AF), decode-and-forward (DF), selective DF (SDF) and incremental DF (IDF) relaying systems are investigated. For a more realistic scenario, the power consumption profile of the PLC modems is assumed to consist of both dynamic and static power. For each system, we derive accurate analytical expressions for the outage probability and the minimum energy-per-bit performance. For the sake of comparison and completeness as well as to quantify the achievable gains, we also analyze the performance of a single-hop PLC system. Monte Carlo simulations are provided throughout this paper to validate the theoretical analysis. Results reveal that AF relaying over the non-Gaussian PLC channel does not always enhance the performance and that the IDF PLC system offers the best performance compared to all other schemes considered. It is also shown that increasing the channel variance, which is related to the PLC network branching, and impulsive noise probability can considerably deteriorate the system performance. Furthermore, when the end-to-end distance is relatively small, it is found that the single-hop PLC approach can perform better than AF relaying.

Non-Gaussian power grid frequency fluctuations characterized by Lévy-stable laws and superstatistics

Multiple types of fluctuations impact the collective dynamics of power grids and thus challenge their robust operation. Fluctuations result from processes as different as dynamically changing demands, energy trading, and an increasing share of renewable power feed-in. Here we analyze principles underlying the dynamics and statistics of power grid frequency fluctuations. Considering frequency time series for a range of power grids, including grids in North America, Japan and Europe, we find a substantial deviation from Gaussianity best described as L\'evy-stable and q-Gaussian distributions. We present a coarse framework to analytically characterize the impact of arbitrary noise distributions as well as a superstatistical approach which systematically interprets heavy tails and skewed distributions. We identify energy trading as a substantial contribution to today's frequency fluctuations and effective damping of the grid as a controlling factor enabling reduction of fluctuation risks, with enhanced effects for small power grids.

Probabilistic load flow with versatile non-Gaussian power injections

A probabilistic load flow distinguished by the versatility in the way in which input data can be provided is presented. The main contribution of the proposal involves taking advantage of the data available as well as completing any missing information. This enables the proposal to be applied at any voltage level, even in medium-voltage networks where there is a glaring lack of systematic data collection. The use of the Gaussian Mixture Model is also a key feature of the proposed solution, and determinant in the final solution. Not only does the detailed and thorough analysis through numerous tests demonstrate the good performance of the proposed procedure, but it also confirms the accuracy of the results.

Experimental Approbation of Selective Laser Melting of Powders by the Use of Non-Gaussian Power Density Distributions

Experimental results on laser beam modulation at selective laser melting (SLM) are presented. The modulation is a possible way to improve the efficiency of the SLM process. The optical diagnostics shows the energy loss up to 30%. This could be a consequence of high thermal gradients in the melt pool resulted by the Gaussian power density distribution. The Gaussian distribution can be changed to the flat-top one or to the inverse-Gaussian (donut) one. An experimental stand with a 200W laser source was developed. Twenty single tracks for each of the three modes were obtained for a Co-Cr alloy. The samples were studied by scanning electronic microscopy (SEM) on irregularity. Optical diagnostics by high velocity camera (HVC) shows that the use of the non-Gaussian laser beam distributions can significantly reduce the width of the free-of-powder consolidation zone, which is considered as the main reason for irregularity of single tracks. A better metallurgical bonding of the single tracks with the substrate was obtained by the use of the flat-top laser beam. Both of these facts show a significant influence of the laser beam energy distribution on the energy loss at SLM, especially for high power laser sources. The observed escape of granules shows a possible influence of the dynamic factor. The use of the non-Gaussian distributions can enhance 3D parts, for example, improve the geometrical tolerance and decrease the residual porosity.

A simple and efficient methodology to approximate a general non-Gaussian stationary stochastic process by a translation process

Some widely used methodologies for simulation of non-Gaussian processes rely on translation process theory which imposes certain compatibility conditions between the non-Gaussian power spectral density function (PSDF) and the non-Gaussian probability density function (PDF) of the process. In many practical applications, the non-Gaussian PSDF and PDF are assigned arbitrarily; therefore, in general they can be incompatible. Several techniques to approximate such incompatible non-Gaussian PSDF/PDF pairs with a compatible pair have been proposed that involve either some iterative scheme on simulated sample functions or some general optimization approach. Although some of these techniques produce satisfactory results, they can be time consuming because of their nature. In this paper, a new iterative methodology is developed that estimates a non-Gaussian PSDF that: (a) is compatible with the prescribed non-Gaussian PDF, and (b) closely approximates the prescribed incompatible non-Gaussian PSDF. The corresponding underlying Gaussian PSDF is also determined. The basic idea is to iteratively upgrade the underlying Gaussian PSDF using the directly computed (through translation process theory) non-Gaussian PSDF at each iteration, rather than through expensive ensemble averaging of PSDFs computed from generated non-Gaussian sample functions. The proposed iterative scheme possesses two major advantages: it is conceptually very simple and it converges extremely fast with minimal computational effort. Once the underlying Gaussian PSDF is determined, generation of non-Gaussian sample functions is straightforward without any need for iterations. Numerical examples are provided demonstrating the capabilities of the methodology.

INCREASING THE FISHER INFORMATION CONTENT IN THE MATTER POWER SPECTRUM BY NONLINEAR WAVELET WIENER FILTERING

We develop a purely mathematical tool to recover some of the information lost in the non-linear collapse of large-scale structure. From a set of 141 simulations of dark matter density fields, we construct a non-linear Weiner filter in order to separate Gaussian and non-Gaussian structure in wavelet space. We find that the non-Gaussian power is dominant at smaller scales, as expected from the theory of structure formation, while the Gaussian counterpart is damped by an order of magnitude on small scales. We find that it is possible to increase the Fisher information by a factor of three before reaching the translinear plateau, an effect comparable to other techniques like the linear reconstruction of the density field.

Ultrasonic intensity power spectrum estimation by using the bispectral reconstruction technique

The application of bispectrum technique to ultrasonic scattering data obtained in a turbulent, particle-laden flow is investigated. The bispectrum of the scattering intensity signal is firstly estimated by using an outlier-excluded-mean estimation. It is then used to reconstruct the Fourier magnitude, and therefore the power spectrum, of this scattering intensity data by the least square method. Simulations show that the recovered non-Gaussian power spectrum is reasonably good. For experimental investigations under different turbulent conditions, agreement between the recovered intensity power spectrum and the microstructure scattering theory has been obtained.

Noise simulations of an Ising model of glassy kinetics.

Non-Gaussian properties of equilibrium ``magnetization'' fluctuations in a two-spin facilitated Ising model of the glass transition were studied by Monte Carlo simulation. The noise had properties unlike parallel distributed kinetics and similar to those expected for hierarchical kinetic schemes. The power spectrum had a region of slightly steeper than 1/f frequency dependence consistent with a highly stretched exponential response function. Fluctuations of noise power in different octaves were found to be correlated over a frequency range much larger than would be expected for systems in which each site contributed noise with one time constant. Further, higher-order non-Gaussian power spectra had a similar 1/f behavior demonstrating an overall order invariance in higher moments of the noise. This behavior might be used to identify similar kinetic cooperativity in real systems.

Measurement of mode field radius in axially nonsymmetrical single-mode fibers with arbitrary power distribution

A new method for the measurement of mode field radius in axially symmetrical and nonsymmetrical single-mode fibers with non-Gaussian power distribution is proposed. It is based on monitoring the power passed through a variable-width slit in the far field. A few results for different types of single-mode optical fiber are presented.

Normalized Power Spectra for FDM/FM Signals and Noise Power Ratio Calculations

By using normalized power spectra to represent frequency division multiplex/frequency modulation (FDM/FM) signals, the noise-power ratio (NPR) resulting from interference between FM signals can be easily calculated. These non-Gaussian power spectra are shown for CCIR preemphasis, rms frequency modulation ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> ) indexes from 0.1 to 1.2 and for top-to-bottom baseband frequency ratios ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> ) from 2 to 20. Proof is given that signals with the same <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> index, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> ratio, and preemphasis characteristics will have the same power spectrum. Also given is the value of the index <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> for a given <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> ratio that can be used to represent the FDM/FM signals by a Gaussian power spectrum.