Probability Density Function Distribution Research Articles

Stochastic ground motion simulation and seismic damage performance assessment of a 3-D subway station structure based on stochastic dynamic and probabilistic analysis

In practical engineering of underground structures, it is well-known that the randomness of earthquake generation and a structure's seismic response need to be considered for the safety of the project. The dynamic response and reliability level of a subway station are analyzed from the perspective of random vibration and the effect of stochastic ground motions on seismic responses is considered in this paper. This research proposes a model that can generate completely nonstationary ground motions and combines it with the probability density evolution method (PDEM) to analyze the seismic response and reliability level of a subway station. First, combining random function spectrum representation and evolutionary power spectrum density function, a probability model of nonstationary ground motion is established. According to the code (Gb 50909-2014), the model parameters of 5 sites are suggested, and the efficiency and accuracy are verified. A series of intensity–frequency nonstationary stochastic earthquakes are generated for the seismic response analysis. Then, a 3-D finite element analysis is performed for the deterministic dynamic response of the Daikai subway station, and the second-order statistics of three evaluation indicators (interstory drift, relative settlement, damage area ratio) are attained. Finally, the stochastic dynamic response and reliability level of the station structure are obtained by processing the data with the PDEM. The limit state related to the three evaluation indicators is established through probability density function (PDF) and cumulative distribution function (CDF). In addition, this probability evaluation system is more intuitive and effective than conventional methods. The statistical and probabilistic results show that the characteristics of earthquakes have a significant influence on the seismic response of subway stations, furthermore, the stochastic earthquakes generated by the intensity–frequency nonstationary ground motion model match well with the target value and the PDEM can comprehensively evaluate the seismic performance and reliability level of subway stations. Therefore, combing the stochastic earthquake and PDEM is an effective way to analyze the seismic response of subway stations.

Surface Dynamic Damage Prediction Model of Horizontal Coal Seam Based on the Idea of Wave Lossless Propagation.

According to traditional concepts, the movement of overlying strata and surface damage caused by coal mining in horizontal coal seams are symmetrical in terms of spatial distribution. However, in a lot of engineering practices, this symmetry has not been discovered. We often use the symmetry function to establish the profile prediction function of the surface damage, which results in a large difference between the prediction result and the actual situation. To solve this problem, this paper takes subsidence velocity as an example. Firstly, the spatial distribution functions of subsidence velocity on both sides were deduced theoretically. Through comparison, it is found that the change rate of the spatial distribution curve of the coal pillar side subsidence velocity is smoother than that of the goaf side and the subsidence velocity curves are skewed to the left. Secondly, based on the idea of lossless propagation of harmonic waves and idealizing the propagation environment, the spatial propagation relationship of surface subsidence velocity in the time domain is established. Then, the Box–Cox transform function is introduced to improve the normal distribution probability density function, and a new dynamic subsidence prediction model based on the Box–Cox transformation is obtained, which is suitable for the full mining stage. The model is tested by practical cases, the prediction accuracy is better than 7%, and the prediction results can meet the needs of engineering prediction accuracy (10%). The results of this research can enrich the existing subsidence prediction theory and provide theoretical and technical support for the prediction of dynamic surface damage caused by similar mining.

Statistical channel modelling of full‐angle 3.5 GHz OAM in indoor corridor scenario

A 3.5 GHz OAM antenna with ±1 mode was designed and fabricated, and the channel measurement in frequency domain was used to measure the channel in full angle domain under indoor multipath scenarios. Then, the modified probability density function of Gaussian distribution, Laplacian distribution, and Von Mises distribution are used to establish the OAM full-angle statistical channel model. The fitting curve results well represent the statistical characteristics of OAM in indoor multipath scenario.

On the maximal-ratio combining of FSO and RF links over generalized distributions and its applications in hybrid FSO/RF systems

Recently, generalized fading distributions have gained substantial research interest due to their applicability and adaptability in practical scenarios. In this paper, we derive the exact closed-form expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the maximal-ratio combining (MRC) of the free space optics (FSO) and radio frequency (RF) links over the most generalized Malaga and α-η-κ-μ fading models. Further, with the help of derived MRC channel statistics (i.e. PDF and CDF), closed-form expressions for outage probability and average symbol error rate (SER) for the hybrid FSO/RF system based on MRC and adaptive combining schemes are derived. In an adaptive combining scheme, the MRC of FSO and RF links is employed at the receiver, if the operating FSO link quality is unacceptable for transmission. Further, the derived expressions are unified for both intensity modulation/direct detection (IM/DD) and heterodyne detection (HD) techniques. We have considered the effects of atmospheric path loss, atmospheric turbulence-induced fading and non-zero boresight pointing errors in case of the FSO link. We have also determined the optimum switching threshold and beam waist values for the adaptive combining scheme to achieve optimal performance. Moreover, computationally less complex asymptotic expressions are calculated and the diversity gains of MRC and adaptive combining schemes are determined. Finally, the performances of single-link FSO, hard-switching-based hybrid FSO/RF, MRC, and adaptive-combining-based hybrid FSO/RF systems are compared and the inferences are reported under various turbulence conditions, link distances, weather conditions, and severity of the pointing errors.

A Generalized Family of Exponentiated Composite Distributions

In this paper, we propose a new family of distributions, by exponentiating the random variables associated with the probability density functions of composite distributions. We also derive some mathematical properties of this new family of distributions, including the moments and the limited moments. Specifically, two special models in this family are discussed. Three real datasets were chosen, to assess the performance of these two special exponentiated-composite models. When fitting to these three datasets, these three special exponentiated-composite distributions demonstrate significantly better performance, compared to the original composite distributions.

On the Existence of Global Minima and Convergence Analyses for Gradient Descent Methods in the Training of Deep Neural Networks

In this article we study fully-connected feedforward deep ReLU ANNs with an arbitrarily large number of hidden layers and we prove convergence of the risk of the GD optimization method with random initializations in the training of such ANNs under the assumption that the unnormalized probability density function of the probability distribution of the input data of the considered supervised learning problem is piecewise polynomial, under the assumption that the target function (describing the relationship between input data and the output data) is piecewise polynomial, and under the assumption that the risk function of the considered supervised learning problem admits at least one regular global minimum. In addition, in the special situation of shallow ANNs with just one hidden layer and one-dimensional input we also verify this assumption by proving in the training of such shallow ANNs that for every Lipschitz continuous target function there exists a global minimum in the risk landscape. Finally, in the training of deep ANNs with ReLU activation we also study solutions of gradient flow (GF) differential equations and we prove that every non-divergent GF trajectory converges with a polynomial rate of convergence to a critical point (in the sense of limiting Fr\'echet subdifferentiability). Our mathematical convergence analysis builds up on ideas from our previous article Eberle et al., on tools from real algebraic geometry such as the concept of semi-algebraic functions and generalized Kurdyka-Lojasiewicz inequalities, on tools from functional analysis such as the Arzel\`a-Ascoli theorem, on tools from nonsmooth analysis such as the concept of limiting Fr\'echet subgradients, as well as on the fact that the set of realization functions of shallow ReLU ANNs with fixed architecture forms a closed subset of the set of continuous functions revealed by Petersen et al.

Performance Analysis of Multi-Hop Underwater Wireless Optical Communication Systems Over Exponential-Generalized Gamma Turbulence Channels

Underwater wireless optical communication (UWOC) has generated a much interest in the research community thanks to its wide range of applications, including submarine navigation, seafloor exploration, and military operation. Turbulence, scattering, and absorption phenomena generally can severely affect the performance of UWOC systems. The relaying methods have been shown as a promising technique to alleviate turbulence-induced and short-range fading. In this paper, under varying channel conditions, we present a unified framework to analyze the performance of multi-hop UWOC systems with both amplify-and-forward and decode-and-forward relaying techniques. In particular, for the end-to-end signal-to-noise ratio (SNR), we use the univariate and bivariate Fox-H functions to derive both the probability density function (PDF) and cumulative distribution function (CDF). Furthermore, for the performance of systems under both types of relaying techniques, we derive tight mathematical expressions for the outage probability and bit error probability. Besides, to highlight and provide some prominent engineering insights into the diversity order, an asymptotic analysis for the outage likelihood and average bit error rate (BER) is given. Consequently, results from Monte-Carlo simulations are used to verify the correctness of our derived mathematical expression.

Research on cross-region interconnection detection method of abnormal data in power monitoring system

When conventional methods collect cross region interconnection detection data of power monitoring system, one data has multiple signal strengths, resulting in low detection rate and high false alarm rate of cross region interconnection abnormal data. Therefore, a cross region interconnection detection method of abnormal data in power monitoring system is proposed. Divide the cross region interconnection scene types of power monitoring system, make the detection end send data packets to the security partition corresponding to different scenes, and Gaussian filter the signals received in the security partition, and only retain the signals with probability distribution density function of [0.5,1]; Convert multiple signal strength indications of MAC address into one, so that the security partition only responds to the MAC address, obtains the MAC address detection data, and inputs it into the deep confidence network after preprocessing. If the MAC address matches successfully, it is judged that there is cross region interconnection in the power monitoring system, and the cross region interconnection detection of abnormal data is completed. The cross region interconnection alarm data provided by a power Co., Ltd. is selected as the experimental data, and the comparative experiment is set. The results show that the design method improves the abnormal data detection rate, reduces the false alarm rate, and ensures the detection accuracy of large-scale cross region interconnection data.

The microstructure characterization and elastic properties of closed-cell foams

The present study investigated the microstructure and compressive response of ALPORAS closed-cell foam. The micro-CT technique was adopted to study the interior structure of the foam. Image processing algorithms were then incorporated to measure the geometrical characteristics of the foam cellular microstructure, including the size, surface area, geometrical shape, aspect ratio, and sphericity of cells. The cells were irregular polyhedra with approximately 14–15 faces and 5 edges per face; they had longer dimensions in the gravitational direction than in the other two mutually orthogonal directions. Three statistical measurements were employed in the development of random 3D numerical models based on the Laguerre tessellation of random close packed spheres along with three probability density functions for sphere size distributions. The effects of parameters such as probability density functions, sphere volume fraction, and coefficient of variation in sphere volume on the geometrical characteristics of the developed numerical models were examined. The difference in geometric characteristics of the numerical models generated using different probability functions was nonsignificant. Numerical simulations were subsequently conducted using the developed numerical models. The comparison between numerical and experimental results revealed that numerical models overestimated the experimental value by a factor of approximately 3. This discrepancy can be attributed to the difference in detailed microstructural features between the generated numerical models and real foam.

Phase-resolved characteristics of bubbles in cloud cavitation shedding cycles

To investigate the complex physics and mixture properties inside sheet/cloud cavities, we performed an experimental study that used phase-resolved optical probes and high-speed photography to acquire both vapor-water information and large-scale flow structures. We found distinctive characteristics at different scales: the large scale of cavitation shedding, and the small scale of bubbles. Large-scale cavitation behavior was examined using the mean void fraction profile and raw signal history, characterized by the shedding of cavity cloud. Small-scale information on bubbles and probability density function of bubble size distributions were quite similar at various measurement locations. Based on the experimental data, we proposed an empirical bubble size distribution and discussed the corresponding statistical quantities for cloud cavitation.

Statistical analysis of radio frequency interference in microwave links using frequency scan measurements from multi-transmitter environments

Radio frequency interference (RFI) comprises undesirable distortions from other frequency bands overlapping the specified band of operation. RFI contributes to increased detected power and temperature of the baseband processing units, resulting in considerable discrepancies in the received measurements. RFI mitigation techniques have been proposed to tackle this problem. However, the cost of implementing most RFI removal algorithms can be highly prohibitive. Removing such unwanted signals would involve removing a portion of the desired intelligence, leading to a potential loss of valuable information. The requirement for experimental frequency scans to detect, and mitigate RFI becomes important. Field measurements were conducted across three microwave links located in urban environments. The spectrum analyzer and a 0.6-meter antenna dish were used for the frequency scans at 6–8 GHz in horizontal and vertical polarizations. The availability of the desired frequencies, which is key to frequency planning is presented. The cumulative distribution functions, histogram and probability density functions of the measured data, which are crucial to determining the impact of interference on the microwave links, are presented. The statistical characteristics of the measured data are highlighted. Overall, the RFI data would provide insights on RFI detection and mitigation, leading to a high quality of service for the benefit of mobile subscribers.

Probabilistic analysis of long-term loss incorporating maximum entropy method and analytical higher-order moments

Quantifying economic losses of civil infrastructures subjected to various hazards under a life-cycle context is of vital importance for risk assessment and management. In previous studies, the expected long-term loss has been widely applied as a standard decision criterion during the life-cycle analysis. However, the expectation may not be informative enough to illustrate uncertainties associated with the long-term loss. Therefore, the higher-order moments and the probability distribution should be investigated. In this paper, a probabilistic analysis framework is proposed to construct the probability density function and cumulative distribution function of long-term loss by assessing the analytical statistical moments. The stochastic renewal process is utilized to assess the long-term loss by considering uncertainties associated with stochastic occurrence and frequency of the hazards. Based on the maximum entropy method, the proposed approach shows superior efficiency to assess the probability distribution of long-term loss than crude Monte Carlo simulation. The probability distribution can be essential information for decision-making process of risk management. An illustrative example is investigated to show the probability density function of long-term loss of civil infrastructure subjected to hurricane hazards. A good agreement of results obtained by the proposed approach and Monte Carlo simulation has verified the accuracy and effectiveness of the proposed method.

Nature of Supersonic Turbulence and Density Distribution Function in the Multiphase Interstellar Medium

Supersonic flows in the interstellar medium (ISM) are believed to be a key driver of the molecular cloud formation and evolution. Among molecular clouds’ properties, the ratio between the solenoidal and compressive modes of turbulence plays important roles in determining the star formation efficiency. We use numerical simulations of supersonic converging flows of the warm neutral medium (WNM) resolving the thermal instability to calculate the early phase of molecular cloud formation, and we investigate the turbulence structure and the density probability distribution function (density PDF) of the multiphase ISM. We find that both the solenoidal and compressive modes have their power spectrum similar to the Kolmogorov spectrum. The solenoidal (compressive) modes account for ≳80% (≲20%) of the total turbulence power. When we consider both the cold neutral medium (CNM) and the thermally unstable neutral medium (UNM) up to T ≲ 400 K, the density PDF follows the lognormal distribution, whose width is well explained by the known relation from the isothermal turbulence as (where b is the parameter representing the turbulence mode ratio and is the turbulent Mach number). The density PDF of the CNM component alone (T ≤ 50 K), however, exhibits a narrower by a factor of ∼2. These results suggest that observational estimations of b based on the CNM density PDF requires the internal turbulence within each CNM clump but not the interclump relative velocity, the latter of which is instead powered by the WNM/UNM turbulence.

Extension of Transmuted Halfnormal Distribution Properties and Application

This research extended the HalfNormal distribution to Transmuted HalfNormal Distribution (THND) and it generalizes the classical HalfNormal by expanding its scope to modeling high-class random process that cannot be easily modeled with existing probability distribution. The THND was compounded and its statistical properties were obtained such as moment, hazard function, reliability, probability density and cumulative distribution function. This hybrid model can be used to capture non-normal data with highly skewed, heavily tailed and leptokurtic distribution. Keywords: Transmuted HalfNormal Distribution, Moment, Reliability Function, hazard function and cumulative distribution function DOI: 10.7176/DCS/12-5-03 Publication date: May 31 st 2022

Performance Analysis of Dual-Hop RF/FSO Relaying Systems With Imperfect CSI

This paper proposes a unified performance analysis framework of asymmetric dual-hop radio-frequency/free-space optical (RF/FSO) transmission systems. The generalized independent and identically distributed (i.i.d.) <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha -\mu$</tex-math></inline-formula> distribution is considered for each branch in the RF link, whereas the FSO link is assumed to experience Málaga ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {M}$</tex-math></inline-formula> ) fading with pointing error impairments. With amplify-and-forward (AF) and decode-and-forward (DF) relaying, the imperfect channel state information (CSI) of both links is considered. For the considered system, we first derive exact and novel analytical expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the end-to-end signal-to-noise ratio (SNR). Furthermore, we obtain key performance metrics such as outage probability (OP), average bit-error-rate (ABER), and effective capacity (EC). The important effects of channel parameters on the system are disclosed from mathematical analysis and are further validated by Monte-Carlo simulations.

A revised density split statistic model for general filters

Context. Studying the statistical properties of the large-scale structure in the Universe with weak gravitational lensing is a prime goal of several current and forthcoming galaxy surveys. The power that weak lensing has to constrain cosmological parameters can be enhanced by considering statistics beyond second-order shear correlation functions or power spectra. One such higher-order probe that has proven successful in observational data is density split statistics (DSS), in which one analyses the mean shear profiles around points that are classified according to their foreground galaxy density. Aims. In this paper, we generalise the most accurate DSS model to allow for a broad class of angular filter functions used for the classification of the different local density regions. This approach is motivated by earlier findings showing that an optimised filter can provide tighter constraints on model parameters compared to the standard top-hat case. Methods. As in the previous DSS model we built on large deviation theory approaches and approximations thereof to model the matter density probability distribution function, and on perturbative calculations of higher-order moments of the density field. The novel addition relies on the generalisation of these previously employed calculations to allow for general filter functions and is validated on several sets of numerical simulations. Results. It is shown that the revised model fits the simulation measurements well for many filter choices, with a residual systematic offset that is small compared to the statistical accuracy of current weak lensing surveys. However, by use of a simple calibration method and a Markov chain Monte Carlo analysis, we studied the expected sensitivity of the DSS to cosmological parameters and find unbiased results and constraints comparable to the commonly used two-point cosmic shear measures. Hence, our DSS model can be used in competitive analyses of current cosmic shear data, while it may need refinements for forthcoming lensing surveys.

Microscopic phase identification of cement-stabilized clay by nanoindentation and statistical analytics

The mechanical properties of cement-stabilized clay (CSC), are significantly inferior to that of concrete and mortar due to the presence of clay minerals. To reveal the interaction between clinker's functional constituent and clay minerals within the CSC matrix to promote the performance of stabilized clay, large data sets cross-scale nanoindentation coupled with thermogravimetric analyses were involved. Two commercially available clays (i.e., kaolin and bentonite, whose dominant clay minerals are kaolinite and montmorillonite, respectively) stabilized by two key functional constituents of cement clinker, i.e., tricalcium silicate (C3S) and tricalcium aluminate (C3A), were investigated to unravel the primary hydration of the cement clinker, pozzolanic reactions with clay minerals, and the microstructure's evolution. Large data sets of elastic modulus and hardness derived from the nanoindentation tests were statistically investigated by two deconvolution methods: probability density function and cumulative distribution function. The results indicated that calcium silicate hydrates (C-S-H) and calcium aluminate hydrates (C-A-H) were the major reaction products for C3S-stabilized and C3A-stabilized clays. However, during the secondary pozzolanic reaction process, the dissolved colloids from the clay minerals and the hydrated products would react and accumulate at the interface between the primary cementitious paste and clay particles to form a new soft, porous phase. The deconvolution and thermogravimetric analysis also indicate that more pozzolanic products and high-density (HD) porous phases were detected in bentonite than in kaolin, suggesting that the pozzolanic reaction in montmorillonite is much more pronounced than that in kaolinite. This understanding can clarify the interaction between clay minerals, hydration products, and distinct CSC for traditional cementitious materials.

Clustering-enhanced stock price prediction using deep learning

In recent years, artificial intelligence technologies have been successfully applied in time series prediction and analytic tasks. At the same time, a lot of attention has been paid to financial time series prediction, which targets the development of novel deep learning models or optimize the forecasting results. To optimize the accuracy of stock price prediction, in this paper, we propose a clustering-enhanced deep learning framework to predict stock prices with three matured deep learning forecasting models, such as Long Short-Term Memory (LSTM), Recurrent Neural Network (RNN) and Gated Recurrent Unit (GRU). The proposed framework considers the clustering as the forecasting pre-processing, which can improve the quality of the training models. To achieve the effective clustering, we propose a new similarity measure, called Logistic Weighted Dynamic Time Warping (LWDTW), by extending a Weighted Dynamic Time Warping (WDTW) method to capture the relative importance of return observations when calculating distance matrices. Especially, based on the empirical distributions of stock returns, the cost weight function of WDTW is modified with logistic probability density distribution function. In addition, we further implement the clustering-based forecasting framework with the above three deep learning models. Finally, extensive experiments on daily US stock price data sets show that our framework has achieved excellent forecasting performance with overall best results for the combination of Logistic WDTW clustering and LSTM model using 5 different evaluation metrics.

Weighted p-norm distance t kernel SVM classification algorithm based on improved polarization

The kernel function in SVM enables linear segmentation in a feature space for a large number of linear inseparable data. The kernel function that is selected directly affects the classification performance of SVM. To improve the applicability and classification prediction effect of SVM in different areas, in this paper, we propose a weighted p-norm distance t kernel SVM classification algorithm based on improved polarization. A t-class kernel function is constructed according to the t distribution probability density function, and its theoretical proof is presented. To find a suitable mapping space, the t-class kernel function is extended to the p-norm distance kernel. The training samples are obtained by stratified sampling, and the affinity matrix is redefined. The improved local kernel polarization is established to obtain the optimal kernel weights and kernel parameters so that different kernel functions are weighted combinations. The cumulative optimal performance rate is constructed to evaluate the overall classification performance of different kernel SVM algorithms, and the significant effects of different p-norms on the classification performance of SVM are verified by 10 times fivefold cross-validation statistical comparison tests. In most cases, the results using 6 real datasets show that compared with the traditional kernel function, the proposed weighted p-norm distance t kernel can improve the classification prediction performance of SVM.

Seismic Vulnerability Assessment of Thermal Power Plants Based on Beta Function

The study on seismic vulnerability of thermal power plants is of great significance for seismic damage prevention of the electrical power system and risk analysis of seismic economic loss. This paper presented a study method for obtaining the seismic vulnerability matrix and vulnerability curve of thermal power plants based on beta function. Firstly, according to the study results of thermal power plants in American ATC-25 report, the expected values of seismic damage indexes of thermal power plants under different seismic intensities were calculated. It was assumed that the expected values of seismic damage indexes of substation and thermal power plant were the same, and the dispersion of distribution of seismic damage indexes was the same. Then, based on the actual seismic vulnerability matrix of substations obtained from statistics in Wenchuan earthquake, the seismic vulnerability matrix of thermal power plants under different seismic intensities was fitted by using the probability density function of beta distribution. Finally, the seismic vulnerability curve of the thermal power plant based on peak ground acceleration (PGA) was obtained by using the cumulative function curve of standard lognormal distribution. The calculation results show that the seismic damage ratio calculated from the seismic vulnerability matrix of thermal power plants is basically consistent with the seismic damage ratio in ATC-25 report.