Probability Density Function Of Samples Research Articles

Efficient Methods for Reliability Sensitivity Analysis of Distribution Parameters and Their Application

To efficiently evaluate the influence of the distribution parameters of the input variables on the failure probability of engineering structures and improve the reliability and safety of engineering structures in a targeted manner, new methods for the global reliability sensitivity analysis (RSA) of distribution parameters are proposed in this study based on the cubature formula (CF), surrogate sampling probability density function (SSPDF), and quasi-Monte Carlo (QMC) method. By introducing CF, the proposed methods can effectively improve the computational efficiency of the nested expectation and variance operators in the reliability sensitivity indices of the distribution parameters. Based on the concept of SSPDF, a surrogate importance sampling probability density function was developed. This not only overcomes the problem of the computational effort of propagating parameter uncertainty to the failure probability function (FPF), which depends on the dimensionality of the parameters; it also further improves the efficiency of the RSA of the parameters in the case of a small failure probability. Finally, by incorporating the idea of the QMC method, the process of calculating the reliability sensitivity indices of the parameters is reduced from a double-loop to a single-loop one. Three engineering examples are used in this study to demonstrate the efficiency and accuracy of the new algorithms.

Spatial network disintegration based on kernel density estimation

The problem of network disintegration, such as suppression of an epidemic spread and destabilization of terrorist networks, possesses extensive applications and has lately been the focus of growing interest. Many real-world complex systems are represented by spatial networks in which nodes and edges are spatially embedded. However, existing disintegration approaches for spatial network disintegration focus on singular aspects such as geospatial information or network topography, with insufficient modeling granularity. In this paper, we propose an effective and computationally efficient virtual node model that essentially integrates the geospatial information and topology of the network by modeling edges as virtual nodes with weights. Moreover, we employ Kernel Density Estimation, a well-known non-parametric technique for estimating the underlying probability density function of samples, to fit all nodes, comprising both network and virtual nodes, to identify the critical region of the spatial network, which is also the circular geographic region where disintegration occurs. Extensive numerical experiments on synthetic and real-world networks demonstrate that our method outperforms existing methods in terms of both effectiveness and efficiency, which provides a fresh perspective for modeling spatial networks.

Wide dynamic range signal detection for underwater optical wireless communication using a PMT detector.

In the underwater optical wireless communication (UOWC) scenario, a photomultiplier tube (PMT) with higher sensitivity, lower noise, and a larger receiver area is employed as the photon detector to further extend the transmission distance. Due to the complex underwater environment, the high directionality of the light beam, and the vibration of a transceiver, the incident optical power usually spans a very wide dynamic range, and the PMT may operate in any one of the three regimes: pulse, transition, and waveform. While it is difficult to obtain the analytical characterization of the output electric signals across these regimes, this paper resorts to experimental measurements of the upsampled discrete samples within a training symbol duration. Among different statistical distribution fitting options, generalized extreme value (GEV) distribution is found to show excellent performance in fitting the probability density function (PDF) of either multiple samples or the superimposition of all samples within a symbol duration. Then joint sample distribution (JSD) based and superimposed sample distribution (SSD) based symbol detection methods are proposed by adopting the GEV distribution and log-likelihood ratio (LLR) testing criterion. The proposed methods are experimentally evaluated under different received signal optical powers, data rates, and sampling rates. They are shown to outperform the Poisson and Gaussian based maximum likelihood detection methods which are employed for the pulse regime and waveform regime respectively. Furthermore, the effectiveness of the proposed methods in alleviating strong ambient radiation is experimentally verified.

Research on blade statistics measurement registration planning technology

Blade has the characteristics of high machining quality of complex curved surface. If there is no benchmark before machining, it is impossible to judge whether the blades before machining are qualified. Therefore, it is necessary to analyze the blade measurement data. Due to the large measurement point error and disordered distribution, it is necessary to optimize the blade registration. Therefore, blade model registration and positioning is particularly important in blade shape detection and analysis. First, preregistration is carried out based on the six point optimization selection of the blade. After preregistration, the selection method of registration control point set based on theoretical model and statistical error is proposed, planning the registration datum point set on the blade model. The registration control point set is obtained through the registration operation between the measurement data and the registration reference point set. Finally, based on the stability and reliability of important sampling sensitivity and statistical error, obtain the probability density function of error normal distribution statistics samples and important samples. The selection of statistical control points and the rationality of the objective function were verified. The stability/reliability of the statistical alignment point selection is proved to be feasible. The statistical registration deviation is [0.015,0.026] mm, and the ICP registration deviation is [0.031,0.035] mm. The average deviation of statistics registration is about 0.013 mm smaller than the average deviation of ICP registration. The deviation of statistical sampling points is about 0.0214 mm, and that of traditional sampling points is about 0.0275 mm. The deviation of statistical sampling points is about 0.0061 mm smaller than that of traditional sampling points. It meets the requirements of rapid, high efficiency and high precision measurement for aeroengine blades.

Efficient importance analysis methods for structures with distribution parameter uncertainty based on cubature formula

The importance analysis of a structural system with distribution parameter uncertainty can identify key parameters that significantly affect its output performance, thus providing importance guidance for its design and optimization. However, the traditional importance analysis method requires the three-loop Monte Carlo sampling to estimate the importance measurement index of a distribution parameter with such output characteristic values as mean and variance, whose computational cost is too large. To solve this problem, two efficient cubature formula methods based on the surrogate sampling probability density function (SSPDF) for the importance analysis of distribution parameters are proposed: ①the double-loop cubature formula based on the surrogate sampling probability density function (S-DLCF); ②the single-loop cubature formula based on the surrogate sampling probability density function (S-SLCF). The two methods use cubature formulas to efficiently compute the nested mean and variance in the importance measurement index of a distribution parameter, thus solving the problem that the computational effort of propagating parameter uncertainty to output characteristic values depends on parameter dimensionality because of SSPDF. The S-DLCF makes full use of the efficiency and accuracy of the cubature formula to estimate output statistical moments; the S-SLCF simplifies the integral to calculate output moments by expanding the dimensionality of the distribution parameter. The numerical and engineering examples verify the efficiency and accuracy of the two methods for the importance analysis of distribution parameters.

Combined Non-parametric and Parametric Classification Method Depending on Normality of PDF of Training Samples

Classification method with combined nonparametric and parametric classifications which depends on the normality of Probability Density Function of training samples is proposed. The proposed classification method is also based on spatial information for high spatial resolution of satellite based optical sensor images is proposed. Also, a classification method which takes into account not only spectral but also spatial features for LANDSAT-4 and 5 Thematic Mapper (TM) data is proposed. Treatment of the spatial-spectral variability existing within a region is more important for such high spatial resolution of satellite imagery data. Standard deviations in small cells, such as 2x2, 3x3 and 4x4 pixels, were used as measures to represent the spatial-spectral variabilities. This information can be used together with conventional spectral features in a unified way, for the traditional classifier such as the pixelwise Maximum Likelihood Decision Rule (MLHDR). The classification performance of new clear cuts and alpine meadows which are very close in spectral space characteristics and difficult to distinguish them by conventional methods are focused. Through experiments, it is found that there is a substantial improvement in overall classification accuracy for TM forestry data. The Probability of Correct Classification (PCC) for the new clear cuts and the alpine meadows classes rose by 7% to 97% correct. The confusion between alpine meadows and new clear cuts was reduced from 9% to 3%.

AK-PDF: An active learning method combining kriging and probability density function for efficient reliability analysis

An important challenge in structural reliability is to reduce the number of calls to evaluate the performance function, especially the complex implicit performance functions. To reduce the computational burden and improve the reliability analysis efficiency, a new active learning method is developed to consider the probability density function of samples based on the learning function U in an active learning reliability method that combines the kriging and Monte Carlo simulation. In the proposed method, the proposed active learning function contains two parts: part A is based on function U, and part B is based on the probability density function and function U. By changing the weights of parts A and B, the sample points close the limit-state function, and those in the region with a higher probability density function have more weight to be selected compared to the others. Subsequently, the kriging model can be constructed more effectively. The proposed method avoids a large number of time-consuming function evaluations, and the recommended weight is also reported. The performance of the proposed method is evaluated through three numerical examples and one engineering example. The results demonstrate the efficiency and accuracy of the proposed method.

Study of a New Imaging Strategy Based on Compressed Sensing to Shorten the Imaging Time of a Fourier Telescope

A Fourier telescope (FT) is an imaging system based on laser illumination and optical aperture synthesis and is suitable for imaging distant targets with high resolution through the atmosphere. The imaging time of a conventional FT is so long (about two hours) as to limit its practical applications. In order to shorten the imaging time of a Fourier telescope, we propose a new imaging system called the compressed sensing Fourier telescope (CS-FT). The image quality, the imaging time and the effects of noise of the new system are studied in detail. Based on the analysis, we find that by reasonably choosing the total sampling rate (TSR), the imaging time of the CS-FT is obviously shorter than that of the conventional FT while the image quality of the CS-FT is near that of the conventional FT. Worth noting is that with the smaller values of the low-frequency sampling rate (LSR) and the order of the sampling probability density function (OPDF), the CS-FT can achieve a better image quality. The reconstruction results of the field data show that the CS-FT can reduce the imaging time of a FT in an actual noise environment.

Ultrasonic Measurement of Temperature Rise in Breast Cyst and in Neighbouring Tissues as a Method of Tissue Differentiation

Abstract Texture of ultrasound images contain information about the properties of examined tissues. The analysis of statistical properties of backscattered ultrasonic echoes has been recently successfully applied to differentiate healthy breast tissue from the benign and malignant lesions. We propose a novel procedure of tissue characterization based on acquiring backscattered echoes from the heated breast. We have proved that the temperature increase inside the breast modifies the intensity, spectrum of the backscattered signals and the probability density function of envelope samples. We discuss the differences in probability density functions in two types of tissue regions, e.g. cysts and the surrounding glandular tissue regions. Independently, Pennes bioheat equation in heterogeneous breast tissue was used to describe the heating process. We applied the finite element method to solve this equation. Results have been compared with the ultrasonic predictions of the temperature distribution. The results confirm the possibility of distinguishing the differences in thermal and acoustical properties of breast cyst and surrounding glandular tissues.

A new algorithm for importance analysis of the inputs with distribution parameter uncertainty

ABSTRACTImportance analysis is aimed at finding the contributions by the inputs to the uncertainty in a model output. For structural systems involving inputs with distribution parameter uncertainty, the contributions by the inputs to the output uncertainty are governed by both the variability and parameter uncertainty in their probability distributions. A natural and consistent way to arrive at importance analysis results in such cases would be a three-loop nested Monte Carlo (MC) sampling strategy, in which the parameters are sampled in the outer loop and the inputs are sampled in the inner nested double-loop. However, the computational effort of this procedure is often prohibitive for engineering problem. This paper, therefore, proposes a newly efficient algorithm for importance analysis of the inputs in the presence of parameter uncertainty. By introducing a ‘surrogate sampling probability density function (SS-PDF)’ and incorporating the single-loop MC theory into the computation, the proposed algorithm can reduce the original three-loop nested MC computation into a single-loop one in terms of model evaluation, which requires substantially less computational effort. Methods for choosing proper SS-PDF are also discussed in the paper. The efficiency and robustness of the proposed algorithm have been demonstrated by results of several examples.

Complete sampling distribution for primary sampling, sample preparation and analysis

Author Summary: Following from the author’s recent paper at Sampling 2014 which presented a method for calculation of the sampling probability density function due to the particulate heterogeneity (density function of the fundamental sampling uncertainty), it is possible to apply the same characteristic function method to arrive at the overall sampling distribution for any sampling protocol and analysis method. This paper develops the application of the method of characteristic functions to the overall sampling problem including the uncertainty which derives from the primary sampling from a process stream. The assay distribution in a process stream or of impurities in the flow of a final product can be governed by non-Gaussian, serially correlated distributions. The paper shows how such circumstances can be dealt with to arrive at robust solutions. The paper represents an end-point in the theory of sampling as it provides a means of determining the entire distribution function for a sampling system. Such a determination has not previously been possible and having determined the entire distribution function, the statistics of the sampling process are completely determined.

The Distribution of the Concentration Ratio for Samples from a Uniform Population

In the present paper we derived, with direct method, the exact expressions for the sampling probability density function of the Gini concentration ratio for samples from a uniform population of size n = 6, 7, 8, 9 and 10. Moreover, we found some regularities of such distributions valid for any sample size.

Statistical computation of Boltzmann entropy and estimation of the optimal probability density function from statistical sample

In this work, we investigate the statistical computation of the Boltzmann entropy of statistical samples. For this purpose, we use both histogram and kernel function to estimate the probability density function of statistical samples. We find that, due to coarse-graining, the entropy is a monotonic increasing function of the bin width for histogram or bandwidth for kernel estimation, which seems to be difficult to select an optimal bin width/bandwidth for computing the entropy. Fortunately, we notice that there exists a minimum of the first derivative of entropy for both histogram and kernel estimation, and this minimum point of the first derivative asymptotically points to the optimal bin width or bandwidth. We have verified these findings by large amounts of numerical experiments. Hence, we suggest that the minimum of the first derivative of entropy be used as a selector for the optimal bin width or bandwidth of density estimation. Moreover, the optimal bandwidth selected by the minimum of the first derivative of entropy is purely data-based, independent of the unknown underlying probability density distribution, which is obviously superior to the existing estimators. Our results are not restricted to one-dimensional, but can also be extended to multivariate cases. It should be emphasized, however, that we do not provide a robust mathematical proof of these findings, and we leave these issues with those who are interested in them.

Generating cotton fiber length probability density function from fiber length parameters by finite mixture model

This paper generated the fiber length probability density function of different cotton samples on the basis of Finite Mixture Model. With the mixture of two Weibull distributions, the commonly used and discrete fiber length parameters were employed to obtain the function. Curve fitting of fiber length frequency histograms was carried out and the differences between the model proposed by this paper and previous model were compared. The results show that when fiber length measures were modified by the related linear relational expressions, this model, compared with other models, could fit the test data very well with the errors all less than 0.0045.

Discussion on: Applications of asymptotic sampling on high dimensional structural dynamic problems: M.T. Sichani, S.R.K. Nielsen and C. Bucher, Structural Safety, 33 (2011) 305–316

Article history: Available online 19 October 2013 2013 Elsevier Ltd. All rights reserved. C. Bucher originally proposed the asymptotic sampling (AS) for high dimensional reliability analysis with lower failure probability [1]. Further, M.T. Sichani, S.R.K. Nielsen and C. Bucher developed the AS for the evaluation of high dimensional structural dynamic problems [2]. In their method, the lower failure probability is extrapolated by several high failure probabilities on the asymptotic behavior of the failure probability in n-dimensional independent and identically distributed (i.i.d) Gaussian space with respect to the standard deviation of the variables. The detailed steps of their methods can be summarized as follows: Step 1. Transform the random variables X of the original problem into the i.i.d Gaussian U space, i.e. N, using Rosenblatt transformation or Nataf transformation T: X ? U. Step 2. Choose the parameters fi 2 [0, 1] (i = 1, 2, . . . ,m), and increase the standard deviation of N but keep the mean of N invariant, and then obtain the scaled random variables Nfi . If N NðlN;r2Þ, then Nfi N lN;rN=f 2 i . Step 3. Perform the standard Monte Carlo simulation with Nfi and calculate the scaled failure probability Pf(fi). Step 4. Evaluate the scaled reliability index as b(fi) = U (1 Pf(fi)). Step 5. Obtain all the b(fi) (i = 1, 2, . . . ,m), for different fi to construct a set of (fi, b(fi)) which will be regarded as ‘‘support points’’. Step 6. Estimate coefficients A and B of Eq. (1) using the set of (fi, b(fi)) (i = 1, . . . ,m) by regression analysis. bðfiÞ 1⁄4 Af i þ B fi ð1Þ Step 7. Estimate the un-scaled reliability index b(1) = A + B and the lower failure probability Pf(1) = 1 U(b(1)). As revealed by Steps 1–7, we can find that the main computational cost of the AS lies in the sum of the time for all the b(fi) (i = 1, 2, . . . ,m) by the standard Monte Carlo simulation. Admittedly, the sum of time for b(fi) is generally less than that of the direct evaluation of b(1) and the AS consequentially reduces the computational cost. In addition, this method behaves well when it comes to problems with high nonlinearity and lower failure probability. However, after conducting a detailed investigation into the AS method, we find that there are still three issues necessary to be discussed. The first is that the computational cost in AS may be still prohibitive due to the fact that m independent evaluations of b(fi) are required to obtain the ‘‘support points’’. The second is that sometimes the individual regression analysis using Eq. (1) leads to remarkable discrepancies compared with the accurate result. The third is that the AS is not able to give the confidence interval of the result. In this paper, we discuss these issues for further improving the accuracy and efficiency of the AS method. 1. Discussion to the first issue It is assumed that the performance function is y = G(x). For Pf(fi) (i = 1, 2, . . . ,m), we first construct the same importance sampling probability density function (IS PDF) f (x), then Pf(fi) (i = 1, 2, . . . ,m) can be estimated by the same importance sampling (IS) method through only one set of input–output samples (xk,G(xk)) (k = 1, . . . ,N), where N is the number of samples. Pf 1⁄4 Z F f ðxÞdx 1⁄4 Z IFðxÞ f ðxÞ f ðxÞ f ðxÞdx 1⁄4 E IFðxÞxðxÞ 1⁄2 ð2Þ

Energy-guided learning approach to compressive FD-OCT

High quality, large size volumetric imaging of biological tissue with optical coherence tomography (OCT) requires large number and high density of scans, which results in large data acquisition volume. This may lead to corruption of the data with motion artifacts related to natural motion of biological tissue, and could potentially cause conflicts with the maximum permissible exposure of biological tissue to optical radiation. Therefore, OCT can benefit greatly from different approaches to sparse or compressive sampling of the data where the signal is recovered from its sub-Nyquist measurements. In this paper, a new energy-guided compressive sensing approach is proposed for improving the quality of images acquired with Fourier domain OCT (FD-OCT) and reconstructed from sparse data sets. The proposed algorithm learns an optimized sampling probability density function based on the energy distribution of the training data set, which is then used for sparse sampling instead of the commonly used uniformly random sampling. It was demonstrated that the proposed energy-guided learning approach to compressive FD-OCT of retina images requires 45% fewer samples in comparison with the conventional uniform compressive sensing (CS) approach while achieving similar reconstruction performance. This novel approach to sparse sampling has the potential to significantly reduce data acquisition while maintaining image quality.

The applications of an importance sampling method to reliability analysis of the inside flap of an aircraft

Based on the cross-entropy method and modified Metropolis–Hastings algorithm, an efficient importance sampling method is proposed to perform reliability analysis for the inside flap of an aircraft. In the proposed method, the cross-entropy method is utilized to estimate the optimal importance sampling probability density function. The derivation shows that the distribution parameters of the optimal importance sampling probability density function can be estimated by the failure sample points generated according to the original probability density function. The crude Monte Carlo method can be employed to simulate the failure sample points but its computational cost is prohibitive. In order to circumvent the difficulties, the modified Metropolis–Hasting algorithm is employed to generate failure sample points distributed as the original probability density function. After the distribution parameters of the optimal importance sampling probability density function have been obtained, the importance sampling probability density function can be constructed, then the proposed importance sampling method combined with the quasi-Monte Carlo simulation can be utilized to perform the reliability analysis. Meanwhile, the confidence interval and its length for the results have also been derived. Various numerical examples demonstrate the efficiency and accuracy of the presented importance sampling method. Finally, the reliability analysis and sensitivity analysis of an inside flap are solved by the proposed importance sampling method.

Modelling of patterns and estimation of characteristic parameters

A two-dimensional pattern represents a fingerprint of the process that generated it. It is therefore expected that the information about the production process can be extracted from the pattern. In this paper, a non-parametric statistical method for modelling chaotic two-dimensional patterns and the estimation of the characteristic parameters is proposed. It is based on the joint probability density function of samples taken from known two-dimensional patterns representing a database. A new pattern with an unknown production process is reproduced by comparing parts of the new pattern with samples taken from the database. Because the samples in the database also include information about the production process, relevant parameters and the type of production process can be estimated simultaneously with the reproduction of patterns.

Stochastic BER estimation for coherent QPSK transmission systems with digital carrier phase recovery

We propose a stochastic bit error ratio estimation approach based on a statistical analysis of the retrieved signal phase for coherent optical QPSK systems with digital carrier phase recovery. A family of the generalized exponential function is applied to fit the probability density function of the signal samples. The method provides reasonable performance estimation in presence of both linear and nonlinear transmission impairments while reduces the computational intensity greatly compared to Monte Carlo simulation.

Novel Method Based on Sequence Shepard Interpolation for Structual Reliability Analysis

A novel method based on sequence Shepard interpolation is presented for reliability analysis of structure with implicit limit state equation.Using experiment points of uniform distribution generated by hammersly point set,the response values of samples from importance sampling probability density function are predicted by Shepard interpolation method,and the failure probability of structure is estimated simultaneously.In the process of importance sampling,the most probable failure point searched in current iteration is used as the center of the experiment point set and the density center of the importance sampling probability density function in the next iteration.Comparing with Monte Carlo method,three numerical examples show that the sequence Shepard interpolation is more efficient.The reliability analysis of flap mechanism verifies the engineering applicability of this method.