Oversampling Method Based on Gaussian Distribution and K-Means Clustering

Learning from imbalanced data is one of the greatest challenging problems in binary classification,and this problem has gained more importance in recent years.When the class distribution is imbalanced,classical machine learning algorithms tend to move strongly towards the majority class and disregard the minority.Therefore,the accuracy may be high,but the model cannot recognize data instances in the minority class to classify them,leading to many misclassifications.Different methods have been proposed in the literature to handle the imbalance problem,but most are complicated and tend to simulate unnecessary noise.In this paper,we propose a simple oversampling method based on Multivariate Gaussian distribution and K-means clustering,called GK-Means.The new method aims to avoid generating noise and control imbalances between and within classes.Various experiments have been carried out with six classifiers and four oversampling methods.Experimental results on different imbalanced datasets show that the proposed GK-Means outperforms other oversampling methods and improves classification performance as measured by F1-score and Accuracy.

STRUCTURES OF SYSTEMS WITH EXPONENTIAL LIFE DISTRIBUTIONS

Structures of monotone systems and cold standby systems with exponen-tial life distributions and dependent components are studied. It is shown that a mono-tone system composed of components with multivariate HNBUE life distributions isessentially a series system composed of components with multivariate exponential lifedistributions. Also, it is proved that for cold standby systems composed of componentswith multivariate NBU life distributions, all but oue of the components are degenerateat zero while the remaining one is exponential. In addition, several equivalent char-acterizations of multivariate exponential distribution are provided in the multivariateHNBUE life distribution class which include many existing results as special cases.

Research on Gaussian distribution preprocess method of infrared multispectral image background clutter

This paper introduces a sliding-window mean removal high pass filter by which background clutter of infrared multispectral image is obtained. The method of selecting the optimum size of the sliding-window is based on the skewness-kurtosis test. In the end, a multivariate Gaussian distribution mathematical expression of background clutter image is given.

Probabilistic analysis of tunnel displacements based on correlative recognition of rock mass parameters

Displacement is vital in the evaluations of tunnel excavation processes,as well as in determining the postexcavation stability of surrounding rock masses.The prediction of tunnel displacement is a complex problem because of the uncertainties of rock mass properties.Meanwhile,the variation and the correlation relationship of geotechnical material properties have been gradually recognized by researchers in recent years.In this paper,a novel probabilistic method is proposed to estimate the uncertainties of rock mass properties and tunnel displacement,which integrated multivariate distribution function and a relevance vector machine(RVM).The multivariate distribution function is used to establish the probability model of related random variables.RVM is coupled with the numerical simulation methods to construct the nonlinear relationship between tunnel displacements and rock mass parameters,which avoided a large number of numerical simulations.Also,the residual rock mass parameters are taken into account to reflect the brittleness of deeply buried rock mass.Then,based on the proposed method,the uncertainty of displacement in a deep tunnel of CJPL-II laboratory are analyzed and compared with the in-situ measurements.It is found that the predicted tunnel displacements by the RVM model closely match with the measured ones.The correlations of parameters have significant impacts on the uncertainty results.The uncertainty of tunnel displacement decreases while the reliability of the tunnel increases with the increases of the negative correlations among rock mass parameters.When compared to the deterministic method,the proposed approach is more rational and scientific,and also conformed to rock engineering practices.

Study of Vertical Breakwater Reliability Based on Copulas

The reliability of a vertical breakwater is calculated using direct integration methods based on joint density functions.The horizontal and uplifting wave forces on the vertical breakwater can be well fitted by the lognormal and the Gumbel distributions,respectively.The joint distribution of the horizontal and uplifting wave forces is analyzed using different probabilistic distributions,including the bivariate logistic Gumbel distribution,the bivariate lognormal distribution,and three bivariate Archimedean copulas functions constructed with different marginal distributions simultaneously.We use the fully nested copulas to construct multivariate distributions taking into account related variables.Different goodness fitting tests are carried out to determine the best bivariate copula model for wave forces on a vertical breakwater.We show that a bivariate model constructed by Frank copula gives the best reliability analysis,using marginal distributions of Gumbel and lognormal to account for uplifting pressure and horizontal wave force on a vertical breakwater,respectively.The results show that failure probability of the vertical breakwater calculated by multivariate density function is comparable to those by the Joint Committee on Structural Safety methods.As copulas are suitable for constructing a bivariate or multivariate joint distribution,they have great potential in reliability analysis for other coastal structures.

MULTIVARIATE EXTREME VALUE DISTRIBUTION AND ITS FISHER INFORMATION MATRIX

The paper is concerned with the basic properties of multivariate extreme value distribution (in the Logistic model). We obtain the characteristic function and recurrence formula of the density function. The explicit algebraic formula for Fisher information matrix is indicated. A simple and accurate procedure for generating random vector from multivariate extreme value distribution is presented.

Design Code Calibration of Offshore Platform Against Typhoon/Hurricane Attacks

Hurricanes Katrina and Rita resulted in the largest number of platforms destroyed and damaged in the history of Gulf of Mexico operations. With the trend of global warming, sea level rising and the frequency and intensity of typhoon increase. How to determine a reasonable deck elevation against the largest hurricane waves has become a key issue in offshore platforms design and construction for the unification of economy and safety. In this paper, the multivariate compound extreme value distribution （MCEVD） model is used to predict the deck elevation with different combination of tide, surge height, and crest height. Compared with practice recommended by American Petroleum Institute （API）, the prediction by MCEVD has probabilistic meaning and universality.

Unseen head pose prediction using dense multivariate label distribution

Accurate head poses are useful for many face-related tasks such as face recognition, gaze estimation,and emotion analysis. Most existing methods estimate head poses that are included in the training data(i.e.,previously seen head poses). To predict head poses that are not seen in the training data, some regression-based methods have been proposed. However, they focus on estimating continuous head pose angles, and thus do not systematically evaluate the performance on predicting unseen head poses. In this paper, we use a dense multivariate label distribution(MLD) to represent the pose angle of a face image. By incorporating both seen and unseen pose angles into MLD, the head pose predictor can estimate unseen head poses with an accuracy comparable to that of estimating seen head poses. On the Pointing'04 database, the mean absolute errors of results for yaw and pitch are 4.01?and 2.13?, respectively. In addition, experiments on the CAS-PEAL and CMU Multi-PIE databases show that the proposed dense MLD-based head pose estimation method can obtain the state-of-the-art performance when compared to some existing methods.

Superiority of empirical Bayes estimator of the mean vector in multivariate normal distribution

In this paper, the Bayes estimator and the parametric empirical Bayes estimator(PEBE) of mean vector in multivariate normal distribution are obtained. The superiority of the PEBE over the minimum variance unbiased estimator(MVUE) and a revised James-Stein estimators(RJSE) are investigated respectively under mean square error(MSE) criterion. Extensive simulations are conducted to show that performance of the PEBE is optimal among these three estimators under the MSE criterion.

Seismic reliability analysis of large electric power systems

Based on the De.Morgan laws and Boolean simplification, a recursive decomposition method is introduced in this paper to identify the main exclusive safe paths and failed paths of a network. The reliability or the reliability bound of a network can be conveniently expressed as the summation of the joint probabilities of these paths. Under the multivariate normal distribution assumption, a conditioned reliability index method is developed to evaluate joint probabilities of various exclusive safe paths and failed paths, and, finally, the seismic reliability or the reliability bound of an electric power system. Examples given in the paper show that the method is very simple and provides accurate results in the seismic reliability analysis.

An Advanced Probabilistic Neural Network for the Design of Breakwater Armor Blocks

In this study, an advanced probabilistic neural network （APNN） method is proposed to reflect the global probability density function （PDF） by summing up the heterogeneous local PDF which is automatically determined in the individual standard deviation of variables. The APNN is applied to predict the stability number of armor blocks of breakwaters using the experimental data of＇ van der Meet, and the estimated results of the APNN are compared with those of an empirical formula and a previous artificial neural network （ANN） model. The APNN shows better results in predicting the stability number of armor bilks of breakwater and it provided the promising probabilistic viewpoints by using the individual standard deviation in a variable.

Efficient Algorithms for Generating Truncated Multivariate Normal Distributions

Sampling from a truncated multivariate normal distribution （TMVND） constitutes the core computational module in fitting many statistical and econometric models. We propose two efficient methods, an iterative data augmentation （DA） algorithm and a non-iterative inverse Bayes formulae （IBF） sampler, to simulate TMVND and generalize them to multivariate normal distributions with linear inequality constraints. By creating a Bayesian incomplete-data structure, the posterior step of the DA Mgorithm directly generates random vector draws as opposed to single element draws, resulting obvious computational advantage and easy coding with common statistical software packages such as S-PLUS, MATLAB and GAUSS. Furthermore, the DA provides a ready structure for implementing a fast EM algorithm to identify the mode of TMVND, which has many potential applications in statistical inference of constrained parameter problems. In addition, utilizing this mode as an intermediate result, the IBF sampling provides a novel alternative to Gibbs sampling and elimi- nares problems with convergence and possible slow convergence due to the high correlation between components of a TMVND. The DA algorithm is applied to a linear regression model with constrained parameters and is illustrated with a published data set. Numerical comparisons show that the proposed DA algorithm and IBF sampler are more efficient than the Gibbs sampler and the accept-reject algorithm.

Nonlinear industrial process fault diagnosis with latent label consistency and sparse Gaussian feature learning

With the increasing complexity of industrial processes, the high-dimensional industrial data exhibit a strong nonlinearity, bringing considerable challenges to the fault diagnosis of industrial processes. To efficiently extract deep meaningful features that are crucial for fault diagnosis, a sparse Gaussian feature extractor(SGFE) is designed to learn a nonlinear mapping that projects the raw data into the feature space with the fault label dimension. The feature space is described by the one-hot encoding of the fault category label as an orthogonal basis. In this way, the deep sparse Gaussian features related to fault categories can be gradually learned from the raw data by SGFE. In the feature space,the sparse Gaussian(SG) loss function is designed to constrain the distribution of features to multiple sparse multivariate Gaussian distributions. The sparse Gaussian features are linearly separable in the feature space, which is conducive to improving the accuracy of the downstream fault classification task. The feasibility and practical utility of the proposed SGFE are verified by the handwritten digits MNIST benchmark and Tennessee-Eastman(TE) benchmark process,respectively.

Construction of <i>k</i>-Variate Survival Functions with Emphasis on the Case <i>k</i>= 3

The purpose of this paper is to present a general universal formula for k-variate survival functions for arbitrary k = 2, 3, ..., given all the univariate marginal survival functions. This universal form of k-variate probability distributions was obtained by means of “dependence functions” named “joiners” in the text. These joiners determine all the involved stochastic dependencies between the underlying random variables. However, in order that the presented formula (the form) represents a legitimate survival function, some necessary and sufficient conditions for the joiners had to be found. Basically, finding those conditions is the main task of this paper. This task was successfully performed for the case k = 2 and the main results for the case k = 3 were formulated as Theorem 1 and Theorem 2 in Section 4. Nevertheless, the hypothetical conditions valid for the general k ≥ 4 case were also formulated in Section 3 as the (very convincing) Hypothesis. As for the sufficient conditions for both the k = 3 and k ≥ 4 cases, the full generality was not achieved since two restrictions were imposed. Firstly, we limited ourselves to the, defined in the text, “continuous cases” (when the corresponding joint density exists and is continuous), and secondly we consider positive stochastic dependencies only. Nevertheless, the class of the k-variate distributions which can be constructed is very wide. The presented method of construction by means of joiners can be considered competitive to the copula methodology. As it is suggested in the paper the possibility of building a common theory of both copulae and joiners is quite possible, and the joiners may play the role of tools within the theory of copulae, and vice versa copulae may, for example, be used for finding proper joiners. Another independent feature of the joiners methodology is the possibility of constructing many new stochastic processes including stationary and Markovian.

Dynamic prediction of over-excavation gap due to posture adjustment of shield machine in soft soil

The probability analysis of ground deformation is becoming a trend to estimate and control the risk brought by shield tunnelling.The gap parameter is regarded as an effective tool to estimate the ground loss of tunnelling in soft soil.More specifically,x,which is a gap parameter component defined as the over(or insufficient)excavation due to the change in the posture of the shield machine,may contribute more to the uncertainty of the ground loss.However,the existing uncertainty characterization methods for x have several limitations and cannot explain the uncertain correlations between the relevant parameters.Along these lines,to better characterize the uncertainty of x,the multivariate probability distribution was developed in this work and a dynamic prediction was proposed for it.To attain this goal,1523 rings of the field data coming from the shield tunnel between Longqing Road and Baiyun Road in Kunming Metro Line 5 were utilized and 44 parameters including the construction,stratigraphic,and posture parameters were collected to form the database.According to the variance filter method,the mutual information method,and the value of the correlation coefficients,the original 44 parameters were reduced to 10 main parameters,which were unit weight,the stoke of the jacks(A,B,C,and D groups),the pressure of the pushing jacks(A,C groups),the chamber pressure,the rotation speed,and the total force.The multivariate probability distribution was constructed based on the Johnson system of distributions.Moreover,the distribution was satisfactorily verified in explaining the pairwise correlation between x and other parameters through 2 million simulation cases.At last,the distribution was used as a prior distribution to update the marginal distribution of x with any group of the relevant parameters known.The performance of the dynamic prediction was further validated by the field data of 3 shield tunnel cases.

SOME MULTIVARIATE DMRL AND NBUE DEFINITIONS BASED ON CONDITIONAL STOCHASTIC ORDER

Two classes of multivariate DMRL distributions and a class of multivariate NBUE distributions are introduced in this paper by using conditional stochastic order.That is, a random vector belongs to a multivariate DMRL class of life distributions if its residual life(defined as a conditional random vector)is decreasing in time under convex or linear order.Some conservation properties of these classes are studied.

New Results on Truncated Elliptical Distributions

Truncated elliptical distributions occur naturally in theoretical and applied statistics and are essential for the study of other classes of multivariate distributions.Two members of this class are the multivariate truncated normal and multivariate truncated t distributions.We derive statistical properties of the truncated elliptical distributions.Applications of our results establish new properties of the multivariate truncated slash and multivariate truncated power exponential distributions.

Information Divergence and the Generalized Normal Distribution:A Study on Symmetricity

This paper investigates and discusses the use of information divergence,through the widely used Kullback–Leibler(KL)divergence,under the multivariate(generalized)γ-order normal distribution(γ-GND).The behavior of the KL divergence,as far as its symmetricity is concerned,is studied by calculating the divergence of γ-GND over the Student’s multivariate t-distribution and vice versa.Certain special cases are also given and discussed.Furthermore,three symmetrized forms of the KL divergence,i.e.,the Jeffreys distance,the geometric-KL as well as the harmonic-KL distances,are computed between two members of the γ-GND family,while the corresponding differences between those information distances are also discussed.

DESIGNING THE OPTIMUM CONFIGURATIONS OF CIRCULAR AND SPHERICAL PRODUCT SPECIFICATIONS FOR MULTIPLE QUALITY CHARACTERISTICS

As an integral part of tolerance design in the context of design for six sigma, determining optimal product specifications has become the focus of increased activity, as manufacturing industries strive to increase productivity and improve the quality of their products. Although a number of research papers have been reported in the research community, there is room for improvement. Most existing research papers consider determining optimal specification limits for a single quality characteristic. In this paper, we develop the modeling and optimization procedures for optimum circular and spherical configurations by considering multiple quality characteristics. The concepts of multivariate quality loss function and truncated distribution are incorporated. This has never been adequately addressed, nor has been appropriately applied in industry. A numerical example is shown and comparison studies are made.

Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets

The probability hypothesis density(PHD) filter has been recognized as a promising technique for tracking an unknown number of targets. The performance of the PHD filter, however, is sensitive to the available knowledge on model parameters such as the measurement noise variance and those associated with the changes in the maneuvering target trajectories. If these parameters are unknown in advance, the tracking performance may degrade greatly. To address this aspect, this paper proposes to incorporate the adaptive parameter estimation(APE) method in the PHD filter so that the model parameters, which may be static and/or time-varying, can be estimated jointly with target states. The resulting APE-PHD algorithm is implemented using the particle filter(PF), which leads to the PF-APE-PHD filter. Simulations show that the newly proposed algorithm can correctly identify the unknown measurement noise variances, and it is capable of tracking multiple maneuvering targets with abrupt changing parameters in a more robust manner, compared to the multi-model approaches.