• East Asian mathematical journal
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• 제39권5호
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• pp.537-547
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• 2023

In this paper, we introduce the optimal approximation by a Gaussian function for a probability density function. We show that the approximation can be obtained by solving a non-linear system of parameters of Gaussian function. Then, to understand the non-normality of the empirical distributions observed in financial markets, we consider the nearly Gaussian function that consists of an optimally approximated Gaussian function and a small periodically oscillating density function. We show that, depending on the parameters of the oscillation, the nearly Gaussian functions can have fairly thick heavy tails.

• Structural Engineering and Mechanics
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• 제28권4호
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• pp.373-386
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• 2008

In this study stochastic analysis of non-linear dynamical systems under ${\alpha}$-stable, multiplicative white noise has been conducted. The analysis has dealt with a special class of ${\alpha}$-stable stochastic processes namely sub-Gaussian white noises. In this setting the governing equation either of the probability density function or of the characteristic function of the dynamical response may be obtained considering the dynamical system forced by a Gaussian white noise with an uncertain factor with ${\alpha}/2$- stable distribution. This consideration yields the probability density function or the characteristic function of the response by means of a simple integral involving the probability density function of the system under Gaussian white noise and the probability density function of the ${\alpha}/2$-stable random parameter. Some numerical applications have been reported assessing the reliability of the proposed formulation. Moreover a proper way to perform digital simulation of the sub-Gaussian ${\alpha}$-stable random process preventing dynamical systems from numerical overflows has been reported and discussed in detail.

• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 2005년도 ICCAS
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• pp.408-413
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• 2005

A new methodology for discrete non-Gaussian probability density estimation is investigated in this paper based on a dynamic Bayesian network (DBN) and kernel functions. The estimator consists of a DBN in which the transition distribution is represented with kernel functions. The estimator parameters are determined through a recursive learning algorithm according to the maximum likelihood (ML) scheme. A discrete-type Poisson distribution is generated in a simulation experiment to evaluate the proposed method. In addition, an unknown probability density generated by nonlinear transformation of a Poisson random variable is simulated. Computer simulations numerically demonstrate that the method successfully estimates the unknown probability distribution function (PDF).

• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 2001년도 ICCAS
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• pp.49.1-49
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• 2001

The skin color model is a very important concept in face detection, face recognition and face tracking. Usually, this model is obtained by estimating a probability density function of skin color distribution. In many cases, it is assumed that the underlying density function follows a Gaussian distribution. In this paper, a new method for non-parametric estimation of the probability density function, by using feed-forward neural network, is used to estimate the underlying skin color model. By using this method, the resulting skin color model is better than the Gaussian estimation and substantially approaches the real distribution. Applications to face detection and face ...

• Structural Engineering and Mechanics
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• 제86권3호
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• pp.349-359
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• 2023

This paper presents a technique for determining the optimal number of elements in stochastic finite element analysis based on reliability analysis. Using the change-of-variable perturbation stochastic finite element approach, the probability density function of the dynamic responses of stochastic structures is explicitly determined. This method combines the perturbation stochastic finite element method with the change-of-variable technique into a united model. To further examine the relationships between the random fields, discretization of the random field parameters, such as the variance function and the scale of fluctuation, is also performed. Accordingly, the reliability index is calculated based on the explicit probability density function of responses with Gaussian or non-Gaussian random fields in any number of elements corresponding to the random field discretization. The numerical examples illustrate the effectiveness of the proposed method for a one-dimensional cantilever reinforced concrete column and a two-dimensional steel plate shear wall. The benefit of this method is that the probability density function of responses can be obtained explicitly without the use simulation techniques. Any type of random variable with any statistical distribution can be incorporated into the calculations, regardless of the restrictions imposed by the type of statistical distribution of random variables. Consequently, this method can be utilized as a suitable guideline for the efficient implementation of stochastic finite element analysis of structures, regardless of the statistical distribution of random variables.

• 한국해안해양공학회지
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• 제6권1호
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• pp.88-93
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• 1994

유한수심에서의 불규칙파에 적용할 수 있는 파고의 확률분포함수를 2가지 해석적 방법으로 유도하였다. 첫번째 방법으로 새로이 유도된 확률분포함수는 Rayleigh 확률분포함수에 대한 직교 다항식을 유도함으로써 급수형태로 표시된다. 유도된 확률밀도함수를 비정규성이 강한 천해에서 측정한 파랑자료와 비교하였다. 확률밀도함수가 자료의 막대그래프와 잘 일치하였으나, 확률밀도함수가 급수로 표시되어 있기 때문에 파고가 큰 부분에서 음의 확률값이 된다. 비록 음의 확률값의 크기가 작다 하더라도 파고의 극치분포함수를 구하기에 부적절하다고 판단된다. 두번째 방법은 최대 엔트로피 법(maximum entropy method)을 적용하여 파고 분포와 매우 잘 일치하며, 극치파고분포와 파고의 통계적인 특성 등을 추정하는 데 매우 유용함을 알 수 있다. 그러나 최대 엔트로피 법을 사용했을 경우, 비정규분포 특성을 나타내는 변위의 분포함수와 파고의 분포함수 사이의 함수관계를 구할 수 없었다.

• Journal of applied mathematics & informatics
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• 제28권1_2호
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• pp.257-273
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• 2010

One can construct a theory of probability of fractional order in which the exponential function is replaced by the Mittag-Leffler function. In this framework, it seems of interest to generalize some useful classical mathematical tools, so that they are more suitable in fractional calculus. After a short background on fractional calculus based on modified Riemann Liouville derivative, one summarizes some definitions on probability density of fractional order (for the motive), and then one introduces successively fractional Euler's integrals (first and second kind) and fractional Hermite polynomials. Some properties of the Gaussian density of fractional order are exhibited. The fractional probability so introduced exhibits some relations with quantum probability.

• 해양환경안전학회지
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• 제21권3호
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• pp.253-258
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• 2015

The purpose of this study is to find suitable probability distribution function of complex distribution data like multimodal. Normal distribution is broadly used to assume probability distribution function. However, complex distribution data like multimodal are very hard to be estimated by using normal distribution function only, and there might be errors when other distribution functions including normal distribution function are used. In this study, we experimented to find fit probability distribution function in multimodal area, by using AIS(Automatic Identification System) observation data gathered in Mokpo port for a year of 2013. By using chi-squared statistic, gaussian mixture model(GMM) is the fittest model rather than other distribution functions, such as extreme value, generalized extreme value, logistic, and normal distribution. GMM was found to the fit model regard to multimodal data of maritime traffic flow distribution. Probability density function for collision probability and traffic flow distribution will be calculated much precisely in the future.

• 대한조선학회논문집
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• 제52권1호
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• pp.88-95
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• 2015

For the frequency-domain spectral fatigue analysis, the probability density function of stress range needs to be estimated based on the stress spectrum only, which is a frequency domain representation of the response. The probability distribution of the stress range of the narrow-band spectrum is known to follow the Rayleigh distribution, however the PDF of wide-band spectrum is difficult to define with clarity due to the complicated fluctuation pattern of spectrum. In this paper, efforts have been made to figure out the links between the probability density function of stress range to the structural response of wide-band Gaussian random process. An artificial neural network scheme, known as one of the most powerful system identification methods, was used to identify the multivariate functional relationship between the idealized wide-band spectrums and resulting probability density functions. To achieve this, the spectrums were idealized as a superposition of two triangles with arbitrary location, height and width, targeting to comprise wide-band spectrum, and the probability density functions were represented by the linear combination of equally spaced Gaussian basis functions. To train the network under supervision, varieties of different wide-band spectrums were assumed and the converged probability density function of the stress range was derived using the rainflow counting method and all these data sets were fed into the three layer perceptron model. This nonlinear least square problem was solved using Levenberg-Marquardt algorithm with regularization term included. It was proven that the network trained using the given data set could reproduce the probability density function of arbitrary wide-band spectrum of two triangles with great success.

• The Journal of the Acoustical Society of Korea
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• 제17권4E호
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• pp.3-10
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• 1998

This paper concerns the dynamical behavior, in probabilistic sense, of a feedforward neural network performing auto association for novelty. Networks of retinotopic topology having a one-to-one correspondence between and output units can be readily trained using back-propagation algorithm, to perform autoassociative mappings. A novelty filter is obtained by subtracting the network output from the input vector. Then the presentation of a "familiar" pattern tends to evoke a null response ; but any anomalous component is enhanced. Such a behavior exhibits a promising feature for enhancement of weak signals in additive noise. As an analysis of the novelty filtering, this paper shows that the probability density function of the weigh converges to Gaussian when the input time series is statistically characterized by nonsymmetrical probability density functions. After output units are locally linearized, the recursive relation for updating the weight of the neural network is converted into a first-order random differential equation. Based on this equation it is shown that the probability density function of the weight satisfies the Fokker-Planck equation. By solving the Fokker-Planck equation, it is found that the weight is Gaussian distributed with time dependent mean and variance.