Monte Carlo evaluation of biological variation: Random generation of correlated non-Gaussian model parameters

When modelling the behaviour of horticultural products, demonstrating large sources of biological variation, we often run into the issue of non-Gaussian distributed model parameters. This work presents an algorithm to reproduce such correlated non-Gaussian model parameters for use with Monte Carlo simulations. The algorithm works around the problem of non-Gaussian distributions by transforming the observed non-Gaussian probability distributions using a proposed SKN-distribution function before applying the covariance decomposition algorithm to generate Gaussian random co-varying parameter sets. The proposed SKN-distribution function is based on the standard Gaussian distribution function and can exhibit different degrees of both skewness and kurtosis. This technique is demonstrated using a case study on modelling the ripening of tomato fruit evaluating the propagation of biological variation with time.     

Data Clustering Based on Maximization of Outlier Factor

There exist many data clustering algorithms, but they can not adequately handle the number of clusters or cluster shapes. Their performance mainly depends on a choice of algorithm parameters. Our approach to data clustering and algorithm does not require the parameter choice; it can be treated as a natural adaptation to the existing structure of distances between data points. The outlier factor introduced by the author specifies a degree of being an outlier for each data point. The outlier factor notion is based on the difference between the frequency distribution of interpoint distances in a given dataset and the corresponding distribution of uniformly distributed points. Then data clusters can be determined by maximizing the outlier factor function. The data points in dataset are divided into clusters according to the attractor regions of local optima. An experimental evaluation of the proposed algorithm shows that the proposed method can identify complex cluster shapes. Key advantages of the approach are: good clustering properties for datasets with comparatively large amount of noise (an additional data points), and an absence of important parameters which adequate choice determines the quality of results.     

A Bayesian Motivated Laplace Inversion for Multivariate Probability Distributions

The paper introduces a recursive procedure to invert the multivariate Laplace transform of probability distributions. The procedure involves taking independent samples from the Laplace transform; these samples are then used to update recursively an initial starting distribution. The update is Bayesian driven. The final estimate can be written as a mixture of independent gamma distributions, making it the only methodology which guarantees to numerically recover a probability distribution with positive support. Proof of convergence is given by a fixed point argument. The estimator is fast, accurate and can be run in parallel since the target distribution is evaluated on a grid of points. The method is illustrated on several examples and compared to the bivariate Gaver–Stehfest method.     

Multivariate Gaussian Simulation Outside Arbitrary Ellipsoids

Methods for simulation from multivariate Gaussian distributions restricted to be from outside an arbitrary ellipsoidal region are often needed in applications. A standard rejection algorithm that draws a sample from a multivariate Gaussian distribution and accepts it if it is outside the ellipsoid is often employed; however, this is computationally inefficient if the probability of that ellipsoid under the multivariate normal distribution is substantial. We provide a two-stage rejection sampling scheme for drawing samples from such a truncated distribution. Experiments show that the added complexity of the two-stage approach results in the standard algorithm being more efficient for small ellipsoids (i.e., with small rejection probability). However, as the size of the ellipsoid increases, the efficiency of the two-stage approach relative to the standard algorithm increases indefinitely. The relative efficiency also increases as the number of dimensions increases, as the centers of the ellipsoid and the multivariate Gaussian distribution come closer, and as the shape of the ellipsoid becomes more spherical. We provide results of simulation experiments conducted to quantify the relative efficiency over a range of parameter settings.     

混合变异粒子群优化算法及其应用

为改善粒子群优化算法在解决复杂优化问题时收敛质量不高的不足,提出了一种改进的粒子群优化算法,即混合变异粒子群优化算法(HMPSO).HMPSO算法采用了带有随机因子的惯性权重取值更新策略,降低了标准粒子群优化算法中由于粒子飞行速度过大而错过最优解的概率,从而加速了算法的收敛速度.此外,通过混合变异进化环节的引入,缓解了粒子种群在进化过程中的多样性与收敛性这一矛盾,使得算法的全局探索与局部开发得到有效平衡.利用经典的基准测试函数和平面冗余机械臂逆运动学问题的求解来验证提出算法的有效性,试验结果表明:与其他算法相比,HMPSO算法具有更快的收敛速度、更高的收敛精度、更强的收敛稳定性以及更低的计算成本.     

A local limit theorem for a transient chaotic walk in a frozen environment

This paper studies particle propagation in a one-dimensional inhomogeneous medium where the laws of motion are generated by chaotic and deterministic local maps. Assuming that the particle’s initial location is random and uniformly distributed, this dynamical system can be reduced to a random walk in a one-dimensional inhomogeneous environment with a forbidden direction. Our main result is a local limit theorem which explains in detail why, in the long run, the random walk’s probability mass function does not converge to a Gaussian density, although the corresponding limiting distribution over a coarser diffusive space scale is Gaussian.     

Clustering Multivariate Longitudinal Observations: The Contaminated Gaussian Hidden Markov Model

The Gaussian hidden Markov model (HMM) is widely considered for the analysis of heterogenous continuous multivariate longitudinal data. To robustify this approach with respect to possible elliptical heavy-tailed departures from normality, due to the presence of outliers, spurious points, or noise (collectively referred to as bad points herein), the contaminated Gaussian HMM is here introduced. The contaminated Gaussian distribution represents an elliptical generalization of the Gaussian distribution and allows for automatic detection of bad points in the same natural way as observations are typically assigned to the latent states in the HMM context. Once the model is fitted, each observation has a posterior probability of belonging to a particular state and, inside each state, of being a bad point or not. In addition to the parameters of the classical Gaussian HMM, for each state we have two more parameters, both with a specific and useful interpretation: one controls the proportion of bad points and one specifies their degree of atypicality. A sufficient condition for the identifiability of the model is given, an expectation-conditional maximization algorithm is outlined for parameter estimation and various operational issues are discussed. Using a large-scale simulation study, but also an illustrative artificial dataset, we demonstrate the effectiveness of the proposed model in comparison with HMMs of different elliptical distributions, and we also evaluate the performance of some well-known information criteria in selecting the true number of latent states. The model is finally used to fit data on criminal activities in Italian provinces. Supplementary materials for this article are available online     

Rare Event Probability Estimation in the Presence of Epistemic Uncertainty on Input Probability Distribution Parameters

The accurate estimation of rare event probabilities is a crucial problem in engineering to characterize the reliability of complex systems. Several methods such as Importance Sampling or Importance Splitting have been proposed to perform the estimation of such events more accurately (i.e., with a lower variance) than crude Monte Carlo method. However, these methods assume that the probability distributions of the input variables are exactly defined (e.g., mean and covariance matrix perfectly known if the input variables are defined through Gaussian laws) and are not able to determine the impact of a change in the input distribution parameters on the probability of interest. The problem considered in this paper is the propagation of the input distribution parameter uncertainty defined by intervals to the rare event probability. This problem induces intricate optimization and numerous probability estimations in order to determine the upper and lower bounds of the probability estimate. The calculation of these bounds is often numerically intractable for rare event probability (say 10^?5), due to the high computational cost required. A new methodology is proposed to solve this problem with a reduced simulation budget, using the adaptive Importance Sampling. To this end, a method for estimating the Importance Sampling optimal auxiliary distribution is proposed, based on preceding Importance Sampling estimations. Furthermore, a Kriging-based adaptive Importance Sampling is used in order to minimize the number of evaluations of the computationally expensive simulation code. To determine the bounds of the probability estimate, an evolutionary algorithm is employed. This algorithm has been selected to deal with noisy problems since the Importance Sampling probability estimate is a random variable. The efficiency of the proposed approach, in terms of accuracy of the found results and computational cost, is assessed on academic and engineering test cases.     

Generation of three-dimensional random rotations in fitting and matching problems

An algorithm is developed to generate random rotations in three-dimensional space that follow a probability distribution arising in fitting and matching problems. The rotation matrices are orthogonally transformed into an optimal basis and then parameterized using Euler angles. The conditional distributions of the three Euler angles have a very simple form: the two azimuthal angles can be decoupled by sampling their sum and difference from a von Mises distribution; the cosine of the polar angle is exponentially distributed and thus straighforward to generate. Simulation results are shown and demonstrate the effectiveness of the method. The algorithm is compared to other methods for generating random rotations such as a random walk Metropolis scheme and a Gibbs sampling algorithm recently introduced by Green and Mardia. Finally, the algorithm is applied to a probabilistic version of the Procrustes problem of fitting two point sets and applied in the context of protein structure superposition.     

风电功率波动特性的概率分布模型研究

为保证电网安全稳定运行,在大规模风电并网运行控制过程中,准确构建风电出力波动特性的概率分布模型具有重要意义.基于数据驱动的方法,采用加权高斯混合概率分布模型来拟合大规模风电基地的波动特性,模型拟合参数可采用基于期望最大化(Expectation Maximization,EM)的极大似然估计算法来获得,并提出了拟合评价指标来与其它多种概率分布模型进行对比,结果验证了加权高斯混合概率模型的有效性和适用性.