量化分形图形的新指数——简便分形指数

分形的广泛存在性已被普遍接受,然而分形维数的现有定义计算得到的结果是:不同的分形维数定义得到不同的分形维数值,甚至会出现不同的变化趋势,且在应用时使用的最小二乘回归结果不稳定,导致数值应用也会受影响,出现这些现象主要归咎于现有分形维数定义的严格性、抽象性以及分形图形的码尺效应.为避免这些问题,本文结合分形图形的长尾分布特征及自相似性提出一个新的分形量化形式——简便分形指数,并阐述了该定义背后的分形原理及计算方法,简便分形指数越大,形状复杂程度越高.最后本文利用岩石裂隙图像说明简便分形指数对不同裂隙网络复杂性描述的准确性,验证其作为分形图形量化方法的合理性及便利性.     

粗糙面分形计算理论研究进展

为提出一种工程上适用可靠的粗糙面分形维数计算方法,在分形曲线的维数计算方法(码尺法,盒维法)基础上,先后提出了星积分形曲面的维数计算方法、三角形棱柱表面积法、投影覆盖法、立方体覆盖法、改进的立方体覆盖法、分形的增变量描述法等曲面分形维数理论.鉴于上述方法的共有缺陷——获取三维坐标的激光表面仪器的扫描尺度限制,研究者提出了粗糙面图像维数计算理论,包括二值化图像维数、灰度图像维数、RGB图像维数计算理论.最后,本文展望了分形维数计算理论领域内亟待解决的三大问题.     

星积分形曲面及其维数

通过分形曲线定义了一类分形曲面(被称为星积分形曲面),讨论了这类分形曲面的分形维数,得出了分形曲线的维数与它们所构造出的分形曲面维数之间的关系。     

异位孔爆破破碎块度与设计孔的分形匹配研究

针对露天矿炮区工作面中穿爆作业不规律导致爆破效果不佳现象,通过在胜利露天矿进行爆破试验得到实测数据,利用分形理论中分维数与爆破破碎块度的关系,计算出异位孔连续装药结构岩石爆破破碎块度的分维区间,与其设计孔爆破破碎块度分维区间进行对比分析,对异位孔爆破破碎块度分维区间进行优化,利用工程实测方法与分形理论相结合找到最佳的爆破参数,以达到不规则作业面最佳爆破效果.     

基于双论域量化模糊粗糙集的公共卫生应急决策模型

公共卫生应急决策是公共卫生管理的核心,也是有效应对公共卫生事件的关键.近年来频发的公共卫生事件已对公众安全和身体健康造成严重的威胁.因此,亟待研究应对突发公共卫生事件的应急决策理论与方法.文章以公共卫生应急决策为研究背景讨论双论域量化模糊粗糙集理论与方法.首先定义了基于模糊相容关系的双论域量化模糊粗糙集模型,进而充分考虑公共卫生应急决策问题的特征,并将公共卫生应急决策过程转化为具有模糊决策对象的模糊近似决策问题,构建基于突发公共卫生应急决策的双论域量化模糊粗糙集应用模型.通过在双论域量化近似空间上计算模糊决策对象的上下近似,并结合经典非确定型决策的思想给出公共卫生应急处置的决策规则.该模型提出了一种在不完全信息环境下公共卫生应急决策的方法,以及在充分考虑决策者个人偏好信息基础上的最优决策规则.最后,将理论模型应用于两个不同的公共卫生应急决策问题以说明文章所构建模型的应用过程及其有效性,同时结合数值算例应用结果提出处理实际问题时相应的决策建议.     

齐次集及其拟Lipschitz等价

本文定义并研究一类齐次分形,该类分形包含所有的(拟)Ahlfors-David正则集和许多非正则的Moran集,这里如果一个分形的Hausdorff维数与packing维数不相等,则称它是非正则的.对于这类齐次分形,本文得出它们的分形维数,并且给出在适当分离条件下两个齐次分形拟Lipschitz等价的充要条件.随后,本文将这些结果应用到非正则的Moran集上.     

基于分形流形学习的支持向量机空气污染指数预测模型北大核心CSCD

针对目前北京、上海和广州地区较严重空气污染问题,建立了基于分形流形学习的支持向量机空气污染指数预测模型.首先采用分形理论计算出空气污染数据集分形维数;其次根据分形维数,采用流形学习将高维空气污染数据集通过非线性映射嵌入到低维空间中,对空气污染数据集进行降维;最后建立基于高斯核的支持向量机预测模型对三地区空气污染指数进行预测.北京、上海和广州三地空气污染指数预测结果表明,该模型较传统预测模型,预测性能更优,具有良好的稳定性和有效性.     

基于分形法的岩石断裂面粗糙度研究

从分形几何角度详细综述了近年来国内外在研究岩石断裂面粗糙度方面的成果.分别阐述了分形中的盒维数法、小岛法、分形插值法和多重分形法,并且剖析了每种方法在理论与试验方面的优势与不足.在此基础上进行总结与归纳,并提出了对今后岩石断裂面形貌学研究的三点展望.     

基于分形流形学习的支持向量机空气污染指数预测模型

针对目前北京、上海和广州地区较严重空气污染问题,建立了基于分形流形学习的支持向量机空气污染指数预测模型.首先采用分形理论计算出空气污染数据集分形维数;其次根据分形维数,采用流形学习将高维空气污染数据集通过非线性映射嵌入到低维空间中,对空气污染数据集进行降维;最后建立基于高斯核的支持向量机预测模型对三地区空气污染指数进行预测.北京、上海和广州三地空气污染指数预测结果表明,该模型较传统预测模型,预测性能更优,具有良好的稳定性和有效性.     

融合基于MapReduce并行改进二元蚁群算法与分形维数的属性选择方法

属性选择是数据挖掘领域用于降低数据特征维度的预处理方法.针对大数据环境下高维数据的属性约简问题,提出了融合基于MapReduce并行改进二元蚁群算法与分形维数的属性选择方法.首先,引入了参数控制的位置更新策略、对蚂蚁个体与种群进行交叉变异、重新定义阻塞机制的信息素更新,提出了并行改进的二元蚁群算法MRIBACO.其次,以并行二元蚁群算法作为离散解空间的搜索策略,结合分形维数提出了属性选择模型.在6个UCI数据集上的实验结果表明,较其他方法计算效率更优,同时表明了其有效性与稳定性.     

Fractal analysis of time series in oil and gas production

The peculiarities of fractal characteristics’ calculations for time series are described in this article. An algorithm for calculation of fractal dimension is suggested. It has been proved that the suggested method possesses high accuracy and the rapidity of convergence on the limited number of measurements compared to the methods of covering.The criteria of early diagnosis for changes in the condition of hydrodynamic processes, which do not vary by fractal dimension, have been recommended.The presented method is applicable for practical engineering calculations with self-affine, chaotic data, usually with relatively limited number of measurements. It is quite a simple method for calculation of fractal dimension, algorithm can be easily realized and it should be useful for engineers.The applicability of the proposed algorithm for fractal dimension calculation and early diagnosis criteria of qualitative changes in the behaviour of various dynamic systems has been tested both on simulated as well as practical examples of oil and gas production.     

Analytical variance based global sensitivity analysis for models with correlated variables

In order to quantitatively analyze the variance contributions by correlated input variables to the model output, variance based global sensitivity analysis (GSA) is analytically derived for models with correlated variables. The derivation is based on the input-output relationship of tensor product basis functions and the orthogonal decorrelation of the correlated variables. Since the tensor product basis function based simulator is widely used to approximate the input-output relationship of complicated structure, the analytical solution of the variance based global sensitivity is especially applicable to engineering practice problems. The polynomial regression model is employed as an example to derive the analytical GSA in detail. The accuracy and efficiency of the analytical solution of GSA are validated by three numerical examples, and engineering application of the derived solution is demonstrated by carrying out the GSA of the riveting and two dimension fracture problem.     

Investigating the scale effects in strength of fractured rock mass

Based on the method of fractal geometry, the relationship between quantity and size of fracture in a rock mass can be studied. It was found that this relationship is power: N_δ=N₀δ^-D. Based on this result and theoretical analysis, the nonlinear relationship between rock mass strength and fracture quantity is found: Rc=Rc_m+ΔRcExp[-BN_δ].the formula includes the residual strength of fractured rock mass Rc_m, the difference in value (ΔRc) between the intact rock strength Rc₀ and the residual strength, and fracture quantity. Then, combining the relationship between fracture quantity and the size of rock mass, the formula of rock mass strength is found: . The formula includes fracture quantity, the fractal dimension of fracture quantity distribution and rock mass size. The results show that the attenuation index of rock mass strength is related to the fractal dimension of fracture quantity distribution: α=2-D.     

基于分形理论的金融系统波动行为分数维理论计算

运用分形理论中分数维的定义和方法,对金融系统的波动行为进行了描述和研究,并且对金融系统中的时间序列数据介绍了两种分数维理论计算方法.     

On the Parameter Identification Problem in the Plane and the Polar Fractal Interpolation Functions

Fractal interpolation functions provide a new means for fitting experimental data and their graphs can be used to approximate natural scenes. We first determine the conditions that a vertical scaling factor must obey to model effectively an arbitrary function. We then introduce polar fractal interpolation functions as one fractal interpolation method of a non-affine character. Thus, this method may be suitable for a wider range of applications than that of the affine case. The interpolation takes place in polar coordinates and then with an inverse non-affine transformation a simple closed curve arises as an attractor which interpolates the data in the usual plane coordinates. Finally, we prove that this attractor has the same Hausdorff dimension as the polar one.     

Adaptive reliability analysis based on a support vector machine and its application to rock engineering

The response surface method (RSM), a simple and effective approximation technique, is widely used for reliability analysis in civil engineering. However, the traditional RSM needs a considerable number of samples and is computationally intensive and time-consuming for practical engineering problems with many variables. To overcome these problems, this study proposes a new approach that samples experimental points based on the difference between the last two trial design points. This new method constructs the response surface using a support vector machine (SVM); the SVM can build complex, nonlinear relations between random variables and approximate the performance function using fewer experimental points. This approach can reduce the number of experimental points and improve the efficiency and accuracy of reliability analysis. The advantages of the proposed method were verified using four examples involving random variables with different distributions and correlation structures. The results show that this approach can obtain the design point and reliability index with fewer experimental points and better accuracy. The proposed method was also employed to assess the reliability of a numerically modeled tunnel. The results indicate that this new method is applicable to practical, complex engineering problems such as rock engineering problems.     

Connectivity evaluation of fracture networks considering the correlation between trace length and aperture

Connectivity evaluation of fracture networks is important in the design, assessment, and development of reservoirs in various engineering applications involving geothermal exploitation and the petroleum industry. By employing nonhomogeneous Poisson distribution and annealing arithmetic, this study generates fracture networks that fit well with actual outcrop fracture data. Based on Allard's definition of the connectivity index, a weighted factor is introduced, and an extended connectivity evaluation method is proposed to consider the effects of the aperture and its correlation with trace length. The results of the analysis show that the extended method improves the accuracy and reliability of connectivity evaluation compared with the traditional method. Moreover, the extended method is effective and accurate at predicting potential preferential flow paths in a practical example of the Dragon and Tiger Mountain (Jiangxi, China), and can better show the anisotropy with a change in the aperture. Hence, the proposed method extends the function of connectivity analysis and can benefit well location optimization in geothermal or petroleum exploration.