Nonlinear regression for dependent variables

Nonlinear regression methods can be used to fit functions for two related variables where both variables are subject to error. A computer program for nonlinear estimation described previously has been modified to fit such functions for a given set of data. A numerical example is provided for a second-degree equation in xand y.A closer fit to an observed set of data is possible if the error structure for the variables is specified.     

Efficiency of regression techniques for ore valuation at Kolar Gold Fields, India

The Kolar Gold Fields are some of the best known gold deposits in India. An example of ore valuation utilizing 49 ore blocks of the Oriental lode of the West Reefs, explored and developed in the Nundydroog mines, is given. In this reef system, there are large ore reserves of sulfidebearing quartz reefs, and the gold distribution is erratic both along strike and downdip. Ore valuation at present is based on the arithmetic mean of samples taken at peripheral positions of the blocks. Samples taken from internal portions of the blocks give a totally different picture of the value. To correct this discrepancy, normal regression and lognormal regression of internal block and total block values, over peripheral block values have been used to evaluate the deposits. The valuation efficiency criterion shows the logarithmic variance for distribution of ratios of unregressed and regressed block values with the corresponding arithmetic mean of internal stope values as observed inside the blocks. The studies have shown that the logarithmic variance is minimum if the logarithmic regression is used, thereby indicating maximum efficiency. Further, the undervaluation and overvaluation of low- and high-grade blocks is less for the logarithmic example. With help of the logarithmic regression equation an effective pay limit of 177.8 in.-dwt has been found for selective mining, for peripheral block values corresponding to the official pay limit of 240 in.-dwt.NGRI contribution number 71-281.     

Efficient reliability method for implicit limit state surface with correlated non‐Gaussian variables

In contrast to the traditional approach that computes the reliability index in the uncorrelated standard normal space (u‐space), the reliability analysis that is simply realized in the original space (x‐space, non‐Gaussian type) would be more efficient for practical use, for example, with the Low and Tang's constrained optimization approach. On the other hand, a variant of Hasofer, Lind, Rackwits and Fiessler algorithm for first‐order reliability method is derived in this paper. Also, the new algorithm is simply formulated in x‐space and requires neither transformation of the random variables nor optimization tools. The algorithm is particularly useful for reliability analysis involving correlated non‐Gaussian random variables subjected to implicit limit state function. The algorithm is first verified using a simple example with closed‐form solution. With the aid of numerical differentiation analysis in x‐space, it is then illustrated for a strut with complex support and for an earth slope with multiple failure modes, both cases involving implicit limit state surfaces. Copyright © 2015 John Wiley & Sons, Ltd.     

初始地应力场多方程回归分析

在初始地应力场分析问题中,采用了多方程回归方法。该模型同时考虑了一组变量(包括自变量与因变量)之间的所有关系。在此基础上,采用逐步回归法进行计算,进一步提高了计算精度。由此,得到了回归效果更显著的初始地应力场。     

Statistical analysis of tectonic patterns in areal volcanism: The bunyaruguru volcanic field in West Uganda

A quantitative analysis was made of the spatial arrangement of 149explosion craters in the western rift of Uganda. A variety of methods demonstrate that the spatial pattern of the craters reveals significant structural patterns that have guided volcanism to the surface. It is shown that the east-west elements in the field affected location, and the main rift fault is resolved into two main components. Tentatively, a possible dextral transform fault is identified that affected the relative location of the two main zones of activity. Grouping techniques demonstrate that crater groups obey an exponential rank-size rule and allow a mapping of the craters into energy classes that reveals a concentric pattern of energy in the field. The effect of the topography on energy levels and crater size show that only topography greater than 11,000ft could have prevented all eruptive activity, but the smaller energies and craters are sensitive to height differences on the order of the height of the rift wall, about 1000ft. Total energy in each crater class size is roughly constant, and the field energy could create one or two single craters comparable in size to small central volcanoes.     

主成分-逐步回归在滑坡稳定性分析中的应用

多元回归分析是概率统计理论滑坡稳定性研究中的一种重要应用。然而,一般多元回归方法存在局限性,本文引入主成分-逐步回归模型,既尽可能全面考虑到各影响因素,又能解决因素之间多重相关性可能带来的误差。在理论探讨的基础上,本文结合巴东县赵树岭滑坡研究进行了工程实例分析。     

多元线性回归和逐步回归分析在白石泉Cu-Ni硫化物矿床研究中的应用

新疆东天山的白石泉Cu-Ni硫化物矿床一直受到众多构造学家、岩石学家和矿床学家的关注,但是数学地质在此地的研究尚属空白。文中在系统的野外地质调查基础上,进行了系统采样和岩石地球化学分析,应用多元线性回归和逐步回归分析相结合的方法,借助SPSS数学统计软件系统的研究Cu-Ni变量与造岩元素变量的关系,建立Cu-Ni变量与其他变量的回归方程,验证与解释白石泉Cu-Ni硫化物矿床的成因、岩浆来源等问题,对指导找矿勘探有重要作用,对应用数学地质方法有指引作用。     

Stochastic response surface method for reliability analysis of rock slopes involving correlated non-normal variables

This paper proposes a stochastic response surface method for reliability analysis involving correlated non-normal random variables, in which the Nataf transformation is adopted to effectively transform the correlated non-normal variables into independent standard normal variables. Transformations of random variables that are often used in reliability analyses in terms of standard normal variables are summarized. The closed-form expressions for fourth to sixth order Hermite polynomial chaos expansions involving any number of random variables are formulated. The proposed method will substantially extend the application of stochastic response surface method for reliability problems. An example of reliability analysis of rock slope stability with plane failure is presented to demonstrate the validity and capability of the proposed stochastic response surface method. The results indicate that the proposed stochastic response surface method can evaluate the reliability of rock slope stability involving correlated non-normal variables accurately and efficiently. Its accuracy is shown to be higher than that for the first-order reliability method, and it is much more efficient than direct Monte-Carlo simulation. The results also show that the number of collocation points selected should ensure that the Hermite polynomial matrix has a full rank so that different order SRSMs can produce a robust estimation of probability of failure for a specified performance function. Generally, the accuracy of SRSM increases as the order of SRSM increases.     

Efficiency of local polynomials in contour mapping

On the basis of samples taken from a known topographic surface, the parameters of two types of linear models are estimated. The first category is defined by polynomials or trigonometric functions, whose parameters are simultaneously computed from available data. In the second category a set of local centers is defined, and in the neighborhood of each center a fixed-degree polynomial is developed. An approximative resemblance index is calculated, and contour maps corresponding to various models are compared with the topographic map. It is found that with an increasing number of grid points, maps of local polynomials are converging both in continuity and in resemblance. For a sufficient number of grid points, this resemblance is always higher than those produced by models of the first category.     

Trend removal in spatially correlated datasets

Typically, datasets originated from mining exploration sites, industrially polluted and hazardous waste sites are correlated spatially over the region under investigation. Ordinary kriging (OK) is a well-established geostatistical tool used for predicting variables, such as precious metal contents, biomass, species counts, and environmental pollutants at unsampled spatial locations based on data collected from the neighboring sampled locations at these sites. One of the assumptions required to perform OK is that the mean of the characteristic of concern is constant for the entire region under consideration (e.g., there is no spatial trend present in the contaminant distribution across the site). This assumption may be violated by dalasets obtained from environmental applications. The occurrence of spatial trend in a dataset collected from a polluted site is an indication of the presence of two or more statistical populations (strata) with significantly different mean concentrations. Use of OK in these situations can result in inaccurate kriging estimates with higher SDs which, in turn, can lead to incorrect decisions regarding all subsequent environmental monitoring and remediation activities. A univariate and a multivariate approach have been described to identify spatial trend that may be present at the site. The trend then is removed by subtracting the respective means from the corresponding populations. The results of OK before and after trend removal are being compared. Using a real dataset, it is shown that standard deviations (SDs) of the kriging estimates obtained after trend removal are uniformly smaller than the corresponding SDs of the estimates obtained without the trend removal.     

煤与瓦斯突出预报的理想灰贴近函数聚类关联分析

从预报煤与瓦斯突出所获取的数据信息知，其信息多数为灰色特征。为突出其灰色特性，本文采用理想灰贴近函数聚类关联分析预测预报煤与瓦斯突出的危险程度，并将此方法同模糊聚类预测预报法和模糊聚类关联分析预报法及模糊聚类相似分析预报法进行了对比分析。由对比分析可知，采用4种数学方法预测预报煤与瓦斯是否突出，给出的结论是一致的，只是预测预报是否发生的可能程度略有差别。其中关联分析比相似分析计算过程较简单，且相似分析在确定编号时，有时受截值λ取值影响。尽管如此，但不影响预测预报结果。所以，不论是采用模糊聚类，还是灰色聚类数学方法预报煤与瓦斯突出均是可行的，可在实践中应用。     

Application of logistic regression model for slope instability prediction in Cuyahoga River Watershed,Ohio, USA

High incidences of slope movement are observed throughout Cuyahoga River watershed in northeast Ohio, USA. The major type of slope failure involves rotational movement in steep stream walls where erosion of the banks creates over-steepened slopes. The occurrence of landslides in the area depends on a complex interaction of natural as well as human induced factors, including: rock and soil strength, slope geometry, permeability, precipitation, presence of old landslides, proximity to streams and flood-prone areas, land use patterns, excavation of lower slopes and/or increasing the load on upper slopes, alteration of surface and subsurface drainage. These factors were used to evaluate the landslide-induced hazard in Cuyahoga River watershed using logistic regression analysis, and a landslide susceptibility map was produced in ArcGIS. The map classified land into four categories of landslide susceptibility: low, moderate, high, and very high. The susceptibility map was validated using known landslide locations within the watershed area. The landslide susceptibility map produced by the logistic regression model can be efficiently used to monitor potential landslide-related problems, and, in turn, can help to reduce hazards associated with landslides.     

Mineral variation in anorthositic, troctolitic, and adamellitic rocks of the Barth Island layered structure in the Nain anorthosite complex, Labrador

The Barth Island layered structure is an oval, 6 by 9 km body, consisting of rhythmically layered adamellitic rock in the center which grades outward through jotunite into troctolite. Farther outward the sequence repeats itself in reversed order, strongly reduced in magnitude and finer grained; the adamellitic zone is followed by jotunite which grades into coarse-grained leuconorite and into anorthosite of the Nain complex. The Barth Island structure, having an inverted conical base topped by a hemispherical depression, seems to represent a distorted sequence of rock layers with troctolite at the bottom, grading upward into adamellitic rocks which grade into anorthosite at the top. Trend-surface analysis demonstrates the regional variation of plagioclase and orthopyroxene compositions in the troctolite—adamellite sequence of the central part of the structure. The fits for the second- and third-degree surfaces are good and significant at the 99 percent level. The regression line for compositional variation in coexisting plagioclase and orthopyroxene in all analyzed rocks has a correlation coefficient of r² = 0.78. The difference between the trends in the troctolite—adamellite sequence and the anorthosite—adamellite sequence is insignificant. The regression curve for compositional variation in coexisting orthopyroxene and olivine has a correlation coefficient of r² = 0.98. The curve shows good correlation with the experimentally established partitioning curve of Medaris, which indicates that equilibrium conditions prevailed during formation of the olivine—ortho-pyroxene pairs. The results suggest that the troctolite—adamellite sequence and the anorthosite—adamellite sequence are products of fractional crystallization, possibly from the same parental magma.     

Spline-surface interpolation, spatial filtering, and trend surfaces for geological mapped variables

Spline surfaces are interpolated for top of the Dundee Limestone of the central Michigan Basin, USA. The requirement of gridded data render spline functions inappropriate tools for representing many types of geological mapped data. Comparisons are drawn with maps for the same Michigan data based on trend surfaces and spatial filtering.     

On the misuse of regression in earth science

Simple least-squares regression analysis is applied to almost all empirical curve-fitting problems in earth science (and related fields). Its use, however, should be restricted to predictive situations. For comparisons with theory or among fitted lines, the related technique termed functional analysisshould be employed. To apply this method, the ratio of the random components of the variances of the variables must be estimated. Principles are illustrated with examples from geomorphometry, especially the stream frequency-drainage density relation.     

滑坡预报的多元回归分析方法

在回顾了滑坡预报的概念、类型及已有滑坡预报的几种主要模型的基础上，利用多元回归分析方法，根据最小二乘法原理，建立了非线性回归预报模型，提出滑坡破坏的时间为位移．时间曲线的拐点，此方法一般适合于临滑预报。结合实例，利用黄龙西村滑坡位移．时间监测资料，采用该模型进行滑坡时间预报，和其他滑坡预报模型预报的结果接近，说明该模型具有一定的合理性。     

机器学习及其相关算法综述

机器学习已经在滑坡易发性评价中大量应用且取得了较好的表现,但在进行大区域评价时,仍存在数据库样本需求量大,算力要求高;影响因素分级机械化,未考虑其与滑坡机理的相关性等情况。为减少数据库样本需求,本文提出了构建包含3种坡体状态的滑坡的数据库：已经发生过失稳的坡体、正处于失稳状态的坡体、失稳概率小的坡体,该数据库可以在数值上划分出临界值,便于模型更准确地识别滑坡,较大幅度地减小了数据量。针对影响因素分级机械化的问题,提出了基于频数分布图、累计曲线及其导数图的数理统计方式,更精细地描述因数与滑坡易发性的关系。以新疆滑坡灾害为例,验证了“包含3种坡体状态的数据库”与“基于数理统计的描述方法”的适用性,获得了新疆滑坡灾害易发性分区图。结果表明与传统数据库对比,在不明显改变精度的前提下,减少了90%以上的样本量;基于数理统计的描述方法可以绘制出更加细致的滑坡危险性分区图;活动性断裂和地形起伏度对新疆滑坡易发性起到重要的控制作用。     

On the resolution of mixed frequency distributions into normal components

The resolution of mixed frequency distributions into normal components may be accomplished graphically on probability paper. By means of nonlinear regression, a method for such a resolution in the probability net is given and exemplified. It is based on the observed cumulative frequencies.     

鹤岗煤田新华勘探区地质构造定量分析

鹤岗煤田新华勘探区地质构造复杂,断层较为发育,对煤层赋存形态和煤层厚度产生了较强烈的改变运动。通过收集钻孔资料及断层数据,对该区断层数据开展数理统计与断层密度等值线分析,并对矿区30煤底板标高进行四次趋势面计算,以此研究断层构造发育规律及对该区煤炭开采造成的影响。研究表明：区内断层走向为NE、NW和近EW向,以北西向为主;断层密度等值线表明矿区内存在三个密闭的等值线圈;趋势面分析表明区中存在三处异常带且呈NW向展布,主要分布在矿区西北部的F2、F3,东北部的F25,F34,南部F10、F14。、F27断层处;30煤的煤层在矿区中部较厚,在东南处和西北处煤层较薄,且西北处煤层厚度变化较为急剧,东南处煤层厚度变化较平稳。