Multiple regression with correlated independent variables

Multiple linear regression analysis may be used to describe the relation of one geologic variable to a number of other (independent) variables, and also may be used to fit a trend surface to geographically distributed variables. The leastsquares estimates of the regression coefficients differ unpredictably from the true coefficients if the independent variables are correlated. The estimates can be too large in absolute value, and may have the wrong sign. Also, the least-squares solution may be unstable in that replicate samples can give widely differing values of the regression coefficients. Ridgeregression analysis is a technique for removing the effect of correlations from the regression analysis. The procedure involves addition of a small constant K to the diagonal elements of the standardized covariance matrix. The estimates obtained are biased but have smaller sums of squared deviations between the coefficients and their estimates. The ridge trace, a plot of the coefficients versus K, helps determine the value of K that stabilizes the estimates. Correlations between geologic variables are common, and regression coefficients based on these data may be suspect. In trendsurface analysis, correlations between the geographic coordinates may differ widely, and extreme correlations may be introduced if higher order terms are used in the trend. Ridgeregression analysis serves to guide the geologist to a more reliable interpretation of the results of multiple regression if the independent variables are correlated.     

Long-term analysis of rainfall-induced landslides in Umbria,central Italy

Natural Hazards - Analyses of historical records of landslides and climate variables are useful tools to search for correlations between damaging landslide events and their triggers. In this work,...     

地球化学计算中误差传递分析的应用

 通过两个实例讨论了地球化学计算中的误差传递问题,对照了四种不同方法的潜力和限制,拓广了蒙特卡洛方法在相关变量分析中的应用,指出其在不同条件下总能用有限机时逼近合理结果,揭示出协方差矩阵法的特殊效益,可相对于独立初始变量展现中间变量的相关关系。     

地球化学计算中误差传递分析的应用

通过两个实例讨论了地球化学计算中的误差传递问题,对照了四种不同方法的潜力和限制,拓广了蒙特卡洛方法在相关变量分析中的应用,指出其在不同条件下总能用有限机时逼近合理结果,揭示出协方差矩阵法的特殊效益,可相对于独立初始变量展现中间变量的相关关系。     

Some properties of principal component scores

Commonly used methods for calculating component scores are reviewed. Means, variances, and the covariance structures of the resulting sets of scores are examined both by calculations based on a large set of electron microprobe analyses of melilite (supplied by D. Velde)and by a survey of recent geological applications of principal component analysis. Most of the procedures used to project raw data into the new vector space yield uncorrelated scores. In exceptions so far encountered, correlations between scores seem to have been occasioned by the use of unstandardized variables with components calculated from a correlation matrix. In a number of cases substantive interpretations of such correlations have been proposed. A different set of correlations results for the same data if scores are computed from standardized variables and components based on the covariance matrix. If unscaled components are rotated by the varimax procedure, the result is a return to the original space. In the work reported here, nevertheless, scores calculated from varimax-rotated scaled vectors are uncorrelated.     

Applications of Error Propagation Analyses to Geochemical Calculation

Two examples are given for comparing applications and limitations of four methods which can be used to deal with error propagation in geochemical calculations.The examples indicate that the Monte Carlo method can also be employed to evaluate the effect of covariance.A special function of the method for covariance matrix shown here can reveal the correlations of middle variables relative to the independent primary variables.     

Provincialism and correlations between some properties of vitrinites

Optical and chemical properties for hand-picked samples of vitrinite from a number of British coalfields are correlated and the correlations compared with previously published data. The form of the relationship found for the properties of British vitrinites is generally similar to that exhibited by a number of other sets of analyses, but some significant differences also exist in several of the correlations. For maximum reflectance as a function of carbon content, the present data indicate that at medium and low ranks, maximum reflectance is lower than the level suggested by most previous studies. The relationships of volatile-matter yield to reflectance and carbon content suggest that at low ranks, volatile-matter yield is strongly dependent upon the nature of the coalification history of the vitrinite. Furthermore, these correlations are likely to show provincialism, in that correlations which hold for one coalfield may not be representative of the relationship in other coalfields. Similarly, major differences in relationships involving bireflectance are associated with the tectonic setting at the time of effective coalification. The data presented here indicate that for low-rank coals at least, correlations between properties of vitrinites must take account of provincialism if they are to be sufficiently reliable to be useful. The measurement of a number of rank-sensitive variables can yield additional information about rank, as compared with the use of a single rank-sensitive variable.     

A comparative study of hydraulic conductivity estimations using geostatistics

Three approaches for estimating the hydraulic conductivity (K) of the Trifa aquifer, Morocco were investigated: (1) kriging of the K values obtained from pumping tests, (2) cokriging of the pumping test data with electrical resistivity data as a secondary variable, and (3) cokriging of the pumping test data with the slope of the water table. Gauss-transformed values of the variables are used because they provide more robust variograms and transformed values of the primary and secondary variables show correlations higher than the raw values, which is beneficial in cokriging. In cokriging with electrical resistivity, two zones are considered since the geological deposits are different from the north to the south of the aquifer, which is reflected in different correlations between the variables. Comparison of the three approaches is based mainly on the estimation errors, and to a lesser degree on the cross-validations of the corresponding variogram models and general considerations, like the measurements’ reliability and aquifer make-up. The best-estimated K is given by cokriging with the slope of the water table and is therefore preferred for further use in groundwater flow modeling. Thus, electrical resistivity or the slope of the water table can both be used as secondary variables to estimate K, especially in heterogeneous aquifers with lateral variations in lithology, as is the case of the Trifa aquifer.     

Comparison of approaches to spatial estimation in a bivariate context

The problem of estimating a regionalized variable in the presence of other secondary variables is encountered in spatial investigations. Given a context in which the secondary variable is known everywhere (or can be estimated with great precision), different estimation methods are compared: regression, regression with residual simple kriging, kriging, simple kriging with a mean obtained by regression, kriging with an external drift, and cokriging. The study focuses on 19 pairs of regionalized variables from five different datasets representing different domains (geochemical, environmental, geotechnical). The methods are compared by cross-validation using the mean absolute error as criterion. For correlations between the principal and secondary variable under 0.4, similar results are obtained using kriging and cokriging, and these methods are superior slightly to the other approaches in terms of minimizing estimation error. For correlations greater than 0.4, cokriging generally performs better than other methods, with a reduction in mean absolute errors that can reach 46% when there is a high degree of correlation between the variables. Kriging with an external drift or kriging the residuals of a regression (SKR) are almost as precise as cokriging.     

Geometallurgical Modeling at Olympic Dam Mine, South Australia

Modeling of geometallurgical variables is becoming increasingly important for improved management of mineral resources. Mineral processing circuits are complex and depend on the interaction of a large number of properties of the ore feed. At the Olympic Dam mine in South Australia, plant performance variables of interest include the recovery of Cu and U₃O₈, acid consumption, net recovery, drop weight index, and bond mill work index. There are an insufficient number of pilot plant trials (841) to consider direct three-dimensional spatial modeling for the entire deposit. The more extensively sampled head grades, mineral associations, grain sizes, and mineralogy variables are modeled and used to predict plant performance. A two-stage linear regression model of the available data is developed and provides a predictive model with correlations to the plant performance variables ranging from 0.65–0.90. There are a total of 204 variables that have sufficient sampling to be considered in this regression model. After developing the relationships between the 204 input variables and the six performance variables, the input variables are simulated with sequential Gaussian simulation and used to generate models of recovery of Cu and U₃O₈, acid consumption, net recovery, drop weight index, and bond mill work index. These final models are suitable for mine and plant optimization.     

Cross-covariances and localization for EnKF in multiphase flow data assimilation

The ensemble Kalman filter has been successfully applied for data assimilation in very large models, including those in reservoir simulation and weather. Two problems become critical in a standard implementation of the ensemble Kalman filter, however, when the ensemble size is small. The first is that the ensemble approximation to cross-covariances of model and state variables to data can indicate the presence of correlations that are not real. These spurious correlations give rise to model or state variable updates in regions that should not be updated. The second problem is that the number of degrees of freedom in the ensemble is only as large as the size of the ensemble, so the assimilation of large amounts of precise, independent data is impossible. Localization of the Kalman gain is almost universal in the weather community, but applications of localization for the ensemble Kalman filter in porous media flow have been somewhat rare. It has been shown, however, that localization of updates to regions of non-zero sensitivity or regions of non-zero cross-covariance improves the performance of the EnKF when the ensemble size is small. Localization is necessary for assimilation of large amounts of independent data. The problem is to define appropriate localization functions for different types of data and different types of variables. We show that the knowledge of sensitivity alone is not sufficient for determination of the region of localization. The region depends also on the prior covariance for model variables and on the past history of data assimilation. Although the goal is to choose localization functions that are large enough to include the true region of non-zero cross-covariance, for EnKF applications, the choice of localization function needs to balance the harm done by spurious covariance resulting from small ensembles and the harm done by excluding real correlations. In this paper, we focus on the distance-based localization and provide insights for choosing suitable localization functions for data assimilation in multiphase flow problems. In practice, we conclude that it is reasonable to choose localization functions based on well patterns, that localization function should be larger than regions of non-zero sensitivity and should extend beyond a single well pattern.     

Improving the Ensemble Estimate of the Kalman Gain by Bootstrap Sampling

Using a small ensemble size in the ensemble Kalman filter methodology is efficient for updating numerical reservoir models but can result in poor updates following spurious correlations between observations and model variables. The most common approach for reducing the effect of spurious correlations on model updates is multiplication of the estimated covariance by a tapering function that eliminates all correlations beyond a prespecified distance. Distance-dependent tapering is not always appropriate, however. In this paper, we describe efficient methods for discriminating between the real and the spurious correlations in the Kalman gain matrix by using the bootstrap method to assess the confidence level of each element from the Kalman gain matrix. The new method is tested on a small linear problem, and on a water flooding reservoir history matching problem. For the water flooding example, a small ensemble size of 30 was used to compute the Kalman gain in both the screened EnKF and standard EnKF methods. The new method resulted in significantly smaller root mean squared errors of the estimated model parameters and greater variability in the final updated ensemble.     

Correlation analysis of NDVI dynamics and hydro-meteorological variables in growth period for four land use types of a water scarce area

As the characterization of primary productivity of wetland ecosystem, the Normalized Difference Vegetation Index (NDVI) plays an important role in local ecosystem conservation for environmental management. In this paper, the correlations of NDVI and hydro-meteorological variables were studied in a water scarce area with emphasis on different land use types, namely water, wetland, residential land and farmland, during the growing seasons of 1999 and 2000. The significant NDVI changes were detected between spring and summer for all land use types. The correlation analysis revealed that the NDVI-temperature correlation (P?P?P?farmland > R_wetland > R_{residential land} > R_water for NDVI and precipitation correlations (P?water > R_wetland > R_residential land > R_farmland for NDVI and temperature correlations (P?相似文献   

An evaluation of the relationship between physical/mechanical properties and mineralogy of landscape rocks as determined by hyperspectral reflectance

We investigated the relationships between mineral content and the physical and mechanical properties of landscape rock using a non-destructive remote sensing method applied in the laboratory. Using this technique, the spectral properties of the landscape rock could be collected at different wavelengths without harming the samples. Differences in spectral reflectance were compared with the physical and mechanical properties of the stone. Significant correlations were observed between reflectance values and the rocks’ mechanical and physical properties, with correlation coefficients of 95 to 99 %. However, establishing a correlation between two variables is not a sufficient condition to establish a causal relationship. Mineral densities and mineral content are characteristics used for the classification of landscape rock. We have concluded that although spectral signatures from landscape rock can be used for predicting which stones might have similar features when comparing two batches of stone, the high correlations we discovered cannot confirm a cause and effect relationship that would allow for the prediction of a rock’s physical and mechanical properties. Although this conclusion is disappointing, the mineral content and the significant correlations discovered by hyperspectral reflectance scanning can be used as supplementary information when comparing two samples of landscape rock.     

A numerical method for obtaining multiple linear regression parameters with physically realistic signs and magnitudes: Applications to the determination of equilibrium constants from solubility data

A common problem in experimental geochemistry is the derivation of equilibrium constants from solubility experiments. A simple method of deriving these equilibrium constants, multiple linear regression, often results in the appearance of negative values. This has been a significant obstacle to continuing research in this field. The problem occurs for the most part because of significant correlations among the “independent” concentration variables. These correlations are an inescapable result of the nature of the experiments and the physical model being fitted. Ridge regression is an appropriate modification to simple linear regression which overcomes this difficulty. Ridge regression results in a simple procedure to obtain physically plausible, yet statistically rigorous stability constants. Of course, other problems may further degrade the quality of derived equilibrium constants, e.g. uncertainty in activity coefficients and no purely statistical method can overcome these types of problems. However, ridge regression is an effective procedure to overcome the multicolinearity which is the main cause of negative equilibrium constants. We demonstrate the use of ridge regression with a general mathematical model and then illustrate its use in the determination of iron-chloro complex equilibrium constants from solubility studies of pyrite-pyrrhotite-magnetite in NaCl solutions at 250°C. Ridge regression may also be of use in other geochemical problems where one must estimate parameters with a physical interpretation and where the independent variables are significantly intercorrelated.     

Related information measures for the associations of earth-science variables

A basic task in earth-science data integration is to quantify variable associations. Although manv measures have been used to determine the associations between quantitative variables, the ability to quantify qualitative attributes (e.g., categorical) is limited. Moreover, most traditional association measures are restricted to linear correlations or similarities, for example, correlation coefficient. The measures proposed in this report are designed on the basis of Shannon's entropy concepts, including directional related information, ordinary related information, and partial related information. The directional related information quantifies the association of one variable in terms of another. The ordinary related information determines the mutual association of two variables. The partial related information characterizes the association of an individual stale of one variable in terms of another variable. The properties of these measures are discussed and their sample estimates are derived from both maximum likelihood and Bayesian methods. The relations between these measures are illustrated by using synthetic examples. Two applications of these measures also are developed, including the selection of variables and evaluation of mineral resources. Finally, a case study is given to demonstrate the use of the measures in mineral resources evaluation.     

Evaluation and error analysis: Kalman gain regularization versus covariance regularization

Ensemble size is critical to the efficiency and performance of the ensemble Kalman filter, but when the ensemble size is small, the Kalman gain generally cannot be well estimated. To reduce the negative effect of spurious correlations, a regularization process applied on either the covariance or the Kalman gain seems to be necessary. In this paper, we evaluate and compare the estimation errors when two regularization methods including the distance-dependent localization and the bootstrap-based screening are applied on the covariance and on the Kalman gain. The investigations were carried out through two examples: 1D linear problem without dynamics but for which the true Kalman gain can be computed and a 2D highly nonlinear reservoir fluid flow problem. The investigation resulted in three primary conclusions. First, if localizations of two covariance matrices are not consistent, the estimate of the Kalman gain will generally be poor at the observation location. The consistency condition can be difficult to apply for nonlocal observations. Second, the estimate of the Kalman gain that results from covariance regularization is generally subject to greater errors than the estimate of the Kalman gain that results from Kalman gain regularization. Third, in terms of removing spurious correlations in the estimation of spatially correlated variables, the performance of screening Kalman gain is comparable as the performance of localization methods (applied on either covariance or Kalman gain), but screening Kalman gain outperforms the localization methods in terms of generality for application, as the screening method can be used for estimating both spatially correlated and uncorrelated variables, and moreover, no assumption about the prior covariance is required for the screening method.     

Examination of the Chayes-Kruskal procedure for testing correlations between proportions

The Chayes-Kruskal procedure for testing correlations between proportions uses a linear approximation to the actual closure transformation to provide a null value,p _ij , against which an observed closed correlation coefficient,r _ij , can be tested. It has been suggested that a significant difference betweenp _ij andr _ij would indicate a nonzero covariance relationship between theith andjth open variables. In this paper, the linear approximation to the closure transformation is described in terms of a matrix equation. Examination of the solution set of this equation shows that estimation of, or even the identification of, significant nonzero open correlations is essentially impossible even if the number of variables and the sample size are large. The method of solving the matrix equation is described in the appendix.     

Empirical behavior of Chayes' null model

Chayes' t″-test for closure correlations is developed from various approximations and assumptions. The empirical behavior of this test is observed through the use of random-sampling number matrices of order to thirteen. The experiments demonstrate a lower limit for the reliability of Chayes' model of about 91/100. That is, about one chance in ten remains that an error due to the statistical testing is committed on some product-moment correlations in a given matrix of correlation, given a confidence level of 90 percent. With increase in the number of variables, the test for departure from zero correlation can be used provided the sample size remains small.     

Minres factor analysis of hypersthene data, Strathcona Mine (Ontario, Canada)

This is the first application of minimum residuals (minres),a type of factor analysis, in the study of hypersthene minerals from a mafic norite formation at the Strathcona Mine near Sudbury, Ontario. Minres, because it yields highest communalities for some variables, is preferred to other types of factoring solutions including a common factor model with Chayes' null correlations as factor input. Oblique rotation of factors is rejected as a model for statistical and geochemical reasons. A five oxide-variable model that reasonably well determines hypersthene is reduced by minres to a two-factor model which is statistically significant. Because of the small number of variables in the analysis, it is difficult to interpret the isolated factors in terms of specific geologic processes. The factors, however, even if surrogate, are linked with substitution phenomena in the hypersthene.