A Note on Robust Estimates

It is well known that the mean can be an unsatisfactory estimator of a true value if data are a poor approximation to a normal distribution or are too few in number for a normal distribution to be adequately delineated. Unfortunately, this knowledge is not always implemented in the evaluation of analytical data. The median may be used occasionally because it is recognised to be less affected by discrepent data. However, there are other 'robust' estimators which can be better than the simple median. This note complements a previous paper by the writer (1).     

A review of lognormal estimators forin situ reserves

The term lognormal kriging does not correspond to a single well defined estimator. In fact, several types of lognormal estimators forin situ reserves are available, and this may cause confusion. These estimators are based on different assumptions—that is, different models. This paper presents a review of these estimators.     

A distribution-free alternative to least-squares regression and its application to Rb/Sr isochron calculations

A distribution-free estimator of the slope of a regression line is introduced. This estimator is designated S_m and is given by the median of the set of n(n – 1)/2 slope estimators, which may be calculated by inserting pairs of points (X _i, Y_i)and (X _j, Y_j)into the slope formula S _i = (Y _i – Y_j)/(X _i – X_j),1 i < j n Once S _m is determined, outliers may be detected by calculating the residuals given by R_i = Y_i – S_mX_i where 1 i n, and chosing the median R_m. Outliers are defined as points for which |R_i – R_m| > k (median {|R _i – R_m|}). If no outliers are found, the Y-intercept is given by R_m. Confidence limits on R_m and S_m can be found from the sets of R_i and S_i, respectively. The distribution-free estimators are compared with the least-squares estimators now in use by utilizing published data. Differences between the least-squares and distribution-free estimates are discussed, as are the drawbacks of the distribution-free techniques.     

多参数统计定值模式及其在标样定值中的应用

本文用多参数统计定值的新模式进行标样定值。对离群值的处理提出了新方案。采用11种统计参数对原始数据进行处理,并将这11种参数的算术平均值、中位值、几何平均值、选择平均值、Hampel M估计值和主族众数的平均值作为最佳定值。在NCR TOWER-1632多用户高档微机UNIX操作系统下开发了多参数模式定值软件MPDPS,对6个新研制的标样进行了定值。     

On solution of the inverse problem for confined aquifer flow via maximum likelihood

Joint estimation of transmissivity (T) and storativity (S) in a confined aquifer is done via maximum likelihood (ML). The differential equation of groundwater flow is discretized by the finite-element method, leading to equation _t+x _t=u _t. Elements of matrices and , as well as estimated covariance matrix of noise termu _t, are functions of T and S. By minimizing the negative loglikelihood function corresponding to discretized groundwater flow equation with respect to T and S, ML estimators are obtained. The ML approach is found to yield accurate estimates of T and S (within 9 and 10% of their actual values, respectively) and showed quadratic convergence in Newton's search technique. Prediction of aquifer response, using ML estimators, results in estimated piezometric heads accurate to ±0.5 m from their actual, exact values. Statistical properties of ML estimators are derived and some basic results for statistical inference are given.     

Recursive parameter estimation with applications in earth sciences

Parameter estimation has become increasingly interesting the last few decades for a variety of engineering topics. In such situations, one may face problems like (a) how to estimate parameters for which erroneous measurements are available (direct estimation), or (b) how to estimate coefficients of some process model governing a geological phenomenon when these coefficients are inaccessible or difficult to access by direct investigation (inverse estimation or identification). Both these problems are examined in this presentation from a modern stochastic viewpoint, where parameters sought are interpretated mathematically as random functions, generated and estimated in space or time with the aid of recursive models. Advantages of this methodology are remarkable, from both theoretical and physical points of view, as compared to conventional statistics or nonrecursive estimators. Particularly it may offer more accurate estimators, better representation of spatial variation, and a means of overcoming difficulties such as excessive computational time or computer storage. To test effectiveness of this type of estimation, a series of representative case studies from geotechnical practice have been computed in detail.     

Quality control in geochemistry from a comparison of four central tendency and five dispersion estimators and example of a geochemical reference material

Data quality control in geochemistry constitutes a fundamental problem that is still to be solved from the application of statistics and computation. We used refined Monte Carlo simulations of 10,000 replications and 190 independent experiments for sample sizes of 5 to 100. Statistical contaminations of 1 to 4 observations were used to compare 9 statistical parameters (4 central tendency—mean, median, trimean, and Gastwirth mean, and 5 dispersion estimates—standard deviation, median absolute deviation, S _n , Q _n , and \( {\widehat{\sigma}}_n \)). The presence of discordant observations in the data arrays rendered the outlier-based and robust parameters to disagree with each other. However, when the mean and standard deviation (outlier-based parameters) were estimated from censored data arrays obtained after the identification and separation of outlying observations, they generally provided a better estimate of the population than the robust estimates obtained from the original data arrays. This inference is contrary to the general belief, and therefore, reasons for the better performance of the outlier-based methods as compared to the robust methods are suggested. However, when all parameters were estimated from censored arrays and appropriate precise and accurate correction factors put forth in this work were applied, all of them became fully consistent, i.e., the mean agreed with the median, trimean and Gastwirth mean, and the standard deviation with the median absolute deviation, S _n , Q _n , and \( {\widehat{\sigma}}_n \). An example of inter-laboratory chemical data for a Hawaiian reference material BHVO-1 included sample sizes from 5 to 100, which showed that small samples of up to 20 provide inconsistent estimates, whereas larger samples of 20–100, especially >40, were more appropriate for estimating statistical parameters through robust or outlier-based methods. Although all statistical estimators provided consistent results, our simulation study shows that it is better to use the censored sample mean and population standard deviation as the best estimates.     

通用同位素标准化程序及在Sr、Nd同位素分析上的应用

固体元素的同位素在热表面电离质谱计上分析时,会产生同位素分馏,使得同位素比值测量的重复性和测量精度都受到局限。而在同位素地质学中,有时所遇到的样品之间的同位素比值差别很小,以致被测量过程中的同位素分馏所掩盖,使同位素地质方法的应用受到限制。     

Estimation of Areal Frequency and Mean Trace Length of Discontinuities Observed in Non-Planar Surfaces

Summary. Mauldon (1998) suggested end-point estimators of areal frequency and mean trace length for a planar sampling window which were recently proved to be unbiased maximum likelihood estimators by Lyman (2003). The present paper is to expand the concept and applicability of the end-point estimators to those for a general non-planar sampling window. The generalized end-point estimators are verified and its applicability for variable discontinuity orientation is checked by Monte Carlo simulation. Standard deviation of estimation error and estimation efficiency of areal frequency and mean trace length are also considered.     

Power spectrum analysis of geomagnetic indices

Using the method of maximum entropy spectral analysis (MESA) by Burg and the least-squares linear prediction (lslp) (also calledFABNE) by Barrodale and Erickson, the spectra of geomagnetic indices Ap, AN, AS, AE, AU, AL, Dst and cosmic ray neutron intensity at Deep River are obtained for 1965 and 1969 from daily means and for longer periods from monthly means. A large number of peaks of periodicities from 2 days to several years is obtained, many of which are shown by all the indices. Some of these are probably harmonics of the 27 days solar rotation period and the 20–22 years double sunspot cycle period. Comparison is made with results of earlier workers who reported fewer peaks.     

A robust-symmetric mean: A new way of mean calculation for environmental data

Environmental data seldom follow normal distributions, so how to calculate their means is a very important problem. Commonly used methods for mean calculation, such as arithmetic mean, geometric mean, and median, were evaluated. Arithmetic means should only be used when datasets follow normal distributions. Geometric means are suitable for datasets which follow log-normal distributions. Medians are a kind of robust treatment. However, their efficiency is very low. Based on the methods described, two new ideas are developed: robust and symmetric, for calculating means. As far as the symmetric feature is concerned, Box-Cox power transformation is better than logarithmic transformation. Robust statistics and Box-Cox transformation are combined to produce the robust-symmetric mean. As environmental data often follow log-normal or skewed distributions, this method is better than the previous ones and also is appropriate.     

The influence of materials on the breaching process of natural dams

In this study, a series of natural dam overtopping laboratory tests are reported. In these tests, the effect of seven different sediment mixtures on the breaching process was investigated. According to the test results, three stages of the breaching process of natural dams made of different materials were observed. Backward erosion was the primary cause for the incising slopes. The effects of backward erosion became stronger with the larger fines contents of the materials. With an increase in the median diameter (d ₅₀) of particles, the breaching time became longer. However, the peak discharge became smaller. With an increase in the fines contents (p), the median diameter of the particles and the void ratio were changed, which resulted in a decrease in the breaching time and an increase in the peak discharge. The breaching time and peak discharge were more sensitive to the median diameter than to the fines contents. The relation between breach width and depth was found to follow a logistic function \( W\kern0.5em =\kern0.5em \frac{\zeta }{1\kern0.5em +\kern0.5em {e}^{\left(-k\left(D\kern0.5em -\kern0.5em {D}_0\right)\right)}} \). The parameters ζ, k, and D ₀ are defined by a linear relationship with the median diameter and fines content. A breach of the side slope occurred as a tensile failure when the fines contents of the materials were large; otherwise, shear failure occurred. Furthermore, when the materials had fewer fines contents, the volume of the collapsed breach side slope became larger.     

Summary statistics in geochemistry: A study of the performance of robust estimates

Numerical data summaries in many geochemical papers rely on arithmetic means, with or without standard deviations. Yet the mean is the worst average (estimate of location) for those extremely common geochemical data sets which are non-normally distributed or include outliers. The widely used geometric mean, although allowing for skewed distributions, is equally susceptible to outliers. The superior performance of 19 robust estimates of location (simple median, plus various combined, adaptive, trimmed, and skipped,L, M, andW estimates) is illustrated using real geochemical data sets varying in sources of error (pure analytical error to multicomponent geological variability), modality (unimodal to polymodal), size (20 to >2000 data values), and continuity (continuous to truncated in either or both tails). The arithmetic mean tends to overestimate location of many geochemical data sets because of positive skew and large outliers; robust estimates yield consistent smaller averages, although some (e.g., Hampel's and Andrew's) do perform better than others (e.g., Shorth mean, dominant cluster mode). Recommended values for international standard rocks, and for such important geochemical concepts as average chondrite, can be reproduced far more simply via robust estimation on complete interlaboratory data sets than via the rather complicated and subjective methods (e.g., laboratory ratings) so far used in the literature. Robust estimates also seem generally less affected by truncation than the mean; for example, if values below machine detection limits are alternatively treated as missing values or as real values of zero, similar averages are obtained. The standard (and mean) deviations yield consistently larger values of scale for many geochemical data sets than the hinge width (interquartile range) or median absolute deviation from the median. Therefore, summaries of geochemical data should always include at least the simple median and hinge width, to complement the often misleading mean and standard deviation.     

Spatial Breakdown Point of Variogram Estimators

In the context of robust statistics, the breakdown point of an estimator is an important feature of reliability. It measures the highest fraction of contamination in the data that an estimator can support before being destroyed. In geostatistics, variogram estimators are based on measurements taken at various spatial locations. The classical notion of breakdown point needs to be extended to a spatial one, depending on the construction of most unfavorable configurations of perturbation. Explicit upper and lower bounds are available for the spatial breakdown point in the regular unidimensional case. The difficulties arising in the multidimensional case are presented on an easy example in IR² , as well as some simulations on irregular grids. In order to study the global effects of perturbations on variogram estimators, further simulations are carried out on data located on a regular or irregular bidimensional grid. Results show that if variogram estimation is performed with a 50% classical breakdown point scale estimator, the number of initial data likely to be contaminated before destruction of the estimator is roughly 30% on average. Theoretical results confirm the previous statement on data in IR^d , d 1.     

Evaluation and comparison of spatial interpolators

This study evaluates 15 different estimators to determine their relative merits in estimating block concentrations at contaminant waste sites. The evaluation was based on 54 subsets of data drawn from an exhaustive set of 19,800 data. For each subset, 198 block estimates were made with each estimator. The measurements of estimation quality were a linear loss function and a more standard statistic, the mean square error. The linear loss function showed that seven of the estimators produced scores close enough to be within the same statistical population. Results based on the mean square error were similar. The surprising results of this study were that inverse distance and inverse distance squared both produced better scores than kriging.     

A review of lognormal estimators forin situ reserves

The term “lognormal kriging” does not correspond to a single well defined estimator. In fact, several types of lognormal estimators forin situ reserves are available, and this may cause confusion. These estimators are based on different assumptions—that is, different models. This paper presents a review of these estimators.     

PRAGMATISM vs RIGOUR: A Debate on the Resolution Of Disparate Analytical Data on Four Canadian Iron - Formation Reference Samples

The analytical data received in an inter-laboratory study of four Canadian reference samples were treated by two different methods in order to arrive at "usable values" for the concentration of each constituent. The first method ("Select Laboratories"), proposed by S. Abbey (S.A.), is based on the fact that some laboratories consistently produce better results than others. S.A. suggests an approach which permits isolation of results from such laboratories, calculation of a mean or a median of such results and use of one or the other as the concentration of a constituent. That approach involves a degree of subjectivity and S.A. attempts to demonstrate the validity of his approach by means of several tests applied to the results obtained for the four samples.     

Seismogeological and hydrological criteria for the New European Macroseismic Scale (MSK-92)

In 1988, an ESC Working Group Macroseismic Scales started upgrading the MSK-81 Intensity Scale. This paper presents the background and decisions made with respect to the so-called seismogeological effects. Discussion has pointed out that they cannot be treated and used in the same way as the effects on humans, objects and buildings, for many reasons. Therefore, the WG adopted the solution of using such effects as a side tool for intensity assessment, providing a comprehensive table where the experimental relations between seismogeological effects and intensity degrees - assessed by means of other effects - are presented.     

On the equivalence of kriging and maximum entropy estimators

This study compares kriging and maximum entropy estimators for spatial estimation and monitoring network design. For second-order stationary random fields (a subset of Gaussian fields) the estimators and their associated interpolation error variances are identical. Simple lognormal kriging differs from the lognormal maximum entropy estimator, however, in both mathematical formulation and estimation error variances. Two numerical examples are described that compare the two estimators. Simple lognormal kriging yields systematically higher estimates and smoother interpolation surfaces compared to those produced by the lognormal maximum entropy estimator. The second empirical comparison applies kriging and entropy-based models to the problem of optimizing groundwater monitoring network design, using six alternative objective functions. The maximum entropy-based sampling design approach is shown to be the more computationally efficient of the two.