STATISTICAL METHODS IN ROCK MAGNETISM

Summary. This paper describes the statistical techniques available to the experimenter in palaeomagnetic work. The theory of these methods is based on an assumed probability distribution of errors. It is shown that the mathematical requirements of this distribution are obeyed by the observations from rock samples which are known to possess a stable magnetization; observations on rocks with unstable magnetization however do not conform to it. A theoretical derivation is given for this probability distribution.
The problem of estimating the mean direction of magnetization of a geological formation has in recent years become a matter of the greatest geophysical interest since it is from such estimates that the position of the pole of the Earth in past geological ages is determined. This problem is largely one of the judicious choice of samples and a procedure is suggested whereby such estimates may be achieved with the greatest sample economy.     

STANDARD ERRORS IN THE EIGENVALUES OF A CROSS-PRODUCT MATRIX:THEORY AND APPLICATIONS

New expressions are derived for the standard errors in the eigenvalues of a cross-product matrix by themethod of error propagation.Cross-product matrices frequently arise in multivariate data analysis,especially in principal component analysis (PCA).The derived standard errors account for the variabilityin the data as a result of measurement noise and are therefore essentially different from the standarderrors developed in multivariate statistics.Those standard errors were derived in order to account for thefinite number of observations on a fixed number of variables,the so-called sampling error.They can beused for making inferences about the population eigenvalues.Making inferences about the populationeigenvalues is often not the purposes of PCA in physical sciences,This is particularly true if themeasurements are performed on an analytical instrument that produces two-dimensional arrays for onechemical sample:the rows and columns of such a data matrix cannot be identified with observations onvariables at all.However,PCA can still be used as a general data reduction technique,but now the effectof measurement noise on the standard errors in the eigenvalues has to be considered.The consequencesfor significance testing of the eigenvalues as well as the usefulness for error estimates for scores andloadings of PCA,multiple linear regression (MLR) and the generalized rank annihilation method(GRAM) are discussed.The adequacy of the derived expressions is tested by Monte Carlo simulations.     

空间多维经济统计数据的降维方法——以四川省经济统计数据为例

经济统计信息往往包含多维属性,需要采用降维方法将多维信息转换到三维以内的空间来实现多维信息可视化,这有助于研究其内在空间分布规律。在评价线性方法 (PCA)、非线性方法 (NLM和SOFM),以及监督分类方法 (SVM)等四种降维方法的基础上,以2007年四川省区县尺度为研究单元,运用不同分类方法针对区县社会经济发展现状进行聚类(分类)处理,并对成果的差异性展开了深入讨论,主要结论如下:PCA虽然能在整体上揭示经济发展趋势,但结果与实际情况差异较大;NLM能很好地展现出四川经济发展的区域态势和核心区域,准确反映了四川经济发展现状;SOFM的分类结果与发展现状较吻合,但局部地区存在一定的错分情况,且不能进行类内目标的比较;SVM是监督分类,需要已知样本来训练分类过程,在样本的选择上存在较大的主观性,且最优参数的搜索过程较为复杂。本文对几种降维方法的比较,并在经济统计领域中的应用,可以为相关的空间多维信息降维研究提供参考。     

Slump folds as estimators of palaeoslope: a case study from the Fisherstreet Slump of County Clare, Ireland

Folds and faults preserved within slump horizons are thought to be the only sedimentary structures that primarily reflect palaeoslope direction. By evaluating methods for the analysis of slump folds, the robustness of many palaeoslope and palaeogeographical reconstructions can be tested. Five methods for estimating palaeoslope direction from slump fold orientation data are tested with reference to the Upper Carboniferous Fisherstreet Slump horizon of County Clare, Ireland. These methods are the Mean Axis Method, the Separation Arc Method, the Downslope Average axis Method, the Axial–planar Intersection Method and the Fold Hinge Azimuth and Interlimb Angle Method. Transport determination techniques have mixed success when applied to this example and reveal a mean downslope transport direction of 067°. This result reflects the varied boundary conditions that are possible during slumping, resulting from variations in initial fold geometries, progressive fold formation, spatial and temporal changes in velocity, material properties, pore fluid pressures and the local slope. Without a comprehensive analysis of slump data – using all available techniques – large errors in interpreted transport direction can exist and may invalidate palaeogeographic reconstructions. It is essential to use as many transport determination techniques as possible to determine slip direction accurately. With this in mind, a general procedure for determining palaeoslope directions is suggested and used to examine existing palaeogeographic models for the Clare Basin.     

荒漠植物枝系构型影响因素分析

荒漠植物的枝系构型的影响因素包括各级分枝角度、各级分枝长度、枝径比、逐步分枝率和总体分枝率、树冠分维数、分枝分维数和计盒维数等17个指标。采用主成分分析（PCA）方法对50种典型荒漠植物的17个构型指标进行分析,结果表明：第一主成分的贡献率十分高,达到77.0％,其中2级分枝长度和3级分枝长度是影响枝系构件的主要因素;第二主成分的贡献率只有11.3％,主要综合了3级分枝角度、2级分枝角度和1级分枝角度三个因素。对枝系构型因素进行PCA排序可将其划分为三类：第一类包括各级分枝长度;第二类包括逐步分枝率和分形维数;第三类包括各级分枝角度和枝径比等。通过第一和第二主成分分析体现了荒漠植物的两种不同的生态适应机制,即在分枝长度上和分形维数上,趋于增加和减小两种趋势,但在分枝角度方面,都有趋于增大的趋向。     

Attribute error and sensitivity analysis of map operations in geographical informations systems: suitability analysis

The theory and methods for attribute error and sensitivity analysis associated with map-based suitability analysis are developed. In particular, this paper delineates the underlying types of sensitivities for suitability analysis and derives ways to measure these sensitivities. Additionally, it shows how to undertake geographical sensitivity analyses given a generic geographical information system. The uses of these methods include understanding the relationship of the attribute errors in the output map generated by errors in the input maps for a given geographical analysis. This information provides a means to assess the quality and reliability of conclusions inferred from the output map created by such an analysis.     

A BOOTSTRAP RESAMPLING SCHEME FOR USING THE CANONICAL CORRELATION TECHNIQUE IN RANK ESTIMATION

Rank estimation by canonical correlation analysis in multivariate statistics has been proposed as analternative approach for estimating the number of components in a multicomponent mixture.Amethodological turning point of this new approach is that it focuses on the difference in structure ratherthan in magnitude in characterizing the difference between the signal and the noise.This structuraldifference is quantified through the analysis of canonical correlation,which is a well-established datareduction technique in multivariate statistics.Unfortunately,there is a price to be paid for having thisstructural difference:at least two replicate data matrices are needed to carry out the analysis.In this paper we continue to explore the potential and to extend the scope of the canonical correlationtechnique.In particular,we propose a bootstrap resampling method which makes it possible to performthe canonical correlation analysis on a single data matrix.Since a robust estimator is introduced to makeinference about the rank,the procedure may be applied to a wide range of data without any restrictionon the noise distribution.Results from real as well as simulated mixture samples indicate that when usedin conjunction with this resampling method,canonical correlation analysis of a single data matrix isequally efficient as of replicate data matrices.     

ROBUST PRINCIPAL COMPONENT ANALYSIS BY PROJECTION PURSUIT

Principal component analysis (PCA) is a widely used technique in chemometrics.The classical PCAmethod is,unfortunately,non-robust,since the variance is adopted as the objective function.In thispaper,projection pursuit (PP) is used to carry out PCA with a criterion which is more robust than thevariance.In addition,the generalized simulated annealing (GSA) algorithm is introduced as anoptimization procedure in the process of PP calculation to guarantee the global optimum.The resultsfor simulated data sets show that PCA via PP is resistant to the deviation of the error distribution fromthe normal one.The method is especially recommended for use in cases with possible outlier(s) existingin the data.     

可吸入颗粒物(PM10)浓度时空变异性及影响因素分析

本文以厦门市为例,首先对PM10平均浓度数据进行趋势性、周期性和空间变异性进行分析,然后分析了气温、降水强度、降水日数、风向等气候因子对PM10浓度变化在时间上的影响,同时使用了交叉相关分析的方法对月平均降水日数和PM10大气污染指数(API)为优(PM10浓度<0.05mg/m3)的月内天数百分比的相关性进行分析,最后利用从遥感影像获取的地表覆盖数据,分析了土地利用类型与局部PM10年平均浓度的关系。分析结果表明：厦门市PM10浓度年内变化无明显的上升和下降的趋势,但有显著的周期性,分别为3、7和29天,气候因素对PM10浓度的变化有显著的影响,其中月平均降水日数对API为优的月内天数百分比的影响有明显的滞后性,滞后周期大约为3个月,局部PM10年均浓度与该地区的土地覆盖类型有明显的相关性,并且植被覆盖的比例越大,该地区的PM10年平均浓度就越小。     

Diatom-based water chemistry reconstructions from northern Sweden: a comparison of reconstruction techniques

The diatom composition in surface sediments from 119 northern Swedish lakes was studied to examine the relationship with lake-water pH, alkalinity, and colour. Diatom-based predictive models, using weighted-averaging (WA) regression and calibration, partial least squares (PLS) regression and calibration, and weighted-averaging partial least squares (WA-PLS) regression and calibration, were developed for inferences of water chemistry conditions. The non-linear response between the diatom assemblages and pH and alkalinity was best modelled by weighted-averaging methods. The lowest prediction error for pH was obtained using weighted averaging, with or without tolerance downweighting. For alkalinity there was still some information in the residual structure after extracting the first weighted-averaging component, which resulted in a slight improvement of predictions when using a two component WA-PLS model. The best colour predictions were obtained using a two component PLS model. Principal component analysis (PCA) of the prediction errors, with some characteristics of the training set included as passive variables, was performed to compare the results for the different alkalinity predictive models. The results indicate that calibration techniques utilizing more than one component (PLS and WA-PLS) can improve the predictions for lakes with diatom taxa that have broad tolerances. Furthermore, we show that WA-PLS performs best compared with the other techniques for those lakes that have a high relative abundance of the most dominant taxa and a corresponding low sample heterogeneity.     

主成分分析和熵值法相结合的水质综合评价模型及其在郑州市金水河的应用

水质评价对决策者决定水的使用功效尤为重要。水质综合评价系统中涉及到大量因子与指标,因子之间相互作用,致使水质的评价工作相对困难。主成分分析法可以消除因子间的相关性,因而被广泛应用于水质评价,但其忽略了数据离散程度的问题。熵值法则考虑了数据的离散特点。为更好地进行水质的综合评价,本文提出把主成分分析法和熵值法结合起来确定指标权重的方法,建立了水质评价模型,并采用该模型对郑州市金水河再生水2009年的水质情况进行评价,将评价结果与单独采用主成分分析或熵值法的结果进行了比较。结果表明了该方法的可行性与实用性,能够为非常规水资源利用提供理论依据和决策参考。     

Polarization analysis in the wavelet domain based on the adaptive covariance method

We present an improved method for computing polarization attributes of particle motion from multicomponent seismic recordings in the time–frequency domain by using the continuous wavelet transform. This method is based on the analysis of the covariance matrix. We use an approximate analytical formula to compute the elements of the covariance matrix for a time window which is derived from an averaged instantaneous frequency of the multicomponent record. The length of the time-window is automatically and adaptively set to match the dominant period of the analysing wavelet at each time–frequency point. Then the eigenparameters are estimated for each time–frequency point without interpolation. With these key features, our method provides a suitable approach for polarization analysis of dispersive signals or overlapping seismic arrivals in multicomponent seismic data. For polarization analysis in the time domain, we show that the proposed method is consistent with existing polarization analysis methods. We apply the method to real data sets from exploration and earthquake seismology to illustrate some filtering applications and wave type characterizations.     

反射光谱技术:研究南极企鹅古生态演化的一种新方法

为探求快速、廉价、无损和同步的光谱技术在南极生态环境研究中的可能性,本文利用南极阿德雷岛的四根企鹅粪土沉积柱样品的反射光谱,通过逐步多元线性回归和主成分回归两种数学运算方法,建立了反射光谱数据与企鹅粪九种标型元素浓度之间的关系,并探讨了南极企鹅粪土沉积物光谱数据的古生态意义。结果表明:南极粪土沉积样品反射率光谱与企鹅粪九种标型元素(P、Ca、Cu、F、Ba、S、Zn、Sr、Se)含量之间存在良好的相关性,预测值与实测值之间的相关系数R都达到了0.9以上,在深度剖面上预测浓度与实测浓度具有非常一致的变化趋势;南极粪土沉积物光谱数据包含有明确的古生态变化信息,可利用主成分分析快速恢复历史时期企鹅数量演化过程。本研究结果为在偏远的南极地区开展古生态环境研究提供了一种新的快捷方法和技术途径。     

沙拐枣属植物果实性状的数值分类研究

筛选出我国沙拐枣属(calligonum L.)植物15种,测定了果实刺/翅毛排数、刺/翅毛质地、扭曲方向、扭曲程度等11项指标。应用主成分分析法对15种沙拐枣属植物果实性状的各项指标进行了分析,结果表明,第一主成分贡献率为29.704%,其中沟槽深度、肋棱突起程度、刺毛稀疏度是影响沙拐枣果实性状的主要因素;第二主成分贡献率为25.598%,主要包括了基部加宽度、果组。对沙拐枣果实性状指标进行PCA排序可将其划分为4类:第一类包括翅/刺毛排数、刺毛稀疏度、肋棱宽度;第二类包括果组、果实扭曲方向、扭曲程度;第三类包括刺毛/翅质地、刺毛/翅粗硬、沟槽深度、肋棱突起程度;第四类包括果实基部是否加宽。通过第一、第二和第三主成分分析体现了沙拐枣果实趋向于沟槽浅、肋棱浅,而果实刺毛密度趋向于增加。     

Special Issue on parallel processing in GIS

This paper explores three theoretical approaches for estimating the degree of correctness to which the accuracy figures of a gridded Digital Elevation Model (DEM) have been estimated depending on the number of checkpoints involved in the assessment process. The widely used average‐error statistic Mean Square Error (MSE) was selected for measuring the DEM accuracy. The work was focused on DEM uncertainty assessment using approximate confidence intervals. Those confidence intervals were constructed both from classical methods which assume a normal distribution of the error and from a new method based on a non‐parametric approach. The first two approaches studied, called Chi‐squared and Asymptotic Student t, consider a normal distribution of the residuals. That is especially true in the first case. The second case, due to the asymptotic properties of the t distribution, can perform reasonably well with even slightly non‐normal residuals if the sample size is large enough. The third approach developed in this article is a new method based on the theory of estimating functions which could be considered much more general than the previous two cases. It is based on a non‐parametric approach where no particular distribution is assumed. Thus, we can avoid the strong assumption of distribution normality accepted in previous work and in the majority of current standards of positional accuracy. The three approaches were tested using Monte Carlo simulation for several populations of residuals generated from originally sampled data. Those original grid DEMs, considered as ground data, were collected by means of digital photogrammetric methods from seven areas displaying differing morphology employing a 2 by 2 m sampling interval. The original grid DEMs were subsampled to generate new lower‐resolution DEMs. Each of these new DEMs was then interpolated to retrieve its original resolution using two different procedures. Height differences between original and interpolated grid DEMs were calculated to obtain residual populations. One interpolation procedure resulted in slightly non‐normal residual populations, whereas the other produced very non‐normal residuals with frequent outliers. Monte Carlo simulations allow us to report that the estimating function approach was the most robust and general of those tested. In fact, the other two approaches, especially the Chi‐squared method, were clearly affected by the degree of normality of the residual population distribution, producing less reliable results than the estimating functions approach. This last method shows good results when applied to the different datasets, even in the case of more leptokurtic populations. In the worst cases, no more than 64–128 checkpoints were required to construct an estimate of the global error of the DEM with 95% confidence. The approach therefore is an important step towards saving time and money in the evaluation of DEM accuracy using a single average‐error statistic. Nevertheless, we must take into account that MSE is essentially a single global measure of deviations, and thus incapable of characterizing the spatial variations of errors over the interpolated surface.     

河南省县域经济实力时空差异研究

综合考虑经济发展水平指标、居民生活水平指标和经济结构指标,利用加权主成分分析方法、自然断点法及趋势分析法对多时点数据进行分析,揭示河南省县域经济实力时空差异特征。研究表明:经济发展水平与经济结构及居民生活水平具有较高的相关度,河南省县域经济实力的空间地域差异及空间分异显著,但近10年来总体变化趋势不大;近十年河南省各类型县域的数量波动较大,其空间分布格局的演变由原来典型的"单一核心"结构发展成明显的"多核心"结构;河南省县域总体经济实力普遍较低,且发达县域和不发达县域两极分化显著,两者在经济实力排名上虽有波动,但变化不大;核心城市的带动效应显著,河南省县域经济实力增长呈现明显的"双核心"驱动。     

CHARACTERIZATION OF THE EFFECT OF PEAK SHIFTS ON THE PERFORMANCE OF THE KALMAN FILTER IN MULTICOMPONENT ANALYSES

The effect of peak shifts on the performance of a Kalman filter multicomponent analysis algorithm hasbeen investigated.A series of Gaussian test systems were employed to characterize the concentrationestimation errors and the morphology of the on-line residuals(the innovations sequence).Both forwardand reverse filters were used in the generation of the innovations sequences.It was found that thedifference between the forward and reverse innovations sequences gave an accurate indication of thedirection and magnitude of the peak shift.A series of overlapped two-component systems were alsoinvestigated.Again,a correlation between the difference innovations and the degree of the responseshift was observed.The behavior of the Katman filter in fitting the shifted fluorescence emissionspectrum of benzo[a]pyrene was also examined.The response for henzo[a]pyrene in cyclohexanesolution was compared to that obtained on the surface of a reversed phase thin layer chromatographyplate.A red shift of 4 nm was detected for the surface spectrum by observing the difference innovationssequence.This approach holds promise for correction of response shifts in multicomponent systems.     

基于混合光谱理论的太湖水体叶绿素a浓度提取

TM图象多波段数据作为遥感监测水体叶绿素a浓度的数据源,已有多种遥感定量模型与之对应,但主要还是以经验模型为主。利用TM数据首先采用特征波段比值方法,建立太湖水体合适的叶绿素a浓度反演的遥感定量模型。由于经验模型的缺陷性,还提出了基于TM数据的水体混合光谱分析模型,同时分析了端元光谱提取方法对模型求解的影响。通过计算叶绿素a浓度模型估算结果与实地测量数据的相关系数和均方根误差(RMSE),可以发现混合光谱分析模型也是水体叶绿素a浓度遥感估算的另一条佳径。     

Graphical methods for rapid vector analysis of demagnetization data

Summary. Working graphs are presented which, when used in conjunction with a stereographic projection, enable (a) magnitude and direction of the difference between two vectors to be estimated, and (b) a vector to be resolved into two or three components of prescribed directions. Applications of the methods to multicomponent natural remanent magnetizations are illustrated.     

A Comparison of Unstructured and Structured Principal Component Analyses and their Interpretation

Multivariate analysis is employed to investigate the structure of variations within highly heterogeneous data. Traditionally, principal component analysis (PCA) is run by analyzing the entire wireline log and using PCA scores to characterize variability within and between lithologies. In this paper, we propose a technique using only specific subsets of all well records to quantify reservoir heterogeneity due to second order lithological variability. These subsets are chosen from uniform lithofacies parts of the wireline log in order to reduce the variability in the correlation matrix that otherwise would cause lithological changes. The purpose is to assess the efficiency of structured PCA in analyzing small-scale heterogeneity that is captured by wireline logs but often masked by traditional PCA approaches. This paper shows that a structured PCA procedure based upon special lithological units is superior to an unstructured PCA, when the focus is within lithology variations. This structured procedure is applied to data from the Heidrun field, offshore mid-Norway. The results demonstrate clear benefits from added insight into the variability of a complex fluviodeltaic heterolithic sequence that poses great challenges to hydrocarbon development.