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.     

AN IMPROVED ALGORITHM FOR THE GENERALIZED RANK ANNIHILATION METHOD

An improved algorithm for the generalized rank annihilation method(GRAM)is presented.GRAM isa method for multicomponent calibration using two-dimensional instruments,such as GC-MS.In thispaper an orthonormal base is first computed and used to project the calibration and unknown sampleresponse matrices into a lower-dimensional subspace.The resulting generalized eigenproblem is thensolved using the QZ algorithm.The result of these improvements is that GRAM is computationally morestable,particularly in the case where the calibration sample contains chemical constituents not present inthe unknown sample and the unknown contains constituents not present in the calibration(the mostgeneral case).     

TENSORIAL RESOLUTION:A DIRECT TRILINEAR DECOMPOSITION

Modern instrumentation in chemistry routinely generates two-dimensional(second-order)arrays of data.Considering that most analyses need to compare several samples,the analyst ends up with a three-dimensional(third-order)array which is difficult to visualize or interpret with the conventional statisticaltools.Some of these data arrays follow the so-called trilinear model,(?)These trilinear arrays of data are known to have unique factor analysis decompositions which correspondto the true physical factors that form the data,i.e.given the array (?),a unique solution can be found inmany cases for each order X,Y and Z.This is in contrast to the well-known second-order bilinear datafactor analysis,where the abstract solutions obtained are not unique and at best cannot be easilycompared with the underlying physical factors owing to a rotational ambiguity.Trilinear decompositions have had the disadvantage,however,that a non-linear optimization withmany parameters is necessary to reach a least-squares solution.This paper will introduce a method forreducing the problem to a rectangular generalized eigenvalue-eigenvector equation where the eigenbectorsare the contravariant form(pseudo-inverse)of the actual factors.It is shown that the method works wellwhen the factors are linearly independent in at least two orders(e.g.X_(jr),and Y_(jr) are full rank matrices).Finally,it is shown how trilinear decompositions relate to multicomponent calibration,curve resolutionand chemical analysis.     

COMMENTS ON THE NIPALS ALGORITHM

The Non-linear lterative Partial Least Squares(NIPALS)algorithm is used in principal componentanalysis to decompose a data matrix into score vectors and eigenvectors(loading vectors)plus a residualmatrix.N1PALS starts with some guessed starting vector.The principal components obtained by NIPALSdepends on the starting vector;the first principal component could not always be computed.Wold hassuggested a starting vector for NIPALS,but we have found that even if this starting vector is used,thefirst principal component cannot be obtained in all cases.The reason why such a situation occurs isexplained by the power method.A simple modification of the original NIPALS procedure to avoid gettingsmaller eigenvalues is presented.     

Sensitivities of seismic traveltimes and amplitudes in reflection tomography

Seismic traveltimes and amplitudes in reflection-seismic data show different dependences on the geometry of reflection interfaces, and on the variation of interval velocities. These dependences are revealed by eigenanalysis of the Hessian matrix, defined in terms of the Fréchet matrix and its adjoint associated with different norms chosen in the model space. The eigenvectors and eigenvalues of the Hessian clearly show that for reflection tomographic inversion, traveltime and amplitude data contain complementary information. Both for reflector-geometry and for interval-velocity variations, the traveltimes are sensitive to the model components with small wavenumbers, whereas the amplitudes are more sensitive to the components with high wavenumbers. The model resolution matrices, after the rejection of eigenvectors corresponding to small eigenvalues, give us some insight into how the addition of amplitude information could potentially contribute to the recovery of physical parameters.
In order to cooperatively invert seismic traveltimes and amplitudes simultaneously, we propose an empirical definition of the data covariance matrix which balances the relative sensitivities of different types of data. We investigate the cooperative use of both data types for, separately, interface-geometry and 2-D interval-velocity variations. In both cases we find that cooperative inversions can provide better solutions than those using traveltimes alone. The potential benefit of including amplitude-data constraints in seismic-reflection traveltime tomography is therefore that it may be possible to resolve the known ambiguity between the reflector-depth uncertainty and the interval-velocity uncertainty better.     

Testing spatial heterogeneity in geographically weighted principal components analysis

We propose a method to evaluate the existence of spatial variability in the covariance structure in a geographically weighted principal components analysis (GWPCA). The method, that is extensive to locally weighted principal components analysis, is based on performing a statistical hypothesis test using the eigenvectors of the PCA scores covariance matrix. The application of the method to simulated data shows that it has a greater statistical power than the current statistical test that uses the eigenvalues of the raw data covariance matrix. Finally, the method was applied to a real problem whose objective is to find spatial distribution patterns in a set of soil pollutants. The results show the utility of GWPCA versus PCA.     

Statistics and geometry of the eigenspectra of three-dimensional second-rank symmetric random tensors

Geoscientists have undertaken mapping of the Earth's crustal strain (or stress) fields using a great variety of field data. The output can be represented by a 3-D second-rank symmetric random strain tensor. The random principal strains-land rotations of the random tensor are frequently computed. The accuracy is calculated using a first-order approximation. The distribution aspects of the random principal strains and rotations have received almost no attention in Earth Sciences. A first-order approximation of accuracy may not be sufficient if the signal-to-noise ratio is small, as is often the case for geodetically derived random strain tensors. Therefore, the purpose of this paper is to investigate the distribution and estimation problems of the general 3-D second-rank tensor equation GΛG ^T= T , where T is a given 3-D second-rank symmetric random tensor, Λ a diagonal (3 × 3) random eigenvalue matrix, and G a (3 × 3) random orientation matrix, which is also orthogonal. Λ and G are to be estimated (or solved) from T . If some eigenvalues coincide, additional conditions are imposed on the eigenvectors so that they can be chosen uniquely. The joint probability density function (pdf) of the random eigenvalues and rotations will be worked out, given a joint pdf of the elements of random tensors T. Because the rotations are of special interest in Earth Sciences, we shall also derive the joint marginal pdf of random rotations. The geometry of eigenspectra will be studied. The biases of random eigenvalues and rotations will be derived, which have been neglected in the past. They can be very crucial in interpreting the pattern of a derived strain field, however, when applied to a real Earth Science problem. The variance-covariance matrices will be computed using a second-order approximation.     

On covariance propagation of eigenparameters of symmetric n-D tensors

Symmetric tensors are typically encountered during investigations associated with stress and strain analysis and, thus, they are of particular interest to geophysicists and geodesists. Furthermore, symmetric tensors are studied using eigentheory analysis which provides the decomposition of the tensor on its principal components (n independent eigenvalues and the corresponding eigenvectors). In this paper, an analytical expression of the covariance matrix of the eigenvalues and eigenvectors of an n-D symmetric tensor is derived based on the principles of linear algebra and differential calculus. Through numerical tests, the proposed formulation is proven to give realistic uncertainty estimates of the determined eigenparameters. The methodology also reveals the significant impact on uncertainty assessments when the parameter dependencies between principal components are neglected.     

Horizontal layered structure with heterogeneity beneath continents and island arcs from particle orbits of long-period P waves

We investigate the particle orbits of long-period (about 20 s) P waves observed with the global seismic network. By analysing 84 three-component seismograms recorded at 25 stations from 60 earthquakes occurring beneath 300 km, we quantitatively evaluate the orbits by three sets of eigenvalues and eigenvectors, using a covariance matrix method. The eigenvalues for P waves recorded at stations located on continents are explained by the standard horizontal layered structure model (iasp91). On the other hand, the orbits observed at stations close to island arcs are affected not only by the horizontal layered structure but also by heterogeneity due to subducting plates, mantle diapirs and so on. On the basis of a single-scattering model for a plane P wave, we quantify the heterogeneities by an isotropic scattering coefficient g0. Fitting the theoretical eigenvalues to the observed ones, we estimate g0 for the crust and upper mantle beneath continents to be less than 0.0005 km^-1, and the mean g₀ for the structure beneath island arcs to be about 0.0015 to 0.003 km^-1.     

基于无尺度网络的中国城市医疗网络分析

根据城市统计中存在大量属性数据的特点,基于无尺度网络方法,根据2002~2014年中国城市医疗统计数据变化的相似性来建立城市之间的联系,得到中国城市医疗软件网络和硬件网络。在对两个网络对比和测试分析后得到：① 中国城市医疗网络是由20%高度城市代表的具有圈层聚类结构的无尺度复杂网络。② 中国城市医疗硬件网络比软件网络复杂和稳定。③ 中国城市医疗网络的结构和功能主要由强关系城市决定,弱关系城市对网络没有实质影响。可见,以无尺度网络方法来分析城市网络可以发现很多隐藏的问题。     

Mapping Geochemical Anomalies Through Integrating Random Forest and Metric Learning Methods

Extracting geochemical anomalies from geochemical exploration data is one of the most important activities in mineral exploration. Geochemical anomaly detection can be regarded as a binary classification problem. The similarity between geochemical samples can be measured by their distance. The key issue of this classification is to find the intrinsic relationship and distance between geochemical samples to separate geochemical anomalies from background. In this paper, a hybrid method that integrates random forest and metric learning (RFML) is used to identify geochemical anomalies related to Fe-polymetallic mineralization in Southwest Fujian Province of China. RFML does not require any specific statistical assumption on geochemical data, nor does it depend on sufficient known mineral occurrences as the prior knowledge. The geochemical anomaly map obtained by the RFML method showed that the known Fe deposits and the generated geochemical anomaly area have strong spatial association. Meanwhile, the receiver operating characteristic curves for the results of RFML and another method, namely maximum margin metric learning, indicated that the RFML method exhibited better performance, suggesting that RFML can be effectively applied to recognize geochemical anomalies.

     

柴达木盆地东缘祁连圆柏轮宽序列标准化的方法研究

利用在青海柴达木盆地东缘山地获取的大量祁连圆柏树轮资料,对拟合并去除树木生长趋势即标准化方法进行探讨,提出了一种总体曲线方法。该方法用包含完整树木髓心、并且在40~60年间达到生长顶峰的树轮资料拟合柴达木盆地东缘祁连圆柏的总体生长趋势曲线,并用广义的负指数函数来描述树木自茎的次生生长开始以来的径向生长过程。用相同的树轮资料建立估算缺失轮数的最初径向生长模型,其方差解释量高达90.9%。建立年表时包含髓心的样芯认定生理年龄为1年,不包含髓心的样芯用最初径向生长模型估算缺失轮数,然后全部样芯用生长趋势曲线对应部分进行去趋势标准化。该方法对建立可靠、准确的长年表有重要意义,所建年表比用传统负指数函数方法建立的年表保留了更多的低频变化信息。     

First-order corrections to the radiation patterns of P-and S-waves in a prestressed medium

Summary. The long wavelength radiation patterns of P - and S -waves are determined for an elastic prestressed earth model. The prestress is treated as a perturbation of an isotropic medium. Non-zero first-order corrections are found to the eigenvalues and eigenvectors of the deviatoric seismic moment tensor. The radiation pattern given by a purely tangential dislocation is equivalent to a double couple contaminated by a dipolar term linear in the perturbation.     

一种基于地理环境相似度的滑坡负样本可信度度量方法

滑坡负样本在统计型滑坡危险度制图中具有重要作用,能抑制统计模型对滑坡危险度的高估。当前滑坡负样本采样方法采集的负样本可信度未知,在负样本采样过程中,极有可能将那些潜在滑坡点错选为负样本,这些假的负样本会降低负样本集的质量和训练样本集的质量,进而影响统计模型的精度。本文基于“地理环境越相似、地理特征越相似”的地理学常识,认为与正样本有着相似地理环境的点极有可能是未来发生滑坡的点;与正样本的地理环境越不相似的点,则越有可能是负样本。基于此假设提出一种基于地理环境相似度的负样本可信度度量方法,将该方法应用于滑坡灾害频发的陇南山区油房沟流域,对油房沟进行滑坡负样本可信度评价制图;使用油房沟流域的滑坡发生初始面来验证该方法的有效性。结果发现：滑坡发生初始面上所有栅格点的负样本可信度平均值为0.26,超过95%的栅格点的负样本可信度都小于0.5,说明本文提出的负样本可信度度量方法合理。     

DETERMINATION OF A MULTIVARIATE DETECTION LIMIT AND LOCAL CHEMICAL RANK BY DESIGNING A NON-PARAMETRIC TEST FROM THE ZERO-COMPONENT REGIONS

In this paper we redefine the term detection limit to embrace the inherent multivariate nature of samples,instrumental measurements and chemometrics resolution procedures. The so-called zero-componentregions, i.e. parts with no chemical components eluting, are used as repeated analytical blanks to estimatea statistical multivariate detection limit for determining the number of chemical species in local regionsof a single two-way chromatogram or a collection of synchronized one-way chromatograms. For two-waychromatography the detection limit is determined from the distribution of the first eigenvalues obtainedfrom all possible combinations of spectra in the zero-component regions. The number of spectra in eachcalculation should correspond to the number included in the later examination of the local retention timeregions. For one-way chromatography on a collection of samples with similar chemical components atvarying concentrations the same procedure is used, with the samples taking the role of the spectra intwo-way chromatography. The detection limit can be chosen at various confidence levels depending onwhether false positive or negative detection of minor components is most critical. The results obtainedfrom the zero-eigenvalue distribution are more robust than those obtained by a previously developedF-test.     

Extremal inversion of static earth displacements due to volume sources

Summary. The inverse problem of using static displacements observed at the surface to infer volume changes within the Earth is considered. This problem can be put in a form such that the method of ideal bodies and the method of positivity constraints may both be applied. Thus all of the techniques previously developed for the gravity inverse problem can be extended to the static displacement problem. Given bounds on the depth, the greatest lower bound on the fractional volume change can be estimated, or, given bounds on the fractional volume change, the least upper bound on the depth can be estimated. Methods of placing bounds on generalized moments of the perturbing body are also developed, and techniques of handling errors in the data are discussed.
Examples are given for both two- and three-dimensional problems. The ideal body method is suited for both 2- and 3-D problems when only two data points are considered, but is unwieldy for more data points. The method of positivity constraints is more versatile and can be used when there are many data points in the case of 2-D problems, but it may lead to an excessive amount of computation in 3-D problems.     

BOX-COX TRANSFORMATIONS IN THE ANALYSIS OF COMPOSITIONAL DATA

The statistical analysis of compositional data is of fundamental importance to practitioners in generaland to chemists in particular.The existing methodology is principally due to Aitchison,who effectivelyuses two transformations,a ratio followed by the logarithmic,to create a useful,coherent theory thatin principle allows the plethora of normal-based multivariate techniques to be used on the transformeddata.This paper suggests that the well-known class of Box-Cox transformations can be employed inplace of the logarithmic to significantly improve the existing methodology.This is supported in part byshowing that one of the most basic problems that Aitchison managed to overcome,namely thespecification of an interpretable covariance structure for compositional data,can be resolved,or nearlyresolved,once the ratio transformation has been applied.Hence the resolution is not directly dependenton the logarithmic transformation.It is then verified that access to the general Box-Cox family will allowa more accurate use of the normal-based multivariate techniques,simply because better fits to normalitycan be achieved.Finally,maximum likelihood estimation and some associated asymptotics are employedto construct confidence intervals for ratios of the true,unknown compositional constituents.Heretoforethis had not been done even in the context of the logarithmic transformation.Applications to real dataare presented.     

An integrated data-directed numerical method for estimating the undiscovered mineral endowment in a region

An integrated data-directed numerical method has been developed to estimate the undiscovered mineral endowment within a given area. The method has been used to estimate the undiscovered uranium endowment in the San Juan Basin, New Mexico, U.S.A. The favorability of uranium concentration was evaluated in each of 2,068 cells defined within the Basin. Favorability was based on the correlated similarity of the geologic characteristics of each cell to the geologic characteristics of five area-related deposit models. Estimates of the undiscovered endowment for each cell were categorized according to deposit type, depth, and cutoff grade. The method can be applied to any mineral or energy commodity provided that the data collected reflect discovered endowment.     

Quality assessment of building footprint data using a deep autoencoder network

Volunteered geographic information (VGI), OpenStreetMap (OSM), has been used in many applications, especially when official spatial data are unavailable or outdated. However, the quality of VGI remains a valid concern. In this paper, we use the matched results between OSM building footprints and official data as the samples for training an autoencoder network, which encodes and reconstructs the sample populations according to unknown complex multivariate probability distributions. Then, the OSM data are assessed based on the theory that small probability samples contribute little to the autoencoder network and that they can be recognized by the higher reconstructed errors during training. In the method described here, the selected measures, including data completeness, positional accuracy, shape accuracy, semantic accuracy and orientation consistency between OSM and official data, are used as the inputs for a deep autoencoder network. Finally, building footprint data from Toronto, Canada, are evaluated, and experiments show that the proposed method can assess the OSM data comprehensively, objectively and accurately.