共查询到20条相似文献，搜索用时 31 毫秒
1.
Ronald Christensen 《Mathematical Geology》1993,25(5):541-558
This article illustrates the use of linear and nonlinear regression models to obtain quadratic estimates of covariance parameters. These models lead to new insights into the motivation behind estimation methods, the relationships between different methods, and the relationship of covariance estimation to prediction. In particular, we derive the standard estimating equations for minimum norm quadratic unbiased translation invariant estimates (MINQUEs) from an appropriate linear model. Connections between the linear model, minimum variance quadratic unbiased translation invariant estimates (MIVQUEs), and MINQUEs are examined and we provide a minimum norm justification for the use of one-step normal theory maximum likelihood estimates. A nonlinear regression model is used to define MINQUEs for nonlinear covariance structures and obtain REML estimates. Finally, the equivalence of predictions under various models is examined when covariance parameters are estimated. In particular, we establish that when using MINQUE, iterative MINQUE, or restricted maximum likelihood (REML) estimates, the choice between a stationary covariance function and an intrinsically stationary semivariogram is irrelevant to predictions and estimated prediction variances. 相似文献
2.
A procedure is proposed that employs first-moment estimation (kriging), cross-validation, and response surface analysis to estimate parameters of a generalized covariance function. Results from application of this procedure to two data sets are given. 相似文献
3.
Structural analysis of data displaying trends may be performed with the help of generalized increments, the variance of these
increments being a function of a generalized covariance. Generalized covariances are estimated primarily by parametric methods
(i. e., methods searching for the best coefficients of a predetermined function), but also may be computed by one known nonparametric
alternative. In this paper, a new nonparametric method is proposed. It is founded on the following principles: (1) least-squares
residues are generalized increments; and (2) the generalized covariance is not a unique function, but a family of functions
(the system is indeterminate). The method is presented in a general context of a k order trend in Rd, although the full solution is given only fork = I in Ri. In Ri, higher order trends may be developed easily with the equations included in this paper. For higher dimensions in space, the
problem is more complex, but a research approach is proposed. The method is tested on soil pH data and compared to a parametric
and nonparametric method. 相似文献
4.
Peter K. Kitanidis 《Mathematical Geology》1985,17(2):195-208
The parameters of covariance functions (or variograms) of regionalized variables must be determined before linear unbiased estimation can be applied. This work examines the problem of minimum-variance unbiased quadratic estimation of the parameters of ordinary or generalized covariance functions of regionalized variables. Attention is limited to covariance functions that are linear in the parameters and the normality assumption is invoked when fourth moments of the data need to be calculated. The main contributions of this work are (1) it shows when and in what sense minimum-variance unbiased quadratic estimation can be achieved, and (2) it yields a well-founded, practicable, and easy-to-automate methodology for the estimation of parameters of covariance functions. Results of simulation studies are very encouraging. 相似文献
5.
Noel Cressie 《Mathematical Geology》1987,19(5):425-449
Fitting trend and error covariance structure iteratively leads to bias in the estimated error variogram. Use of generalized increments overcomes this bias. Certain generalized increments yield difference equations in the variogram which permit graphical checking of the model. These equations extend to the case where errors are intrinsic random functions of order k, k=1, 2, ..., and an unbiased nonparametric graphical approach for investigating the generalized covariance function is developed. Hence, parametric models for the generalized covariance produced by BLUEPACK-3D or other methods may be assessed. Methods are illustrated on a set of coal ash data and a set of soil pH data. 相似文献
6.
Quadratic estimators of components of a nested spatial covariance function are presented. Estimators are unbiased and possess a minimum norm property. Inversion of a covariance matrix is required but, by assuming that spatial correlation is absent, a priori, matrix inversion can be avoided. The loss of efficiency that results from this assumption is discussed. Methods can be generalized to include estimation of components of a generalized polynomial covariance assuming the underlying process to be an intrinsic random function. Particular attention is given to the special case where just two components of spatial covariance exist, one of which represents a nugget effect. 相似文献
7.
Michael L. Stein 《Mathematical Geology》1986,18(7):625-633
Marshall and Mardia (1985) and Kitanidis (1985) have suggested using minimum norm quadratic estimation as a method to estimate parameters of a generalized covariance function. Unfortunately, this method is difficult to use with large data sets as it requires inversion of an n × n matrix, where n is number of observations. These authors suggest replacing the matrix to be inverted by the identity matrix, which eliminates the computational burden, although with a considerable loss of efficiency. As an alternative, the data set can be broken into subsets, and minimum norm quadratic estimates of parameters of the generalized covariance function can be obtained within each subset. These local estimates can be averaged to obtain global estimates. This procedure also avoids large matrix inversions, but with less loss in efficiency. 相似文献
8.
On the estimation of the generalized covariance function 总被引：1，自引：0，他引：1
The estimation of the generalized covariance function, K, is a major problem in the use of intrinsic random functions of order k to obtain kriging estimates. The precise estimation by least-squares regression of the parameters in polynomial models for K is made difficult by the nature of the distribution of the dependent variable and the multicollinearity of the independent variables. 相似文献
9.
Generalized covariance functions in estimation 总被引：3，自引：0，他引：3
Peter K. Kitanidis 《Mathematical Geology》1993,25(5):525-540
I discuss the role of generalized covariance functions in best linear unbiased estimation and methods for their selection. It is shown that the experimental variogram (or covariance function) of the detrended data can be used to obtain a preliminary estimate of the generalized covariance function without iterations and I discuss the advantages of other parameter estimation methods. 相似文献
10.
Roussos Dimitrakopoulos 《Mathematical Geology》1990,22(3):361-380
Conditional simulation of intrinsic random functions of orderk is a stochastic method that generates realizations which mimic the spatial fluctuation of nonstationary phenomena, reproduce their generalized covariance and honor the available data at sampled locations. The technique proposed here requires the following steps: (i) on-line simulation of Wiener-Levy processes and of their integrations; (ii) use of the turning-bands method to generate realizations in Rn; (iii) conditioning to available data; and (iv) verification of the reproduced generalized covariance using generalized variograms. The applicational aspects of the technique are demonstrated in two and three dimensions. Examples include the conditional simulation of geological variates of the Crystal Viking petroleum reservoir, Alberta, Canada. 相似文献
11.
Multivariable spatial prediction 总被引：1，自引：0，他引：1
For spatial prediction, it has been usual to predict one variable at a time, with the predictor using data from the same type of variable (kriging) or using additional data from auxiliary variables (cokriging). Optimal predictors can be expressed in terms of covariance functions or variograms. In earth science applications, it is often desirable to predict the joint spatial abundance of variables. A review of cokriging shows that a new cross-variogram allows optimal prediction without any symmetry condition on the covariance function. A bivariate model shows that cokriging with previously used cross-variograms can result in inferior prediction. The simultaneous spatial prediction of several variables, based on the new cross-variogram, is then developed. Multivariable spatial prediction yields the mean-squared prediction error matrix, and so allows the construction of multivariate prediction regions. Relationships between cross-variograms, between single-variable and multivariable spatial prediction, and between generalized least squares estimation and spatial prediction are also given. 相似文献
12.
Roussos Dimitrakopoulos 《Mathematical Geosciences》1990,22(3):361-380
Conditional simulation of intrinsic random functions of orderk is a stochastic method that generates realizations which mimic the spatial fluctuation of nonstationary phenomena, reproduce their generalized covariance and honor the available data at sampled locations. The technique proposed here requires the following steps: (i) on-line simulation of Wiener-Levy processes and of their integrations; (ii) use of the turning-bands method to generate realizations in Rn; (iii) conditioning to available data; and (iv) verification of the reproduced generalized covariance using generalized variograms. The applicational aspects of the technique are demonstrated in two and three dimensions. Examples include the conditional simulation of geological variates of the Crystal Viking petroleum reservoir, Alberta, Canada. 相似文献
13.
Computational power poses heavy limitations to the achievable problem size for Kriging. In separate research lines, Kriging algorithms based on FFT, the separability of certain covariance functions, and low-rank representations of covariance functions have been investigated, all three leading to drastic speedup factors. The current study combines these ideas, and so combines the individual speedup factors of all ideas. This way, we reduce the mathematics behind Kriging to a computational complexity of only $\mathcal{O}(dL^{*} \log L^{*})$ , where L ? is the number of points along the longest edge of the involved lattice of estimation points, and d is the physical dimensionality of the lattice. For separable (factorized) covariance functions, the results are exact, and nonseparable covariance functions can be approximated well through sums of separable components. Only outputting the final estimate as an explicit map causes computational costs of $\mathcal{O}(n)$ , where n is the number of estimation points. In illustrative numerical test cases, we achieve speedup factors up to 108 (eight orders of magnitude), and we can treat problem sizes of up to 15 trillion and two quadrillion estimation points for Kriging and spatial design, respectively, within seconds on a contemporary desktop computer. The current study assumes second-order stationarity and simple Kriging on a regular, equispaced lattice, without working with restricted neighborhoods. Extensions to many other cases are straightforward. 相似文献
14.
C. R. Dietrich 《Mathematical Geology》1993,25(4):439-451
The generation over two-dimensional grids of normally distributed random fields conditioned on available data is often required in reservoir modeling and mining investigations. Such fields can be obtained from application of turning band or spectral methods. However, both methods have limitations. First, they are only asymptotically exact in that the ensemble of realizations has the correlation structure required only if enough harmonics are used in the spectral method, or enough lines are generated in the turning bands approach. Moreover, the spectral method requires fine tuning of process parameters. As for the turning bands method, it is essentially restricted to processes with stationary and radially symmetric correlation functions. Another approach, which has the advantage of being general and exact, is to use a Cholesky factorization of the covariance matrix representing grid points correlation. For fields of large size, however, the Cholesky factorization can be computationally prohibitive. In this paper, we show that if the data are stationary and generated over a grid with regular mesh, the structure of the data covariance matrix can be exploited to significantly reduce the overall computational burden of conditional simulations based on matrix factorization techniques. A feature of this approach is its computational simplicity and suitability to parallel implementation. 相似文献
15.
Looking at kriging problems with huge numbers of estimation points and measurements, computational power and storage capacities
often pose heavy limitations to the maximum manageable problem size. In the past, a list of FFT-based algorithms for matrix
operations have been developed. They allow extremely fast convolution, superposition and inversion of covariance matrices
under certain conditions. If adequately used in kriging problems, these algorithms lead to drastic speedup and reductions
in storage requirements without changing the kriging estimator. However, they require second-order stationary covariance functions,
estimation on regular grids, and the measurements must also form a regular grid. In this study, we show how to alleviate these
rather heavy and many times unrealistic restrictions. Stationarity can be generalized to intrinsicity and beyond, if decomposing
kriging problems into the sum of a stationary problem and a formally decoupled regression task. We use universal kriging,
because it covers arbitrary forms of unknown drift and all cases of generalized covariance functions. Even more general, we
use an extension to uncertain rather than unknown drift coefficients. The sampling locations may now be irregular, but must
form a subset of the estimation grid. Finally, we present asymptotically exact but fast approximations to the estimation variance
and point out application to conditional simulation, cokriging and sequential kriging. The drastic gain in computational and
storage efficiency is demonstrated in test cases. Especially high-resolution and data-rich fields such as rainfall interpolation
from radar measurements or seismic or other geophysical inversion can benefit from these improvements. 相似文献
16.
In kriging, parametric approaches to covariance (or variogram) estimation require that unknown parameters be inferred from a single realization of the underlying random field. An approach to such an estimation problem is to assume the field to be Gaussian and iteratively minimize a (restricted) negative loglikelihood over the parameter space. In doing so, the associated computational burden can be considerable. Also, it is usually not easy to check whether or not the minimum achieved is global. In this note, we show that in many practical cases, the structure of the covariance (or variogram) function can be exploited so that iterative minimizing algorithms may be advantageously replaced by a procedure that requires the computation of the roots of a simple rational function and the search for the minimum of a function depending on one variable only. As a consequence, our approach allows one to observe in a straightforward fashion the presence of local minima. Furthermore, it is shown that insensitivity of the likelihood function to changes in parameter value can be easily detected. The note concludes with numerical simulations that illustrate some key features of our estimation procedure. 相似文献
17.
George Christakos 《Mathematical Geology》1985,17(5):489-515
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. 相似文献
18.
Positive definiteness is not enough 总被引：2，自引：0，他引：2
M. Armstrong 《Mathematical Geology》1992,24(1):135-143
Geostatisticians know that the mathematical functions chosen to represent spatial covariances and variograms must have the appropriate type of positive definiteness, but they may not realize that there are restrictions on the types of covariances and variograms that are compatible with particular distributions. This paper gives some examples showing that (1) the spherical model is not compatible with the multivariate lognormal distribution if the coefficient of variation is 2.0 or more (even in 1-D), and (2) the Gaussian covariance and several other models are not compatible with indicator random functions. As these examples concern quite different types of random functions, it is clear that there is a general problem of compatibility between spatial covariance models (or variograms) and a specified multivariate distribution. The problem arises with all distributions except the multivariate normal, and not just the two cited here. The need for a general theorem giving the necessary and sufficient conditions for a covariance or a variogram to be compatible with a particular distribution is stressed. 相似文献
19.
陆地生态系统CO2和水热通量的长期观测研究一直是国际上关注的热点问题。截止目前,利用微气象学原理的涡度协方差技术是唯一能直接测定生物圈与大气间物质与能量通量的标准方法,成为国际通量观测网络的主要技术。但是涡度协方差技术的测定仍然是一种小尺度观测方法,其观测结果难于直接外推到更大尺度。同时,缺乏区域、跨尺度生态系统及其时空动态观测数据一直是限制碳循环研究的主要障碍,而遥感技术的发展可望在不远的将来使大尺度、高分辨生态系统变化的长期定量观测成为可能。这些问题在当今集中体现在如何建立通量—遥感的跨尺度观测体系,并有效地将有限的通量站点测量数据与大尺度遥感资料以及生态模型有机地结合。总结过去耦合涡度协方差技术与遥感技术的工作,主要在以下3个层面展开：①涡度协方差技术与遥感技术对碳通量估算的相互验证;②涡度协方差技术为遥感反演提供地面参数;③遥感观测解译辅助分析通量贡献区（footprint）。集中在这3个方面进行探讨,通过总结各方面的研究特点与进展,可望为未来在这个领域开展工作理顺思路。 相似文献
20.
The Second-Order Stationary Universal Kriging Model Revisited 总被引：3，自引：0，他引：3
Universal kriging originally was developed for problems of spatial interpolation if a drift seemed to be justified to model the experimental data. But its use has been questioned in relation to the bias of the estimated underlying variogram (variogram of the residuals), and furthermore universal kriging came to be considered an old-fashioned method after the theory of intrinsic random functions was developed. In this paper the model is reexamined together with methods for handling problems in the inference of parameters. The efficiency of the inference of covariance parameters is shown in terms of bias, variance, and mean square error of the sampling distribution obtained by Monte Carlo simulation for three different estimators (maximum likelihood, bias corrected maximum likelihood, and restricted maximum likelihood). It is shown that unbiased estimates for the covariance parameters may be obtained but if the number of samples is small there can be no guarantee of good estimates (estimates close to the true value) because the sampling variance usually is large. This problem is not specific to the universal kriging model but rather arises in any model where parameters are inferred from experimental data. The validity of the estimates may be evaluated statistically as a risk function as is shown in this paper. 相似文献