Geostatistical modelling of spatial uncertainty using p -field simulation with conditional probability fields

This paper presents a variant of p-field simulation that allows generation of spatial realizations through sampling of a set of conditional probability distribution functions (ccdf) by sets of probability values, called p-fields. Whereas in the common implementation of the algorithm the p-fields are nonconditional realizations of random functions with uniform marginal distributions, they are here conditional to 0.5 probability values at data locations, which entails a preferential sampling of the central part of the ccdf around these locations. The approach is illustrated using a randomly sampled (200 observations of the NIR channel) SPOT scene of a semi-deciduous tropical forest. Results indicate that the use of conditional probability fields improves the reproduction of statistics such as histogram and semivariogram, while yielding more accurate predictions of reflectance values than the common p-field implementation or the more CPU-intensive sequential indicator simulation. Pixel values are then classified as forest or savannah depending on whether the simulated reflectance value exceeds a given threshold value. In this case study, the proposed approach leads to a more precise and accurate prediction of the size of contiguous areas covered by savannah than the two other simulation algorithms.     

Combining spatial transition probabilities for stochastic simulation of categorical fields

Categorical spatial data, such as land use classes and socioeconomic statistics data, are important data sources in geographical information science (GIS). The investigation of spatial patterns implied in these data can benefit many aspects of GIS research, such as classification of spatial data, spatial data mining, and spatial uncertainty modeling. However, the discrete nature of categorical data limits the application of traditional kriging methods widely used in Gaussian random fields. In this article, we present a new probabilistic method for modeling the posterior probability of class occurrence at any target location in space-given known class labels at source data locations within a neighborhood around that prediction location. In the proposed method, transition probabilities rather than indicator covariances or variograms are used as measures of spatial structure and the conditional or posterior (multi-point) probability is approximated by a weighted combination of preposterior (two-point) transition probabilities, while accounting for spatial interdependencies often ignored by existing approaches. In addition, the connections of the proposed method with probabilistic graphical models (Bayesian networks) and weights of evidence method are also discussed. The advantages of this new proposed approach are analyzed and highlighted through a case study involving the generation of spatial patterns via sequential indicator simulation.     

Estimating Population Attribute Values in a Table: “Get Me Started in” Iterative Proportional Fitting

Iterative proportional fitting (IPF) is a technique that can be used to adjust a distribution reported in one data set by totals reported in others. IPF is used to revise tables of data where the information is incomplete, inaccurate, outdated, or a sample. Although widely applied, the IPF methodology is rarely presented in a way that is accessible to nonexpert users. This article fills that gap through discussion of how to operationalize the method and argues that IPF is an accessible and transparent tool that can be applied to a range of data situations in population geography and demography. It offers three case study examples where IPF has been applied to geographical data problems; the data and algorithms are made available to users as supplementary material.     

The Reliability of Using the Iterative Proportional Fitting Procedure*

This paper demonstrates the utility of the iterative proportional fitting procedure (IPF) in generating disaggregated spatial data from aggregated data and evaluates the performance of the procedure. Estimates of individual level data created by IPF using data of equal-interval categories are reliable, but the performance of the estimation can be improved by increasing sample size. The improvement usually is enough to offset the increase in error created by other factors. If the two variables defining the cross-classification have a significant interaction effect and the number of categories in each variable is larger than two, then IPF is preferred over an independent model.     

Combining Sample and Census Data in Small Area Estimates: Iterative Proportional Fitting with Standard Software*

The combination of detailed sample data with less detailed but fully enumerated marginal subtotals is the focus of a wide range of research. In this article we advocate careful modeling of sample data, followed by Iterative Proportional Fitting (IPF). The modeling aims to estimate accurately the interaction or odds ratios of complex tables, which is information not contained in the marginal subtotals. IPF ensures consistency with the subtotals. We advance this work in three practical ways. First, we show that detailed small‐area estimates of both counts and proportional distributions usually gain accuracy by combining data for larger areas containing the small areas, and we illustrate the multilevel framework to achieve these estimates. Second, we find that a general classification or socioeconomic typology of the small areas is even more associated with the within‐area interactions than is membership of the larger area. Third, we show how the Statistical Package for the Social Sciences (SPSS) can be used for IPF in any number of dimensions and with any structure of constraining marginal subtotals. Throughout, we use an example taken from the 1991 U.K. Census. These data allow us to evaluate various methods combining 100 percent tabulations and the Samples of Anonymised Records.     

Development of a technique to decompose heterogeneous mapping units in a categorical map of the biotic environment using correlated categorical data

In this article a technique is presented to estimate the proportions of different map categories in a series of heterogeneous mapping units, using information on the degree of spatial correlation with other categorical data. The technique has been applied to decompose ecotope complexes in a categorical map of the biotic environment in Flanders, using secondary information on land cover and soil type. Because the conditional probability of an ecotope occurring given a certain soil type depends on the frequency with which the ecotope occurs in an area, determining the probability of occurrence of an ecotope from the conditional probabilities can lead to predictions that contradict prior knowledge about the composition of the different mapping units. A measure expressing the affinity of an ecotope for a soil type is proposed and is used as an alternative to conditional probability in the estimation procedure. The proposed method has been tested in a study area for which detailed field observations were collected, and proves to work well if reliable a priori knowledge about the composition of complex mapping units is available.     

Fuzzy Weights of Evidence Method and Its Application in Mineral Potential Mapping

This paper proposes a new approach of weights of evidence method based on fuzzy sets and fuzzy probabilities for mineral potential mapping. It can be considered as a generalization of the ordinary weights of evidence method, which is based on binary or ternary patterns of evidence and has been used in conjunction with geographic information systems for mineral potential mapping during the past few years. In the newly proposed method, instead of separating evidence into binary or ternary form, fuzzy sets containing more subjective genetic elements are created; fuzzy probabilities are defined to construct a model for calculating the posterior probability of a unit area containing mineral deposits on the basis of the fuzzy evidence for the unit area. The method can be treated as a hybrid method, which allows objective or subjective definition of a fuzzy membership function of evidence augmented by objective definition of fuzzy or conditional probabilities. Posterior probabilities calculated by this method would depend on existing data in a totally data-driven approach method, but depend partly on expert's knowledge when the hybrid method is used. A case study for demonstration purposes consists of application of the method to gold deposits in Meguma Terrane, Nova Scotia, Canada.     

Response to Commentaries by Craig Campbell,Lyle Courtney and Walter G. Hardwick,and John E. Chappell,Jr.

Abstract

The combination of detailed sample data with less detailed but fully enumerated marginal subtotals is the focus of a wide range of research. In this article we advocate careful modeling of sample data, followed by Iterative Proportional Fitting (IPF). The modeling aims to estimate accurately the interaction or odds ratios of complex tables, which is information not contained in the marginal subtotals. IPF ensures consistency with the subtotals. We advance this work in three practical ways. First, we show that detailed small-area estimates of both counts and proportional distributions usually gain accuracy by combining data for larger areas containing the small areas, and we illustrate the multilevel framework to achieve these estimates. Second, we find that a general classification or socioeconomic typology of the small areas is even more associated with the within-area interactions than is membership of the larger area. Third, we show how the Statistical Package for the Social Sciences (SPSS) can be used for IPF in any number of dimensions and with any structure of constraining marginal subtotals. Throughout, we use an example taken from the 1991 U.K. Census. These data allow us to evaluate various methods combining 100 percent tabulations and the Samples of Anonymised Records.

     

台风灾害多元致灾因子联合分布研究

以沿海城市海口为例,运用三维Copula函数构建台风灾害多元致灾因子联合分布模型,开展多元致灾因子相互作用下的台风灾害事件联合重现期和失效概率的分析,提出全面评估台风灾害影响的研究思路。结果表明：三维Gumbel Copula函数能够合理描述台风灾害多元致灾因子之间的联合分布,以单变量作为设计依据会低估具有一定严重程度的台风灾害发生频次,相对于单变量重现期和二维联合重现期,三变量联合重现期的计算结果更加贴近实际情况。防台措施设计标准的制定应全面考虑台风灾害多元致灾因子,且应充分考虑各致灾因子间的相互作用以及设计期内的失效概率。研究成果可为中国沿海省市的可持续发展以及减灾、防灾政策的制定等提供重要的科学依据。     

Bayesian inversion with Markov chains—II. The one-dimensional DC multilayer case

In this paper, we will report on the application of Bayesian inference to DC resistivity inversion for 1-D multilayer models. The posterior probability distribution is explored through a Markov process based upon a Gibbs's sampler. The process would lead to unrealistic estimates without additional prior information, which takes the form of a second Markov chain where the transition kernel corresponds to a smoothness constraint. The outcomes are posterior marginal probabilites for each parameter, as well as, if required, joint probabilities for pairs of parameters. We will discuss the main properties of the method in the light of a theoretical example and illustrate its capabilities with some field examples taken from various contexts.     

Extended weights of evidence modeling for the pseudo-estimation of metal grades

The weights of evidence modeling (WEM) for binary patterns is extended to take account of general categorical variables. The extension makes it possible to use the weights of evidence model in estimating the conditional probability distributions of metal grades. First, the target feature is converted into a set of binary target indicators. Second, the posterior probabilities are estimated for each of the target categories. Third, the estimates are combined to yield the posterior probability distribution of the target feature. Finally, the pseudometal estimates are derived from the probability distribution. The metal grade estimates are prefixed with pseudo, because the estimates are created from indirect evidence (explanatory variables). The pseudo-estimates provide a unique quantitative means to the delineation of exploration targets. This advantage reduces the ambiguities of target selection based solely on probability estimates. In order to use the generalized WEM, continuous geoscience attributes must be converted into categorical variables by means of optimal segmentation based on the target attribute of interest. The segmentation may be viewed as a process of defining evidence of the target feature. The extended weights of evidence model is demonstrated on a case study to select gold targets of Carlin type. The dataset used in the modeling includes apparent resistivity fields, soil geochemical samples, lithological and alteration information, and structural data.     

车辆GPS监控数据的自适应采集算法研究

在大规模GPS车辆监控中,当精度要求高时终端与中心端的通信负荷及中心端的数据处理负荷都很大,自适应采集技术可以有效解决这一问题。该文提出了车辆GPS监控数据的自适应采集算法,采用基于端点速度约束和里程约束的拟合多项式表达速度函数,综合考虑终端有地图和无地图的情况,采用偏差距离表示速度的方向偏差,并提出了数据的恢复算法、算法评价指标以及算法的应用模式。最后基于北京市的真实路网和实测GPS数据对算法进行了验证,恢复后的数据在速度和位置上都逼近真实数据,数据压缩率达3%,证明了算法的有效性。     

Random encounters in probabilistic time geography

Probabilistic time geography considers the encounter of moving agents to be random; therefore, a quantitative time geography analysis must consider the actual encounter probability. The existing algorithm of encounter probability is oriented over a discrete space and is sensitive to the unit definition of a virtual grid; thus, it is not suitable for continuous space. For this reason, a new method is presented in this paper for the encounter of two moving agents in continuous space. When the encounters are less than a specified distance threshold apart, an encounter event occurs based on the probability of the product, which is calculated by their respective probability density functions over their respective potential location areas. This probability provides a quantitative basis for predicting the likelihood of two agents meeting, as well as the location of this meeting point. Finally, the validity of the proposed model is verified by an experiment, which uses tracking data to calculate the encounter probabilities of three zebras and analyse the distribution characteristics of these probabilities over time and space.     

基于有限比较法研究万州区滑带土干重度的概率分布

对万州区滑坡滑带土干重度的86个测试数据进行拟合优度检验,发现多种传统的检验方法都既可以接受正态分布也可以接受对数正态分布.在传统检验方法失效的情况下,引入了有限比较法,确定了干重度的最优概率分布为正态分布,并用统计模拟的方法分析了有限比较法选中真实分布的概率.     

Bayesian and multivariate methods applied to favorability quantification in Recôncavo Basin and Ribeira Belt, Brazil

A methology to define favorable areas in petroleum and mineral exploration is applied, which consists in weighting the exploratory variables, in order to characterize their importance as exploration guides. The exploration data are spatially integrated in the selected area to establish the association between variables and deposits, and the relationships among distribution, topology, and indicator pattern of all variables. Two methods of statistical analysis were compared. The first one is the Weights of Evidence Modeling, a conditional probability approach (Agterberg, 1989a), and the second one is the Principal Components Analysis (Pan, 1993). In the conditional method, the favorability estimation is based on the probability of deposit and variable joint occurrence, with the weights being defined as natural logarithms of likelihood ratios. In the multivariate analysis, the cells which contain deposits are selected as control cells and the weights are determined by eigendecomposition, being represented by the coefficients of the eigenvector related to the system’s largest eigenvalue. The two techniques of weighting and complementary procedures were tested on two case studies: 1. Recôncavo Basin, Northeast Brazil (for Petroleum) and 2. Itaiacoca Formation of Ribeira Belt, Southeast Brazil (for Pb-Zn Mississippi Valley Type deposits). The applied methdology proved to be easy to use and of great assistance to predict the favorability in large areas, particularly in the initial phase of exploration programs.     

Prediction of categorical spatial data via Bayesian updating

This study introduces a transition probability-based Bayesian updating (BU) approach for spatial classification through expert system. Transition probabilities are interpreted as expert opinions for updating the prior marginal probabilities of categorical response variables. The main objective of this paper is to provide a spatial categorical variable prediction method which has a solid theoretical foundation and yields relatively higher classification accuracy compared with conventional ones. The basic idea is to first build a linear Bayesian updating (LBU) model that corresponds to an application of Bayes’ theorem. Since the linear opinion pool is intrinsically suboptimal and underconfident, the beta-transformed Bayesian updating (BBU) model is proposed to overcome this limitation. Another type of BU approach, conditional independent Bayesian updating (CIBU), is derived based on conditional independent experts. It is shown that traditional Markovian-type categorical prediction (MCP) is equivalent to a particular CIBU model with specific parameters. As three variants of the BU method, these techniques are illustrated in synthetic and real-world case studies, comparison results with both the LBU and MCP favor the BBU model.     

Statistic-empirical modelling of mountain permafrost distribution in the Abisko region,sub-Arctic northern Sweden

The mountain permafrost distribution in the Abisko region in northern Sweden has been assessed using a combination of empirical and statistical analysis. The empirical data was obtained using the bottom temperature of snow cover (BTS) method, supported by continuous ground temperature measurements. The statistical analysis was based on 148 data points in total and used logistic regression to model the probability of permafrost occurrence. Further, Geographically Weighted Regression (GWR) was introduced as an exploratory tool for detecting non-stationarity in the relationships between permafrost and the independent variables models and showed to be a useful tool in the statistical analysis. As a result of the GWR analysis the region was divided into two subregions. The models show probabilities >0.8 for permafrost at elevations above 1300 m a.s.l. in the western part of the region. In the eastern part, the probabilities are likely to be influenced by the potential incoming shortwave summer radiation, indicating a probability >0.8 above 850 m a.s.l. on the north-east and east-facing slopes, above 1000 m on the west-facing slopes and above 1100 m a.s.l. on the south-facing slopes. Permafrost conditions throughout the region were found to be marginal and sensitive to current warming trends.     

Conditional Independence Test for Weights-of-Evidence Modeling

Weights-of-evidence modeling is a GIS-based technique for relating a point pattern for locations of discrete events with several map layers. In general, the map layers are binary or ternary. Weights for presence, absence or missing data are added to a prior logit. Updating with two or more map layers is allowed only if the map layers are approximately conditionally independent of the point pattern. The final product is a map of posterior probabilities of occurrence of the discrete event within a small unit cell. This paper contains formal proof that conditional independence of map layers implies that T, the sum of the posterior probabilities weighted according to unit cell area, is equal to n, being the total number of discrete events. This result is used in the overall or omnibus test for conditional independence. In practical applications, T generally exceeds n, indicating a possible lack of conditional independence. Estimation of the standard deviation of T allows performance of a one-tailed test to check whether or not T-n is significantly greater than zero. This new test is exact and simpler to use than other tests including the Kolmogorov-Smirnov test and various chi-squared tests adapted from discrete multivariate statistics.     

东北气候和生态过渡区近50年来降水和温度概率分布特征变化

文章从中国160个站的观测资料中选取位于东北气候和生态过渡区内9个测站的冬、夏季降水和温度资料,分析该地区近50年来冬夏季降水和温度的年际变化及其概率分布特征,结果表明,东北气候和生态过渡区的冬夏季降水和温度有明显的年代际变化特征,在不同的年代际变化阶段,降水和温度的总体概率分布特征差异较大,这种概率分布形式的差异与高温、干旱等极端天气气候事件的频繁发生具有密切关系。20世纪80年代以来降水处于平均值减小的总体分布中,温度则处于平均值增加的总体分布中,因此该地区冬季发生暖冬和少雨(雪)的机会增大,夏季出现严重干旱和高温的可能性增大。