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.     

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.     

广州市社会空间的因子生态再分析

运用因子分析、聚类分析方法对广州市中心区1990年人口普查数据进行了研究.从9类47个变量中抽取出5个形成广州市社会区类型的主因子,根据各主因子的得分,把广州市中心区划分为七类社会区,并把形成广州市社会空间分异现象的原因归结为五类.通过与1985年类似研究结果的比较,发现广州市社会空间结构的分异现象更趋明显,具体反映在人口“外溢”、居住条件改善及新开发区的形成等方面.最后就如何引导广州市社会空间结构演变进行了讨论.     

新时期国土空间治理的价值传导与目标演进

基于对国土空间及空间规划内涵的梳理分析,深入剖析新时期国土空间治理的内在价值传导关系,探讨面向高质量发展的市县国土空间总体规划指标体系构建逻辑。研究认为：1）国土空间规划目标与国土空间治理指标之间有着严密的价值传导关系,不同层级政府的战略目标和空间治理理念在传递中分化衍变,导致规划指标体系在关注对象、控制性、统筹性等方面的差异;2）市县国土空间总体规划指标体系的建构,需要基于价值传导和指标控制的协同,以及基于战略目标导向和国土空间治理的统筹,成为规划导向落实的重要抓手;3）多规并行时代的规划指标体系构建共识正在打破重组,构建面向高质量发展的规划指标体系需要认清国土空间规划的战略性和基础性作用,对国家战略的逐级落实和国土空间治理能力提升做出响应。     

Modelling of geographic cancer risk factor disparities in US counties

The goal of this research is to create a theoretical framework for the identification of cancer risk factor disparities and address the recognition of geographic patterns in these factors. 34 secondary variables covering the entire US at the county level in 2010 were analyzed, both individually and grouped (theoretically and statistically), in relation to the mortality to incidence ratio (MIR) for all cancer sites. An a priori assessment and a principal components analysis (PCA) were used to group variables to test societal constructs. OLS and geographically weighted regressions (GWRs) were used to assess influence of both individual and grouped variables against the MIR. The theoretical grouping of variables showed little change in predictive capability of OLS models. In GWR model, there was marked improvement over the OLS. Maps produced using local R2 showed clear regional patterns of influence between the indicators and the MIR. Both the theoretical model and the justification for a spatial approach to cancer risk factor disparities were shown to be effective in this paper. The link between this suite of indicators and the health outcomes is clear, and supports the idea that a full representation of the SES landscape should be used to both predict health outcomes and to assess policy options for improving these outcomes. With the presence of definitive regional patterns and clear connections between the MIR and societal groupings, the findings from this research suggest a need to shift to a more comprehensive and spatial approach to cancer disparities research.     

Geographically weighted regression with parameter-specific distance metrics

Geographically weighted regression (GWR) is an important local technique to model spatially varying relationships. A single distance metric (Euclidean or non-Euclidean) is generally used to calibrate a standard GWR model. However, variations in spatial relationships within a GWR model might also vary in intensity with respect to location and direction. This assertion has led to extensions of the standard GWR model to mixed (or semiparametric) GWR and to flexible bandwidth GWR models. In this article, we present a strongly related extension in fitting a GWR model with parameter-specific distance metrics (PSDM GWR). As with mixed and flexible bandwidth GWR models, a back-fitting algorithm is used for the calibration of the PSDM GWR model. The value of this new GWR model is demonstrated using a London house price data set as a case study. The results indicate that the PSDM GWR model can clearly improve the model calibration in terms of both goodness of fit and prediction accuracy, in contrast to the model fits when only one metric is singly used. Moreover, the PSDM GWR model provides added value in understanding how a regression model’s relationships may vary at different spatial scales, according to the bandwidths and distance metrics selected. PSDM GWR deals with spatial heterogeneities in data relationships in a general way, although questions remain on its model diagnostics, distance metric specification, and computational efficiency, providing options for further research.     

WEIGHTED ANALYSIS OF PRINCIPAL COMPONENTS: TWO APPROXIMATIONS TO STATISTICAL WEIGHTS

Two approximate methods for weighted principal components analysis (WPCA) were devised and testedin numerical experiments using either empirical variances (obtained from replicated data) or assumedvariances (derived from unreplicated data). In the first ('spherical') approximation each data vector wasassigned a weight proportional to the geometrical mean of its variances in all dimensions. Thearithmetical mean of variances was used instead in the other approximation. Both the numericalexperiments with artificial data containing random errors of various kinds (constant, proportional,constant plus proportional, Poisson) and the analysis of two sets of Raman spectra clearly indicated thenecessity of introducing statistical weights. The spherical approximation was found to be slightly betterthan the arithmetical one. The application of statistical weighting was found to improve the performanceof PCA in estimation problems.     

天津表土PAHs的空间主成分与污染源分析

在多元空间结构分析的基础上,应用空间主成分分析方法研究了天津表层土壤16种多环芳烃(PAHs)的空间主成分特征,并在此基础上探讨了空间污染源问题。分析结果显示,从总体来看,PAHs主要的污染源是燃烧源和石油源。从不同空间尺度来看,石油源的影响范围一般在5km以内,而燃烧源的主要影响范围在5~10km之间,在更大的范围上,可能是天然源,或者区域性的大气沉降。     

Nagmati River Sub-watershed Prioritization Using PCA,Integrated PCWS,and AHP: A Case Study

Prioritization of potential regions that are severely threatened by soil erosion is a prerequisite for formulating and implementing conservation measures and management practices, particularly in fragile semiarid regions. The present study prioritized eight delineated Nagmati sub-watersheds located in the Kutch District of Gujarat State, India, based on three approaches, namely principal component analysis (PCA), integrated PCA with weighted sum (I-PCWS), and analytical hierarchy process (AHP), and on 10 morphometric erosion risk parameters (ERPs). Sub-watersheds were categorized into three priority classes, namely high, medium, and low. PCA retrieved the three most significant ERPs (i.e., length of overland flow, Lo; drainage texture, Dt; and compactness coefficient, Cc) explaining 86.876% of the variance and exhibiting the highest correlation, i.e., r?=?0.961, 0.986, and 0.934 for the first three principal components. Sub-watersheds SW2 and SW7 were rated high priority, SW1 was rated low priority, and the rest were rated medium priority. The I-PCWS approach revealed that sub-watersheds SW2, SW6, and SW7 were in high-priority zone, followed by SW3, SW4, and SW8 in medium-priority zone and SW1 and SW5 in the low-priority zone. The AHP assigned the highest and lowest ranks to “Lo” and “Cc,” respectively, with consistency ratio of 8.1% and principal eigenvalue of 11.075. Results from AHP revealed sub-watershed SW2 to be the highest priority and sub-watersheds SW1 and SW5 to be the lowest priority. Out of eight prioritized sub-watersheds from three approaches, five were found to be the common priority classes, with SW2, SW6, and SW7 demanding urgent implementation of efficient soil conservation measures to prevent further degradation of the identified regions. Results from I-PCWS approach closely complied with the existing landforms within the study area, and this approach was considered more reliable and robust than the other two approaches. The methodology adopted in this study can be applied to different vulnerable, data-scarce regions to sustainably manage and conserve soil erosion through efficiently framed strategies.

     

用主分量方法分析广东春季低温阴雨年景

本文利用主分量方法对广东47站1954～1991年2～3月平均温度和广东2～3月间低温阴雨出现年景进行统计分析，根据主分量原理，计算该时期温度的时空分布特征，直接评价低温阴雨出现年景：①广东2～3月温度时空分布极为集中，第1主分量已占埸的总方差的95．1％；③用前4个主分量及其对应的特征向量配合划分温度分布类型；③广东2～3月温度分布主要由2个类型控制，即全省一致的偏低（或高）分布和南暖北冷或南冷北暧分布．由主分量极大值（正）和极小值（负）表明：1957、1968、1969年为全省性温度偏低年，1973和1987年为全省性温度偏高年。这些年份恰好对应广东2～3月低温阴雨严重和轻微（或无）的年份。     

SOIL AND VEGETATION PATTERNS IN BARRIER-ISLAND DUNE ENVIRONMENTS

Plant species distributions often have been attributed to landform characteristics or their associated geomorphic processes. This complicates interpretation of vegetation patterns in that geomorphic processes shape, and are shaped by, landforms. To characterize the biogeographic impacts of this interaction, I used principal components analysis (PCA) to examine hypotheses regarding the structure of variation among soil properties in active barrier-island dune systems. Dune soils and vegetation were sampled on two well-recognized barrier-island morphologies. On low-profile, wave-dominated microtidal barrier islands (South Core Banks, North Carolina) frequent overwash exerts a greater control on the distribution of soil properties. On mixed-energy mesotidal barrier islands (Sapelo Island, Georgia), overwash is less frequent, and the distribution of soil properties is shaped by a complex dune topography. Nontrivial principal components on both islands captured an equivalent amount of variance in the soil data. However, there were inter-island differences in the dimensionality of these nontrivial principal components, and differences in the distribution of variance and factor loadings. Suites of topography-modifying species, unique to each island, were uniform in the strength of their individual correlation with local edaphic variability. I posit that soil variance structure is a useful criterion to distinguish the relative influence on vegetation patterns of soil properties expressed through landforms (Sapelo Island) versus sediment transport processes (South Core Banks). [Key words: dune vegetation, barrier islands, principal components analysis (PCA), overwash.]     

基于主成分分析的关中地区农业粮食生产变化的影响因素研究

针对陕西省关中区域1978—2017年的农业生产数据,在分析关中40 a农业粮食生产的趋势变化后,运用主成分分析法,对影响关中农业生产中的地理环境和生产投入等主要因素进行了评价研究。结果表明：（1） 关中农业粮食生产的趋势变化呈现周期为3～7 a的循环增长方式,平均每周期峰值增长率为4.5%。（2） 主成分分析研究后得出,第一主成分全是地理因素指标,方差贡献率达到0.554,对关中地区农业粮食生产起着非常显著的决定影响作用,包括受灾农田面积（不含病虫害）、主要粮食作物播种面积、成灾农田面积（不含病虫害）、有效灌溉耕地面积、耕地面积;第二主成分方差贡献率为0.25,是影响粮食生产的重要因素和农业生产的生命补给。包括农业用电量、化肥、农用机械等生产资料投入和主要粮食作物稳产面积、劳动力投入因素指标;第三主成分为农药应用量,方差贡献率为0.068,影响较小。主成分累计方差贡献为0.872。通过对关中地区农业粮食生产变化的影响因素分析,可以为政府部门提出数据支撑和相关性的建议。     

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.     

Forthcoming events

Spatial data uncertainty models (SDUM) are necessary tools that quantify the reliability of results from geographical information system (GIS) applications. One technique used by SDUM is Monte Carlo simulation, a technique that quantifies spatial data and application uncertainty by determining the possible range of application results. A complete Monte Carlo SDUM for generalized continuous surfaces typically has three components: an error magnitude model, a spatial statistical model defining error shapes, and a heuristic that creates multiple realizations of error fields added to the generalized elevation map. This paper introduces a spatial statistical model that represents multiple statistics simultaneously and weighted against each other. This paper's case study builds a SDUM for a digital elevation model (DEM). The case study accounts for relevant shape patterns in elevation errors by reintroducing specific topological shapes, such as ridges and valleys, in appropriate localized positions. The spatial statistical model also minimizes topological artefacts, such as cells without outward drainage and inappropriate gradient distributions, which are frequent problems with random field-based SDUM. Multiple weighted spatial statistics enable two conflicting SDUM philosophies to co-exist. The two philosophies are ‘errors are only measured from higher quality data’ and ‘SDUM need to model reality’. This article uses an automatic parameter fitting random field model to initialize Monte Carlo input realizations followed by an inter-map cell-swapping heuristic to adjust the realizations to fit multiple spatial statistics. The inter-map cell-swapping heuristic allows spatial data uncertainty modelers to choose the appropriate probability model and weighted multiple spatial statistics which best represent errors caused by map generalization. This article also presents a lag-based measure to better represent gradient within a SDUM. This article covers the inter-map cell-swapping heuristic as well as both probability and spatial statistical models in detail.     

生态补偿效益、标准——国际经验及对我国的启示

随着环境压力的不断增大,生态补偿作为解决环境问题的创新手段日益受到重视,补偿效益是生态补偿的核心。通过综合分析近年来国内外生态补偿研究的相关文献,发现优化选择补偿区域和合理的补偿标准是提高补偿效益的关键,机会成本是应用较广的确定生态补偿标准的方法,不同区域提供的生态服务以及损失的机会成本有差异。我国对区域差异关注较少,补偿标准未能反映农户的真实成本,存在补偿不足或对不需补偿就能提供生态服务的区域实施补偿等问题,补偿资金利用效益较低。国际上比较注重生态补偿的区域差异,在生态补偿效益、促进环境保护积极性等方面效果较好。针对我国目前生态补偿中存在的问题,借鉴国际经验,提出了确定生态补偿标准的理论框架。     

2002-2015年中国社会保障水平时空分异及驱动机制

运用主成分分析法对2002-2015年的中国社会保障水平进行测算,分析其时空分异特征,同时运用地理加权回归模型分析其影响因素及驱动机制。研究表明：① 中国社会保障总体、地区水平在逐年提高,但区域相对保障水平差距有扩大趋势。② 中国社会保障水平发展不均衡,社会保障水平的“东—中—西”的格局与中国地区发展的格局相吻合。社会保障水平的热点区和冷点区表现出较明显的空间演变特征,高热点区在东部地区扩散并向中部地区辐射,冷点区在西部地区分布并不断加深和强化。③ 人均GDP、农村人均纯收入、城镇化率、教育水平、财政转移支付5个因素,形成经济、教育、财政和社会四大驱动力引致社会保障水平的时序变化和空间布局。     

乌鲁木齐市职住空间组织特征及影响因素

本文借助职住分离指数分析法从宏观层面分析了乌鲁木齐市辖区的职住空间匹配状况,利用问卷调查数据,从微观角度分析了不同类型居住区居民的通勤距离和通勤时间。研究发现：职住空间分离是乌鲁木齐城市空间结构的突出特征;全市职住空间不匹配程度较大,居住主导区的街道有35个,占街道总数的44.3%;就业主导区的街道有25个,占街道总数的31.65%;基本匹配区的街道仅有7个。居住区类型、居民受教育程度、职业类型、收入水平、住房形式等社会经济属性对其职住分离程度均有一定程度的影响;采用GWR模型定量分析了职住空间关系的影响因素,认为学历构成、住房产权性质、职业类型、流动人口以及住房面积是影响乌鲁木齐市职住空间关系的主要因素。     

TEMPORAL CHARACTERISTICS OF PENNSYLVANIA SNOWFALL, 1950–1951 THROUGH 1989–1990

While speculation exists on global and regional climate change for temperature and precipitation, relatively little research is available on snowfall and its changes. Twenty-six sites were selected to describe and analyze various characteristics of snowfall in Pennsylvania from 1950–1951 through 1989–1990. Overall, the state experienced a significant stepwise change in seasonal snowfall total during this period. Abnormally high seasonal totals prevailed from 1957–1958 through 1971–1972. This rise in snowfall was accompanied by colder-than-normal temperatures and a dramatic increase in large daily snow events. Principal components analysis (PCA) revealed that the seasonal temporal patterns were not uniform across the state. The PCA revealed four distinct seasonal regions. These regions exhibited everything from nearly linear increases and decreases over time to cyclical formations. PCA performed on the months of November through April each unveiled between three and five separate temporal regions. PCA analyses generally identified an eastern region, a north-central region, and a western region across Pennsylvania. [Key words: regionalization, principal components, variability, snowfall, climate change, Pennsylvania.]     

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。     

Approximation of Gaussian spatial autoregressive models for massive regular square tessellation data

Most of the literature to date proposes approximations to the determinant of a positive definite n × n spatial covariance matrix (the Jacobian term) for Gaussian spatial autoregressive models that fail to support the analysis of massive georeferenced data sets. This paper briefly surveys this literature, recalls and refines much simpler Jacobian approximations, presents selected eigenvalue estimation techniques, summarizes validation results (for estimated eigenvalues, Jacobian approximations, and estimation of a spatial autocorrelation parameter), and illustrates the estimation of the spatial autocorrelation parameter in a spatial autoregressive model specification for cases as large as n = 37,214,101. The principal contribution of this paper is to the implementation of spatial autoregressive model specifications for any size of georeferenced data set. Its specific additions to the literature include (1) new, more efficient estimation algorithms; (2) an approximation of the Jacobian term for remotely sensed data forming incomplete rectangular regions; (3) issues of inference; and (4) timing results.