空间分析文献综述

1、空间分析课程文献阅读综述空间分析是基于地理对象的位置和形态特征的空间数据分析技术，其目的在于提取和传输空间信息。一个地理信息系统应当具有完备的空间分析功能，也就是说空间分析是地理信息系统的核心。本课程涉及空间数据，空间位置，空间分布，空间形态，空间关系和地统计学的内容，由于阅读有限，故只对局部阅读过的内容做出综述。一、 用加权多项式回归进行球状模型变差图的最优拟合【1】中国地质大学的王仁铎教授提出了加权回归多项式法拟合参数，推进了地质统计学计算过程自动化的研究。在此之前，对于变差函数的拟合一般通过做实验变差函数图，通过肉眼观察来进行人工拟合。在地质出版社1981年出版的?地质统计学及其在矿产储

2、量计算中的运用?候景儒，黄竞先一书中介绍了两种理论变异曲线的构制，即手工拟合和利用最小二乘法拟合，在最小二乘法拟合中，由于实验变异曲线的头几个点的可靠性要比尾部的点大得多，如果不考虑这一因素，所得到的理论曲线势必产生偏移。最小二乘法是一种纯数学的方法，它不能充分反映地质特征【2】。而人工拟合中，那么是通过对实验变异函数散点图的观察来确定各参数的，过程耗时，费力，结果因人而异，主观性强，缺乏统一、客观的标准【3】。地质统计学有个根本工具就是变差图r(h),在二阶平稳或本征内蕴假设下，其定义为： 1其中h为沿一定方向的间隔(或称根本滞后),x 为空间点。由于变差图的性状能够同时反映区域化变量的结构

3、特征和随机特性, 在计算估计方差、离散方差、正那么化变量的变差函数和克立格方程组中的系数时都是以变差函数的计算为根底的。因此, 变差图确定得好坏就成为应用地质统计学方法能否取得成效的关键之一。球状模型spherical model由地统计学理论奠基者法国学者G.Matheron提出，也称马特隆模型。实践证明，实际工作中95%以上的实验变异函数散点图都可以用该模型拟合。其一般公式为：其中c0为块金常数,a为变程,c为拱高,c0+c为基台值。要拟合出一条最优的球状模型变差函数曲线, 就是要确定最优的球状模型中的三个参数c0,a和c。用加权多项式回归拟合球状模型变差图。设通过1式已算出n个实验变差函

4、数值r*(hi)(i=1,2,，n)，它是用Ni对数据对计算出来的，那么球状模型可写成： (2)要用2式对实验变差函数值点进行最小二乘意义下的最优拟合，可用加权多项式回归。令 3那么2式化为 4于是，拟合就化为加权二元回归了。由于点对数越多这些点往往是变异函数的头几个点，根据它们算出的变异函数值代表性就越强，故以数据点对数目为权进行加权二元线性回归。根据加权二元线性回归理论可得4式在最小二乘意义下的三个参数b0,b1,b2 如下:其中：但在最终拟合结果中，当b0<0时，人为规定b0=0，再进行重新拟合。此处理论模型参数的正负号问题没有解决,这是该方法的薄弱处。对于加权多项式回归拟合方法，

5、在许多论文中都提及，是使用较为普遍，运用较为广泛的，在科学出版社2021年出版的?地统计学概论?刘爱利，王培法，丁园圆一书中，介绍的变异函数理论模型的最优拟合就是主要介绍了这种方法，可见其是比拟成熟的。但对于上述问题，也有人提出过其它方法来进行拟合，如长春地质学院的矫希国和刘超提出用线性规划拟合理论模型的参数,由线性方程组非负解理论解决了理论模型参数的正负号问题。二、 球状模型的最优参数估计【4】查恩斯与库伯于1961 年提出了目标规划问题。目标规划通过引入偏差变量、绝对约束、相对约束、优先因子( 优先等级) 与权系数等概念, 结合线性规划, 使得求解最优化问题更加灵活。偏差变量设原问题第i个

6、约束条件为 5以变量作为该约束条件的正偏差，表示决策值超过目标值的局部；以变量作为该约束条件的负偏差, 表示决策值未到达目标值的局部。绝对约束和目标约束 绝对约束是指必须严格满足的约束, 如线性规划问题的所有约束条件, 不能满足这些条件的解为非可行解, 所以他们是硬约束。目标约束是目标规划特有的, 可把约束右端项看作要追求的目标值。在得到此目标值时允许发生正或负偏差。因此, 在这些约束中参加正、负偏差变量, 它们是软约束。原问题的绝对约束变为下式: 6优先因子( 优先等级)与权系数 凡要求第一位到达的目标赋予优先因子P 1, 次位的目标赋予优先因子P 2, , P k, 并规定：Pk>

7、> Pk+ 1 ( k= 1, 2, , K ) , 表示Pk 比Pk+ 1有更大的优先权。即首先满足目标Pk , 不考虑其它目标, 然后在满足目标P k 的根底上考虑目标P k+ 1, 依次类推。目标规划的目标函数目标规划中的目标函数是按各目标约束的正、负偏差变量和赋予相应的优先因子而构成的。当某一目标确定后，决策者的要求是尽可能小的偏离目标值。因此目标规划的目标函数只能是 7对于有n 个决策变量、m 个目标约束、目标函数中有K 个优先级的目标规划问题，其一般数学模型为:其中：在进行变差函数理论模型的拟合时, 可以根据经验知识、实验变差函数值点的可靠程度确定目标值、优先等级、权系数等,

8、 建立相应的线性目标规划数学模型, 以得到最满意的变差函数参数值。目标规划模型中的变量都必须满足非负约束, 不可能计算出负的参数值, 不存在理论模型参数正负号问题。目标规划法的约束条件是软约束, 不管约束条件之间和约束条件与目标函数的关系如何, 都能求出满意解, 不会出现线性规划中无可行解的情况。目标规划法拟合实验变差函数同样地，将球状模型变化为4式的线性形式，令实验变差函数值为vi( i= 1, 2, , m) , 根据目标规划原理列出约束条件为: 8根据专家经验可列出具有不同层次和权重的目标函数, 其一般形式为: 9在实际操作中，根据理论变差函数拟合的目标规划一般数学模型, 建立实例的数学

9、模型，用目标规划单纯形法求解该模型，即得到拟合后的球状模型。目标规划法能够结合专家经验知识, 提高理论变差函数的可靠性, 在实现地质统计学计算过程自动化方面解决了加权回归多项式法和线性规划法线性规划模型完全由实验变差函数决定, 给予各实验变差函数值相同的权重分别存在的问题。可见该方法还是有一定优越性的。但是，目标规划那么缺乏简便性,不便于用计算机进行自动化拟合【5】。显然，该方法具有很大优越性，只要对其计算机实现进行改造，使之更加简便，还是有一定的可能的。三、 理论变异函数球状模型的加权线性规划法拟合【6】前面提到，线性规划法由于没有考虑实验变差函数的头几个点的重要性而在球状模型拟合中存在缺乏

10、，这里提出加权线性规划法拟合法。同样地，将球状模型变化为4式的线性形式，鉴于C0 、C、a 均有非负要求，所以也要求b0 0 , b1 0 , b2 0。如果在计算实验变异函数时有m对数据: hj , r( hj) ( j = 1 ,2 , ., m) , 可先将这m 对数据按(4)式变换成: yj , x1 j , x2 j ( j = 1 ,2 , .,m) , 并令tj = |yj - b0 - b1 x1j - b2 x2j |( j = 1 ,2 , ., m)。同样对不同的实验变差函数值应赋予不同的权重，按照最小二乘原理，最正确拟合为： 10其中Wj为权重，如一中所介绍。所以，球状

11、模型的拟合就变成了一个线性规划问题，上述所有可写成： 11可将11式写成矩阵形式： 12求出b0，b1，b2之后便可求出C，C0，a。 由上所介绍可以看出，该方法综合加权多项式拟合法及线性规划拟合法的各自优点，且在计算上较目标规划拟合法更为简单。以上介绍了三种变差函数球状模型的拟合方法，各有优缺，综合来看，我认为加权多项式拟合的方法是比拟成熟的，运用也比拟多，对于其余两种方法也有很明显的优势。四、Spatial interpolation of monthly climate data for Finland:comparing the performance of kriging and g

12、eneralized additive models【7】The object of this paper is to compare three different methods for spatial prediction: kriging with external drift (KED), GAM, and GAM combined with residual kriging (GK).Geostatistical methods include ordinary kriging, universal kriging, co-kriging, and indicator krigin

13、g (Goovaerts 1999). In many cases, natural processes show a curvilinear rather than linear relationship (e.g.,Leathwick et al. 2006; Hjort and Luoto 2021), and the possibility of relaxing the assumptions about linearity to let the data determine the shape of the response is often appreciated (Kamman

14、n and Wand 2003) Generalized additive models (GAM) are a semiparametric extension of generalized linear models (GLM) (Hastieand Tibshirani 1990), with the benefit of no strict para-metric form, allowing a more flexible response (Hastieand Tibshirani 1984).文档来自于网络搜索1.Study area and dataThe mean surfa

15、ce temperature and annual accumulation of Finland. And a spatial grid with a resolution of 1 × 1 km is constructed for the entire country, containing a total of 336,897 points include precipitation observations and temperature observations.2. MethodsKrigingThe trend surfaces were fitted using t

16、he least-squares method, and the same terms were used for all months.The final trend model for monthly temperature wasand, for monthly precipitation,where x and y are the coordinates in meters (Euref Fin TM35 coordinate reference system), e is the mean elevation, l is the lake percentage, and s is t

17、he sea percentage.Generalized additive modelsKriging with external drift assumes linearity between the response and the covariates, but this assumption rarely holds true in environmental studies.GAM are semiparametric extensions of GLM (McCullagh and Nelder 1989)where no parametric coefficients are

18、estimated for predictors. GAM allow a nonlinear relationship between the response and explanatory variables (Hastie and Tibshirani 1990). GAM have the form (Hastie and Tibshirani 1990;Wood and Augustin 2002):where g() is the link function, 0 is a constant, sk is the smoothing parameter to be estimat

19、ed, and xk is the explanatory variable. GAM aim to minimize the generalized cross validation (GCV) criterion (Wood 2021),where D is the deviation, n the number of data points, and DoF is the effective degrees of freedom in the mode.GAM combined with residual krigingBy combining the GAM-modeled trend

20、 with geostatistics, we obtain geoadditive models (Kammann and Wand 2003; Kneib et al. 2021). The large-scale trend is modeled with GAM, and kriging is then performed on the spatially autocorrelated residuals.The final interpolation result is a combination of both elements (Ginsbourger et al. 2021)

21、referred to as GK.3. EvaluationThe most important part of any interpolation task is to evaluate the concordance between predictions and reality. Three statistical measures were calculated to assess the performance: root mean squared error (RMSE), mean absolute difference (MAD), and Pearsons correlat

22、ion coefficient (COR) (Hofstra et al.2021). According to Wilmott (1982), RMSE and MAD provide good overall measures of model performance, as they summarize the mean error in the same units as the observed and predicted values. RMSE is defined here as文档收集自网络，仅用于个人学习where Oi is the observation and Pi

23、the predicted value. RMSE is sensitive to outliers and can be used as indicator of the magnitude of extreme errors (Price et al 2000). MAD is less sensitive to extreme values and is defined here asStandardized prediction error (SPE) is also calculated:where Vari is the kriging variance.The standard

24、deviation of SPE, SPE, is ideally close to one indicating that themagnitude of the kriging variance is correct, and the variogram model is realistic (Bivand et al. 2021). Also included in the output datafiles are the lower (2.5 %) and upper (97.5 %) quantiles of SPE, which, if close to 2 and 2, indi

25、cating normally distributed prediction errors as assumed.文档收集自网络，仅用于个人学习 These measures provide a quick overview of the success of the interpolation routine.4. ResultsTable 1 Example of statistical measures included with the KED output file for September 2021:Table 2 Interpolation results for monthl

26、y mean temperature over the period of 19812021:Table 3 Interpolation results for monthly mean precipitation over the period of 198120215.ConclusionsAccording to the cross-validation statistics for the 30-year period, we have shown that there was little difference between the methods tested, especial

27、ly for mean temperatures, where all three methods produced results with small errors and high correlation between observed and predicted values. GAM was the best overall method for predicting monthly mean temperature. The same interpolation methods showed larger differences for monthly mean precipit

28、ation, with KED and GK showing best overall performance. It is possible to improve interpolation accuracy using nonparametric methods such as GAM for climate variables which show little or no spatial autocorrelation between observations (e.g., temperature). When combined with residual kriging, these

29、 methods can be extended to variables with spatially autocorrelated errors (e.g., precipitation). KED proved accurate and stable and was selected for interpolating monthly interpolations because of the robustness of kriging. 资料个人收集整理，勿做商业用途五、Point-Set Topological Spatial Relations【8】The model of top

30、ological spatial relations is based on the point-set topological notions of interior and boundary. And there are two basic results in point-set topology:(1)Let X be a set. A topology on X is a collection Y of subsets of A that satisfies the three conditions: the empty set and X are in Y ;Y is closed

31、 under arbitrary unions, and Y is closed under finite intersections.(2)Closed set. The collection of closed sets: contains the empty set and X,is closed under arbitrary intersections, and is closed under finite unions.Besides, there are three fundamental concepts and three crucial concepts. They are

32、 Interior, Closure, Boundary and Separation, Connectedness, Topological Equivalence.Empty/non-empty is the simplest and most general invariant so that any other invariant may be considered a more restrictive classifier. So, we assign the appropriate value of empty andnon-empty to the entries in the

33、four-tuple, so there are sixteen possibilities.The framework for the spatial relations between point-sets carries over to spatial regions, however, not all of the sixteen relations between arbitrary point-sets exist between two spatial regions. we conclude that at least the relations r0, r1 , r3 , r6 , r7, r10, r11, r14, and r15 exist between two spatial regions. And these nine topol