熵理论在确定点位不确定性指标上的应用

分析了传统点位不确定性指标的局限性,基于信息论中的联合熵和最大熵定理导出了n维随机点熵不确定指标以及落入其内概率的统一公式;提出了以熵误差椭圆与熵误差椭球作为2维、3维GIS中点元的位置不确定性度量指标.提出的熵指标具有唯一确定、不受置信水平选取的主观性影响等特点,适合于度量未知分布的点位不确定性.     

二维随机点的熵误差指标

根据信息论基本原理，提出了二维随机点的熵误差指标，该指标与以往的误差指标不同，它是一个在熵意义下唯一确定的，与置信概率无关的客观指标，把它作为二维点位置不确定性的度量指标具有特殊优越性。     

光斑影响下激光点云的不确定性评价

利用光斑的特性确定激光点位在光斑中的不确定性,将误差熵引入到激光点位不确定性的评价中。根据激光反射特性,确定了激光点位不确定性的概率密度函数,利用信息熵的定义推导了激光点位的信息熵,同时,利用信息熵与误差熵的关系进行了激光点位误差熵的推导,根据误差熵关系式确定了误差熵与光斑面积的线性关系。根据点云光斑实际面积,得到了点云误差熵及每个激光点位的平均误差熵。利用入射角与误差熵之间的关系,分析了入射角对激光点位不确定性的影响程度,确定了扫描的最佳入射角范围。通过设置不同扫描间隔得到的点云数据,验证了利用误差熵对点云不确定性进行评价的可行性。     

利用误差熵评价光斑中点云不确定性

三维激光扫描点位精度受光斑影响较大,激光点在光斑中呈现了不确定性,该不确定性的准确描述关系到激光点位精度的评价。将误差熵模型引入到点位不确定性的评价中,利用激光点位在光斑中不确定性的概率密度函数,推导了激光点位信息熵,并依据误差熵与信息熵的关系得到了激光点位的误差熵。通过分析误差熵与光斑面积的关系,得到点云光斑平均误差熵,实现了将平均误差熵引入到点云不确定性的评价中。通过设置不同扫描间隔得到的点云数据,分析了平均熵模型进行基于光斑影响下的点云精度评价的可行性,最终实现了对光斑中点云不确定性的准确评价。     

GIS中三维空间直线的误差熵模型

从信息熵的角度提出了三维空间直线的误差熵模型，该模型由以垂直直线的平面误差熵为半径的圆柱体和两端点的误差球组成，是一种完全确定的度量空间线元不确定性的模型。理论分析与实验表明，本文所提出的模型具有较好的效果。     

未知分布误差的熵不确定度

分析现有估计方法的不足，提出基于最大熵的不确定度估计。所得的指标不受置信水平选取时的主观性影响，适合于GIS中未知分布误差的不确定性度量。     

关于“新的点位误差度量”的讨论

多维点位误差度量应该反映点的位置的不确定度,它可以是方差或方差的某种函数.实践中,有多种点位误差的表示形式,不同的表示形式有不同的几何意义或概率意义.文献[1]提出采用坐标分量方差的均值作为点位误差的度量.为了将其与现有点位误差度量进行比较,本文列出4种点住误差度量表达式,并讨论了它们的几何意义.强调指出,点位误差度量的具体数值应大于等于最大可能的误差,以保证用户使用数据的安全性.     

利用误差熵确定激光点云变形可监测指标

利用三维激光扫描确定变形区域的主要方法是对相同区域的点云进行对比分析,根据对比值确定变形区域及变形量,这种方法虽然能够简单地实现变形监测,但对于监测结果的可靠性并没有进行评价。因此,为了提高变形监测结果的可靠性,对点云误差及点云配准误差进行分析,并由此确定点云变形监测的可监测指标。为了避免相邻点位误差之间的相互影响及误差空间大小的不确定性影响,利用误差熵来确定点云误差空间,并根据其实际大小和误差极值的关系来确定变形可监测指标。通过不同距离和入射角下模拟的平面板变形来验证其可行性,并将该方法应用于某个滑坡场景,以确定该滑坡的变形区域和变形大小。     

卫星导航的不确定性、不确定度与精度若干注记

卫星导航定位必然有误差。其误差可分为偶然误差、系统误差、异常误差、有色噪声等。误差存在多种不同的度量模型和度量方法,如,精密度（precision）、精确度（accuracy）、可靠性（reliability）、不确定度（uncertainty）等。实践中,经常有学者和工程技术人员将精度指标描述成误差,也有人将不确定性与不确定度概念相混淆。尤其是在描述精度指标或误差指标时,经常将精度指标描述成误差,将误差指标描述成精度,如在卫星导航定位中,用户距离误差经常被描述成用户距离精度。本文基于不确定度概念将卫星导航中的用户距离误差重新作了定义;给出了用户距离误差与用户距离精度的关系;并提出将不确定性与不确定度进行区别;对测量平差中常用的可靠性概念进行了描述;最后给出了几点注记。     

线元点位误差带的“纺锤形”模型

对当前GIS界流行的以点位误差描述线元位置不确定性的误差带理论提出相反的观点。最早提出的线元误差带理论为"ε-带"模型,后来又提出了"E-带"模型和在其基础上发展的"G-带"模型。后两者均认为以控制点点位误差描述的线元的误差带的基本形状呈"哑铃"形,即认为线元上端点的位置不确定性大于端点之间的点的位置不确定性。笔者的看法与此相反,笔者认为线元上两控制点之间的点的位置不确定性应大于控制点的位置不确定性,且在两控制点的中间达到最大,即线元误差带的基本形状应为"纺锤形"而不是"哑铃形"。     

General Entropy Model of Line Segments Uncerta,nty in GIS

Spatial data uncertainty can directly affect the quality of digital products and GIS-based decision making. On the basis of the characteristics of randomicity of positional data and fuzziness of attribute data, taking entropy as a measure, the stochastic entropy model of positional data uncertainty and fuzzy entropy model of attribute data uncertainty are proposed. As both randomicity and fuzziness usually simultaneously exist in linear segments, their omnibus effects are also investigated and quantified. A novel uncertainty measure, general entropy, is presented. The general entropy can be used as a uniform measure to quantify the total uncertainty caused by stochastic uncertainty and fuzzy uncertainty in GIS.     

General entropy model of line segments uncertainty in GIS

Spatial data uncertainty can directly affect the quality of digital products and GIS-based decision making. On the basis of the characteristics of randomicity of positional data and fuzziness of attribute data, taking entropy as a measure, the stochastic entropy model of positional data uncertainty and fuzzy entropy model of attribute data uncertainty are proposed. As both randomicity and fuzziness usually simultaneously exist in linear segments, their omnibus effects are also investigated and quantified. A novel uncertainty measure, general entropy, is presented. The general entropy can be used as a uniform measure to quantify the total uncertainty caused by stochastic uncertainty and fuzzy uncertainty in GIS.     

General Entropy Model of Line Segments Uncertainty in GIS

Spatial data uncertainty can directly affect the quality of digital products and GIS-based decision making. On the basis of the characteristics of randomicity of positional data and fuzziness of attribute data, taking entropy as a measure, the stochastic entropy model of positional data uncertainty and fuzzy entropy model of attribute data uncertainty are proposed. As both randomic-ity and fuzziness usually simultaneously exist in linear segments, their omnibus effects are also investigated and quantified. A novel uncertainty measure, general entropy, is presented. The general entropy can be used as a uniform measure to quantify the total un-certainty caused by stochastic uncertainty and fuzzy uncertainty in GIS.     

GIS中线元的误差熵带研究

基于现有的线元位置不确定性模型大多与置信水平的选取有关,而置信水平的选取带有一定程度的主观性,因而不能惟一确定,引入信息熵理论,提出了线元的误差熵带模型,并将它与“E－带”进行了比较,计算了落入其内的概率。该模型根据联合熵惟一确定,与置信水平的选取无关。     

Entropy error model of planar geometry features in GIS

Positional error of line segments is usually described by using “g-band”, however, its band width is in relation to the confidence level choice. In fact, given different confidence levels, a series of concentric bands can be obtained. To overcome the effect of confidence level on the error indicator, by introducing the union entropy theory, we propose an entropy error ellipse index of point, then extend it to line segment and polygon, and establish an entropy error band of line segment and an entropy error donut of polygon. The research shows that the entropy error index can be determined uniquely and is not influenced by confidence level, and that they are suitable for positional uncertainty of planar geometry features.     

Entropy error model of planar geometry features in GIS

Positional error of line segments is usually described byusing “g-band”,however,its band width is in relation to the confidence level choice.In fact,given different confidence levels,a series of concentric bands can be obtained.To overcome the effect of confidence level on the error indicator,by introducing the union entropy theory,we propose an entropy error ellipse index of point,then extend it to line segment and polygon.and establish an entropy error band of line segment and an entropy error do-nut of polygon.The research shows that the entropy error index can be determined uniquely and is not influenced by confidence level,and that they are suitable for positional uncertainty of planar geometry features.     

GIS中空间数据不确定性的混合熵模型研究

基于信息理论和模糊集合理论，针对GIS中部分空间数据既具有随机性又具有模糊性的特点，建立了空间数据不确定性的混合熵模型。以GIS中线元不确定性为例，讨论了线元不确定性的统计熵、模糊熵和混合熵估计方法，并针对特例给出了线元不确定性的熵带分布。     

GIS中基于εσ模型的随机折线元位置不确定性的可视化

首先研究基于εσ模型单一折线段不确定性误差带,导出误差带边界线的解析表达式;然后通过算例分析,针对开折线和闭折线两种情况,由单一折线段误差带边界线的解析表达式,编程绘出位置不确定性随机折线的可视化图形。理论分析和可视化图形表明,在两条相邻折线的公共端点处,前一线段的右误差半圆的半径和后一线段的左误差半圆的半径未必相等,实际分析中需考虑到这种情况。