误差椭球的性质及其在置信域问题中的应用

The Properties of Error Ellipsoid and Its Applications in Confidence Regions of Points

研究了空间点定位误差的误差椭球的性质和空间点误差的表示.以概率论和多元分析为工具,证明空间点误差落入椭球内的概率可以由标准正态分布的分布函数和密度函数表示;证明了空间点误差可以沿误差椭球主轴方向分解为3个独立的分量,并给出了分解方法和算例;证明点误差落在误差椭球一侧内的概率为常数,该常数与切点坐标无关且可以由标准正态分布函数表示;讨论了点误差的可视化表达问题,为空间点建立了椭球形及长方体形置信域,并给出了置信水平的计算公式.

The positional errors of points is assumed to be distributed with 3 dimensions normal distributions, on the basis of which, the error ellipsoid is defined as the isoplethic surface of the probability density function. By employing the theory of the quadratic form and multivariate analysis, three properties concerning error ellipsoid are proved ： （1） The errors of points can be expressed in independent variables; （2） The error drops into error ellipsoid with a constant probability, and this probability can be＇determined by probability density function and distribution functions of the standard normal distribution. （3） The error drops into one side of the plane tangent to error ellipsoid with a constant probability, and this probability is independent of the coordinates of the tangent point. The properties mentioned above can be used to construct the confidence regions, which is a convenient way to express the uncertainly of points. Numerical example shows the process of establishing confidence regions which satisfies the given confidence level.