Abstract
Standard deviational ellipse (SDE) has long served as a versatile GIS tool for delineating the geographic distribution of concerned features. This paper firstly summarizes two existing models of calculating SDE, and then proposes a novel approach to constructing the same SDE based on spectral decomposition of the sample covariance, by which the SDE concept is naturally generalized into higher dimensional Euclidean space, named standard deviational hyper-ellipsoid (SDHE). Then, rigorous recursion formulas are derived for calculating the confidence levels of scaled SDHE with arbitrary magnification ratios in any dimensional space. Besides, an inexact-newton method based iterative algorithm is also proposed for solving the corresponding magnification ratio of a scaled SDHE when the confidence probability and space dimensionality are pre-specified. These results provide an efficient manner to supersede the traditional table lookup of tabulated chi-square distribution. Finally, synthetic data is employed to generate the 1-3 multiple SDEs and SDHEs. And exploratory analysis by means of SDEs and SDHEs are also conducted for measuring the spread concentrations of Hong Kong’s H1N1 in 2009.
Highlights
Standard deviation arises as one of the classical statistical measures for depicting the dispersion of univariate features around its center
Rigorous mathematical derivations attempt to figure out the relationship between the confidence levels characterizing the probabilities of random scattered points falling inside a scaled standard deviational hyper-ellipsoid (SDHE) and the corresponding magnification ratio under the assumption that samples follow Gaussian distribution
Confidence analysis of standard deviational ellipse (SDE) and its extension into higher dimensional Euclidean space has been comprehensively explored from origin, formula derivations to algorithm implementation and applications
Summary
Standard deviation arises as one of the classical statistical measures for depicting the dispersion of univariate features around its center. Its evolution in two dimensional space arrives at the standard deviational ellipse (SDE), which was firstly proposed by Lefever [1] in 1926. SDE has long served as a versatile GIS tool for delineating the bivariate distributed features. It is typically employed for sketching the geographical distribution trend of the features concerned by summarizing both of their dispersion and orientation. SDE’s arrival once aroused great attention, a certain amount of consequent criticism followed as well, mainly due to the fact that Lefever’s defined curve is not an ellipse [2], but the standard deviation curve (SDC) as nominated by Gong [3]. Smith and Cheeseman [4] employ it for estimating
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
