The Research On The Uncertainty Of Position Data In GIS In General Distribution Mode

The uncertainty of GIS is now one of the academic focuses in the community of GIS all over the world. The uncertainty of the position of spatial point and line segment underlies the theoretical research in the uncertainty of the position of GIS. For a long time, persistent efforts have been devoted to the research in the uncertainty in GIS. Great achievements and abundant experience have been harvested satisfactorily. It is noticed that the errors of the GIS position data are probably not normally distributed while most of the academic research in the uncertainty of GIS position data is being conducted on the basis of normal distribution of errors. Better achievements can be expected if the actual distribution mode of errors is taken into consideration in the research of uncertainty in GIS position data.This paper centers on the assumption that the actual distribution of the errors in GIS position data might be non-normal. A series of problem will arise without the assumption that the distribution of the errors is normal, such as: how to reach a reasonable assumption on the distribution of the errors of GIS position data? How to estimate the actual distribution of the errors? How to describe the differences or similarities between the actual distribution of the errors and the normal distribution? How to measure the quality of the non-normally distributed data? How to model the error propagation for non-normally distributed data? This paper presents a thorough systematic study of the preceding problems. The distribution of the errors of digitized data and the related issues are researched in the second chapter . In the first section the author deduces the distribution density of position errors in relation to coordinate error of points in the hope that it can serve as an answer to the frequent question from GIS users how the errors of points vary onthe average. In order to diminish and eliminate the systematic error of spatial data inGIS caused by the deformation of the map and digitized scanning, correction should be done to a digitized map. The second section discusses the conceptual model of thecorrection of a digitized map. Similarity transformation, affine transformation and polynomial regression model are all mathematical models describing the conceptual model. This section focuses on test and optimization of the correction models and a case study is conducted to test the optimization of a correction model and the effect of the correction. The third section presents an easier way of generating P-normdistributed sample with the help of pseudo-random number form computers. Based on the P-norm distributed samples, academic research is conducted on the methods of collocation testing of error distribution, with further research of the estimation of error distribution.Oriented to further study of Error Distribution Estimation, Chapter Three dwells on theories of Information Diffusion Estimation. The first section presents the concept and algorithm of Optimal Window-width of Diffusion Estimation. The second section puts forward the concept, theory and estimation techniques of overall optimal window-width, which enriches the theories of Diffusion Estimation with much more reliable estimating results as an essential solution to ascertaining the window-width and lays a solid theoretical foundation in making Diffusion Estimation the best-preferred technique in the research of error distribution.The third section presents the Analytic Collocation of Error Distribution based on Diffusion Estimation. Guided by this technique, we can get the unique analytic expression of an error distribution, which works better than the Collocation Testing methods of distribution based on the histogram or the cumulative distribution. The fourth section presents the theory and techniques of Maximum-Likelihood Estimation based on Information Distribution. This technique is able to automatically choose the best estimation method according to the actual error distribution, thus consequently enriches the variety of techniques in the study of error distribution and parameter estimation. In Maximum-Likelihood Estimation of single parameter, an objective standard for judging gross error is suggested independent of a prominent level.Chapter Four advances the similarity theory of error distribution and presents a calculating method of similarity concerning different kinds of problems. The achievements in this chapter offer a logical and natural guide rule for the estimation and analytical collocation of distribution, and serves as a theoretical foundation for the quantitative study of similarity, approximation and graduality of error distribution as well as the study of error of data-processing caused by the approximation of error distribution.In face of the fact that there is currently no covariance matrix information for GIS position data, the second section in Chapter Five offers a new technique of estimating variance by using corresponding points in overlaying of GIS when the weight matrix is unknown. The third section advances Maximum-Likelihood Estimation of the overlay layers' variance components, considering the correlation ofstatistic and its distribution (Wishart Distribution). In this way, the Maximum-Likelihood is adopted to estimating the variance components of each map layer (corresponding points), and ameliorates the achievements mentioned in section two when positional errors are normally distributed. The striking contribution of the second and third sections is that we don't have to know the correlation matrix or weight matrix of the overlay layers when we want to successfully evaluate the quality of the non-normal distributing or unknown distributing data. Entropy is a more appropriate index, especially when the variance of the data's error distribution does not exist. The fourth section offers a research on the uncertainty of entropy of error distribution in general distribution mode. The author points out the issue how to calculate the numerical value of the entropy of a continuous random variable, propounds the practical arithmetic of entropy-uncertainties on the basis of Diffusion Estimation, and educes the relation between calculating entropy's integral interval and interceptive error limit. When error is normally distributed, error ellipse is satisfactory enough to serve us as the standard tool to portray the randomicity of points on a plane. The fifth section presents the concept and theory of error oblate with which the author depicts the randomicity when positional error of points on planes obeys the P-norm distribution. It is the natural extension of error ellipse, and it is of definite statistic significance.The sixth chapter presents a series of study of the propagation of GIS position data error in the mode of general distribution. Currently, there is generally no distribution mode attached to GIS position data. By taking advantages of the contrast of overlaying and the analysis of statistics in the hope of finding out the statistical distribution of the position data, this chapter dwells on error distribution of point coordinates in the analysis of overlaying. The second section offers a deduction of the difference distribution between the positions of corresponding points and offers a detailed study of analytical collocation of error distribution when the similarity of distribution reaches the maximum. Since the difference between the positions of corresponding points is true error, the research based on it is of great significance. The third section studies the estimation of the error distribution of the positions of corresponding points. With the knowledge of general mode of error distribution, by restricting the solution to unknown function is a density function of P-norm distribution, the estimation of the distribution of the corresponding points' positional error is offered and the shortcut approximation solution based on the criterion ofMaximum-Similarity is advanced. Corresponding points represent the spatial-characterized points, thus the distribution from estimation supplies information for reference to the positional error distribution of other characteristic points. P-norm-Maximum-Likelihood Estimation is introduced to estimate the corresponding points' coordinates in the overlaying export map in the Section Four.The error band model of line segments in GIS is the difficult part of significant importance in the study of the uncertainty of position data in GIS. Most existing error bands are constructed based on points, that is to say, constructed in a certain way on the basis of error distribution of a random point on a line segment. The error band model based on points can hardly show us the probability of the line segments' true position falling within the error band. The seventh chapter puts forward the concept and theories of integral equi-density error band of a line segment in GIS, constructs the integral euqi-density error band of line segments, provides the short-cut approximation arithmetic concerning the probability of the line segments' true position falling within the error band, and offers here some relevant diagrams. Furthermore, based on the theories of error oblate, this dissertation constructs an integral equi-density error band model in the mode of general error distribution and lays a foundation for the practical application of error band of a line segment in GIS.