The Hypothesis Testing of Scale Parameter in Coordinate Transformation Model

As we know,the Bursa-Wolf coordinate transformation model is generally used in the transformation between two datum systems.This kind of model has a single scale parameter,which implies a hypothesis that the scale space is isotropic.The single scale parameter actually harmonizes the average scale of the three coordinate axis.When the precision of the three coordinate vectors has a large discrepancy,it is not wise to use the single scale parameter coordinate transformation model. Following this fact,we present a coordinate transformation model of three-scale parameters based on the hypothesis that the scale space is isotropic and analyze how to choose the scale parameter in coordinate transformation model theoretically.We also give its special instances, namely, the two-scale and non-scale parameter coordinate transformation models. From the comparison of external and internal precision,we can make a decision on the choice of scale parameter.This method,however,does not provide an objective standard and rule.The reason is that the results of the external and the internal tests do not match each other sometimes,which brings some difficulties to the choosing.In this paper,an united hypothesis testing model of scale parameter is given and the corresponding statistic of testing is constructed.This thinking way comes from the regression testing with linear bound.We can express the three-scale parameter coordinate transformation model as a linear model,and express its special instances such as two-scale parameter,non-scale parameter as linear bounds.Then we can make hypothesis testing on the choice among the coordinate transformation models with one-scale,two-scale,three-scale and non-scale parameter. There are three main steps in this hypothesis testing:①giving the linear bound equation,namely, the hypothesis testing condition; ②constructing the statistic of testing and calculating its value;and ③ giving the significance level and deciding whether the zero hypothesis is accepted or not.Two numerical examples are given and some helpful conclusions are shown in the paper.