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EBook - Mathematical Methods for Robotics and Vision Part 11

Tham khảo tài liệu 'ebook - mathematical methods for robotics and vision part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 7.4. BLUE ESTIMATORS 91 least squares criterion. In section 7.4.2 we will see that in a very precise sense ordinary least squares solve a particular type of estimation problem namely the estimation problem for the observation equation 7.12 with a linear function and n Gaussian zero-mean noise with the indentity matrix for covariance. An estimator is said to be linear if the function is linear. Notice that the observation function can still be nonlinear. If is required to be linear but is not we will probably have an estimator that produces a worse estimate than a nonlinear one. However it still makes sense to look for the best possible linear estimator. The best estimator for a linear observation function happens to be a linear estimator. 7.4.2 Best In order to define what is meant by a best estimator one needs to define a measure of goodness of an estimate. In the least squares approach to solving a linear system like 7.13 this distance is defined as the Euclidean norm of the residue vector y- x between the left and the right-hand sides of equation 7.13 evaluated at the solution x. Replacing 7.13 by a noisy equation y x n 7.14 does not change the nature of the problem. Even equation 7.13 has no exact solutionwhen there are more independent equations than unknowns so requiring equality is hopeless. What the least squares approach is really saying is that even at the solution x there is some residue n y- x 7.15 and we would like to make that residue as small as possible in the sense of the Euclidean norm. Thus an overconstrained system of the form 7.13 and its noisy version 7.14 are really the same problem. In fact 7.14 is the correct version if the equality sign is to be taken literally. The noise term however can be used to generalize the problem. In fact the Euclidean norm of the residue 7.15 treats all components all equations in 7.14 equally. In other words each equation counts the same when computing the norm of the residue. However different equations can .