Theoryandapplication of Leastabsolute Deviation
|Keywords||Linear model Non-linear model Olgarithm Grey System Model Gray convex related degree|
As early as in 1760, the idea of ??Least Absolute, 1786, Laplace studied the problem, significantly earlier than the least-squares method proposed by Boscovich, l 1 norm minimization problem has continued without interruption since the the l 1 norm minimization problem of the objective function is not differentiable, easy to calculate, until the 1950s, the advent of the computer l 1 mode pole small problem solving easier subsequently been widely used, and so research the l 1 -mode minimization problem is very important. This paper studies the l 1 norm minimization problem, the main work is as follows: Chapter 2, systematically summed up the previous linear model 1 1 mode minimization the work, compare the pros and cons of the different algorithms; and the assumption of certain conditions, olgarithm applied to the first-order autoregressive model to prove that the nature of the parameters; l ∞ regression and quantile regression; and solving problems related procedures. The third chapter analyzes the existing non-linear model l 1 norm minimization problem and prove the the l 1 mode minimization problem with minimax problems equivalence this l 1 norm minimization problem can be sequential quadratic programming method for solving; taking into account the disadvantages of this method, a hybrid genetic algorithm. The fourth chapter, first a brief introduction to the emergence and development of the gray system theory, pointed out the advantages and disadvantages of existing modeling method, the average relative error of the absolute value of the minimum criteria given the index of non-homogeneous model linear programming examples show that compared with other methods, this method can reduce the average absolute relative error, improve the modeling accuracy. Use the second method to directly establish a non-homogeneous exponential model, the examples show that this method can further reduce the average absolute relative error, improve the modeling accuracy, and also shows the robustness of the smallest multiplication, and based on gray system The basic principle is given the smallest multiplication of the solution is not unique countermeasures. Finally, the existing lack of smoothness, continuous smoothness, defines the smoothness of translation invariance transformation invariance, and multiply, and prove continuous smooth translational invariance, and multiply invariance; using discrete data convexity, the gray projection correlation degree and prove the gray projection correlation degree with consistency and several multiply transformation rank preservation, through examples demonstrate the effectiveness and practicality of the gray projection correlation degree. The last chapter gives a summary and discussion of the relevant issues.