Theoryandapplication Of Leastabsolute Deviation

The first idea of l₁ problem is attributed to Boscovich in 1760, Laplace studied this problem in 1786, so the l₁ problem is old than the l₂ problem, the relative study has never interrupted, due to the l₁ problem is not-differentiable, minimization of the sum of absolute deviation is computationally difficult, until 1950s the appearance of computer make the l₁ problem became simple, subsequently, the minimum sum of absolute deviation is widely used too, so the study of l₁ problem is very important. Some l₁ problems are studied:In chapter 2, the previous work about the l₁ problem of linear model is reviewed systematically, different algorithms is compared; then given some assumptions, the minimum sum of absolute deviation is used to model first order autoregressive, the properties of parameters is proved; Finally, the quantile and l∞regressive are introduced, the programming of related problem is given.In chapter 3, the previous work about the l₁ problem of nonlinear model is analyzed, the equivalent of l₁ problem and the problem of minimax optimize is proved, so the l₁ problem is solved by the sequential quadratic programming method, considering the shortcoming of the sequential quadratic programming method and the advantages of genetic algorithm, a hybrid genetic algorithm is given.In chapter 4, the original and the development of grey system theory are introduced simply, the shortcoming and the advantages of the existing grey modeling method are analyzed, a linear programming approach to estimate parameters of non-homogeneous exponential model based on the minimization of mean absolute percentage error is present, this approach is tested for real-world example, the results found by the proposed approach are compared with the results of particle swarm optimization algorithm, the comparisons show that the proposed approach comes with lower prediction error. Secondly, estimation the parameters of non-homogeneous exponential model is formulated as the minimax optimization problem and would be solved using the library function fminimax in MATLAB, this approach is tested for two real-world examples, its results demonstrate this approach con further reduce the mean absolute percentage error and the least absolute deviation is robust, the countermeasures to the multiplicity of solutions to l₁ problem is given based the principle of non-uniqueness. Finally, considering the deficiency of the existing smoothness sequence condition, a new continuous smooth degree is set up, multiple transformation invariance and translation transformation invariance are defined, and multiple transformation invariance and translation transformation invariance of continuous smooth degree are proved; the novel grey convex relation degree is present, it is found that the grey convex relation degree holds many good properties, two examples are given to illustrate that the grey convex relation degree can more really reflect the correlation degree of sequence.Some concluding remarks about relevant problem are given in the end.