Studies On Indefinite Least Squares Problem And Approximate Factor Model

The indefinite least squares problem can be treated as a generalization of ordinary least squares problem. The H?estimation and control problems and risk-sensitive estimation and control, finite memory adaptive filtering can also be treated as indefinite least squares problem. This makes the study of indefinite least squares problem meanful and valuable. Beside the study on indefinite least squares problem, we also investigate the approximate factor model. Since the strict factor model can not give clear characterization on the correlations among the variables, and a good estimate is not available in high-dimensional setting, these all together prompt the development of investigations on approximate factor model.Chapter 2 contains some notation and necessary lemmas, and the unified definition of condition number is given here. We also prove some lemmas which will be frequently used in the following parts.In Chapter 3 and 4, we mainly focus on the condition numbers of indefinite least squares problem and its extentions. A novel generic definition of condition number is given first. Under this new framework, we establish the explicit expression of the condition numbers, we also include the normwise, mixed and componentwise condition numbers as its special cases. Particularly, under 2-norm we present some equivalent forms of the condition number, which has some superiorities in computation and storage in computer. In Chapter 3, we also investigate the relationship between indefinite least squares problem and the total least squares problem, we recover the condition number of total least squares problem from the viewpoint of indefinite least squares problem. In Chapter 4, the unified technique for deducing the 2-norm condition number of least squares problem and its generalizations is given. Our method does not rely on the singular value decompositions and the subtle construction techniques. Meanwhile, we also consider the structured condition numbers of indefinite least squares problem and present its explicit expressions. In the end, since compute the explicit condition number may be time consuming in large scale problems, we propose some probability and statistics based algorithms to estimate the condition number. The numerical experiment is given to check the efficiency of the condition number estimators, and compare the differences among different condition numbers.In Chapter 5, we mainly focus on the estimation of approximate factor model. Due to the current methods for solving the maximum likelihood functions can not guarantee the estimate of error covariance matrix to be positive definite. Moreover, a frequently used method to guarantee the positive definiteness is ADMM. Howerer when using ADMM we need to choose a penalty parameter which has no influence on the theoretical convergence, but can give nonnegliable impact on practical calculation. To overcome these drawbacks, we proposed a new algorithm based on Lagrangian duality. The new algorithm can not only guarantee the positive definiteness of error covariance matrix, but also has comparable convergence speed. The simulations are employed to illustrate this point. We also investigate its influence on forecasting. By the decomposition of covariance matrix, our algorithm can also be used to estimate the covariance matrix.