Limiting Properties Of The Sample Eigenvalues Associated To Distant Spikes Of Large Dimensional Random Matrices And Their Applications

With development of science and technology,high-dimensional data now com-monly arise in many scientific fields such as genomics,image processing,microarray,proteomics and finance,to name but a few.The collection for high-dimensional da-ta is becoming increasingly easier.Acting as one of effective tools of dealing with high-dimensional data,the large dimensional random matrices play a more and more important role.The spiked model is proposed by Johnstone(2001)[31]and becomes a popular model in the study of the large dimensional random matrices.The distant spiked eigenvalue is an important concept of the spiked model.The spiked model is dedicated to investigate the properties of the sample eigenval-ues associated with the spiked eigenvalues in the condition that the population eigen-values have outlier eigenvalues.The properties of the sample eigenvalues associated with the spiked eigenvalues include the condition of phase transition phenomenon,the limits and the limiting distribution.The spiked model was originally built on the sam-ple covariance matrix and has been extended to the Fisher matrix,separable matrix and so on.It has a wide range of applications in principal component analysis(PCA),signal processing,regression analysis,etc.The limiting properties of the sample eigen-values associated with the spiked eigenvalues are more special and have more widely used in the in practice.In this paper,to illustrate the concept and properties of distant spiked eigenvalues better,initially,we will introduce the definition of the spiked model and its devel-opment history in sample covariance matrix,Fisher matrix and canonical correlation matrix.Next,we derive the power functions of the Roy's largest eigenvalue test by the theory of the spiked model for the sample covariance matrix and Fisher matrix.Besides that,we define the generalized spiked Fisher matrix and focus on the limits of the sample eigenvalues associated to spiked eigenvalues and the condition of phase transition phenomenon of the sample eigenvalues of generalized spiked Fisher matrix.Finally,we give a method of estimating the population spiked eigenvalues of the gen-eral spiked Fisher matrixThere are mainly three parts in this thesisIn Chapter 1,we derive spiked model from two practical examples and introduce notations and the structure of the paper.In Chapter 2,we introduce the definition of the spiked model for the sample co-variance matrix,separable matrix,Fisher matrix and canonical correlation matrix,be-sides,we review its development historyIn Chapter 3,initially,we bring up the Roy's largest eigenvalue test from the sig-nal detection model.Next,we introduce its important role in the multivariate analysis Besides,we derive the power functions of Roy's largest eigenvalue test by the theory of the spiked model for sample covariance matrix and Fisher matrix.Finally,we compare our results with the other existing worksIn Chapter 4,we introduce the definition of general Fisher matrix and its relat-ed results.Then,we define the generalized spiked Fisher matrix and derive phase transition and limits for the sample eigenvalues associated to the population spiked eigenvalues.In the end,we provide a way to estimate the population spiked eigenval-ues.In Chapter 5,we summarize the full paper and design and prospect for future work.