Lecture 7 Multivariate Signal Analysis (Processing)and Biomedical Applications Prof.N.Rao
Prof. N. Rao Lecture 7 Multivariate Signal Analysis (Processing) and Biomedical Applications
Part 1:Singular Value Decomposition (SVD)and Applications
Part 1: Singular Value Decomposition (SVD) and Applications
Outline 。Introduction Significance and Motivation Mathematical definition of the SVD .Illustrative applications .SVD analysis of gene expression data ●Discussion
Introduction Significance and Motivation Mathematical definition of the SVD Illustrative applications SVD analysis of gene expression data Discussion Outline
Introduction .The goal of this chapter is to provide precise explanations of the use of SVD and PCA for gene expression analysis; Illustrating methods using simple examples
Introduction The goal of this chapter is to provide precise explanations of the use of SVD and PCA for gene expression analysis; Illustrating methods using simple examples
Mathematical definition of the SVD Let X denote an mx n matrix of real-valued data and rank r,where without loss of generality mn, and therefore r≤n. is the element value of the ith row in the jth column. The equation for singular value decomposition of X is the following: X=USVT (5.1)
Mathematical definition of the SVD Let X denote an m × n matrix of real-valued data and rank r, where without loss of generality m ≥ n, and therefore r ≤ n. xij is the element value of the ith row in the jth column. The equation for singular value decomposition of X is the following: (5.1)
Mathematical definition of the SVD X=USVT where U is an m x n matrix,S is an n x n diagonal matrix, and Ir is also an n x n matrix. *The columns of U are called the left singular vectors,{ug, and form an orthonormal basis,so that u;'u;=1 for i=j,and u财u=0fori≠j. The rows of Vr are called the right singular vectors,vp,and form an orthonormal basis. *The elements of S are only nonzero on the diagonal,and are called the singular values.Thus,S=diag(s1,.s). Furthermore,sk>0for1≤k≤r,and sk=0for(件l)≤k≤n. xo-Eusv! =1 The term "closest"means that X minimizes the sum of the squares of the difference of the elements of X and X
Mathematical definition of the SVD where U is an m x n matrix, S is an n x n diagonal matrix, and VT is also an n x n matrix. *The columns of U are called the left singular vectors, {u k}, and form an orthonormal basis, so that ui·uj = 1 for i = j, and ui·uj = 0 for i ≠ j. The rows of VT are called the right singular vectors, {v k}, and form an orthonormal basis. *The elements of S are only nonzero on the diagonal, and are called the singular values. Thus, S = diag(s1 ,...,s n). Furthermore, sk > 0 for 1 ≤ k ≤ r, and sk = 0 for ( r+1) ≤ k ≤ n. The term “closest” means that X (l) minimizes the sum of the squares of the difference of the elements of X and X(l)
Mathematical definition of the svd The calculation of SVD XTY=VS2VT X=USVT-U=XVS-
Mathematical definition of the SVD The calculation of SVD
llustrative applications for SVD and PCA Image processing and compression compression and noisy reduction Information Retrieval linguistic ambiguity issues.For example,keyword search Java
Ilustrative applications for SVD and PCA Image processing and compression compression and noisy reduction Information Retrieval linguistic ambiguity issues. For example, keyword search “Java
SVD analysis of gene expression data systems biology applications to understand relations among genes diagnostic applications to classify tissue samples from individuals with and without a disease
SVD analysis of gene expression data systems biology applications to understand relations among genes diagnostic applications to classify tissue samples from individuals with and without a disease
微阵列的概念 ·什么是微阵列？ 在支撑物（玻璃片、硅片、胶片）上的一种微小 元素的有序阵列，该阵列允许基因进行特定的结 合。阵列尺寸通常在几百至成千上万。 微阵列是一种新的科学术语，取自于希腊字 “mikro”(smal)和法国字“arayer'”(安排的)
微阵列的概念 • 什么是微阵列？ 在支撑物(玻璃片、硅片、胶片）上的一种微小 元素的有序阵列，该阵列允许基因进行特定的结 合。阵列尺寸通常在几百至成千上万。 微阵列是一种新的科学术语，取自于希腊字 “mikro” (small)和法国字“arayer”(安排的）。 •