Biochemistry 218 BioMedical Informatics 231 Computational Molecular Biology Final Project Multivariate Projection Approaches for Microarray Analysis Gang Yu Winter 2005 Research on microarrays or gene chips presents a challenge for biologists in functional genomics because the images generated from microarray experiments are so visually complex that manual comparisons are infeasible computational tools are required to examine microarrays For the past several years there has been an explosion in the numbers of studies on microarray computational tools Eisen et al 1998 Burgess 2001 Risinger et al 2003 Wouters et al 2003 Yeung et al 2004 Saidi et al 2004 Girolami and Breitling 2004 Stoyanova et al 2004 Tan et al 2004 Busold et al 2005 In general four common themes in microarray analysis can be identified These four themes consist of 1 detection of differential expression 2 pattern discovery 3 class prediction and 4 inference of regulatory pathways and networks Slonim 2002 For the pattern discovery theme computational tools roughly fall into two major categories One is multivariate projection methods based upon projections of highdimensional data in a lower dimensional space and plotting both genes and samples in this lower dimensional space using the biplot Chapman et al 2002 This projection into a subspace of low dimensionality can account for the main variance in the data The other is cluster analysis methods This paper discusses approaches in the first category multivariate projection methods however tools in the second category may be mentioned for comparisons The first part of the paper provides a brief overview of the multivariate methods Then algorithms of several major multivariate approaches are presented in the second part The following part summarizes advantages and drawbacks of multivariate methods with a comparison with cluster tools The final part is a conclusion I A Brief Overview of Multivariate Projection Approaches The multivariate projection methods include principal component analysis PCA correspondence factor analysis CFA spectral map analysis SMA partial least squares PLS method and some other variants Initially all of these methods were developed in either statistics or other academic areas but recently used in microarray analysis Multivariate projection methods help to reduce the complexity dimensions of highly dimensional data n genes versus p samples and provide means to identify gene patterns or subjects in the data Projected data are typically displayed in a biplot genes and samples in a new space PCA is the oldest and best known of the multivariate projection techniques Historically PCA dates back to Pearson 1901 and Hotelling 1933 This approach tries to identify components that explain the variance in the data The central idea of PCA is to reduce the dimensionality or complexity of a data set while retaining as much as possible of the variation present in the data The dimension reduction technique is accomplished by introducing a new set of variables principal components that are linear combinations of 1 the original variables and uncorrelated to each other In other words PCA reproduces the total variance among a large number of variables using a much smaller number of unobservable variables or dimensions called latent factors Principal components can be determined with different methods such as singular value decomposition SVD or some other algorithms For the just past three years numerous papers for example Peterson 2003 Barra 2004 Saidi et al 2004 Tham et al 2003 Girolami and Breitling 2004 and Hubert and Engelen 2004 have applied this method in microarray studies In early 1970s J P Benz ecri developed CFA method for contingency tables and in a sense decomposed the 2 statistic Therefore distances between objects in CFA have a 2 distribution The method has been widely employed to multivariate data analysis in sociology environmental science and marketing research Kishino and Waddell 2000 Fellenberg et al 2001 Peterson 2002 Tham et al 2003 Perelman et al 2003 Wouters et al 2003 Tan et al 2004 and Busold et al 2005 introduced the method to the investigations of microarray data by displaying the associations between genes and experiments Since CFA was primarily designed for analyzing contingency tables it can reveal the association both between and within all the variables genes and experiments simultaneously Like CFA SMA was originally developed in 1970s This method was developed not for biological research either but for the display of activity spectra of chemical compounds Lewi 1976 In the past SMA has been successfully applied to a wide variety of problems ranging from pharmacology Lewi 1976 virology Andries et al 1990 to management and marketing research Faes and Lewi 1987 Thielemans et al 1988 have compared SMA with PCA and CFA using a relatively small data set from the field of epidemiology Recently Wouters et al 2003 and Peeters et al 2004 applied this multivariate projection method to microarray analysis They all argued that SMA would be a promising new tool for microarray data analysis PLS is another well known dimension reduction technique Wold 1975a 1975b developed the PLS approach initially used for modeling information scarce situations in social science but recently employed in biochemistry The method relates the data matrix X to a y response that can be either a single y or multiple Y i e generating a model that predicts y or Y from X In the computer literature jargon PLS is known as a supervised method in that it uses both the independent and the dependent variables whereas PCA is an un supervised method that considers only independent variables Datta 2001 Nguyen and Rocke 2002 Park et al 2002 Johansson et al 2003 P rez Enciso and Tenenhaus 2003 Man et al 2004 Tan et al 2004 and Nguyen 2005 have applied this statistical method to microarray data analysis II Algorithms Due to space limit of this paper exhaustive explanations of these methods will not be presented however a review of the basic elements and major structures of their algorithms is necessary to understand their capabilities in microarray analysis Concise presentations of the algorithms will be given below 2 a Principal Component Analysis PCA To demonstrate the PCA algorithm we start to denote Mr s as the matrix containing the original expression levels mij for r genes rows in each of s different biological samples columns We also define two diagonal matrices with row
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