We describe the use of singular value decomposition in transforming genome-wide expression data from genes ؋ arrays space to reduced diagonalized ''eigengenes'' ؋ ''eigenarrays'' space, where the eigengenes (or eigenarrays) are unique orthonormal superpositions of the genes (or arrays). Normalizing the data by filtering out the eigengenes (and eigenarrays) that are inferred to represent noise or experimental artifacts enables meaningful comparison of the expression of different genes across different arrays in different experiments. Sorting the data according to the eigengenes and eigenarrays gives a global picture of the dynamics of gene expression, in which individual genes and arrays appear to be classified into groups of similar regulation and function, or similar cellular state and biological phenotype, respectively. After normalization and sorting, the significant eigengenes and eigenarrays can be associated with observed genome-wide effects of regulators, or with measured samples, in which these regulators are overactive or underactive, respectively. D NA microarray technology (1, 2) and genome sequencing have advanced to the point that it is now possible to monitor gene expression levels on a genomic scale (3). These new data promise to enhance fundamental understanding of life on the molecular level, from regulation of gene expression and gene function to cellular mechanisms, and may prove useful in medical diagnosis, treatment, and drug design. Analysis of these new data requires mathematical tools that are adaptable to the large quantities of data, while reducing the complexity of the data to make them comprehensible. Analysis so far has been limited to identification of genes and arrays with similar expression patterns by using clustering methods (4-9).We describe the use of singular value decomposition (SVD) (10) in analyzing genome-wide expression data. SVD is also known as Karhunen-Loève expansion in pattern recognition (11) and as principal-component analysis in statistics (12). SVD is a linear transformation of the expression data from the genes ϫ arrays space to the reduced ''eigengenes'' ϫ ''eigenarrays'' space. In this space the data are diagonalized, such that each eigengene is expressed only in the corresponding eigenarray, with the corresponding ''eigenexpression'' level indicating their relative significance. The eigengenes and eigenarrays are unique, and therefore also data-driven, orthonormal superpositions of the genes and arrays, respectively.We show that several significant eigengenes and the corresponding eigenarrays capture most of the expression information. Normalizing the data by filtering out the eigengenes (and the corresponding eigenarrays) that are inferred to represent noise or experimental artifacts enables meaningful comparison of the expression of different genes across different arrays in different experiments. Such normalization may improve any further analysis of the expression data. Sorting the data according to the correlations of the genes (and arrays) with eigenge...

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We describe a comparative mathematical framework for two genome-scale expression data sets. This framework formulates expression as superposition of the effects of regulatory programs, biological processes, and experimental artifacts common to both data sets, as well as those that are exclusive to one data set or the other, by using generalized singular value decomposition. This framework enables comparative reconstruction and classification of the genes and arrays of both data sets. We illustrate this framework with a comparison of yeast and human cell-cycle expression data sets.DNA microarrays ͉ cell cycle ͉ yeast Saccharomyces cerevisiae ͉ human HeLa cell line R ecent advances in high-throughput genomic technologies enable acquisition of different types of molecular biological data, e.g., DNA-sequence and mRNA-expression data, on a genomic scale. Comparative analysis of these data among two or more model organisms promises to enhance fundamental understanding of the universality as well as the specialization of molecular biological mechanisms. It also may prove useful in medical diagnosis, treatment, and drug design. Comparisons of the DNA sequence of entire genomes already give insights into evolutionary, biochemical, and genetic pathways.Comparative analysis of mRNA-expression data requires mathematical tools that are able to distinguish the similar from the dissimilar among two or more large-scale data sets. These tools should provide mathematical frameworks for the description of the data, where the variables and operations may represent some biological reality. Recently we showed that singular value decomposition (SVD) provides such a framework for genome-wide expression data (refs. 1-3; see also refs. 4-7). Now we show that generalized SVD (GSVD) (8) provides a comparative mathematical framework for two genome-scale expression data sets. GSVD is a linear transformation of the two data sets from the two genes ϫ arrays spaces to two reduced and diagonalized ''genelets'' ϫ ''arraylets'' spaces. The genelets are shared by both data sets. Each genelet is expressed only in the two corresponding arraylets, with a corresponding ''angular distance'' indicating the relative significance of this genelet, i.e., its significance, in one data set relative to that in the other.We show that a genelet of equal significance in both data sets may represent a process common to both data sets. The two corresponding arraylets may represent the cellular states in each data set that correspond to this common process. A genelet of no significance in one data set relative to the other may represent a process exclusive to the latter data set. The corresponding arraylet of this data set may represent the cellular state that corresponds to this exclusive process.We also show that mathematical reconstruction of gene expression in a subset of genelets may simulate experimental observation of only the process that these genelets are inferred to represent. Similarly, reconstruction of array expression in the subset of corresponding arr...

We describe the use of a higher-order singular value decomposition (HOSVD) in transforming a data tensor of genes ؋ ''x-settings,'' that is, different settings of the experimental variable x ؋ ''y-settings,'' which tabulates DNA microarray data from different studies, to a ''core tensor'' of ''eigenarrays'' ؋ ''x-eigengenes'' ؋ ''y-eigengenes.'' Reformulating this multilinear HOSVD such that it decomposes the data tensor into a linear superposition of all outer products of an eigenarray, an x-and a y-eigengene, that is, rank-1 ''subtensors,'' we define the significance of each subtensor in terms of the fraction of the overall information in the data tensor that it captures. We illustrate this HOSVD with an integration of genome-scale mRNA expression data from three yeast cell cycle time courses, two of which are under exposure to either hydrogen peroxide or menadione. We find that significant subtensors represent independent biological programs or experimental phenomena. The picture that emerges suggests that the conserved genes YKU70, MRE11, AIF1, and ZWF1, and the processes of retrotransposition, apoptosis, and the oxidative pentose phosphate pathway that these genes are involved in, may play significant, yet previously unrecognized, roles in the differential effects of hydrogen peroxide and menadione on cell cycle progression. A genome-scale correlation between DNA replication initiation and RNA transcription, which is equivalent to a recently discovered correlation and might be due to a previously unknown mechanism of regulation, is independently uncovered.cell cycle ͉ DNA replication initiation ͉ N-mode singular value decomposition ͉ oxidative stress ͉ yeast Saccharomyces cerevisiae

The number of high-dimensional datasets recording multiple aspects of a single phenomenon is increasing in many areas of science, accompanied by a need for mathematical frameworks that can compare multiple large-scale matrices with different row dimensions. The only such framework to date, the generalized singular value decomposition (GSVD), is limited to two matrices. We mathematically define a higher-order GSVD (HO GSVD) for N≥2 matrices , each with full column rank. Each matrix is exactly factored as Di = UiΣiVT, where V, identical in all factorizations, is obtained from the eigensystem SV = VΛ of the arithmetic mean S of all pairwise quotients of the matrices , i≠j. We prove that this decomposition extends to higher orders almost all of the mathematical properties of the GSVD. The matrix S is nondefective with V and Λ real. Its eigenvalues satisfy λk≥1. Equality holds if and only if the corresponding eigenvector vk is a right basis vector of equal significance in all matrices Di and Dj, that is σi,k/σj,k = 1 for all i and j, and the corresponding left basis vector ui,k is orthogonal to all other vectors in Ui for all i. The eigenvalues λk = 1, therefore, define the “common HO GSVD subspace.” We illustrate the HO GSVD with a comparison of genome-scale cell-cycle mRNA expression from S. pombe, S. cerevisiae and human. Unlike existing algorithms, a mapping among the genes of these disparate organisms is not required. We find that the approximately common HO GSVD subspace represents the cell-cycle mRNA expression oscillations, which are similar among the datasets. Simultaneous reconstruction in the common subspace, therefore, removes the experimental artifacts, which are dissimilar, from the datasets. In the simultaneous sequence-independent classification of the genes of the three organisms in this common subspace, genes of highly conserved sequences but significantly different cell-cycle peak times are correctly classified.