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Properties of a singular value decomposition based dynamical model of gene expression data

100%
Simek K.
International Journal of Applied Mathematics and Computer Science
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2003
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tom 13
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nr 3
337-345
EN
Recently, data on multiple gene expression at sequential time points were analyzed using the Singular Value Decomposition (SVD) as a means to capture dominant trends, called characteristic modes, followed by the fitting of a linear discrete-time dynamical system in which the expression values at a given time point are linear combinations of the values at a previous time point. We attempt to address several aspects of the method. To obtain the model, we formulate a nonlinear optimization problem and present how to solve it numerically using the standard MATLAB procedures. We use freely available data to test the approach. We discuss the possible consequences of data regularization, called sometimes ``polishing'', on the outcome of the analysis, especially when the model is to be used for prediction purposes. Then, we investigate the sensitivity of the method to missing measurements and its abilities to reconstruct the missing data. Summarizing, we point out that approximation of multiple gene expression data preceded by SVD provides some insight into the dynamics, but may also lead to unexpected difficulties, like overfitting problems.
2
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Signatura of magic and Latin integer squares: isentropic clans and indexing

88%
Cameron I., Rogers A., Loly P.
Discussiones Mathematicae Probability and Statistics
|
2013
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tom 33
|
nr 1-2
121-149
EN
The 2010 study of the Shannon entropy of order nine Sudoku and Latin square matrices by Newton and DeSalvo [Proc. Roy. Soc. A 2010] is extended to natural magic and Latin squares up to order nine. We demonstrate that decimal and integer measures of the Singular Value sets, here named SV clans, are a powerful way of comparing different integer squares. Several complete sets of magic and Latin squares are included, including the order eight Franklin subset which is of direct relevance to magic square line patterns on chess boards. While early examples suggested that lower rank specimens had lower entropy, sufficient data is presented to show that some full rank cases with low entropy possess a set of singular values separating into a dominant group with the remainder much weaker. An effective rank measure helps understand these issues. We also introduce a new measure for integer squares based on the sum of the fourth powers of the singular values which appears to give a useful method of indexing both Latin and magic squares. This can be used to begin cataloging a "library" of magical squares. Based on a video presentation in celebration of George Styan's 75th at LINSTAT2012 and IWMS-21 on 19 July, 2012 at Będlewo, Poland.
3
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Latent Semantic Indexing using eigenvalue analysis for efficient information retrieval

75%
Kumar C. A., Srinivas S.
International Journal of Applied Mathematics and Computer Science
|
2006
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tom 16
|
nr 4
551-558
EN
Text retrieval using Latent Semantic Indexing (LSI) with truncated Singular Value Decomposition (SVD) has been intensively studied in recent years. However, the expensive complexity involved in computing truncated SVD constitutes a major drawback of the LSI method. In this paper, we demonstrate how matrix rank approximation can influence the effectiveness of information retrieval systems. Besides, we present an implementation of the LSI method based on an eigenvalue analysis for rank approximation without computing truncated SVD, along with its computational details. Significant improvements in computational time while maintaining retrieval accuracy are observed over the tested document collections.
4
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An alternative methodology for imputing missing data in trials with genotype-by-environment interaction: some new aspects

75%
Arciniegas-Alarcón S., García-Peña M., Krzanowski W. J., Santos Dias C. T. d.
Biometrical Letters
|
2014
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tom 51
|
nr 2
75-88
EN
A common problem in multi-environment trials arises when some genotypeby- environment combinations are missing. In Arciniegas-Alarcón et al. (2010) we outlined a method of data imputation to estimate the missing values, the computational algorithm for which was a mixture of regression and lower-rank approximation of a matrix based on its singular value decomposition (SVD). In the present paper we provide two extensions to this methodology, by including weights chosen by cross-validation and allowing multiple as well as simple imputation. The three methods are assessed and compared in a simulation study, using a complete set of real data in which values are deleted randomly at different rates. The quality of the imputations is evaluated using three measures: the Procrustes statistic, the squared correlation between matrices and the normalised root mean squared error between these estimates and the true observed values. None of the methods makes any distributional or structural assumptions, and all of them can be used for any pattern or mechanism of the missing values.
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