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This article is part of the supplement: Proceedings of the 2012 International Conference on Intelligent Computing (ICIC 2012)

Open Access Proceedings

Robust PCA based method for discovering differentially expressed genes

Jin-Xing Liu124, Yu-Tian Wang2, Chun-Hou Zheng3, Wen Sha3, Jian-Xun Mi14 and Yong Xu14*

Author Affiliations

1 Bio-Computing Research Center, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, China

2 College of Information and Communication Technology, Qufu Normal University, Rizhao, China

3 College of Electrical Engineering and Automation, Anhui University, Hefei, China

4 Key Laboratory of Network Oriented Intelligent Computation, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, China

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BMC Bioinformatics 2013, 14(Suppl 8):S3  doi:10.1186/1471-2105-14-S8-S3


The electronic version of this article is the complete one and can be found online at: http://www.biomedcentral.com/1471-2105/14/S8/S3


Published:9 May 2013

© 2013 Liu et al.; licensee BioMed Central Ltd.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

How to identify a set of genes that are relevant to a key biological process is an important issue in current molecular biology. In this paper, we propose a novel method to discover differentially expressed genes based on robust principal component analysis (RPCA). In our method, we treat the differentially and non-differentially expressed genes as perturbation signals S and low-rank matrix A, respectively. Perturbation signals S can be recovered from the gene expression data by using RPCA. To discover the differentially expressed genes associated with special biological progresses or functions, the scheme is given as follows. Firstly, the matrix D of expression data is decomposed into two adding matrices A and S by using RPCA. Secondly, the differentially expressed genes are identified based on matrix S. Finally, the differentially expressed genes are evaluated by the tools based on Gene Ontology. A larger number of experiments on hypothetical and real gene expression data are also provided and the experimental results show that our method is efficient and effective.

Background

One of the challenges in current molecular biology is how to find the genes associated with key cellular processes. Up to date, using microarray technology, these genes associated with a special biological process have been detected more comprehensively than ever before.

DNA microarray technology has enabled high-throughput genome-wide measurements of gene transcript levels [1,2], which is promising in providing insight into biological processes involved in gene regulation [3]. It allows researchers to measure the expression levels of thousands of genes simultaneously in a microarray experiment. Gene expression data usually contain thousands of genes (sometimes more than 10,000 genes), and yet only a small number of samples (usually less than 100 samples). Gene expression is believed to be regulated by a small number of factors (compared to the total number of genes), which act together to maintain the steady-state abundance of specific mRNAs. Some of these factors could represent the binding of one (or more) transcription factor(s) (TFs) to the promoter region(s) of the gene [4]. So, it can be assumed that the genes associated with a biological process are influenced only by a small subset of TFs [5]. Although the expression levels of thousands of genes are measured simultaneously, only a small number of genes are relevant to a special biological process. Therefore, it is important how to find a set of genes that are relevant to a biological process.

Various methods have been proposed for identifying differentially expressed genes from gene expression data. These methods can be roughly divided into two categories: univariate feature selection (UFS) and multivariate feature selection (MFS). The commonest scheme of UFS is utilized as follows. First, a score for each gene is independently calculated. Then the genes with high scores were selected [6]. The main virtues of UFS are simple, interpretable and fast. However, UFS has some drawbacks. For example, if each gene is independently selected from gene expression data, a large part of the mutual information contained in the data will be lost.

To overcome the drawbacks of UFS, the methods of MFS use all the features simultaneously to select the genes. So far, many mathematical methods for MFS, such as principal component analysis (PCA), independent component analysis (ICA), nonnegative matrix factorization (NMF), lasso logistic regression (LLR) and penalized matrix decomposition (PMD), have been devised to analyze gene expression data. For example, Lee et al. applied PCA to analyze gene expression data [7]. Liu et al. proposed a method of weighting principal components by singular values to select characteristic genes [8]. Probabilistic PCA was used to analyze gene expression data by Nyamundanda et al. [9]. Huang et al. used ICA to analyze gene expression data [10]. NMF was used to select the gene by Zheng et al. [11]. Liu et al. used LLR to select characteristic gene using gene expression data [12]. In [13], Witten et al. proposed penalized matrix decomposition (PMD), which was used to extract plant core genes by Liu et al. [14]. However, the brittleness of these methods with respect to grossly corrupted observations often puts its validity in jeopardy.

Recently, a new method for matrix recovery, namely robust PCA, has been introduced in the field of signal processing [15]. The problem of matrix recovery can be described as follows, assume that all the data points are stacked as column vectors of a matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1">View MathML</a>, and the matrix (approximately) have low rank:

<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M2">View MathML</a>

(1)

where <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M3">View MathML</a> has low-rank and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M4">View MathML</a> is a small perturbation matrix. The robust PCA proposed by Candes et al. can recover a low-rank matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M3">View MathML</a> from highly corrupted measurements <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1">View MathML</a>[15]. Here, the entries in <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M4">View MathML</a> can have arbitrary large magnitude, and their support is assumed to be sparse but unknown.

Although the method has been successfully applied to model background from surveillance video and to remove shadows from face images [15], it's validity for gene expression data analysis is still need to be studied. The gene expression data all lie near some low-dimensional subspace [16], so it is natural to treat these genes data of non-differential expression as approximately low rank. As mentioned above, only a small number of genes are relevant to a biological process, so these genes with differential expression can be treated as sparse perturbation signals.

In this paper, based on robust PCA, a novel method is proposed for identifying differentially expressed genes. The differentially and non-differentially expressed genes are treated as perturbation signals <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> and low-rank matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M6">View MathML</a>. Firstly, the matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1">View MathML</a> of expression data is decomposed into two adding matrices <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M6">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> by using RPCA. Secondly, the differentially expressed genes are discovered according to the matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a>. Finally, the differentially expressed genes are evaluated by the tools based on Gene Ontology. The main contributions of our work are as follows: firstly, it proposes, for the first time, the idea and method based on RPCA for discovery of differentially expressed genes; secondly, it provides a larger number of experiments of gene selection.

Methods

The definition of Robust PCA (RPCA)

This subsection simply introduces robust PCA (RPCA) proposed by Candes et al. [15]. Let <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M7">View MathML</a> denote the nuclear norm of the matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M6">View MathML</a>, that is, the sum of its singular values, and let <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M8">View MathML</a> denote the <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M9">View MathML</a>-norm of <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a>. Supposing that <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1">View MathML</a> denotes the observation matrix given by Eq.(1), RPCA solves the following optimization problem:

<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M10">View MathML</a>

(2)

where <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M11">View MathML</a> is a positive regulation parameter. Due to the ability to exactly recover underlying low-rank structure in the data, even in the presence of large errors or outliers, this optimization is referred to as Robust Principal Component Analysis (RPCA).

For the RPCA problem Eq.(2), a Lagrange multiplier Y is introduced to remove the equality constraint. According to [17], the augmented Lagrange multiplier method on the Lagrangian function can be applied:

<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M12">View MathML</a>

(3)

where <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M13">View MathML</a> is a positive scalar and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M14">View MathML</a> denotes the Frobenius norm. Lin et al. gave a method for solving the RPCA problem, which is referred to as the inexact ALM (IALM) method [17]. The details of this algorithm can be seen in [17].

The RPCA model of gene expression data

Considering the matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1">View MathML</a> of gene expression data with size <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M15">View MathML</a>, each row of <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1">View MathML</a> represents the transcriptional responses of a gene in all the <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M16">View MathML</a> samples, and each column of <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1">View MathML</a> represents the expression levels of all the <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M17">View MathML</a> genes in one sample. Without loss of generality, <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M18">View MathML</a>, so it is a classical small-sample-size problem.

Our goal of using RPCA to model the microarray data is to identify these significant genes. As mentioned in Introduction, it is reasonable to view the significant genes as sparse signals, so the differential ones are viewed as the sparse perturbation signals <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> and the non-differential ones as the low-rank matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M6">View MathML</a>. Consequently, the genes of differential expression can be identified according to the perturbation signals <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a>. The RPCA model of microarray data is shown in Figure 1. The white and yellow blocks denote zero and near-zero in Figure 1. Red and blue blocks denote the perturbation signals. As shown in Figure 1, the matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> of differentially expressed genes (red or blue block) can be recovered from the matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1">View MathML</a> of gene expression data.

thumbnailFigure 1. The RPCA model of microarray data. The white and yellow blocks denote zero and near-zero in this figure. Red and blue blocks denote the perturbation signals.

Suppose the matrix decomposition <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M19">View MathML</a> has been done by using RPCA. By choosing the appropriate parameter <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M11">View MathML</a>, the sparse perturbation matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> can be obtained, i.e., most of entries in <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> are zero or near-zero (as white and yellow blocks shown in Figure 1). The genes corresponding to non-zero entries can be considered as ones of differential expression.

Identification of differentially expressed genes

After observation matrix has been decomposed by using RPCA, sparse perturbation matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> can be obtained. Therefore the differentially expressed genes can be identified according to sparse matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a>.

Denote the perturbation vector associated with <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M20">View MathML</a>-th sample as:

<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M21">View MathML</a>

(4)

Then the sparse matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> can be expressed as follows:

<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M22">View MathML</a>

(5)

So the sparse matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> can be denoted as:

<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M23">View MathML</a>

(6)

The differentially expressed genes can be classified into two categories: up-and down-regulated ones [18], which are reflected by the positive and negative entries in the sparse matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a>. Here, to discover the differentially expressed genes, only the absolute value of entries in <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> need to be considered. Then the following two steps are executed: firstly, the absolute values of entries in the sparse matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> are find out; secondly, to get the evaluating vector <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M24">View MathML</a>, the matrix is summed by rows. Mathematically, it can be expressed as follows:

<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M25">View MathML</a>

(7)

Consequently, to obtain the new evaluating vector <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M89">View MathML</a>, which is sorted in descending order. Without loss of generality, suppose that the first <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M26">View MathML</a> entries in <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M89">View MathML</a> are non-zero, that is,

<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M27">View MathML</a>

(8)

Generally, the larger the element in <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M89">View MathML</a> is, the more differential the gene is. So, the genes associated with only the first <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M28">View MathML</a> (<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M29">View MathML</a>) entries in <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M89">View MathML</a> are picked out as differentially expressed ones.

Results and discussion

This section gives the experimental results. Firstly, in the first subsection, hypothetical data are exploited to clarify how to set the parameter<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M11">View MathML</a>. Secondly, in the second subsection, our method is compared with the following methods on the real gene expression data of plants responding to abiotic stresses: (a) PMD method using the left singular vectors <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M30">View MathML</a> to identify the differentially expressed genes (proposed by Witten et al. [13]); (b) SPCA method using all the PCs of SPCA (proposed by Journée et al. [19]) to identify the differentially expressed genes. Finally, in the third subsection, the three methods are compared on the real gene expression data of colon tumor.

Experimental results on hypothetical data

Matrices randomly generated will be used for the simulation experiments. The true solution is denoted by the ordered pairs <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M31">View MathML</a>, which are generated by using the method in [17]. The rank-r matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M32">View MathML</a> is generated as <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M33">View MathML</a>, where <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M34">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M35">View MathML</a> are independent <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M36">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M37">View MathML</a> matrices, respectively. Elements of <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M34">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M35">View MathML</a> are i.i.d. Gaussian random variables with zero mean and unit variance. <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M38">View MathML</a> is a sparse matrix whose support is chosen uniformly at random, and whose non-zero entries are i.i.d. uniformly in the space <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M39">View MathML</a>. <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M13">View MathML</a> denotes the sparse degree of matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M40">View MathML</a>, which is defined as the number of non-zero entries divided by the number of all the entries. The matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M41">View MathML</a> is the input data to the RPCA. To evaluate the identification performance of RPCA, <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a> denotes the recognition accuracy of matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a>, which is defined as follows.

<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M43">View MathML</a>

(9)

where correct identified entries mean that the identified entries in <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> approximately equal to the ones in <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M40">View MathML</a>.

In [17,20], a fixed regulation parameter <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M44">View MathML</a> is used, where <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M45">View MathML</a>. In order to clarify how to set <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M11">View MathML</a>, the following two different cases are considered: first, <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M46">View MathML</a>; second, <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M47">View MathML</a>, the small-size-sample problem.

Results while m=n

In this experiment, let <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M48">View MathML</a>, <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M49">View MathML</a>, <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M50">View MathML</a>. Table 1 lists the recognition results with different <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M51">View MathML</a>. As Table 1 listed, when <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M52">View MathML</a>, the recognition accuracy <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a> can be achieved above 90%. When <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M53">View MathML</a>, the matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> can be completely identified, i.e. <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M54">View MathML</a>.

Table 1. The recognition accuracy <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a> with different <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M51">View MathML</a>

Results while m>n

In this experiment, let <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M55">View MathML</a>,<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M56">View MathML</a>, <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M57">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M16">View MathML</a> increase from 10 to 100 with an interval 10. Table 2, 3, 4, 5 list the results. As tables 2 and 3 listed with <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M58">View MathML</a>, when <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M59">View MathML</a>, the recognition accuracy <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a> can be achieved above 90%. As tables 4 and 5 listed with <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M60">View MathML</a>, when <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M61">View MathML</a>, the recognition accuracy <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a> can be achieved above 90%. In words, to achieve the recognition accuracy <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a> above 90%, <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M16">View MathML</a> must be equal to or larger than three times of <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M62">View MathML</a> (<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M63">View MathML</a>). As tables 2, 3, 4, 5 listed, by rows, the larger the number of column <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M16">View MathML</a> is, the higher the recognition accuracy <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a> can be achieved.

Table 2. The recognition accuracy <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a> with <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M58">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M86">View MathML</a>

Table 3. The recognition accuracy <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a> with <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M58">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M87">View MathML</a>

Table 4. The recognition accuracy <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a> with <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M88">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M86">View MathML</a>

Table 5. The recognition accuracy <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a> with <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M88">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M87">View MathML</a>

Now, we investigate how different <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M51">View MathML</a> influences the recovery accuracy <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a>. For example, when <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M64">View MathML</a>, Figure 2 shows the recognition accuracy <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a> with different <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M51">View MathML</a>. As shown in Figure 2, when <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M65">View MathML</a>, the recognition of matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> can reach highest accuracy. With <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M51">View MathML</a> increasing, the recovery accuracy <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a> drops. For example, when <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M45">View MathML</a>, s3 and s4 are degraded to 90%.

thumbnailFigure 2. The recognition accuracy of matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> with different <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M51">View MathML</a>. s1 denotes the recognition accuracy series with <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M85">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M86">View MathML</a>. s2 denotes the recognition accuracy series with <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M58">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M87">View MathML</a>. s3 denotes the recognition accuracy series with <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M88">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M86">View MathML</a>. s4 denotes the recognition accuracy series with <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M88">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M87">View MathML</a>.

From these experiments, a conclusion can be drawn that when the optimal empirical value of <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M11">View MathML</a> is given as: <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M66">View MathML</a>, where the size of data matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M1">View MathML</a> is <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M15">View MathML</a>, the highest identification accuracy <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M42">View MathML</a> can be obtained.

Experimental results on gene expression data of plants responding to abiotic stresses

Along with other two state-of-the-art methods, namely PMD and SPCA, used as comparison, three methods, including RPCA, are used to discover the differentially expressed genes responding to abiotic stresses based on real gene expression data.

Data source

The raw data were downloaded from NASCArrays [http://affy.arabidopsis.info/ webcite] [21], which include two classes: roots and shoots in each stress. The reference numbers are: control, NASCArrays-137; cold stress, NASCArrays-138; osmotic stress, NASCArrays-139; salt stress, NASCArrays-140; drought stress, NASCArrays-141; UV-B light stress, NASCArrays-144; heat stress, NASCArrays-146. Table 6 lists the sample number of each stress type. There are 22810 genes in each sample. The data are adjusted for background of optical noise using the GC-RMA software by Wu et al. [22] and normalized using quartile normalization. The results of GC-RMA are gathered in a matrix for further processed.

Table 6. The sample number of each stress type in the raw data

Selection of the parameters

In this paper, for PMD method, the <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M67">View MathML</a>-norm of <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M68">View MathML</a> is taken as the penalty function, i.e. <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M69">View MathML</a>. Because of <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M70">View MathML</a>, let <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M71">View MathML</a>, where <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M72">View MathML</a>. For simplicity, let <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M73">View MathML</a>, that is, only one factor is used. The results with <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M67">View MathML</a>-norm (<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M74">View MathML</a>) and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M75">View MathML</a>-norm (<a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M76">View MathML</a>, i.e. the number of nonzero coefficients, or cardinality) penalty in SPCA are similar, which is also shown in [19], so <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M75">View MathML</a>-norm penalty and the parameter <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M77">View MathML</a> are taken in SPCA. For a fair comparison, 500 genes are roughly selected by these methods via choosing appropriate parameters <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M78">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M77">View MathML</a> of the two methods, PMD and SPCA, which are listed in Table 7 for different data set. As the first subsection of experiments mentioned, while <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M65">View MathML</a>, RPCA gives the optimization results. Then, according to methods section, the first 500 genes are selected.

Table 7. The values of <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M78">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M77">View MathML</a> on different data set

Gene ontology (GO) analysis

Recently, many tools have been developed for the functional analysis of large lists of genes [23,24]. Most of them focus on the evaluation of Gene Ontology (GO) annotations. GOTermFinder is a web-based tool that finds the significant GO terms shared among a list of genes, helping us discover what these genes may have in common. The analysis of GOTermFinder provides significant information for the biological interpretation of high-throughput experiments.

In this subsection, the genes identified by these methods, RPCA, PMD and SPCA, are sent to GOTermFinder [24], which is publicly available at http://go.princeton.edu/cgi-bin/GOTermFinder webcite. Its threshold parameters are set as following: minimum number of gene products = 2 and maximum P-value = 0.01. Here, the key results are shown. Table 8 lists the terms of Response to abiotic stimulus (GO:0009628), whose background frequency in TAIR set is 1539/29556 (5.2%). Response to abiotic stimulus is the ancestor term of all the abiotic stresses. In GOTermFinder, a p-value is calculated using the hyper-geometric distribution, its details can be seen in [24]. Sample frequency denotes the number of genes hit in the selected genes, such as 107/500 denotes 107 genes associated with the GO term in 500 ones selected by these methods. As listed in Table 8, all the three experimented methods, PMD, SPCA and RPCA, can extract the significant genes with very lower P-value, as well as very higher sample frequency. In Table 8, the superior results are in bold type. In the twelve items, there is only one of them (cold on root) that PMD is equal to our method. In other items, our method is superior to SPCA and PMD.

Table 8. Response to abiotic stimulus (GO:0009628)

Figure 3 shows the sample frequency of response to abiotic stimulus (GO:0009628) given by the three methods. From Figure 3(a), RPCA method outperforms others in all the data sets of shoot samples with six different stresses. Figure 3(b) shows that only in cold-stress data set of root samples, PMD is equal to our method and they are superior to SPCA. In other data sets, our method is superior to the others.

thumbnailFigure 3. The sample frequency of response to abiotic stimulus.

The characteristic terms are listed in Table 9, in which the superior results are in bold type. As listed in Table 9, PMD method outperforms SPCA and our method in three items, such as drought in shoot, salt in root and cold in root, among the whole items. However, it shows that, on one of the twelve items (osmotic in shoot), our method has the same competitive result as PMD, while both methods are superior to SPCA. In other eight items, our method excels PMD and SPCA methods. In addition, on all the characteristic items, our method has superiority over SPCA.

Table 9. Characteristic terms selected from GO by algorithms

From the results of experiments, it can be concluded that our method is efficient and effective.

Experimental results on colon data

The three methods, SPCA, PMD and RPCA, are compared on colon cancer data set [25]. Colon cancer is the fourth most common cancer for males and females and the second most frequent cause of death.

Data source

The raw data were downloaded from http://genomics-pubs.princeton.edu/oncology/affydata/I2000.html webcite, which include gene expression levels for 2000 gene and contain 40 tumor and 22 normal tissue samples.

Selection of the parameters

In this subsection, for PMD method, the <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M67">View MathML</a>-norm of <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M68">View MathML</a> is taken as the penalty function, i.e. <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M79">View MathML</a>. Let <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M80">View MathML</a>, where <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M81">View MathML</a>. For SPCA method, let <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M82">View MathML</a>, that is, only one factor is used. <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M75">View MathML</a>-norm penalty and the parameter <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M77">View MathML</a> are taken in SPCA. For a fair comparison, 100 genes are roughly selected by these methods via choosing appropriate parameters. PMD and SPCA use the parameters <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M83">View MathML</a> and <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M84">View MathML</a> on colon data set, respectively. As the first subsection of experiments mentioned, while <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M65">View MathML</a>, RPCA gives the optimization results. Then, according to Methods section,the first 100 genes are selected using our method.

Gene ontology (GO) analysis

The genes identified by these methods, RPCA, PMD and SPCA, are evaluated by using AmiGO [26]. Its threshold parameters are set as following: minimum number of gene products = 2 and maximum P-value = 0.1. A number of lines of evidence suggest that immune, stimulus and tumor have affinity, so Table 10 lists the key results: the terms of Response to stimulus (GO:0050896) and Immune system process (GO:0002376). As listed in Table 10, RPCA outperforms its competitive methods with higher sample frequency.

Table 10. Characteristic terms selected from GO on colon data

Function analysis

Table 11 lists the top 30 genes selected by using RPCA. To further study the biology functions of the selected genes, we also make the network analysis of the top 100 genes selected by our algorithm using the GeneMANIA tool [27] on the Web sitehttp://genemania.org/ webcite. The result is listed in Table 12. From the table it can be seen that there are 215 genes of this chip participating in the cytokine-mediated signalling pathway, in which there are 21 genes discovered by our method. This pathway has the lowest p-value. It is considered as the most probable pathway with these top 100 genes. Recent findings also indicate that cytokine receptors can regulate immune cell functions by transcription-independent mechanisms [28]. Some other pathways with the most significance are also listed in Table 12.

Table 11. The top 30 genes of colon data selected by RPCA

Table 12. Pathway analysis of the top 100 genes selected by RPCA on colon data

Conclusion

In this paper, a novel RPCA-based method of discovering differentially expressed genes was proposed. It combined RPCA and sparsity of gene differential expression to provide an efficient and effective approach for gene identification. Our method mainly included the following two steps: firstly, the matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a> of differential expression was discovered from gene expression data matrix by using robust PCA; secondly, the differentially expressed genes were discovered according to matrix <a onClick="popup('http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.biomedcentral.com/1471-2105/14/S8/S3/mathml/M5">View MathML</a>. The experimental results on real gene data showed that our method outperformed the other state-of-the-art methods. In future, we will focus on the biological meaning of the differentially expressed genes.

Competing interests

The authors declare that they have no competing interests.

Acknowledgements

This work was supported by fund for China Postdoctoral Science Foundation Funded Project, No. 2012M510091; Program for New Century Excellent Talents in University (No.NCET-08-0156), NSFC under grant No. 61071179, 61272339, 61202276 and 61203376, and the Key Project of Anhui Educational Committee, under Grant No. KJ2012A005; the Foundation of Qufu Normal University under grant no. XJ200947.

Declarations

The publication costs for this article were funded by fund for China Postdoctoral Science Foundation Funded Project, No. 2012M510091.

This article has been published as part of BMC Bioinformatics Volume 14 Supplement 8, 2013: Proceedings of the 2012 International Conference on Intelligent Computing (ICIC 2012). The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/supplements/14/S8.

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