Null space based feature selection method for gene expression data

Abstract

Feature selection is quite an important process in gene expression data analysis. Feature selection methods discard unimportant genes from several thousands of genes for finding important genes or pathways for the target biological phenomenon like cancer. The obtained gene subset is used for statistical analysis for prediction such as survival as well as functional analysis for understanding biological characteristics. In this paper we propose a null space based feature selection method for gene expression data in terms of supervised classification. The proposed method discards the redundant genes by applying the information of null space of scatter matrices. We derive the method theoretically and demonstrate its effectiveness on several DNA gene expression datasets. The method is easy to implement and computationally efficient.

Appendix

Theorem 1

Let the column vectors of orthogonal matrix W span the null space of within-class scatter matrix S _W and ${\mathbf{w}} \in \mathbb{R}^{d}$ be any column vector of W. Let the j-th sample in i-th class be denoted by ${\mathbf{x}}_{j}^{i} \in \mathbb{R}^{d}$. Then the projection of sample ${\mathbf{x}}_{j}^{i}$ onto the null space of S _W is independent of the sample selection in class.

Proof Since ${\mathbf{w}} \in \mathbb{R}^{d}$ is in the null space of S _W , by definition ${\mathbf{S}}_{W} {\mathbf{w}} = 0$ or ${\mathbf{w}}^{\text{T}} {\mathbf{S}}_{W} {\mathbf{w}} = 0.$ The within-class scatter matrix S _W is a sum of scatter matrices ${\mathbf{S}}_{W} = \sum\nolimits_{i = 1}^{c} {{\mathbf{S}}_{i} ,}$ where c denotes the number of classes and scatter matrix S _i can be represented by [8]:

$$ {\mathbf{S}}_{i} = \sum\nolimits_{j = 1}^{{n_{i} }} {\left( {{\mathbf{x}}_{j}^{i} - {\varvec{\mu}}_{i} } \right)\left( {{\mathbf{x}}_{j}^{i} - {\varvec{\mu}}_{i} } \right)^{\text{T}} } $$

(A1)

where μ _i denotes the mean of class i and n _i denotes the number of samples in class i.

Since ${\mathbf{w}}^{\text{T}} {\mathbf{S}}_{W} {\mathbf{w}} = 0$ (or ${\mathbf{w}}^{T} \sum\nolimits_{i = 1}^{c} {{\mathbf{S}}_{i} {\mathbf{w}}} = 0$) and S _i is positive semi-definite matrix, we can represent ${\mathbf{w}}^{T} {\mathbf{S}}_{i} {\mathbf{w}} = 0$. From Eq. A1, we can say

$$ \begin{gathered} {\mathbf{w}}^{\text{T}} \sum\nolimits_{j = 1}^{{n_{i} }} {\left( {{\mathbf{x}}_{j}^{i} - {\varvec{\mu}}_{i} } \right)\left( {{\mathbf{x}}_{j}^{i} - {\varvec{\mu}}_{i} } \right)^{\text{T}} {\mathbf{w}} = 0} \hfill \\ {\text{or}}\quad \sum\nolimits_{j = 1}^{{n_{i} }} {{\mathbf{w}}^{\text{T}} {\mathbf{X}}_{j}^{i} {\mathbf{X}}_{j}^{{i^{\text{T}} }} {\mathbf{w}}} - \sum\nolimits_{j = 1}^{{n_{i} }} {{\mathbf{w}}^{\text{T}} {\varvec{\mu}}_{i} {\varvec{\mu}}_{i}^{\text{T}} {\mathbf{w}}} \hfill \\ {\text{or}}\quad \sum\nolimits_{j = 1}^{{n_{i} }} {\left( {\left\| {{\mathbf{w}}^{T} {\mathbf{x}}_{j}^{i} } \right\|^{2} - \left\| {{\mathbf{w}}^{T} {\varvec{\mu}}_{i} } \right\|^{2} } \right) = 0} \hfill \\ \end{gathered} $$

(A2)

where $\left\| . \right\|$ is the Euclidean norm. Eq. A2 immediately leads to ${\mathbf{w}}^{T} {\mathbf{x}}_{j}^{i} = {\mathbf{w}}^{T} {\varvec{\mu}}_{i} ;$ i.e., projection of sample ${\mathbf{x}}_{j}^{i}$ onto the null space of S _W is independent of j (or in other words independent of sample selection). This concludes the proof of the Theorem.