Strong/Weak Feature Recognition of Promoters Based on Position Weight Matrix and Ensemble Set-Valued Models

2.1. Materials and objectives

A promoter library of E. coli is considered in this study. It includes 52 \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \delta ^E}$$ \end{document} natural promoters of E. coli, and the promoter sequences are from −35 to +20 relative to the transcription start site (Rhodius and Gross, 2010). The strength was measured in vivo and in vitro using GFP (Green Fluorescent Protein) reporter.

Our goal is to extract the features of the DNA sequence and to establish a model to predict the strong/weak property of the promoters. The promoters are divided into two classes: the strong and the weak promoters. Namely, the promoters whose strength is higher than the defined threshold C are regarded as strong promoters, and the other ones are regarded as weak promoters. Generally, a promoter can be regarded as strong or weak promoter as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \left\{ { \begin{matrix} {{ \rm{the \ promoter \ is }} \ S , { \rm{if \ activity}} \ge C;} \hfill \\ {{ \rm{the \ promoter \ is }} \ W , { \rm{if \ activity}} < C} \hfill \\ \end{matrix} } \right. \tag{1} \end{align*} \end{document}

where S denotes the strong promoters and W denotes the weak promoters, C is the predefined strong threshold. That is, a promoter library is divided into two groups by C.

Based on these assumptions, a promoter library is described as in Table 1, where N denotes the sample number of a promoter library and M denotes the number of the nucleotide of a promoter.

2.2. Position weight matrix

The classical PWM method evaluates the contribution of the nucleotides based on the frequencies of occurrences of the nucleotides. The effectiveness of this method has been demonstrated in many references (Stormo, 1990, 2000; Kiryu et al., 2005). Hence, we also use PWM to describe the effect of the nucleotide type on the promoter strength. The PWM is defined as Stormo (2000) and Rhodius and Gross (2010): \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \omega _ { i , k } } = \ln \left[ { { \frac { ( { n_ { b , i } } + 0.005N ) / ( N + 0.08N ) } { { p_b } } } } \right] \; ( i = 1 , 2 , \cdots , M , k = 1 , 2 , \cdots , N ) \tag { 2 } \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \omega _{i , k}}$$ \end{document} is the weight of the nucleotide at the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${i_{{ \rm{th}}}}$$ \end{document} position, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${n_{b , i}}$$ \end{document} is the number of the promoter whose nucleotide type is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$b \in \{ A , G , C , T \} $$ \end{document} at the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${i_{{ \rm{th}}}}$$ \end{document} position. p_b is the probability of the nucleotide type and is taken as 0.0625 for E. coli. With Equation (2), the DNA sequence is converted into a weight matrix, and then a mathematical model can be established.

2.3. Set-valued model of the promoter strength

The strong/weak feature is the set-valued feature, and the traditional continuous value cannot be used to describe it. In this article, we use a linear set-valued model (Wang et al., 2010; Chen et al., 2012) to describe the relationship between the promoter sequence and the strong/weak feature. In this study, it is the first time to use set-valued model to describe the relationship between the promoter sequence and strength feature. A simple linear model is used to describe the relation between the nucleotide weights and the strength, namely \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {y_k} = {b_k} + \mathop \sum \limits_{i = 1}^M \,{ \omega _{i , k}}{ \theta _i} + {d_k} , \;k = 1 , 2 , \cdots , N \tag{3} \end{align*} \end{document}

where y_k is strength of the promoter of the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${k_{{ \rm{th}}}}$$ \end{document} promoter in a library, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \omega _{i , k}}$$ \end{document} is the weight of the nucleotide at the \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${i_{{ \rm{th}}}}$$ \end{document} position, b_k and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \theta _i}$$ \end{document} are the parameters of the linear model, and d_k is the measurement noise. We assume that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${d_k} \sim ( 0 , { \delta ^2} )$$ \end{document} . Note that y_k is unknown, and we only know the strong/weak property of the promoter, denoted by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {I_k} = \left\{ { \begin{matrix} {1 , { \rm{if}} \;{y_k} \ge C} \hfill \\ {0 , { \rm{if}} \;{y_k} < C} \hfill \\ \end{matrix} } \right. \tag{4} \end{align*} \end{document}

where C is the predefined threshold in Equation (1). \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${I_k} = 1$$ \end{document} means the promoter is a strong promoter and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${I_k} = 0$$ \end{document} means the promoter is a weak promoter. Let \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\bm{\theta} = [ {b_k} , { \theta _1} , { \theta _2} , \cdots , { \theta _M}{ ] ^T}$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \phi _k} = [ 1 , { \omega _{1 , k}} , { \omega _{2 , k}} , \cdots , { \omega _{M , k}}{ ] ^T}$$ \end{document} . The classical promoter strength prediction is to estimate θ given \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\{ {y_k} , { \phi _k} , k = 1 , 2 , \cdots , N \} $$ \end{document} . Here, y_k is unknown, and we estimate θ given \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\{ {I_k} , { \phi _k} \} $$ \end{document} .

We use maximum likelihood estimation to estimate the parameter θ, namely, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {\hat {\bm{\theta}}} = \arg \mathop { \max } \limits_{\bm{\theta}} L ( {\bm{\theta}} ) \tag{5} \end{align*} \end{document}

The likelihood function of θ is given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} L ( \bm{\theta} ) = \prod \limits_{i = 1}^N \,p ( {I_1} , {I_2} , \cdots , {I_k} \vert \bm{\theta} ) = \prod \limits_{i = 1}^N \,p ( {I_1} \vert \bm{\theta} ) \,p ( {I_2} \vert \bm{\theta} ) \cdots p ( {I_k} \vert \bm{\theta} ) \tag{6} \end{align*} \end{document}

where p denotes the probability. If \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${I_k} = 1$$ \end{document} , we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \begin{split} p ( { I_k } = 1 \vert {\bm { \theta}} ) & = p ( { d_k } \ge C - \phi _k^T {\bm {\theta}} \vert {\bm {\theta}} ) \\ & = \int_ { C - \phi _k^T {\bm {\theta}}} ^ \infty \frac { 1 } { { \sqrt { 2 \pi \delta } } } { e^ { - { \frac { { x^2 } } { 2 { \delta ^2 } } } } } { \kern 1pt } dx \\ & = 1 - F ( C - \phi _k^T {\bm {\theta}} ) \\ \end{split} \tag { 7 } \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$ F ( C - \phi _k^T {\bm { \theta}} ) = \int_ { - \infty } ^ { C - \phi _k^T \theta } \frac { 1 } { { \sqrt { 2 \pi \delta } } } { e^ { - { \frac { { x^2 } } { 2 { \delta ^2 } } } } } dx $$ \end{document} .

Similarly, if \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${I_k} = 0$$ \end{document} , we have \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$p ( {I_k} = 0 \vert \bm{\theta} ) = F ( C - \phi _k^T \bm{\theta} )$$ \end{document} . The log-likelihood of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$L ( \bm{\theta} )$$ \end{document} is given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \ln L ( \bm{\theta} ) = \mathop \sum \limits_{k:{I_k} = 1} F ( C - \phi _k^T \bm{\theta} ) + \mathop \sum \limits_{k:{I_k} = 0} ( 1 - F ( C - \phi _k^T \bm{\theta} ) ) \tag{8} \end{align*} \end{document}

The estimated \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${\hat {\bm{\theta}}}$$ \end{document} is obtained as \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\partial \ln L ( \bm{\theta} ) / \partial \bm{\theta} = 0$$ \end{document} . \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\partial \ln L ( \bm{\theta} ) / \partial \bm{\theta}$$ \end{document} is given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {{ \frac { \partial \ln L ( {\bm { \theta}} ) } {\partial {\bm {\theta}} } } = } { \left( { \mathop \sum \limits_ { k: { I_k } = 0 } - { \frac { f ( C - \phi _k^T {\bm { \theta}} ) } { F ( C - \phi _k^T {\bm { \theta}} ) } } + \mathop \sum \limits_ { k: { I_k } = 1 } { \frac { f ( C - \phi _k^T {\bm {\theta}} ) } { 1 - F ( C - \phi _k^T {\bm {\theta}} ) } } } \right) { \phi _k } } \tag{9} \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$f ( C - \phi _k^T \bm { \theta } ) = \frac { 1 } { { \sqrt { 2 \pi \delta } } } { e^ { - { \frac { { { ( C - \phi _k^T \bm { \theta } ) } ^2 } } { 2 { \delta ^2 } } } } } $$ \end{document} . Equation (9) can be solved by numerical equation. Hence, we can obtain the value of θ by solving Equation (9). Compared with other classification methods, the set value-based method is relatively simple, and the parameters are obtained by solving a maximum likelihood estimation problem.

2.4. Ensemble approach

In this study, we assume that the strength of the promoter is the sum of the nucleotide weights. To establish such models, a good training set is needed. As stated above, the training set is divided into two classes by threshold C. The class with more training samples is called as major class, and the class with less training samples is called as minor class. For promoter data, in some cases, the training sample number from the major class is significantly different from the sample number from the minor class, namely the number of the training samples from different classes are imbalanced. In this case, an effective method is to construct a new training set to establish the prediction model. Boosting the minor class training set can make the samples from different classes more balanced. Motivated by this, we first propose a method to construct a boosted training set.

The original training set T is divided into two subsets. The set for the samples from the major class is denoted by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\overline T = \{ { \bar t_1} , { \bar t_2} , \cdots , { \bar t_{ \bar n}} \} $$ \end{document} , and the set for the samples from the minor class is denoted by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\underline T = \{ { \underline t _1} , { \underline t _2} , \cdots , { \underline t _{ \underline n }} \} ( \bar n < \underline n )$$ \end{document} . The testing set is denoted by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$S = \{ {s_1} , {s_2} , \cdots , {s_{{n_s}}} \} $$ \end{document} . First, we construct an auxiliary set \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${T_{Au}} = \{ { \underline t _1} , { \underline t _2} , \cdots , { \underline t _{ \underline n }} , { \underline t _1} , { \underline t _2} , \cdots \} $$ \end{document} , which is composed of the elements of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\underline T$$ \end{document} . The number of the element of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${T_{Au}}$$ \end{document} is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\bar n - \underline n$$ \end{document} . In this study, we use an oversampling method to boost the minor class training set, and the boosted minor class training set is given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\underline {T ^\prime } = \underline T \cup {T ^\prime _{Au}} = \{ \underline t { \prime _1} , \underline t { \prime _2} , \cdots , \underline t { \prime _{ \underline {n ^\prime } }} \} $$ \end{document} , where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${T ^\prime _{Au}}$$ \end{document} is a subset of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${T_{Au}}$$ \end{document} , namely \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${T ^\prime _{Au}} \in {T_{Au}}$$ \end{document} . To avoid the effect of the data imbalance, our goal is to select the training samples from \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${T_{Au}}$$ \end{document} such that the major class training set \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\overline T$$ \end{document} and the boosted minor class training set \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\underline {T ^\prime }$$ \end{document} are more balanced than before. Balance not only means that the number of the training samples from different classes are close, but also means that the sequence information extracted both from the major class training set and the minor class training set are close. To obtain this goal, we hope the similarities between the training samples and the testing samples are close. Hamming distance is the commonly used index to evaluate the similarity of two nucleotide sequences, which is defined as the number of the same nucleotide at the same position. Based on the Hamming distance, we define the similarity between the major class training set and the testing set as follows: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { D_ { \overline T , S } } = \frac { 1 } { { \bar n } } \mathop \sum \limits_ { i = 1 } ^ { \bar n } { \mathop \sum \limits_ { j = 1 } ^ { ns } { { D_ { { { \bar t } _i } , } } _ { { s_j } } } } \tag { 10 } \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${D_{{{ \bar t}_i} , {s_j}}}$$ \end{document} is the Hamming distance of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${ \bar t_i}$$ \end{document} and s_j. Similarity, the similarity between the boosted minor class training set and the testing set is given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { D_ { \underline { T ^\prime } , S } } = \frac { 1 } { { \underline { n ^\prime } } } \mathop \sum \limits_ { i = 1 } ^ { \underline { n ^\prime } } \mathop \sum \limits_ { j = 1 } ^ { { n_S } } { D_ { \underline t { ^\prime _i } , { s_j } } } \tag { 11 } \end{align*} \end{document}

Then, the construction of the optimal training set is converted to minimizing \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\vert {D_{ \overline T , S}} - {D_{ \underline {T ^\prime } , S}}{ \rm{ \vert }}$$ \end{document} . Some population-based optimization approaches, such as particle swarm optimization (PSO) (Eberhart and Kennedy, 1995) can be used to solve this optimization problem. If PSO is used, a particle in PSO can be coded as an indicator function set \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$X = \{ {x_1} , {x_2} , \ldots , {x_{ \bar n - \underline n }} \} $$ \end{document} . That is, the dimension of X is the number of the elements of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${T_{Au}}$$ \end{document} . If x_j takes 1, it means that the corresponding \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${j_{{ \rm{th}}}}$$ \end{document} sample in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${T_{Au}}$$ \end{document} is included in the boosted minor class training set; if x_j takes 0, it means that the corresponding \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${j_{{ \rm{th}}}}$$ \end{document} sample in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${T_{Au}}$$ \end{document} is excluded in the boosted minor class training set. The fitness function is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\vert {D_{ \overline T , S}} - {D_{ \underline {T ^\prime } , S}}{ \rm{ \vert }}$$ \end{document} , and the particles for each iteration can be updated by the discrete PSO algorithm (Clerc, 2004). Algorithm 1 summarizes the procedure of obtaining the optimal training set, where the function MinimizingSimilarity_PSO is used to select some samples in \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${T_{Au}}$$ \end{document} such that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\vert {D_{ \overline T , S}} - {D_{ \underline {T ^\prime } , S}}{ \rm{ \vert }}$$ \end{document} .

Although the minor class training set is boosted, the size of the training samples is still relatively small, which makes the prediction of the strong/weak feature of the promoters difficult. Motivated by this, we propose an ensemble learning approach to improve the prediction accuracy. The basic idea of the ensemble learning is to integrate multiple models into a unified model to improve the accuracy and robustness of the model. That is, we do not directly use the optimal training set to train the prediction model; the training set is first divided into multiple subsets and each subset produce a submodel. The prediction result is the integration of the results provided by these submodels. The key here is how to allocate the training set into some subsets. To the best of our knowledge, no allocation method has been proposed for the promoter data. We propose an ensemble learning method based on the similarities between the optimal training samples and the testing set. The similarity between a training sample and the testing set is defined as, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {D_{{{t ^\prime }_i} , {s_j}}} = \mathop \sum \limits_{j = 1}^{{{n ^\prime }_T}} {D_{{{t ^\prime }_i} , {s_j}}} ( i = 1 , 2 , 3 , \cdots , {n ^\prime _T};j = 1 , 2 , \cdots , {n_s} ) \tag{12} \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${t ^\prime _i}$$ \end{document} is one of the elements of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$T ^\prime$$ \end{document} and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${n ^\prime _T}$$ \end{document} is the element number of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$T ^\prime$$ \end{document} . The training set can be sorted in descending order along with the decrease of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${D_{{{t ^\prime }_i} , {s_j}}}$$ \end{document} , denoted by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\{ {t {^\prime} {^\prime} _1} , {t {^\prime} {^\prime} _2} , \cdots , {t {^\prime} {^\prime} _{{{n ^\prime }_T}}} \} $$ \end{document} . The training set is divided into the following \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${n ^\prime _T} - K + 1$$ \end{document} subsets: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} & {T^1}: \{ {{t {^\prime} {^\prime} }_1} , { \kern 1pt} \;{{t {^\prime} {^\prime} }_2} , \cdots , {{t {{^\prime} {^\prime}} }_K} \} \\ & {T^2}: \{ {{t {^\prime} {^\prime} }_2} , { \kern 1pt} \;{{t {^\prime} {^\prime} }_3} , \cdots , {{t {^\prime} {^\prime} }_{K + 1}} \} \\ & {T^3}: \{ {{t {^\prime} {^\prime} }_3} , \;{ \kern 1pt} {{t {^\prime} {^\prime} }_4} , \cdots , {{t {^\prime} {^\prime} }_{K + 2}} \} \\ & {T^{{{n ^\prime }_T} - K + 1}}: \{ {{t {^\prime} {^\prime} }_{{{n ^\prime }_T} - K + 1}} , {{t {^\prime} {^\prime} }_{n + 1}} , \cdots , {{t {^\prime} {^\prime} }_{{{n ^\prime }_T}}} \} \\ \end{align*} \end{document}

where K denotes the sample number of each subspace.

Based on the ensemble approach, we can establish n submodels denoted by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${Z^1} , {Z^2} , \cdots , {Z^{{{n ^\prime }_T} - K + 1}}$$ \end{document} . For a testing sample s_j, each model generates a prediction result, denoted by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$y_j^{{z_i}}$$ \end{document} , and the ensemble result is given by, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {I_i} = \left\{ { \begin{matrix} {1 , { \rm{if}} \;{1 \over 2} \left( {{1 \over { \sum {_{i = 1}^{{{n ^\prime }_T} - K + 1}} I ( y_j^{{z_i}} ) }} \mathop \sum \limits_{i = 1}^{{{n ^\prime }_T} - K + 1} y_j^{{z_i}}I ( y_j^{{z_i}} ) + {1 \over { \sum {_{i = 1}^{{{n ^\prime }_T} - K + 1}} ( 1 - I ( y_j^{{z_i}} ) ) }} \mathop \sum \limits_{i = 1}^{{{n ^\prime }_T} - K + 1} y_j^{{z_i}} ( 1 - I ( y_j^{{z_i}} ) ) } \right) \ge C} \\ {0 , { \rm{if}} \;{1 \over 2} \left( {{1 \over { \sum {_{i = 1}^{{{n ^\prime }_T} - K + 1}} I ( y_j^{{z_i}} ) }} \mathop \sum \limits_{i = 1}^{{{n ^\prime }_T} - K + 1} y_j^{{z_i}}I ( y_j^{{z_i}} ) + {1 \over { \sum {_{i = 1}^{{{n ^\prime }_T} - K + 1}} ( 1 - I ( y_j^{{z_i}} ) ) }} \mathop \sum \limits_{i = 1}^{{{n ^\prime }_T} - K + 1} y_j^{{z_i}} ( 1 - I ( y_j^{{z_i}} ) ) } \right) < C} \\ \end{matrix} } \right. \tag{13} \end{align*} \end{document}

where \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\usepackage{upgreek}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$I ( \cdot )$$ \end{document} is defined as Equation (4). The procedure of the proposed ensemble approach is shown in Figure 1. The algorithm of the proposed ensemble approach is summarized as Algorithm 2.

OPEN IN VIEWER

FIG. 1.

The procedure of the proposed algorithm.

It has been demonstrated that ensemble approach can improve the prediction accuracy. According to the characteristic of the promoter data, we proposed an ensemble strategy based on the similarity index. To the best of our knowledge, it is the first time to use the ensemble approach for the promoter strength recognition. In the proposed method, K is the only parameter of the algorithm, which is related to the size of the training set.