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Feature selection using neighborhood uncertainty measures and Fisher score for gene expression data classification

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International Journal of Machine Learning and Cybernetics Aims and scope Submit manuscript

Abstract

The classification of gene expression data provides a basis for the study of pathogenesis and treatment. However, this type of data is characterized by high dimensionality and small samples, which seriously affect the classification results. Consequently, it is necessary to use a gene selection algorithm to select key genes from gene expression data to improve the classification results, but the existing gene selection algorithm has the problems of low classification precision and high time complexity. Therefore, this paper proposes a gene selection algorithm using neighborhood uncertainty measures and Fisher score. First, to make full use of the information provided by the neighborhood decision system, the neighborhood fusion coverage and neighborhood fusion credibility are defined based on the neighborhood coverage and neighborhood credibility, and they are used to characterize neighborhood uncertainty measures. Second, the neighborhood uncertainty measures are extended by combining the algebraic and information theory views, and a heuristic nonmonotonic gene selection algorithm is designed based on the neighborhood uncertainty measures. The algorithm makes full use of the information in the neighborhood decision system to evaluate the importance of genes from the algebraic and information theory views, thereby selecting an optimal gene subset and improving classification precision. Third, Fisher score method is introduced into the proposed algorithm to preliminarily eliminate redundant genes to reduce the time cost of calculation and improve the performance of the algorithm. Finally, by comparing the experimental results of our algorithm with those of existing gene selection algorithms on ten gene datasets, it is proved that our algorithm can effectively improve the classification results for gene data.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant (61976082, 62002103).

Author information

Authors and Affiliations

  1. College of Computer and Information Engineering, Henan Normal University, Xinxiang, 453007, China

    Jiucheng Xu, Kanglin Qu, Qincheng Hou & Xiangru Meng

  2. Engineering Lab of Intelligence Business & Internet of Things, Xinxiang, Henan Province, China

    Jiucheng Xu, Kanglin Qu, Qincheng Hou & Xiangru Meng

  3. College of Computer Engineering, Nanjing Institute of Technology, Nanjing, 210000, China

    Kangjian Qu

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  1. Jiucheng Xu
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  2. Kanglin Qu
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  3. Kangjian Qu
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Contributions

JX: Conceptualization, Writing review and editing, Visualization, Project administration. KQ: Methodology, Software, Writing-original draft preparaton. QH: Formal analysis, Writing review and editing, Visualization. KQ: Writing review and editing, Visualization. XM: Formal analysis, Writing review and editing, Visualization. All authors have read and agreed to the published version of the manuscipt.

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Correspondence to Kanglin Qu or Xiangru Meng.

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Appendix

Appendix

Proof of Proposition 1

From Eq. (2) and Eq. (8), we can get \(\left| {n_b^\delta \left( {{u_i}} \right) } \right| \ge \left| {n_c^\delta \left( {{u_i}} \right) } \right|\) and \({P_b}\left( D \right) \le {P_c}\left( D \right)\). According to Definition 4, \(N{H_\delta }\left( c \right) \ge N{H_\delta }\left( b \right)\) holds.

Proof of Property 1

$$\begin{aligned}{} & {} N{H_\delta }\left( {D|c} \right) + N{H_\delta }\left( c \right) \\{} & {} \quad = - \frac{{{P_c}\left( D \right) }}{{\left| U \right| }}\mathop \sum \limits _{i = 1}^{\left| U \right| } \log \left( {\frac{{{{\left| {n_c^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| }^2}}}{{\left| {n_c^\delta \left( {{u_i}} \right) } \right| \left| {{n_{\left( {c,D} \right) }}\left( {{u_i}} \right) } \right| }}} \right) \\{} & {} \quad - \frac{{{P_c}\left( D \right) }}{{\left| U \right| }}\mathop \sum \limits _{i = 1}^{\left| U \right| } \log \left( {\frac{{\left| {n_c^\delta \left( {{u_i}} \right) } \right| }}{{\left| U \right| }}} \right) \\{} & {} \quad = - \frac{{{P_c}\left( D \right) }}{{\left| U \right| }}\mathop \sum \limits _{i = 1}^{\left| U \right| } \log \left( {\frac{{{{\left| {n_c^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| }^2}}}{{\left| {n_c^\delta \left( {{u_i}} \right) } \right| \left| {{n_{\left( {c,D} \right) }}\left( {{u_i}} \right) } \right| }}*\frac{{\left| {n_c^\delta \left( {{u_i}} \right) } \right| }}{{\left| U \right| }}} \right) \\{} & {} \quad = - \frac{{{P_c}\left( D \right) }}{{\left| U \right| }}\mathop \sum \limits _{i = 1}^{\left| U \right| } \mathrm{{log}}\left( {\frac{{{{\left| {n_c^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| }^2}}}{{\left| U \right| \left| {{n_{\left( {c,D} \right) }}\left( {{u_i}} \right) } \right| }}} \right) \end{aligned}$$

According to Definition 6, \(N{H_\delta }\left( {D,c} \right) = N{H_\delta }\left( {D|c} \right) + N{H_\delta }\left( c \right)\) holds.

Proof of Proposition 2

$$\begin{aligned} N{H_\delta }\left( {D,c} \right)= & {} - \frac{{{P_c}\left( D \right) }}{{\left| U \right| }}\mathop \sum \limits _{i = 1}^{\left| U \right| } \mathrm{{log}}\left( {\frac{{{{\left| {n_c^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| }^2}}}{{\left| U \right| \left| {{n_{\left( {c,D} \right) }}\left( {{u_i}} \right) } \right| }}} \right) \\= & {} - \frac{{{P_c}\left( D \right) }}{{\left| U \right| }}\mathop \sum \limits _{i = 1}^{\left| U \right| } \mathrm{{log}}\left( {\frac{{\left| {n_c^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| \left| {n_c^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| }}{{\left| U \right| \left| {{n_{\left( {c,D} \right) }}\left( {{u_i}} \right) } \right| }}} \right) \\= & {} - \frac{{{P_c}\left( D \right) }}{{\left| U \right| }}\mathop \sum \limits _{i = 1}^{\left| U \right| } \mathrm{{log}}\left( {\frac{{\left| {n_c^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| }}{{\left| U \right| }}\frac{{\left| {n_c^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| }}{{\left| {{n_{\left( {c,D} \right) }}\left( {{u_i}} \right) } \right| }}} \right) \\= & {} - \frac{{{P_c}\left( D \right) }}{{\left| U \right| }}\mathop \sum \limits _{i = 1}^{\left| U \right| } \mathrm{{log}}\left( {{\kappa _i}*\,{\alpha _i}} \right) \end{aligned}$$

Proof of Proposition 3

From Eq. (2), we know that \(n_b^\delta \left( {{u_i}} \right) \supseteq n_c^\delta \left( {{u_i}} \right)\), so \(n_b^\delta \left( {{u_i}} \right) \cap {\left[ {{u_i}} \right] _D} \supseteq n_c^\delta \left( {{u_i}} \right) \cap {\left[ {{u_i}} \right] _D}\), \(n_b^\delta \left( {{u_i}} \right) \cup {\left[ {{u_i}} \right] _D} \supseteq n_c^\delta \left( {{u_i}} \right) \cup {\left[ {{u_i}} \right] _D}\) and \({n_{\left( {b,D} \right) }}\left( {{u_i}} \right) \supseteq {n_{\left( {c,D} \right) }}\left( {{u_i}} \right)\). Thus, the numerical relationship between \(\frac{{{{\left| {n_b^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| }^2}}}{{\left| {{n_{\left( {b,D} \right) }}\left( {{u_i}} \right) } \right| }}\) and \(\frac{{{{\left| {n_c^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| }^2}}}{{\left| {{n_{\left( {c,D} \right) }}\left( {{u_i}} \right) } \right| }}\) is unknown, so the numerical relationship between \(- \frac{1}{{\left| U \right| }}\mathop \sum \limits _{i = 1}^{\left| U \right| } \mathrm{{log}}\left( {\frac{{{{\left| {n_b^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| }^2}}}{{\left| U \right| \left| {{n_{\left( {b,D} \right) }}\left( {{u_i}} \right) } \right| }}} \right)\) and \(- \frac{1}{{\left| U \right| }}\mathop \sum \limits _{i = 1}^{\left| U \right| } \mathrm{{log}}\left( {\frac{{{{\left| {n_c^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| }^2}}}{{\left| U \right| \left| {{n_{\left( {c,D} \right) }}\left( {{u_i}} \right) } \right| }}} \right)\) is not clear. From Eq. (8), \({P_b}\left( D \right) \le {P_c}\left( D \right)\) can be known. Therefore, the relation between \(- \frac{{{P_b}\left( D \right) }}{{\left| U \right| }}\mathop \sum \limits _{i = 1}^{\left| U \right| } \mathrm{{log}}\left( {\frac{{{{\left| {n_b^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| }^2}}}{{\left| U \right| \left| {{n_{\left( {b,D} \right) }}\left( {{u_i}} \right) } \right| }}} \right)\) and \(- \frac{{{P_c}\left( D \right) }}{{\left| U \right| }}\mathop \sum \limits _{i = 1}^{\left| U \right| } \mathrm{{log}}\left( {\frac{{{{\left| {n_c^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| }^2}}}{{\left| U \right| \left| {{n_{\left( {c,D} \right) }}\left( {{u_i}} \right) } \right| }}} \right)\) is not clear. According to Eq. (20), Proposition 3 holds.

Example

A Neighborhood decision system \(NS = \left( {U,C,D,\,\delta } \right)\) is shown below, where the universe \(U = \left\{ {{u_1},{u_2},{u_3},{u_4}} \right\}\); the conditional attribute set \(C = \left\{ {{c_1},{c_2},{c_3}} \right\}\); the decision attribute \(D = d\); the neighborhood radius \(\delta = 0.3\). Let initial gene subset \(c = \emptyset\), the base of log is 10, and \(P = 2\) in Eq. (1).

U

\({c_1}\)

\({c_2}\)

\({c_3}\)

d

\({u_1}\)

0.12

0.41

0.61

Y

\({u_2}\)

0.21

0.15

0.14

Y

\({u_3}\)

0.31

0.11

0.26

N

\({u_4}\)

0.61

0.13

0.23

N

From Eq. (3), \({\left[ {{u_1}} \right] _D} = {\left[ {{u_2}} \right] _D} = \left\{ {{u_1},{u_2}} \right\}\), \({\left[ {{u_3}} \right] _D} = {\left[ {{u_4}} \right] _D} = \left\{ {{u_3},{u_4}} \right\}\).

From Eq. (1), when \(c = \left\{ {{c_1}} \right\}\), we know that \(D{F_{\left\{ {{c_1}} \right\} }}\left( {{u_1},\,{u_1}} \right) = 0 \le \delta\), \(D{F_{\left\{ {{c_1}} \right\} }}\left( {{u_1},\,{u_2}} \right) = 0.09 \le \delta\), \(D{F_{\left\{ {{c_1}} \right\} }}\left( {{u_1},\,{u_3}} \right) = 0.19 \le \delta\), \(D{F_{\left\{ {{c_1}} \right\} }}\left( {{u_1},\,{u_4}} \right) = 0.49 > \delta\), \(D{F_{\left\{ {{c_1}} \right\} }}\left( {{u_2},\,{u_2}} \right) = 0 \le \delta\), \(D{F_{\left\{ {{c_1}} \right\} }}\left( {{u_2},\,{u_3}} \right) = 0.1 \le \delta\),      \(D{F_{\left\{ {{c_1}} \right\} }}\left( {{u_2},\,{u_4}} \right) = 0.4 > \delta\), \(D{F_{\left\{ {{c_1}} \right\} }}\left( {{u_3},\,{u_3}} \right) = 0 \le \delta\), \(D{F_{\left\{ {{c_1}} \right\} }}\left( {{u_3},\,{u_4}} \right) = 0.3 \le \delta\), \(D{F_{\left\{ {{c_1}} \right\} }}\left( {{u_4},\,{u_4}} \right) = 0 \le \delta\).

From Eq. (2), \(n_{\left\{ {{c_1}} \right\} }^\delta \left( {{u_1}} \right) = \left\{ {{u_1},{u_2},{u_3}} \right\}\), \(n_{\left\{ {{c_1}} \right\} }^\delta \left( {{u_2}} \right) = \left\{ {{u_1},{u_2},{u_3}} \right\}\), \(n_{\left\{ {{c_1}} \right\} }^\delta \left( {{u_3}} \right) = \left\{ {{u_1},{u_2},{u_3},{u_4}} \right\}\), \(n_{\left\{ {{c_1}} \right\} }^\delta \left( {{u_4}} \right) = \left\{ {{u_3},{u_4}} \right\}\).

From Eq. (14), \({n_{\left( {\left\{ {{c_1}} \right\} ,D} \right) }}\left( {{u_1}} \right) = n_{\left\{ {{c_1}} \right\} }^\delta \left( {{u_1}} \right) \cup {\left[ {{u_1}} \right] _D} = \left\{ {{u_1},{u_2},{u_3}} \right\}\), \({n_{\left( {\left\{ {{c_1}} \right\} ,D} \right) }}\left( {{u_2}} \right) = n_{\left\{ {{c_1}} \right\} }^\delta \left( {{u_2}} \right) \cup {\left[ {{u_2}} \right] _D} = \left\{ {{u_1},{u_2},{u_3}} \right\}\), \({n_{\left( {\left\{ {{c_1}} \right\} ,D} \right) }}\left( {{u_3}} \right) = n_{\left\{ {{c_1}} \right\} }^\delta \left( {{u_3}} \right) \cup {\left[ {{u_3}} \right] _D} = \left\{ {{u_1},{u_2},{u_3},{u_4}} \right\}\), \({n_{\left( {\left\{ {{c_1}} \right\} ,D} \right) }}\left( {{u_4}} \right) = n_{\left\{ {{c_1}} \right\} }^\delta \left( {{u_4}} \right) \cup {\left[ {{u_4}} \right] _D} = \left\{ {{u_3},{u_4}} \right\}\).

From Eq. (6), Eq. (7), and Eq. (8), \(\underline{{N_{\left\{ {{c_1}} \right\} }}} \left( D \right) = \,\left\{ {{u_4}} \right\}\), \(\overline{{N_{\left\{ {{c_1}} \right\} }}} \left( D \right) = \left\{ {{u_1},{u_2},{u_3},{u_4}} \right\}\), \({P_{\left\{ {{c_1}} \right\} }}\left( D \right) = \,\frac{{\left| {\underline{{N_{\left\{ {{c_1}} \right\} }}} \left( D \right) } \right| }}{{\left| {\overline{{N_{\left\{ {{c_1}} \right\} }}} \left( D \right) } \right| }} = \frac{1}{4}\).

From Eq. (20), \(N{H_\delta }\left( {D,\left\{ {{c_1}} \right\} } \right) = - \frac{{{P_{\left\{ {{c_1}} \right\} }}\left( D \right) }}{{\left| U \right| }}\mathop \sum \limits _{i = 1}^{\left| U \right| } \log \left( {\frac{{{{\left| {n_{\left\{ {{c_1}} \right\} }^\delta \left( {{u_i}} \right) \cap {{\left[ {{u_i}} \right] }_D}} \right| }^2}}}{{\left| U \right| \left| {{n_{\left( {\left\{ {{c_1}} \right\} ,D} \right) }}\left( {{u_i}} \right) } \right| }}} \right)\)

\(= - \frac{1/4}{4}\left( {\log \left( {\frac{{{2^2}}}{{4 \times 3}}} \right) + \log \left( {\frac{{{2^2}}}{{4 \times 3}}} \right) + \log \left( {\frac{{{2^2}}}{{4 \times 4}}} \right) + \log \left( {\frac{{{2^2}}}{{4 \times 2}}} \right) } \right) = 0.116\)

Similarly, \(N{H_\delta }\left( {D,\left\{ {{c_2}} \right\} } \right) = 0\), \(N{H_\delta }\left( {D,\left\{ {{c_3}} \right\} } \right) = 0.191\), \(N{H_\delta }\left( {D,\left\{ {{c_1},{c_2}} \right\} } \right) = 0.345\), \(N{H_\delta }\left( {D,\left\{ {{c_1},{c_3}} \right\} } \right) = 0.496\), \(N{H_\delta }\left( {D,\left\{ {{c_2},{c_3}} \right\} } \right) = 0.191\),          \(N{H_\delta }\left( {D,\left\{ {{c_1},{c_2},{c_3}} \right\} } \right) = 0.496\).

From Eq. (21), when \(c = \emptyset\), \(Sig\left( {{c_2},\emptyset ,D} \right) = 0< Sig\left( {{c_1},\emptyset ,D} \right) = 0.116 < Sig\left( {{c_3},\emptyset ,D} \right) = 0.191\), so \({c_3}\) is added into c. Because \(Sig\left( {{c_2},\left\{ {{c_3}} \right\} ,D} \right) = 0 < Sig\left( {{c_1},\left\{ {{c_3}} \right\} ,D} \right) = 0.305\), \({c_1}\) is added into c. Because \(Sig\left( {{c_2},\left\{ {{c_1},{c_3}} \right\} ,D} \right) = 0\) satisfies the termination condition, \(c = \left\{ {{c_1},{c_3}} \right\}\) is an optimal gene subset.

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Xu, J., Qu, K., Qu, K. et al. Feature selection using neighborhood uncertainty measures and Fisher score for gene expression data classification. Int. J. Mach. Learn. & Cyber. 14, 4011–4028 (2023). https://doi.org/10.1007/s13042-023-01878-7

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