﻿ 自动调整样本和特征权值的模糊聚类算法
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 哈尔滨工程大学学报  2018, Vol. 39 Issue (9): 1554-1560  DOI: 10.11990/jheu.201704029 0

### 引用本文

LI Kai, GAO Yan, CAO Zhe. Fuzzy clustering algorithm based on the automatic variable weights of samples and features[J]. Journal of Harbin Engineering University, 2018, 39(9), 1554-1560. DOI: 10.11990/jheu.201704029.

### 文章历史

Fuzzy clustering algorithm based on the automatic variable weights of samples and features
LI Kai, GAO Yan, CAO Zhe
School of Cyber Security and Computer, Hebei University, Baoding 071002, China
Abstract: To improve the sensitivity of the fuzzy c-means clustering algorithm to feature and sample noise, feature and sample weights were introduced to the objective function of the fuzzy c-means clustering algorithm to develop a new fuzzy clustering model. The model was solved using the Lagrange method, and a fuzzy clustering algorithm in which the sample and feature weights could be automatically adjusted was proposed. The kernel strategy was introduced to the new fuzzy clustering model, and a kernel fuzzy clustering algorithm in which the sample and feature weights could be automatically adjusted was proposed. The experimental results showed that the method features excellent processing ability for clustering of data containing feature and sample noise; it provides a feasible approach for feature extraction and sample selection.
Keywords: fuzzy clustering    objective function    sample and feature weighting    sample weighting    feature weighting    kernel strategy    feature noise    sample noise

1 样本与特征加权的模糊聚类

 $\begin{array}{l} \mathop {\min }\limits_{U, V} J\left( {\mathit{\boldsymbol{U}}, \mathit{\boldsymbol{V}}} \right) = \sum\limits_{i = 1}^c {\sum\limits_{j = 1}^n {r_j^p\mathit{\boldsymbol{u}}_{ij}^m\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{({\mathit{\boldsymbol{x}}_{jk}} - {\mathit{\boldsymbol{v}}_{ik}})}^2}} } } \\ {\rm{s}}{\rm{.t}}{\rm{.}}\left\{ \begin{array}{l} \sum\limits_{i = 1}^c {{\mathit{\boldsymbol{u}}_{ij}} = 1, } \;\;\;j = 1, 2, \cdots , n\\ \sum\limits_{j = 1}^n {{\mathit{\boldsymbol{r}}_j} = 1} \\ \sum\limits_{k = 1}^s {{\mathit{\boldsymbol{w}}_k} = 1} \end{array} \right. \end{array}$ (1)

 $\begin{array}{l} L\left( {\mathit{\boldsymbol{U}}, \mathit{\boldsymbol{V}};\mathit{\boldsymbol{\alpha }}, \lambda , \beta } \right) = \sum\limits_{i = 1}^c {\sum\limits_{j = 1}^n {r_j^p\mathit{\boldsymbol{u}}_{ij}^m\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{({\mathit{\boldsymbol{x}}_{jk}} - {\mathit{\boldsymbol{v}}_{ik}})}^2}} } } - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{j = 1}^n {{\mathit{\boldsymbol{\alpha }}_j}(\sum\limits_{i = 1}^c {{\mathit{\boldsymbol{u}}_{ij}} - 1} ) - } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\lambda \left( {\sum\limits_{j = 1}^n {{\mathit{\boldsymbol{r}}_j} - 1} } \right) - \beta \left( {\sum\limits_{k = 1}^s {{\mathit{\boldsymbol{w}}_k} - 1} } \right) \end{array}$ (2)

 $m\mathit{\boldsymbol{r}}_j^p\mathit{\boldsymbol{u}}_{ij}^{m - 1}\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{({\mathit{\boldsymbol{x}}_{jk}} - {\mathit{\boldsymbol{v}}_{ik}})}^2} - {\mathit{\boldsymbol{\alpha }}_j} = 0}$ (3)
 $\sum\limits_{i = 1}^c {{\mathit{\boldsymbol{u}}_{ij}} = 1}$ (4)

 ${\mathit{\boldsymbol{u}}_{ij}} = {\left( {{\alpha _j}/m\mathit{\boldsymbol{r}}_j^p\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{({\mathit{\boldsymbol{x}}_{jk}} - {\mathit{\boldsymbol{v}}_{ik}})}^2}} } \right)^{\frac{1}{{m - 1}}}}$ (5)

 $\alpha _j^{\frac{1}{{m - 1}}} = \frac{1}{{{{\sum\limits_{i = 1}^c {\left( {1/m\mathit{\boldsymbol{r}}_j^p\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{({\mathit{\boldsymbol{x}}_{jk}} - {\mathit{\boldsymbol{v}}_{ik}})}^2}} } \right)} }^{\frac{1}{{m - 1}}}}}}$ (6)

 ${u_{ij}} = \frac{1}{{\sum\limits_{l = 1}^c {{{\left( {\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{({\mathit{\boldsymbol{x}}_{jk}} - {\mathit{\boldsymbol{v}}_{ik}})}^2}/\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{({\mathit{\boldsymbol{x}}_{jk}} - {\mathit{\boldsymbol{v}}_{lk}})}^2}} } } \right)}^{\frac{1}{{m - 1}}}}} }}$ (7)

 $\begin{array}{l} \;\;{\mathit{\boldsymbol{v}}_{ik}} = \frac{{\sum\limits_{j = 1}^n {\mathit{\boldsymbol{r}}_j^p\mathit{\boldsymbol{u}}_{ij}^m{\mathit{\boldsymbol{x}}_{jk}}} }}{{\sum\limits_{j = 1}^n {\mathit{\boldsymbol{r}}_j^p\mathit{\boldsymbol{u}}_{ij}^m} }}, \\ i = 1, 2, \cdots , c;k = 1, 2, \cdots , s \end{array}$ (8)
 $\begin{array}{l} {\mathit{\boldsymbol{r}}_j} = \frac{1}{{\sum\limits_{l = 1}^n {{{\left( {\sum\limits_{i = 1}^c {\mathit{\boldsymbol{u}}_{ij}^m\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{({\mathit{\boldsymbol{x}}_{jk}} - {\mathit{\boldsymbol{v}}_{ik}})}^2}/\sum\limits_{i = 1}^c {\mathit{\boldsymbol{u}}_{il}^m\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{({\mathit{\boldsymbol{x}}_{lk}} - {\mathit{\boldsymbol{v}}_{ik}})}^2}} } } } } \right)}^{\frac{1}{{p - 1}}}}} }}, \\ j = 1, 2, \cdots , n \end{array}$ (9)
 $\begin{array}{l} {\mathit{\boldsymbol{w}}_k} = \frac{1}{{\sum\limits_{l = 1}^s {{{\left( {\sum\limits_{i = 1}^c {\sum\limits_{j = 1}^n {\mathit{\boldsymbol{r}}_j^p\mathit{\boldsymbol{u}}_{ij}^m{{({\mathit{\boldsymbol{x}}_{jk}} - {\mathit{\boldsymbol{v}}_{ik}})}^2}/\sum\limits_{i = 1}^c {\sum\limits_{j = 1}^n {\mathit{\boldsymbol{r}}_j^p\mathit{\boldsymbol{u}}_{ij}^m{{({\mathit{\boldsymbol{x}}_{jl}} - {\mathit{\boldsymbol{v}}_{il}})}^2}} } } } } \right)}^{\frac{1}{{q - 1}}}}} }}, \\ k = 1, 2, \cdots , s \end{array}$ (10)

2 样本和特征加权的核模糊聚类

 $\begin{array}{l} \mathop {\min }\limits_{U, V} J\left( {\mathit{\boldsymbol{U}}, \mathit{\boldsymbol{V}}} \right) = \sum\limits_{i = 1}^c {\sum\limits_{j = 1}^n {\mathit{\boldsymbol{r}}_j^p\mathit{\boldsymbol{u}}_{ij}^m\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{\left\| {\varphi ({\mathit{\boldsymbol{x}}_{jk}}) - {\mathit{\boldsymbol{v}}_{ik}}} \right\|}^2}} } } \\ {\rm{s}}{\rm{.}}\;{\rm{t}}{\rm{.}}\left\{ \begin{array}{l} \sum\limits_{i = 1}^c {{\mathit{\boldsymbol{u}}_{ij}} = 1, \;\;\;j = 1, 2, \cdots , n} \\ \sum\limits_{j = 1}^n {{\mathit{\boldsymbol{r}}_j} = 1} \\ \sum\limits_{k = 1}^s {{\mathit{\boldsymbol{w}}_k} = 1} \end{array} \right. \end{array}$ (11)

 $\begin{array}{l} L\left( {\mathit{\boldsymbol{U}}, \mathit{\boldsymbol{V}};\mathit{\boldsymbol{\alpha }}, \lambda , \beta } \right) = \sum\limits_{i = 1}^c {\sum\limits_{j = 1}^n {\mathit{\boldsymbol{r}}_j^p\mathit{\boldsymbol{u}}_{ij}^m\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q||\varphi ({\mathit{\boldsymbol{x}}_{jk}}) - } } } \\ {\mathit{\boldsymbol{v}}_{ik}}|{|^2} - \sum\limits_{j = 1}^n {{\alpha _j}\left( {\sum\limits_{i = 1}^c {{\mathit{\boldsymbol{u}}_{ij}} - 1} } \right) - } \\ \lambda \left( {\sum\limits_{j = 1}^n {{\mathit{\boldsymbol{r}}_j} - 1} } \right) - \beta \left( {\sum\limits_{k = 1}^s {{\mathit{\boldsymbol{w}}_k} - 1} } \right) \end{array}$ (12)

 $\begin{array}{*{20}{c}} {{{\left\| {\varphi ({\mathit{\boldsymbol{x}}_{jk}}) - {\mathit{\boldsymbol{v}}_{ik}}} \right\|}^2} = {{(\varphi ({\mathit{\boldsymbol{x}}_{jk}}) - {\mathit{\boldsymbol{v}}_{ik}})}^{\rm{T}}}(\varphi ({\mathit{\boldsymbol{x}}_{jk}}) - {\mathit{\boldsymbol{v}}_{ik}}) = }\\ {\varphi {{({\mathit{\boldsymbol{x}}_{jk}})}^{\rm{T}}}\varphi ({\mathit{\boldsymbol{x}}_{jk}}) - 2\varphi {{({\mathit{\boldsymbol{x}}_{jk}})}^{\rm{T}}}{\mathit{\boldsymbol{v}}_{ik}} + {\mathit{\boldsymbol{v}}_{ik}}^{\rm{T}}{\mathit{\boldsymbol{v}}_{ik}} = }\\ {K({\mathit{\boldsymbol{x}}_{jk}}, {\mathit{\boldsymbol{x}}_{jk}}) - 2\frac{{\sum\limits_{b = 1}^n {\mathit{\boldsymbol{u}}_{ib}^mK({\mathit{\boldsymbol{x}}_{jk}}, {\mathit{\boldsymbol{x}}_{bk}})} }}{{\sum\limits_{b = 1}^n {\mathit{\boldsymbol{u}}_{ib}^m} }} + }\\ {\frac{{\sum\limits_{b = 1}^n {\sum\limits_{t = 1}^n {\mathit{\boldsymbol{u}}_{ib}^m\mathit{\boldsymbol{u}}_{it}^mK({\mathit{\boldsymbol{x}}_{bk}}, {\mathit{\boldsymbol{x}}_{tk}})} } }}{{{{\left( {\sum\limits_{t = 1}^n {\mathit{\boldsymbol{u}}_{it}^m} } \right)}^2}}}} \end{array}$

 $m\mathit{\boldsymbol{r}}_j^p\mathit{\boldsymbol{u}}_{ij}^{m - 1}\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{\left\| {\varphi ({\mathit{\boldsymbol{x}}_{jk}}) - {\mathit{\boldsymbol{v}}_{ik}}} \right\|}^2} - {\alpha _j} = 0}$ (13)
 $\sum\limits_{i = 1}^c {{\mathit{\boldsymbol{u}}_{ij}} = 1}$ (14)

 ${\mathit{\boldsymbol{u}}_{ij}} = {\left( {\frac{{{\alpha _j}}}{{mr_j^p\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{\left\| {\varphi ({\mathit{\boldsymbol{x}}_{jk}}) - {\mathit{\boldsymbol{v}}_{ik}}} \right\|}^2}} }}} \right)^{\frac{1}{{m - 1}}}}$ (15)

 $\sum\limits_{i = 1}^c {{{\left( {\frac{{{\alpha _j}}}{{mr_j^p\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{\left\| {\varphi ({\mathit{\boldsymbol{x}}_{jk}}) - {\mathit{\boldsymbol{v}}_{ik}}} \right\|}^2}} }}} \right)}^{\frac{1}{{m - 1}}}} = 1} {\rm{ }}$

 $\alpha _j^{\frac{1}{{m - 1}}} = \frac{1}{{\sum\limits_{i = 1}^c {{{\left( {1/m\mathit{\boldsymbol{r}}_j^p\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{\left\| {\varphi ({\mathit{\boldsymbol{x}}_{jk}}) - {\mathit{\boldsymbol{v}}_{ik}}} \right\|}^2}} } \right)}^{\frac{1}{{m - 1}}}}} }},$

 $\begin{array}{*{20}{c}} {{u_{ij}} = }\\ {\frac{1}{{\sum\limits_{l = 1}^c {{{\left( {\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{\left\| {\varphi ({\mathit{\boldsymbol{x}}_{jk}}) - {\mathit{\boldsymbol{v}}_{ik}}} \right\|}^2}/\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{\left\| {\varphi ({\mathit{\boldsymbol{x}}_{jk}}) - {\mathit{\boldsymbol{v}}_{lk}}} \right\|}^2}} } } \right)}^{\frac{1}{{m - 1}}}}} }}} \end{array}$ (16)

 $\begin{array}{l} {\mathit{\boldsymbol{v}}_{ik}} = \frac{{\sum\limits_{j = 1}^n {\mathit{\boldsymbol{r}}_j^p\mathit{\boldsymbol{u}}_{ij}^m\varphi ({\mathit{\boldsymbol{x}}_{jk}})} }}{{\sum\limits_{j = 1}^n {\mathit{\boldsymbol{r}}_j^p\mathit{\boldsymbol{u}}_{ij}^m} }}, \\ i = 1, 2, \cdots , c;k = 1, 2, \cdots , s \end{array}$ (17)
 $\begin{array}{l} {\mathit{\boldsymbol{r}}_j} = \frac{1}{{{{\sum\limits_{l = 1}^n {\left( {\sum\limits_{i = 1}^c {\mathit{\boldsymbol{u}}_{ij}^m\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{\left\| {\varphi ({\mathit{\boldsymbol{x}}_{jk}}) - {\mathit{\boldsymbol{v}}_{ik}}} \right\|}^2}/\sum\limits_{i = 1}^c {\mathit{\boldsymbol{u}}_{il}^m\sum\limits_{k = 1}^s {\mathit{\boldsymbol{w}}_k^q{{\left\| {\varphi ({x_{lk}}) - {v_{ik}}} \right\|}^2}} } } } } \right)} }^{\frac{1}{{p - 1}}}}}}, \\ j = 1, 2, \cdots , n \end{array}$ (18)
 $\begin{array}{l} {\mathit{\boldsymbol{w}}_k} = \frac{1}{{\sum\limits_{l = 1}^s {{{\left( {\sum\limits_{i = 1}^c {\sum\limits_{j = 1}^n {\mathit{\boldsymbol{r}}_j^p\mathit{\boldsymbol{u}}_{ij}^m{{\left\| {\varphi ({\mathit{\boldsymbol{x}}_{jk}}) - {\mathit{\boldsymbol{v}}_{ik}}} \right\|}^2}/\sum\limits_{i = 1}^c {\sum\limits_{j = 1}^n {\mathit{\boldsymbol{r}}_j^p\mathit{\boldsymbol{u}}_{ij}^m{{\left\| {\varphi ({\mathit{\boldsymbol{x}}_{jl}}) - {\mathit{\boldsymbol{v}}_{il}}} \right\|}^2}} } } } } \right)}^{\frac{1}{{q - 1}}}}} }}, \\ k = 1, 2, \cdots , s \end{array}$ (19)

3 实验结果与分析 3.1 实验数据与性能指标

 Download: 图 1 Data3数据集的二维分布 Fig. 1 Two-dimensional distribution with data set Data3
 Download: 图 2 Ring数据集的二维分布 Fig. 2 Two-dimensional distribution with data set Ring
3.2 UCI标准数据集的聚类

 Download: 图 3 不同算法在数据集的聚类正确率 Fig. 3 Clustering accuracies for different algorithms with data sets
 Download: 图 4 基于核的算法在不同数据集的聚类正确率 Fig. 4 Clustering accuracies for different kernel-based algorithms with data sets

3.3 有噪声数据的聚类

 Download: 图 5 算法使用不同m值在iris数据集的聚类正确率和Jaccard系数 Fig. 5 Clustering accuracy and Jaccard index of algorithm in data set iris using the different value of m
 Download: 图 6 算法使用不同p值在iris数据集的聚类正确率和Jaccard系数 Fig. 6 Clustering accuracy and Jaccard index of algorithm in data set iris using the different value of p

4 结论

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