An extension to fuzzy support vector data description (FSVDD*)

Abstract

The well-known support vector data description (SVDD) is based on precise description of precise data. When we know the features of training samples precisely and we are uncertain about their class labels, the fuzzy SVDD can be used to obtain the data description. But if the features of training samples are fuzzy numbers, the fuzzy SVDD cannot be utilized. In this paper, we extend the fuzzy SVDD for the description of such training samples and then apply our proposed method, called FSVDD*, to real data. The experimental results show the ability of the proposed method in Taiwanese tea evaluation.

Appendix

The Lagrangian dual form of the program (1) is as follows:

$$ \begin{gathered} { \max }_{\delta ,\gamma } L(R,e,\xi ,\delta ,\gamma ) \hfill \\ {\text{subject}}\;{\text{to}}\;\delta_{i} ,\gamma_{i} \ge 0,\quad i = 1, \ldots ,n, \hfill \\ \end{gathered} $$

(29)

where δ = (δ ₁, …, δ _n )^T, γ = (γ ₁, …, γ ₂)^T and

$$ L(R,e,\xi ,\delta ,\gamma ) = { \inf }\left\{ {R^{2} + C\sum\limits_{i = 1}^{n} {w_{i} \xi_{i} } - \sum\limits_{i = 1}^{n} {\delta_{i} (R^{2} + \xi_{i} - g(x_{i} )^{\text{T}} g(x_{i} ) + 2e^{\text{T}} g(x_{i} ) - e^{\text{T}} e)} - \sum\limits_{i = 1}^{n} {\gamma_{i} \xi_{i} } } \right\}. $$

(30)

For the optimal solution, the following conditions are satisfied

$$ {\frac{\partial L}{\partial R}} = 0 \to \sum\limits_{i = 1}^{n} {\delta_{i} = 1} , $$

(31)

$$ {\frac{\partial L}{\partial e}} = 0 \to e = \sum\limits_{i = 1}^{n} {\delta_{i} g(x_{i} )} , $$

(32)

$$ {\frac{\partial L}{\partial \xi }} = 0 \to \delta_{i} = Cw_{i} - \gamma_{i} ,\quad i = 1, \ldots ,n, $$

(33)

$$ \delta_{i} (\left\| {g(x_{i} ) - e} \right\|^{2} - R^{2} - \xi_{i} ) = 0,\quad i = 1, \ldots ,n, $$

(34)

$$ \gamma_{i} \xi_{i} = 0,\quad i = 1, \ldots ,n, $$

(35)

Using the above conditions, L(R, e, ξ, δ, γ) is transformed to

$$ \sum\limits_{i = 1}^{n} {\delta_{i} K(x_{i} ,x_{i} )} - \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {\delta_{i} \delta_{j} K(x_{i} ,x_{j} ),} } $$

(36)

where K(x _i , x _j ) = g(x _i )^T g(x _j ). Since δ _i  ≥ 0 and from (33) we have $0 \le \delta_{i} \le Cw_{i} .$ So, the Lagrangian dual form of (1) can be restated as follows:

$$ \begin{gathered} { \max }_{\delta } \sum\limits_{i = 1}^{n} {\delta_{i} K(x_{i} ,x_{i} )} - \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {\delta_{i} \delta_{j} K(x_{i} ,x_{j} )} } \hfill \\ {\text{subject}}\;{\text{to}}\left\{ {\begin{array}{l} {\sum\nolimits_{i = 1}^{n} {\delta_{i} = 1,} } \hfill \\ {0 \le \delta_{i} \le Cw_{i} ,\quad i = 1, \ldots ,n,} \hfill \\ \end{array} } \right. \hfill \\ \end{gathered} $$

(37)

which is a conventional quadratic program and can be solved easily. From (32), $\left\| {g(x_{i} ) - e} \right\|^{2} = K(x_{i} ,x_{i} ) - 2\sum\nolimits_{j = 1}^{n} {\delta_{j} K(x_{i} ,x_{j} ) + \sum\nolimits_{j = 1}^{n} {\sum\nolimits_{k = 1}^{n} {\delta_{j} \delta_{k} K(x_{j} ,x_{k} )} } }$ and from (34) if $\delta_{i} > 0,$ $K(x_{i} ,x_{i} ) - 2\sum\nolimits_{j = 1}^{n} {\delta_{j} K(x_{i} ,x_{j} ) + \sum\nolimits_{j = 1}^{n} {\sum\nolimits_{k = 1}^{n} {\delta_{j} \delta_{k} K(x_{j} ,x_{k} ) = R^{2} + \xi_{i} .} } }$ From (33) if $\delta_{i} < Cw_{i} ,\;\gamma_{i} > 0.$ So, from (35) we have $\xi_{i} = 0.$ So, if $0 < \delta_{i} < Cw_{i} ,$

$$ R^{2} = K(x_{i} ,x_{i} ) - 2\sum\limits_{j = 1}^{n} {\delta_{j} K(x_{i} ,x_{j} )} + \sum\limits_{j = 1}^{n} {\sum\limits_{k = 1}^{n} {\delta_{j} \delta_{k} K(x_{j} ,x_{k} ).} } $$

(38)

Finally, the unknown datum x is inside the hypersphere if $\left\| {g(x) - e} \right\|^{2} \le R^{2}$ or equivalently if

$$ K(x,x) - 2\sum\limits_{i = 1}^{n} {\delta_{i} K(x,x_{i} )} + \sum\limits_{i = 1}^{n} {\sum\limits_{j = 1}^{n} {\delta_{i} \delta_{j} K(x_{i} ,x_{j} ) \le R^{2} .} } $$

(39)