A paper first surveys some variations of support vector machines and their solution, and then apply to credit risk evaluation.

The standard SVM is formulated as

\[\begin{align} && \min_{w,b,\xi}\ & \frac{1}{2}w^Tw + C\sum_i\xi_i \\ \textrm{subject to}&& y_i(w^T \phi(x_i) + b) & \ge 1-\xi_i \quad \forall i \\ && \xi_i &\ge 0\quad \forall i \end{align}\]

with \(\xi_i\) the error margin. Solving the above optimization problem is to find the saddle point of the Lagrangian function.

Training a SVM requires each training data belongs to one class or the other instead of a fuzzy measure. If we associate a membership grade \(\mu_i\in [0,1]\) to each data point as a measure of membership between two classes, we can modify the above optimization problem for a fuzzy SVM:

\[\begin{align} && \min_{w,b,\xi}\ & \frac{1}{2}w^Tw + C\sum_i\mu_i\xi_i \\ \textrm{subject to}&& y_i(w^T \phi(x_i) + b) & \ge 1-\xi_i \quad \forall i \\ && \xi_i &\ge 0\quad \forall i \end{align}\]

Note the \(\mu_i\xi_i\) summation term in the objective function. In FSVM, the error term \(\xi_i\) is scaled by the membership value \(\mu_i\), to reflect the relative confidence of the training samples.

Borrowing from the LS-SVM paper, we can use least square measure for the error term and replace the constraint with equality:

\[\begin{align} && \min_{w,b,\xi}\ & \frac{1}{2}w^Tw + \frac{1}{2}C\sum_i\mu_i\xi_i^2 \\ \textrm{subject to}&& y_i(w^T \phi(x_i) + b) & = 1-\xi_i \quad \forall i \end{align}\]

which we do not even enforce \(\xi_i\) to be non-negative as the objective function will penalize for the result offset from \(\pm 1\) in both direction. Solving this LS-FSVM equation, we first find the Lagrangian

\[L(w,b,\xi,\alpha) = \frac{1}{2}w^Tw + \frac{1}{2}C\sum_i\mu_i\xi_i^2 - \sum_i\alpha_i\left[y_i\left(w^T\phi(x_i)+b\right)-1+\xi_i\right]\]

The Lagrangian multiplier \(\alpha_i\) can be positive or negative as it associates with a equality constraint. To find the saddle point,

\[\begin{align} \frac{\partial L}{\partial w} &= w - \sum_i \alpha_i y_i \phi(x_i) & \Rightarrow && w &= \sum_i \alpha_i y_i \phi(x_i) \\ \frac{\partial L}{\partial b} &= - \sum_i \alpha_i y_i & \Rightarrow && \sum_i \alpha_i y_i &= 0 \\ \frac{\partial L}{\partial \xi_i} &= \mu_i C\xi_i - \alpha_i & \Rightarrow && \alpha_i &= \mu_i C\xi_i \\ \frac{\partial L}{\partial \alpha_i} &= y_i[w^T\phi(x_i)+b]-1+\xi_i & \Rightarrow && y_i[w^T\phi(x_i)+b]-1+\xi_i &= 0 \end{align}\]

Substituting \(w\) from the last equation above (i.e., the constraint equality) gives

\[\begin{align} y_i[\sum_j\alpha_jy_j\phi(x_j)^T\phi(x_i)+b]+\xi_i &= 1 \\ \sum_j\alpha_jy_iy_j\phi(x_j)^T\phi(x_i)+by_i+\frac{\alpha_i}{\mu_iC} &= 1 \end{align}\]

Which, together with the equation \(\sum_i\alpha_iy_i=0\), we can write in the following matrix form:

\[\begin{bmatrix} \mathbf{\Omega} & \mathbf{Y} \\ \mathbf{Y}^T & \mathbf{0} \end{bmatrix} \begin{bmatrix} \mathbf{\alpha} \\ \mathbf{b} \end{bmatrix} = \begin{bmatrix} \mathbf{1} \\ \mathbf{0} \end{bmatrix}\]


\[\begin{align} \Omega_{ij} &= y_i y_j \phi(x_i)^T\phi(x_j) + \frac{1}{\mu_i C} \\ \mathbf{Y} &= \begin{bmatrix}y_1 & y_2 & \cdots & y_N\end{bmatrix}^T \end{align}\]

(i.e., \(\mathbf{\Omega}\) is positive semi-definite) and the solution to the LS-FSVM are:

\[\begin{align} \mathbf{\Omega\alpha} + \mathbf{Y}^T\mathbf{b} &= \mathbf{1} \\ \mathbf{\alpha} &= \mathbf{\Omega}^{-1}(\mathbf{1} - \mathbf{Y}^T\mathbf{b}) \end{align}\]


\[\begin{align} \mathbf{Y}^T\mathbf{\alpha} &= \mathbf{0} \\ \mathbf{Y}^T\mathbf{\Omega}^{-1}(\mathbf{1} - \mathbf{Y}^T\mathbf{b}) &= \mathbf{0} \\ \mathbf{Y}^T\mathbf{\Omega}^{-1} &= (\mathbf{Y}^T\mathbf{\Omega}^{-1}\mathbf{Y})\mathbf{b} \\ \mathbf{b} &= \frac{ \mathbf{Y}^T\mathbf{\Omega}^{-1} }{ \mathbf{Y}^T\mathbf{\Omega}^{-1}\mathbf{Y} } \mathbf{b} &= \frac{1}{ \mathbf{Y}^T\mathbf{\Omega}^{-1}\mathbf{Y} }\mathbf{Y}^T\mathbf{\Omega}^{-1} \end{align}\]

Note the above that a closed form solution to LS-FSVM is provided by solving a set of linear equations. We did not introduce quadratic programming and thus it has lower computational complexity.

Then the paper proceed to evaluate LS-FSVM with credit data. Evaluation is to compare the accuracy, true positive rate, and true negative rate among different classifiers. The paper used linear nad logistic regression, ANN, and various kind of SVM. RBF kernel is used for all SVM. For ANN, the paper used a 3-layer back-propagation network with 10 sigmoidal neurons in the hidden layer; trained with Levenberg-Marquardt algorithm for 1600 epochs, with learning and momentum rates of 0.1 and 0.15 respectively and accepted average square error of 0.05.

The input data has 14 attributes (year of birth, number of children, income of applicant and spouse, value of home, mortgage balance outstanding, etc.) and interpolation is used to impute missing data. There are some treatments used in the training: (1) Because insolvency is relatively rare, the number of observed bad borrowers are tripled to make the number nearly equal to the number of good borrowers in the training, to avoid the impact of imbalanced samples. (2) FSVM depends on membership grade. It is generated by a transformation function.

The result shows a significant improvement of accuracy in LS-FSVM compared to plain SVM or even ANN and regressions. The advantage over simple LS-SVM or FSVM are not so significant.

Bibliographic data

   author = "Lean Yu",
   title = "Credit Risk Evaluation with a Least Squares Fuzzy Support Vector Machine Classifier"",
   journal = "Discrete Dynamics in Nature and Society",
   volume = "2014",
   year = "2014",
   doi = "http://dx.doi.org/10.1155/2014/564213",