Graphical Data Representation in Bankruptcy Analysis

Graphical Data Representation in Bankruptcy Analysis

the penalty ξi is introduced, which is related to the distance from the hyper-
plane bounding observations of the same class to observation i. ξi > 0 if a
misclassification occurs. All observations satisfy the following two constraints:
yi(Xw + b) ≥ 1 - ξi,                          (1)

ξi ≥ 0.                               (2)

With the normalisation of w, b and ξ_i as in (1) the margin equals to 2/ ∣∣w∣∣.
The convex objective function to be minimised given the constraints (1) and
(2) is:

1ⁿ

2 ∣∣w∣∣² + C∑ξi∙                            (3)

i=1

The parameter C called capacity is related to the width of the margin
zone. The smaller the C is, the bigger margins are possible. Using well estab-
lished theory for optimisation of convex functions [6] we can derive the dual
Lagrangian:

₁      nn   n       n

Ld = 2w(α)^τw(α) - ∑ α - ∑ δiθ_i + ∑ γi (o⅛ - C) - β ∑ α⅛y (4)

for the dual problem:

min max LD,                        (5)

αi,δi,γi,β wk ,b,ξi

Here for a linear SVM:

nn

w(α)^τ w(α) = ΣΣαiαjyiyjx_i^τxj .                (6)

i=1 j=1

For obtaining non-linear classifying functions in the data space a more general
form is applicable:

nn

w(α)^τ w(α) = ΣΣαiαjyiyjK(xi, xj).              (7)

The function K(xi, xj ) is called a kernel function. Since it has a closed form
representation, the kernel is a convenient way of mapping low dimensional data
into a highly dimensional (often infinitely dimensional) space of features. It
must satisfy the Mercer conditions [15], i.e. be symmetric and semipositive
definite or, in other words, represent a scalar product in some Hilbert space
[19].

In our study we applied an SVM with an anisotropic Gaussian kernel

K(xi, xj) = exp -(xi - xj)^τr^-2Σ^-1(xi - xj)/2 ,          (8)

where r is a coefficient and Σ is a variance-covariance matrix. The coefficient
r is related to the complexity of classifying functions: the hgher the r is, the
lower is the complexity. If kernel functions allow for sufficiently rich feature
spaces, the performances of SVMs are comparable in terms of out-of-sample
forecasting accuracy [18].

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