Graphics Reference
In-Depth Information
men (
)). As a result we obtain monotonised probabilities of default PD
(
x
i
)
for
the observations in the training set.
Finally, at step three the PDs are computed for any observation described by x as
an interpolation between the two PDs of neighboring (in terms of the score) obser-
vations in the training set, x
i
and x
i
−
, i
=
,
,...,n:
f
(
x
)−
f
(
x
i
−
)
PD
(
x
)=
PD
(
x
i
)+
)
PD
(
x
i
)−
PD
(
x
i
−
)
(
.
)
f
(
x
i
)−
f
(
x
i
−
If the score for an observation x lies beyond the range of scores for the training set,
then PD
equals the score of the first neighboring observation of the training set.
Figure
.
isanexampleofthecumulative PDcurve(powercurve)andestimated
PDs for a subsample of
companies. he PD curve has a plateau area for observa-
tionswithahighscore.Defaultprobabilities canchangefrom
%to
%depending
on the score.
(
x
)
Colour Coding
4.6
he RGB colour space is based on three primary colours, red, green and blue, that
are mixed to produce others. his is the colour coding scheme that is used in mon-
itors and TVs. It is, however, an inconvenient colour coding scheme here since we
only wish to adjust the channel responsible for colour while keeping lightness and
saturation constant. his can be achieved with the HLS colour space.
We will represent the probability of default (PD) estimated with the SVM using
two-dimensional plots where each colour denotes a specific PD.he PD is a number
that can be represented on a greyscale. For example, in the RGB encoding as
)
where i is in the range
to
(e.g. the colour R=
, G=
, B=
corresponds to red,
and R=
, G=
, B=
to violet, etc.).
HLS stands for hue, lightness (or luminance) and saturation. By adjusting only
the hue and keeping the luminance and saturation fixed, we can generate simulated
colours from the range shown in Fig.
.
. A pure red colour corresponds to H=
or
, a pure green to H=
, and a pure blue to H=
.
Red is oten used in finance to highlight negative information, while green and
blue are used to convey positive information. herefore, we would like to code PDs
withcoloursthatrangefromredforthehighestPDtoblue-greenforthemostsolvent
company. We therefore normalise the PDs such that the lowest PD has a hue of
(green-blue) or
(green) while the highest PD has a hue of
(red). he resulting
graphs that show the data and PDs in the dimensions of variables K
and K
are
shown in Figs.
.
-
.
.
heSVM with the radial basis parameter r
(
i, i, i
=
providesthehighestAR(Fig.
.
).
he near-linear SVM, r
, only uses almost linear classification functions and
has a lower classification power due to this limitation (Fig.
.
). he SVM with an
excessively high complexity, r
=
=
.
, suffers from overfitting and also has a lower
prediction accuracy (Fig.
.
).
