using a quantitative measure of non-gaussianity, like
the kurtosis and the Neg-entropy.
Regardless of the distribution of independent ran-
dom variables, based on the central limit theorem,
their sum converges to a Gaussian distribution for a
sufficiently large sample size. Conversely, making the
measurements as non-Gaussian as possible will return
these independent sources. This implies the assump-
tion that the sources are independent and further, that
at most one of the measurements is Gaussian. The
usage of the kurtosis as evaluation criterion for non-
gaussianity is quite popular because it is computation-
ally easy to implement. A kurtosis value of zero im-
plies a perfect Gaussian distribution of the underlying
data, whereas a nonzero value indicates a deviation
from the Gaussian distribution. Unfortunately, the
kurtosis is very sensitive to outliers. A more robust
criterion is so-called Neg-entropy which is defined
as a measure of gaussianity, reflecting the deviation
of the data from a Gaussian distribution. The disad-
vantage of this method is that the probability density
function of the data has to be known in advance. The
uncertainty about the underlying probability density
function can be compensated using approximations
instead. Another procedure is the Infomax-principle
(Bell and Sejnowski, 1995), which stands for infor-
mation maximization and will be realised by minimiz-
ing mutual information.
Further pre-processing steps, which often result in
dimension reduction techniques, can be performed.
Depending on the data and the application, some
of them use PCA, Projection Pursuit (Friedman,
1987), filtering, stochastic search variable selection
like Bayesian networks and wavelet transformation.
5.1.2 Post-processing after ICA
After the ICA has been performed, the resulting
sources have to be evaluated. This can be again a form
of filtering, e.g. the separation in informative sources
and noise. As a first attempt, the NNR method is used,
which takes spatial dependencies of devices over the
wafer into account, see Equation 9. From each device
value, v(x
i
,y
j
), the median of the surrounding devices
is subtracted, whereas m and n are dependent on the
neighborhood:
NNR(x
i
,y
j
) = v(x
i
,y
j
) − med(v(x
i+m
,y
j+n
)). (9)
The size of the neighborhood (8, 24 or more) for
calculating the NNR is a further topic which will be
investigated in this PhD. Previous investigations have
shown that the nearest 24 surrounding devices (m,n =
{−2,−1,1,2}) are a good choice. However, first eval-
uations of the NNR on the sources have shown that the
number of surrounding devices taken into calculation
needs to be determined for this project.
Figure 6: To calculate the NNR value for each device (here
marked in gray), just the pass devices are taken into account
because for the electrical fail devices (marked in black) no
value is available. The black square shows the involved de-
vices for a 24-based neighborhood.
As visualized in Figure 6, only pass devices are
considered for the NNR calculation and therefore it
happens that not all surrounding neighbors are avail-
able. In this case it might be useful to consider either
just the given values or other significant ones instead.
As a first attempt, the gap can be filled up with de-
vices on the main x-y-directions, starting from device
(x
i
,y
i
). Those from the diagonal directions are only
considered if further devices are needed. Addition-
ally, depending on e.g. the distance, weights can be
incorporated.
5.2 Further Considerations
As outlined in Section 5.1.1, various pre-processing
methods are listed, where the most meaningful one
has to be determined to provide a useful starting po-
sition for the ICA method itself. Generally, the num-
ber of sources is unknown, implying that there can be
more sources than measurements (underdetermined)
or vice versa (overdetermined system of equations)
(Naik and Kumar, 2011). For the first case, the cal-
culation of a pseudo-inverse is necessary. An applica-
tion can be found e.g. in bio-signal processing, where
the number of electrodes are limited compared to the
active muscles involved. For an overdetermined sys-
tem of equations dimension-reducing pre-processing
steps can be performed, see Section 5.1.1. A re-
duction of the measurements to the number of ex-
pected sources is preferable. For ICA itself, different
MATLAB
R
packages, like the FastICA (Hyv
¨
arinen
DeviceLevelMaverickScreening-ApplicationofIndependentComponentAnalysisinSemiconductorIndustry
7