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Algorithm for searching and analyzing abnormal observations of statistical information based on the Arnold – Kolmogorov – Hecht-Nielsen theorem

An algorithm for detecting runouts in statistical information
is proposed in this article. The idea of the algorithm is based
on the property of some neural networks to demonstrate a
large error in examples during training, which are runouts.
For example, if a perceptron-type neural network has a
relatively small number of hidden neurons, and if there are
relatively few runouts in the training sample, then the neural
network usually demonstrates a higher training error after the
training procedure on the examples that are runouts than on
nonrunout examples. However, two extreme cases are
possible. On the one hand, if a neural network has too many
degrees of freedom, it is usually well trained and
demonstrates small values of the training error in all
examples during training, including examples that are
runouts. This is why a neural network with a large number of
hidden neurons is not suitable for detecting runouts. On the
other hand, if a neural network has too few degrees of
freedom, it will demonstrate large values of the error both in
runout examples and in examples that are not runouts after
the training procedure. As such, it is also not suitable for
detecting runouts. According to the proposed algorithm, a
special neural network is designed using the formula obtained
based on the relation derived from the Arnold – Kolmogorov
– Hecht-Nielsen theorem. This special neural network is
designed only for detecting and identifying outliers. Another
neural network is being designed or other analysis methods
are used for further data analysis. The proposed algorithm is
intended for nonlinear subject areas described by small
volumes of statistical samples that do not necessarily satisfy
the normal distribution law. The application of the algorithm
turned out to be efficient in solving a wide range of problems
from various subject areas, such as medicine, economics,
forensics, etc.