Statistical Signal Detection and Artificial Neural Networks
Laboratorium for Network Theory
Researchgroup BSC
Department of Electrical Engineering
University of Twente
PO Box 217
7500 AE Enschede
The Netherlands
Introduction
Three conventional probability density functions (PDFs)
estimators [1] will be discussed, which can benefit from
using Kohonen's Neural Network for estimating PDF
parameters. The Self Organising Feature Map (SOFM)
algorithm is used as an implementation of Kohonen's
Neural Network [2].
With these estimators an optimal detector is built by
calculating the likelihood ratio. These detectors are tested
on a real world problem (spike-wave dectection in EEG
signals) and are compared with conventional methods.
Theory
Parametric Signal Detection
Parametric Signal Detection is used when there is a signal,
which has varying parameters (like frequency and phase).
The PDF is given by the sum of the PDF given these
parameter multiplied by the chance on the realization of
these parameters:
. (1)
The SOFM algorithm is used to estimate N parameter
weight vectors, which are characteristic realizations of
the signal class. The chance on one realization 
is
estimated by the chance that the neuron with this weight
vector is closest to vectors in the train set. No
knowledge about the actual meaning of the signal
parameters is needed. The conditional PDF 
is
assumed to be a Gaussian PDF, with expectation value
and a fixed variance set to 1.
Non-Parametric Density Estimation
If a set of N vectors is made of one class, the PDF can be
approximate by the reciprocal of the volume, which
contains K of N vectors [3]:
(2) 
.(2)
This set of N is optimally reduced using the SOFM
algorithm, reducing the number of calculations significantly.
The PDF is approximated by the reciprocal of the distance
(3), resulting into a sufficient statistic of the likelihood ratio.
Semi-Parametric Density Estimation
The PDF is modeled by a sum of Gaussian PDFs, also
called kernels:
. (4)
The position of the kernels, the expectation values 
, are
estimated by the weight vectors of the SOFM algorithm. An
initial estimation for the variance is a function of the total
variance in the train set and the variance within each cluster
[4]. The mixing parameter Pi is estimated by the chance that
the weight vector is closest to vectors in the train set.
Finally the variance is optimized on the train set using
a maximum likelihood optimizer.
Practical Results
With the previous mentioned estimators a detector is built,
based on the quotient of 2 PDFs (likelihood ratio). These
detectors are tested on their ability to detect spike-waves in
Electro EncephaloGram (EEG) signals [5]. Spike-waves are
specific wave forms, which occur only in EEGs of epilepsy
patients, the diagnoses of epilepsy is therefore based on the
occurrence of these spike-waves.
The performance of 6 statistical detectors is given in
table 1. Three method uses the SOFM algorithm, only for
estimation of the parameters.
| Method | Calculations per window | # False alarm | # False rejected |
| Correlating Receiver | 200 | 37 | 0 |
| Parametric Density Estimation | 999 | 9 | 0 |
| Non-Parametric Density Estimation | 309,000 | 0 | 0 |
| Non-Parametric Density Estimation (SOFM) | 900 | 6 | 0 |
| Semi-Parametric Density Estimation (SOFM) | 900 | 13 | 0 |
| Parametric Signal Detection (SOFM) s=1 | 900 | 4 | 0 |
Table 1:The number of false alarms and false rejected spike-wave for the tested methods. The signal contains 66 windows of spike-
waves out of a total of 120,000 windows (approximately 5 minutes).
Conclusions
Kohonen's Neural Network can speed up the non-
parameteric density estimation, with small increase in the
number of false alarm. Kohonen's Neural Network provides
a good estimation of the PDF parameters for the three
conventional mentioned methods and introduce the
flexibility needed for spike-wave detection.
References
[1] C.W. Helstrom, Elements of signal detection and
estimation, 1995, Prentice Hall London, UK
[2] L.P.J. Veelenturf, Analysis and Applications of Artificial
Neural Networks, 1995, Prentice Hall International, UK
[3] K. Fukunaga, Introduction to Statistical Pattern
Recognition, 1990, Academic Press
[4] S.H. Lokerse, L.P.J. Veelenturf, J.G. Beltman, Density
Estimation using SOFM and Adaptive Kernels in
Proceedings of the Third annual SNN Symposium, 1995,
Nijmegen, The Netherlands
[5] J.R. Smith, Automatic Analysis and Detection of EEG
Spikes, IEEE Transaction on biomedical engineering, Vol
BME-21, 1974