Aerospace and Electronic Systems - August 2018 - 44

Artificial Neural Networks
2. To select the significant source of uncertainty in the input I
(for example, experimental uncertainty is usually determined
by the repetition of the experiment multiple times and by the
computation of the mean and standard deviation of the results).
3. To identify the probability density function corresponding
to the selected uncertainty sources. Many statistical tests are
based on the assumption of normality or have data sets that
may properly be transformed into normal distribution data
sets [30]. Families of transformations may be used to obtain
an approximate symmetry or a more normal-like distribution
[35]-[38].
4. To choose the number M of Monte-Carlo simulations.
5. To extract the M samples for each uncertainty source Ii.
6. To compute the outputs Oi.
Thus, the combined uncertainty u(I) for a generic variable can
be calculated as the standard deviation:

u(I ) 

M

M
 i 1 I i 

I

 i 1  i M 


M

(4)

where b1, b2 are the bias weight of the ANN, W1, W2 are the weight
of the neuron connections, I and O the input and output of the
ANN, respectively.
As for the source of uncertainty, displacement measurements
have been employed as input for the model, therefore some commercial displacement sensors were used as a reference to collect
uncertainty data from their datasheet. In general, experimental uncertainty is influenced by other elements than the sensors, such as
the experimental design, but, in this case, they are not considered
due to the numerical nature of this study.
The sensors considered for the study are "OMRON ZWS7030", the datasheet of which report the data about uncertainty
collected in Table 7:
In this study, the input I of the ANN for the recovery of the
position of the impact are the time delays between sensor signals

Table 7.

Sensor Specifications [39]
Sensor

ZW-S7030

Measuring range

± 2 mm

0.25 μm
± 2.0 μm

( 0.25) + ( 4 )
2

(5)

= 4.01 μ m

As for the uncertainty in the time measurement, the measurement chain is supposed to guarantee an uncertainty equal to ut(Z)
= 0.1 μs.
In this study, Matlab® is employed to generate M = 10,000
pseudorandom numbers and the uncertainty about the values of
the input variables is represented through a multivariate normal
distribution.
Monte-Carlo simulations are used to study the propagation of
sensor uncertainty in two cases:

(3)

2


+ 1
O = b2 + W2 
−2( b +W I )
1 + e 1 1


Linearity

ud ( Z ) =

For this numerical study, the equation of the nonlinear model is
given by the ANN structure for the recovery of the impact position:

Static resolution

with respect to a reference signal, the maximum value of the measured displacement in the Z direction and the instant at which it is
recorded. Therefore, the uncertainty in the displacement measurements ud(Z) and in the temporal acquisition of the signal ut(Z) are
required and from the sensor data estimated as

in the previously trained neural network, to understand how
sensor uncertainty propagates through an established neural
network;
in neural networks during their training, to understand the
influence of disturbed inputs on the training process.

UNCERTAINTY PROPAGATION IN THE TRAINED NEURAL
NETWORK
The neural network shown in Figure 8 was trained with the unperturbed sets of input and output data. Afterwards, the set of input data has been perturbed to create 10,000 samples of the same
disturbed impact. These samples have been used as input for the
already trained ANN for analyzing how sensor uncertainty propagates to output through the meta-model.
The results are shown in Figure 11: the aim of the ANN is to recover the position of the impact, which is (−5; −35) as highlighted
by the red vertical line, while the aim of the analysis is to see how
the results are distributed.
The estimate of the mean outputs, the standard deviation, and
skewness, and the expanded uncertainty at the 95% confidence
level are given in Table 8.

UNCERTAINTY PROPAGATION IN NEURAL NETWORKS
DURING THEIR TRAINING
It's interesting to see how sensor uncertainty propagates through
a neural network and influence its output. The set of input data
has been perturbed to create 10,000 samples of the same disturbed
impact. These samples have been used as input for the training of
ANNs.
Figure 12 shows the probability density function of the outputs
of the recovery of the position (−5; −35) of an impact.
The results of the mean values, the standard deviation, the
skewness, and the expanded uncertainty at the 95% confidence
level are given in Table 9.

IEEE A&E SYSTEMS MAGAZINE

AUGUST 2018

Aerospace and Electronic Systems - August 2018

Table of Contents for the Digital Edition of Aerospace and Electronic Systems - August 2018