Acta IMEKO, Title

ACTA IMEKO
ISSN: 2221-870X
March 2018, Volume 7, Number 1, 42-49

ACTA IMEKO | www.imeko.org March 2018 | Volume 7 | Number 1 | 42

Analysis of the RBF ANN-based classifier for the diagnostics
of electronic circuit
Bartosz Połok, Piotr Bilski

Institute of Radioelectronics and Multimedia Technology, Warsaw University of Technology, ul. Nowowiejska 15/19 00-665 Warsaw, Poland

Section: RESEARCH PAPER

Keywords: fault identification; RBF networks; classification

Citation: Bartosz Połok, Piotr Bilski, Analysis of the RBF ANN-based classifier for the diagnostics of electronic circuit, Acta IMEKO, vol. 7, no. 1, article 8,
March 2018, identifier: IMEKO-ACTA-07 (2018)-01-08

Section Editor: Lorenzo Ciani, University of Florence, Italy

Received October 4, 2017; In final form January 6, 2018; Published March 2018

Copyright: © 2018 IMEKO. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited

Funding: This work was partially financed from the statutory grant, Institute of Radioelectronics and Multimedia Technology, Warsaw University of
Technology

Corresponding author: Piotr Bilski, e-mail: pbilski@ire.pw.edu.pl

1. INTRODUCTION
Artificial Neural Networks (ANN) are currently the most

popular Artificial Intelligence (AI) tools used in the diagnostics
of analog systems [1]-[3]. Their numerous advantages include
the ability to extract generalized knowledge from the available
measurement data, autonomous operation (making them useful
in the automated online diagnostics) and the ability to
accurately process data in uncertainty conditions. Limited
memory usage and high processing efficiency make them useful
in embedded applications (implemented in the digital signal
processor or the FPGA array [4]). Their disadvantage is the
obscure form of stored knowledge (illegible for the human
operator), making the explanation of generated decisions
difficult. However, this is not crucial as long as the diagnostic
system accurately detects and identifies faults. Despite
introducing hybrid approaches (such as hierarchical or

convolutional ANN [5]), traditional architectures are still in use,
due to their simple operation principle and high efficiency
proven in multiple cases.

The most popular ANN is the Multilayered Perceptron
(MLP), successfully applied to solve biomedical, financial and
technical problems [6], [7]. Their implementations in the
diagnostics cover power lines [8] or electrical machines [9].
They are currently widely exploited and work as the standard
diagnostic tool, or the self-diagnostic module [10]. Training the
MLP classifier consists in selecting one of available algorithms
(such as error backpropagation or Levenberg-Marquardt) and
adjusting the available data to the predefined architecture. It is
defined by the specific number of layers that contain the
selected type of neurons (computational units). Optimizing
these parameters is the main design challenge, as the optimal
MLP architecture has to be determined individually for each

ABSTRACT
The paper presents the application of the Radial Basis Function (RBF) Artificial Neural Network (ANN) to the diagnostics of analog
circuits. Such networks are in most cases useful in the approximation tasks as the alternative to multilayered perceptrons (MLP) or
Support Vector Machines (SVM). In this work the analysis of various RBF ANN-based classifier configurations for the fault detection
and identification module is conducted. The considered parameters included the optimal number of neurons in the hidden layer,
coding schemes for the output layer and operation duration during the training and testing of the classifier. The efficiency of the
diagnostic system was verified using the electronic circuit, i.e. the fifth order lowpass filter. It was analyzed in terms of testability,
depending on the set of accessible nodes, confronted against the availability of the output node only. Experiments also covered
accuracy comparison between the RBF, MLP and SVM classifiers. Results show advantages and drawbacks of the RBF ANN-based
diagnostic module, compared to other available solutions.

ACTA IMEKO | www.imeko.org March 2018 | Volume 7 | Number 1 | 43

problem.
Similar issues are encountered during the application of

Support Vector Machines (SVM) to the classification task. They
are considered the optimal ANN-based classifiers in the
uncertainty conditions [11]. Their disadvantage is the time
consuming process of selecting the optimal kernel and its
parameters [12]. Despite such challenges, SVM are also popular
in diagnostics, used to identify faults in electrical machinery
[13], electronic circuits [14] or power plants [15].

On the other hand, Radial Basis Function (RBF) networks
are considered simpler in design. Their training is faster and
structure simpler, as only the single hidden layer is present.
Therefore, the following parameters are to be determined:
• the number of hidden computational units, depending

on the size of training data;
• the number of output neurons, depending on the

number of categories to distinguish and the selected
coding scheme;

• the activation function width (identical for all hidden
neurons).

In most cases the RBF network is used for the
approximation task, as it has linear neurons in the output layer.
Its application to classification may require substituting
computational units with their sigmoidal counterparts. Such
approaches are rare and not well described in the literature [16].
This calls for the thorough investigation of the RBF ANN
classification characteristics during the fault detection and
identification in analog systems.

The aim of this paper is to assess performance of various
RBF ANN configurations during the fault detection and
location in analog circuits. The presented work is an extension
of the research published in [17]. All introduced parameters of
the classifier were tested to find the optimal configuration
maximizing the diagnostic accuracy, verified on the model of
the 5th order electronic lowpass filter. The testability of the
circuit was checked by changing the set of accessible nodes (at
which system responses can be measured). The accuracy of the
AI-based diagnostic module depends on the quality of the
training data extracted from the system. Increasing the number
of accessible nodes leads to the improvement of the diagnostic
module. On the other hand, the number of pins in the circuit
casing should be minimized. Therefore, a compromise between
these two goals must be made.

Finally, the RBF ANN-based classifier is compared to its
well-established counterparts, i.e. MLP and SVM to determine
in which situations it outperforms its rivals and should be
selected for the task.

The structure of the paper is as follows. Section 2 presents
the applied diagnostic architecture. Details of the implemented
network are described here. Training and testing data sets
structure is presented in Section 3. The analysed circuit is
introduced in Section 4, while experimental results are in
Section 5. Conclusions about the conducted analysis are in
Section 6.

2. ANN-BASED DIAGNOSTIC ARCHITECTURE
The typical diagnostic scheme using any type of the ANN-

based classifier is presented in Figure 1. Here the System Under
Test (SUT) is monitored by the hardware/software module,
which performs data acquisition on the accessible nodes. In this
way the extraction of the vector of features (symptoms) e={s1,
…, sm} (also called examples) from measured signals is

performed. They are subsequently processed to make the
diagnostic decision (hypothesis) h. It is assumed that knowledge
stored by the module allows for the automatic fault detection
and identification. This requires implementation of the machine
learning algorithms. During this operation the ANN-based
classifier produces the binary output, in which every fault
category is represented by the unique sequence of values {0,1}
or {-1,1}, depending on the used type of sigmoidal neurons.

Application of the RBF network for the fault classification
task requires adjusting its structure to the specific problem. This
involves selection of the number of neurons in the hidden layer
and the coding scheme for binary units in the output layer. As
opposed to MLP and similarly to SVM, the RBF network
contains only one hidden layer with the Gaussian activation
function φ(si) units (1). The function width (defined as the
standard deviation σ, where c is the centre, usually set to 0 for
all neurons) determines the network performance. These
parameters are optimized during the design stage.

( )





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






⋅
−

−= 2

2
exp

σ
ϕ

cσ
σ ii (1)

2.1. Minimization of the number of hidden neurons
The number of neurons k in the hidden layer (usually greater

for the RBF ANN than for MLP) is automatically adjusted in
the result of the training. Although there are approaches to find
it during one training sweep [18], in most cases the
predetermined network structure is trained and validated, then
the process is repeated for another structure. During this time
consuming (though automated) process, the maximum
acceptable value is determined. It is initially equal to the overall
number of examples in the set L (2), which ensures the high
classification accuracy, but makes the network large and slow.
To avoid this, the number of hidden neurons should be
minimized. The training examples are obtained after simulating
the SUT, which allows for connecting its behaviour with the
internal configuration of parameters.

In the presented work, the clustering of examples was
applied to group the most similar ones. It is assumed that
training data from the SUT simulation is redundant, caused by
the following factors:

• low sensitivity of the SUT for changes of the specific
parameter, which results in multiple examples describing
the same category, as they are close to each other (in
terms of distance). Such a group can easily be substituted
by the single example representing all original members. It
becomes their centroid.

• existence of ambiguity groups (AG) [19], which contain
examples belonging to different categories that are close
to each other. In this case examples cannot be replaced by
the single representative and must all remain in the set.

Figure 1. ANN-based diagnostic system architecture.

ACTA IMEKO | www.imeko.org March 2018 | Volume 7 | Number 1 | 44

Data preprocessing aims at discovering all groups of similar
examples by the cluster analysis, which results in the set of
groups G1 easily distinguishable from each other, based on the
similarity calculations. The detailed scheme is presented in
Figure 2. Redundancy in the set L may be exploited in two
ways. The first one consists in providing the original data set
for the RBF network training, but limiting the maximum
number of neurons in the hidden layer to the number of
clusters nc. Alternatively, the reduced data set L’ may be
provided for training.

The distance-based similarity calculation was used in the
presented case. Its only parameter is the threshold θ, below
which two examples ei and e j are considered close to each other
(2). The measure applied to create clusters exploits Euclidean dE
(3) and cosine dc (4) distances, treating every example as the
point in the m-dimensional space. This way groups located close
to each other are easily identified. Thresholds θ1 and θ2 for both
distances should be selected adaptively to minimize the number
of clusters containing examples belonging to different classes
(i.e. forming AG).

( ) ( ) 21 ,,, θθ <∧<⇔∈ jiEjiElji eedeedGee (2)

( ) ( )∑
=

−=
m

k
jkikjiE σσeed

2, (3)

∑∑

∑

⋅

⋅
=

k
jk

k
ik

k
jkik

σσ

σσ
d

1 (4)

The illustration of the clustering method for m=2 is
presented in Figure 3.

2.2. Output layer coding schemes
The important issue during the design of the ANN-based

classifier is the representation of multiple categories by binary-

valued neurons in the output layer. Among multiple coding
schemes, three were selected for experiments [20]. It is assumed
that only single faults are considered (the most probable case).
Schemes differ in the number of output neurons o:
• One-vs-All (OvA) – here each output unit is responsible

for a separate category. The number of fault codes l
determines the number of neurons o in the output layer
(see Table 1). During the fault identification, only one
neuron should be active. If multiple neurons become
active, they can be interpreted as fault candidates in the
uncertainty conditions. This scheme makes the classifier
suitable for the multiple fault diagnostics (i.e. when more
than one parameter is out of tolerance margins).

• One-vs-One (OvO) – each neuron is responsible for
distinguishing between the pair of categories. This time o
classifiers with one output neuron are created and
simultaneously trained. Their number is determined by the
number of distinct fault code pairs: l·(l-1)/2. In this
scheme both active and inactive neuron determines the
specific fault. The diagnostic decision is made based on
the majority voting – the category pointed by the greatest
number of networks supporting it. Note that each
network in this scheme is trained on a different subset,
extracted from L. In each, only examples belonging to
two fault categories are present. For instance, three fault
codes require the following units (trained on the
corresponding subsets): 1: c1 vs c2, 2: c1 vs c3, 3: c2 vs c3.

• Minimum Output Coding (MOC) – the minimal set of
neurons, which represents each fault code by the unique
combination of active units. The subsequent codes are
represented by the binary representation of integer
numbers (see Table 2).

The more sophisticated coding schemes, such as Error
Correcting Output Coding (ECOC) [21] were excluded from
experiments, as their implementation would complicate the
output layer even more, not necessarily increasing diagnostic
accuracy. ANN was implemented using the Matlab
environment including the Neural Networks Toolbox.

3. DATA SETS DESCRIPTION
Learning L and testing T datasets are required to train the

RBF network classifier and verify its accuracy. Both have the

Figure 2. Learning data set preprocessing for the RBF classifier training.

Table 1. Applied OvA coding scheme.

Category Coding scheme

c1 1,0,0, …,0
c2 0,1,0, …,0
… …
cl 0, …,0,0,1

Table 2. Applied MOC coding scheme.

Category Coding scheme

c1 0,0,0, … ,0,1
c2 0,0,0, … 1,0
c3 0,0,0, … 1,1
… …

Figure 3. Illustration of the clustering method.

L clustering L’

nc
RBF training RBF training

knowledge knowledge

ACTA IMEKO | www.imeko.org March 2018 | Volume 7 | Number 1 | 45

same structure, containing n feature vectors e with m attributes
(symptoms s), supplemented by the fault code, describing the
state of the SUT. The generic form of the set (5) is the table,
whose structure is identical for L and T.

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TL

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1111

(5)

The single example is generated after simulating the SUT
model with the fault introduced to its structure. This way the
effect of the fault on the responses measured in accessible
nodes can be observed. The key issue is the generation of the
fault code for each example, based on the actual value of the
faulty parameter. As before [22], it is the integer number, being
the combination of the parameter identifier and its deviation
from the nominal value. The code contains the identifier of the
faulty parameter and the discrete degree of its change, both
negative and positive (for values greater and smaller than the
nominal one). The number of parameter states depends on the
assumed thresholds for deviations. For instance, if the
parameter value is above ten percent of nominal, the degree is
labeled with “1” or “-1“, indicating it being “larger than” or
“smaller than”, respectively (depending on the direction of
deviation). Additional intervals may be introduced to increase
the resolution of the diagnostic module. For instance, if the
value of the parameter is above fifty percent of the nominal
one, the degree “2” may be assigned. This way the second
parameter with the value larger than the nominal one is
represented by the code “21”, while the first parameter with the
value much smaller than the nominal one obtains the code “-
12”. The exception is the nominal state, encoded with the value
“0”. The number of simulated parameter states depends on the
expected data set size and the accuracy of the SUT modeling. In
the presented work, both sets are of the same size, containing
mutually exclusive examples. This way the generalization of the
ANN RBF-based diagnostic module may be verified.

Diagnostic accuracy acc of the classifier is measured using the
set T, for which the number of incorrectly classified examples
(i.e. the ones for which the hypothesis h(e) is not equal to the
fault code c(e)) determines the sample error es:

( ) ( )
T

hcT
eacc σ

eee ≠∈
−=−=

:
11 (6)

Each SUT simulation was performed after changing the
single parameter value beyond the tolerance margins (here 10%
of the nominal value) with all other parameters remaining
nominal. A subset of fault-free examples was also prepared to
check the resilience of the module to false alarms.

4. ANALYZED SUT
The fifth-order analog filter is a good example to implement

the data processing methods. Similar active circuits are still
popular in military or acoustic applications, therefore their
analysis is justified [23]. It is complex (because of the large
number of nodes and SUT elements) and difficult to diagnose
based only on the analysis of the output node. Therefore,
multiple symptoms are needed to identify the state of all
elements (resistances and capacitances). The circuit in Figure 4
contains ten elements with the following nominal values:
R1=R2=R3=R4=R5=1 kΩ, C1=16 nF, C2=19 nF, C3=13 nF,
C4=51 nF and C5=49 nF. Subsequent resistances were labelled

with numbers from 1 to 5 respectively, while capacitances were
referred to as parameters No. 6 to 10. The cutoff frequency of
the filter for such values is 10 kHz. The model of the circuit
was implemented in the Simulink environment with the
operational amplifiers realized as equivalent circuits with
controlled voltage source. Simulations were performed to
obtain examples of the SUT behaviour for different values of
elements (changed up to 90 percent of the nominal value). The
excitation signal provided at node No. 1 was a sinusoid with 9
kHz frequency (i.e. close to the cutoff frequency). The filter was
analysed in the time domain, where at the accessible nodes
sinusoidal responses were recorded. Measurements were taken
at nodes 2, 3, 5, 6, 8 and 9. From each response the first three
maximal and minimal values of the signal with their time
instants and time instants of zero crossings are extracted
(Figure 5). This gives the total number of 54 symptoms for
each example (nine for each node), assuming all discussed
nodes are accessible.

To evaluate the dependency between the size of the set and
the accuracy of RBF-based classifier, various sizes of the sets
were prepared, starting from 70 examples (7 simulations for
each parameter, including the nominal state) for the set L1 to
180 (18 simulations for each parameter) for the set L2. To
consider tolerances, additional sets were created with results of
simulations affected by the random value added to the actual
parameter. This allows for verifying if random deviations of
parameter values influence the ability of the diagnostic module
to distinguish faults. The number of different fault codes to
distinguish was 41, as not all 5 were used for each parameter
due to their smaller sensitivity.

5. EXPERIMENTAL RESULTS
Conducted experiments consisted of four stages. First,

coding schemes were compared. Next, relation between the size

Figure 5. Examples of responses from the filter for various values of the
resistor R1.

Figure 4. Scheme of the 5th order lowpass filter.

ACTA IMEKO | www.imeko.org March 2018 | Volume 7 | Number 1 | 46

of the data set and the diagnostic accuracy was determined. The
testability of the circuit, depending on the number of analysed
nodes, was also verified. Finally, comparison between RBF,
SVM and MLP (also implemented in Matlab) was performed to
determine efficiency of all classifiers and find their advantages
and drawbacks. The ANN training is a random procedure,
consisting in the adjustment of weights between neurons to the
training data. Knowledge structure depends on the initial
(randomly selected) values of weights, therefore every training
sweep results in different content of weights matrices.
Therefore, to gain the general information about the ANN
efficiency, training and testing experiments were repeated ten
times for each architecture and data set. Results presented in
the following subsections refer to average accuracy values.

5.1. Coding schemes verification
Coding schemes presented in Section 2.2 were implemented

to design the RBF network, train it on the set L and test using
the set T. The first operation is done according to the
Simulation Before Test (SBT) [24], [25] paradigm, i.e. when
knowledge is extracted prior to the diagnostic module
implementation in the industrial environment. Duration of the
operation is of secondary importance here. The trained network
is then used to process measurement data and make decisions.
As the operation duration of ANN in the testing phase is
negligible, there are virtually no restraints, even for their on-line
application (where the diagnostic decision must be made within
the specific time limitations). Optimal results for all
configurations (regarding the width of the RBF σ, number of
output neurons o, number of hidden neurons k, the maximum
achieved training error e and obtained accuracy on the set T) for
the set L2 (180 examples) are in Table 3.

In most considered cases, the OvO coding is the best
scheme, producing the minimum error for all network
configurations. This method is the most complex, requiring
multiple ANN, each responsible for only two faults. This leads
to many networks, but with relatively simple structure. The
average number of hidden neurons for the optimal OvO
network structure was 8. The most complex is MOC, as
multiple output neurons have to participate in producing the
fault category. For each coding the best width of the Gaussian
function was also determined. The full sweep optimization
procedure was applied to obtain it (although more sophisticated
approaches, such as simulated annealing [12] or genetic
algorithm can be used). The optimal RBF width and the
number of hidden neurons are in all cases in the middle of the
verified range (for σ it is between 0.1 and 2.0). Figure 6 shows
the relation between σ and the obtained accuracy (with all other
parameters set as in Table 3). For all coding schemes, there is
the best σ value, for which the minimum error is obtained.
Results show that narrow Gaussian functions are preferred (like
σ =0.5 for OvA). The cost of the highest OvO accuracy is the

greater computation effort, caused by the large number of
trained networks in comparison to the single network used for
remaining coding schemes.

The experiments with the optimization of the number of
hidden neurons show similar effects as during the selection of
the Gaussian function width. The discrete sweep was
implemented here to train and test the network while the
number of hidden neurons was incremented (i.e. increased by
one). At the specific point the maximum accuracy is reached.
Adding more computational units does not improve the ANN
efficiency but increases the training duration. This concludes
the structure optimization stage, with results of the OvO
coding presented in Figure 7. As each network distinguishes
between two categories only, it is simpler than the ones
implementing other coding schemes.

The detailed analysis of the network performance shows that
most diagnostic errors are made because of mistaking the first
element (R1) with the sixth one (C1). In such a case, the
direction of changes is correct (for instance, the typical error is
producing the code “-62” instead of “-12”). Changes in
elements for which the SUT has low sensitivity (especially R2,
R3 and C3) are difficult to detect and the classifier regards the
circuit as nominal. In such cases (when all symptoms are within
tolerance margins) categories of such examples should be
changed to “0”. There are also errors of minor importance, i.e.
the incorrect identification of the intensity of changes, although
the element is correctly located (for instance, when the code “-
52” instead of “-51” is generated). Such mistakes should be
treated as less important and regarded with smaller weights
during the accuracy (6) calculation.

5.2. Data set size
Experiments regarding the size of the training set L were

divided into two steps. In the first one, various numbers of

Figure 6. Influence of the RBF width on the diagnostic accuracy.

Table 3. Comparison of performance of RBF network output coding
schemes.

Coding
scheme σ k o e acc [%]

OvA 0.5 82 41 0.0 81.11
OvO 0.6 8 820 0.0 83.33
MOC 0.3 95 6 0.02 80.56

Figure 7. Influence of the number of hidden neurons on the diagnostic
accuracy (OvO coding).

ACTA IMEKO | www.imeko.org March 2018 | Volume 7 | Number 1 | 47

examples were inserted to the set and then used to train the
RBF classifier (starting from the set L1 with the minimal
number of symptom vectors, through the medium set with
added some new examples, and ending with the maximal set
L2). The initial diagnostic results for L1 are significantly
improved by introducing additional examples, provided they
carry useful information. Table 4 compares the diagnostic
accuracy for data sets of different sizes. As in other
applications, the amount of training data influences the
accuracy of the classifier, which may be a problem if they are
difficult to collect (for example, when the model of the SUT
does not exist or requires large amount of calculations).

During the second stage, the original set is processed to
group the most similar examples (see Section 2.1) to compress
their number. Additionally, dependency between the training
duration and the size of processed data should be established.
This is done using the clustering algorithm. In this way it is
possible to eliminate redundant examples, which in turn would
be represented by the single vector of symptoms. Results of the
diagnostic procedure for the RBF ANN trained on the
minimized version of the set are presented in Table 5.
Increasing the number of examples improves the fault detection
and identification performance, but to obtain the accuracy equal
to the RBF trained on the L2 set, the density of clustering must
not be large. Comparable diagnostic results are obtained on
data sets with size close to L2. Again, the most effective is the
OvO scheme.

Relation between the RBF network structure and training
duration for the OvA and MOC coding is in Figure 8. All
computations were performed on the computer equipped with
the AMD FX-8320 processor (eight cores, 3.50 GHz) and 16
GB of RAM. The processing time for both mentioned coding
schemes depends on the number of hidden neurons in the
network structure. The relation between the size of the data set
and the number of neurons is more complex, as even larger
data sets may be optimally processed by smaller networks. The
training duration of the RBF ANN-based classifier with OvO
coding is significantly longer, requiring between 10 and 20
minutes to complete on the same hardware configuration. The
time curves are close to linear, which is the positive effect. The
OvO RBF classifier requires the greatest amount of time to

train because of a large amount of networks to process. This
calls for the parallel implementation of the training procedure
[26], where each network would be processed by a separate
processor, core or computer. As this procedure is implemented
off-line, its duration is of secondary importance, although
considering the thorough values optimization of σ and k for
each network will complicate the process even more.

Another problem is the testing duration, as it determines the
usability of the diagnostic module in the on-line mode. Average
times required by the RBF ANN with subsequent coding
schemes to process the single set of symptoms are in Figure 9
(on the same computer configuration as before). Although
much faster for all schemes, the processing time for OvO
coding is significantly longer, making it difficult to implement in
the embedded system working under strict time limited
conditions.

5.3. Testability verification
Experiments presented so far were conducted on the

complete data sets, i.e. containing symptoms extracted from all
circuit nodes, as discussed in Section 4. Because of practical
reasons it is rarely possible to use all of them (especially in the
integrated system). Therefore, the number of diagnostic pins
available from the circuit casing should be minimal. The
compromise between the accuracy and the set of accessible
nodes must be made. Figure 10 presents comparison between
different coding schemes of the optimal RBF network structure
for different sets of accessible nodes. Besides the node 9
(always observable, therefore usable in the input-output analysis
[24]), all others can be theoretically made accessible. Because
the filter has the sequential structure, it is reasonable to assume

Figure 8. Training duration of the RBF network (OvA and MOC coding
schemes).

Table 4. Influence of the data set size on the RBF classifier performance.

Data set size accOvA [%] accMOC [%] accOvO [%]
70 68.57 67.14 65.71

120 75.78 73.86 75.54
180 81.12 80.56 83.33

Table 5. Influence of the data clustering on the RBF classifier
performance.

Data set
size

accOvA
[%]

accMOC [%] accOvO [%]

70 68.57 67.14 65.71
88 70.67 69.56 68.23

103 72.23 68.34 71.12
109 70.00 70.56 71.12
136 75.89 73.45 75.78
153 76.67 72.23 77.42
162 81.12 72.78 81.89
180 81.12 80.56 83.33

Figure 9. Duration of processing the single vector of symptoms by the RBF
ANN-based diagnostic module.

ACTA IMEKO | www.imeko.org March 2018 | Volume 7 | Number 1 | 48

that more information can be recovered from the nodes close
to the output of the SUT. Therefore, the order of adding them
to the analysis was as follows: 8, 6, 5, 3, and 2. The key step for
improving the accuracy was the addition of the nodes 5 and 8.
Further extensions led to only small change in diagnostic
outcomes. Again, in all experiments the OvO coding proved to
be the most effective. The procedure for other similar circuits
(such as other filters) should be the same. If the system is more
complex, more sophisticated node selection method should be
used [27].

5.4. Comparison against other classifiers
Besides the RBF-based network classifier, also SVM and

MLP can be used to process the same data sets. In Tab. 6
diagnostic outcomes of the optimal network configurations for
the largest set T2 are presented. Performance of classifiers is
comparable, the main difference lies in the training time, which
is the longest for the MLP and comparable for RBF and SVM.
In the latter case, additional parameters must be determined
(such as the kernel function and its parameters), which
significantly increases the training duration. The RBF ANN-
based classifier is the fastest one to design and has simpler
training algorithm than competitors. Its optimization is also less
complex than SVM. This classifier can then be used during the
fault analysis of SUTs, which are moderately difficult to
diagnose. In the case of wide element tolerances (for cheap
circuits) or high level of noise, more sophisticated ANN (such
as SVM) may be preferred.

6. CONCLUSIONS
The paper presented analysis and optimization of various

RBF network-based classifier in the diagnostics of analog
circuits. Multiple parameters, including the width of the
activation function, number of neurons in the hidden layer and
coding scheme in the output layer were verified. The testability
of the circuit was also analysed.

The networks with OvO scheme were the best in most
cases, but required the greatest amount of time to be trained.
This is because this scheme requires creating a large number of

simple networks, distinguishing between only two fault
categories. The MOC, although requiring the smallest number
of output neurons is more susceptible to random classification
errors. The OvA scheme is the most popular one. Its accuracy
is similar to OvO, but it is much faster trained. Selection of the
scheme is a compromise between the expected accuracy and
available time for training and processing the testing vector of
symptoms (when the on-line diagnostics is considered).

The optimization of the RBF training process is similar to
other ANN-based classifiers. The supervised learning is
implemented here as the standard parameterized algorithm. The
designer’s task is to configure the network (by determining the
number of hidden neurons or values of the activation function
parameters) to ensure the highest diagnostic accuracy. This
process is time consuming, but results in the improvement in
the diagnostic accuracy. Multiple continuous or discrete
optimization methods may be used for this purpose, regarding
both network structure and the SUT testability.

Future research should cover analysis of other systems
belonging to various technical domains. Successful
implementation of the presented diagnostic scheme to their
analysis would confirm usefulness of the RBF ANN in the fault
detection and identification task. Also, more sophisticated
approaches should be developed for the selection of the
accessible nodes, especially in complex systems [10] with many
parameters to determine. Finally, parallel implementations of
the OvO coding scheme would eliminate its main disadvantage
i.e. the long duration of both training and testing phases.

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Figure 10. Testability analysis for various sets of ANN coding schemes.

Table 6. Comparison of the ANN-based diagnostic modules (trained on
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RBF 81.12 80.56 83.33
MLP 80.75 80.56 79.80
SVM 81.12 81.12 79.80

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Analysis of the RBF ANN-based classifier for the diagnostics of electronic circuit

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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
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/OtherNamespaces [
<<
/AsReaderSpreads false
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(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
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/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
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/ConvertColors /ConvertToCMYK
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/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
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>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice