1
Quantification of the uncertainty of pattern
recognition approaches applied to acoustic
emission signals
MARKUS G. R. SAUSE
1
and SIEGFRIED HORN
1
1
University of Augsburg, Institute for Physics, Experimental Physics II, D-86135
Augsburg
+498215983238
+498215983411
markus.sause@physik.uni-augsburg.de
Pattern recognition, cluster validity indices, acoustic emission, carbon fiber
reinforced plastics
CFRP Carbon fiber reinforced plastics
AE Acoustic emission
S
i
Lamb-wave mode, symmetric, i-th order
A
i
Lamb-wave mode, antisymmetric, i-th order
SH
i
Shear-horizontal wave mode, i-th order
C
ij
Elastic coefficients
DB Davies-Bouldin Index
TOU Tou Index
S Rousseuw’s silhouette value
Hubert’s Gamma statistics
DCB Double Cantilever Beam
J Degree of cluster separation
RAND Rand Index
C Cluster validity measure
C
0
Fit parameter (logistics function)
s Fit parameter (logistics function)
h Fit parameter (logistics function)
A
min
Fit parameter (logistics function)
A
max
Fit parameter (logistics function)
A Range of univariate distribution
N Univariate distribution
2
Abstract
Acoustic emission analysis is a nondestructive technique frequently used to assess the integrity of
fiber reinforced plastics. Pattern recognition techniques have shown great potential to identify
microscopic failure mechanisms in plate-like structures. Because every assignment of an acoustic
emission signal to a respective failure mechanism is possibly associated with an error, one key
question is the reliability of the assignment method. It is useful to distinguish between the
uncertainty of the assignment and the false assignment of an acoustic emission signal to a group of
signals. The first is owed to statistical effects and the reliability of the classification method itself.
The second is caused by false conclusions or disputable assumptions on the source mechanisms.
The present study will focus on the first aspect. For this purpose, we propose a model based
algorithm that estimates the uncertainty of a feature based pattern recognition approach based on
cluster validity indices. Further, we demonstrate the application of the algorithm to experimental
acoustic emission data obtained from a double cantilever beam specimens with unidirectional
layup of carbon fiber reinforced polymer. Based on previous investigation we use a pattern
recognition approach to distinguish between different failure mechanisms like matrix cracking,
interfacial failure and fiber breakage based on the frequency features of the acoustic emission
signals. We consider the influence of dispersion and attenuation effects during propagation of
Lamb-waves on the extracted acoustic emission features. This is done by investigating the
influence of source-sensor distance by test sources like pencil lead breaks and piezoelectric
pulsers. Using the model based algorithm it is possible to calculate the uncertainty of the pattern
recognition results as a function of source-sensor distance. It is found that dispersion effects of
Lamb-waves do not seriously affect the distinction between microscopic failure mechanisms for
source-sensor distances up to 375 mm. We demonstrate that the spatial distribution of acoustic
emission sources has a larger impact on the uncertainty of assignment than the absolute source-
sensor distance. Applying the proposed algorithm to the current experimental setup, we obtain an
uncertainty of classification below 7 % for source-sensor distances below 375 mm. Attenuation is
quantified to be 0.165 dB/mm for the A
0
-mode and 0.047 dB/mm for the S
0
-mode. Within the
source-sensor distance of 375 mm this causes severe attenuation of the signal amplitude and thus
prohibits detection of weak acoustic emission signals long before the uncertainty of the
classification method reaches 10 %.
1 Introduction
Fiber reinforced composites show an extraordinary potential for application as
light-weight structure materials due to their high strength-to-weight and high
stiffness-to-weight ratio. Most of the time the investigation of material failure of
fiber reinforced composites by conventional nondestructive techniques occurs
offline, i.e. after loading and unloading of the specimen. In contrast, acoustic
emission (AE) analysis is a powerful nondestructive technique for online
monitoring of material failure during loading of the specimen [1]. Here,
microscopic internal displacements like crack generation or crack propagation
cause stress-waves that are detectable as transient acoustic waves. Within a plate-
type specimen, these acoustic waves are symmetric (S
i
) and antisymmetric (A
i
)
Lamb-wave modes, as well as shear-horizontal (SH
i
) modes.
In addition to the activity of acoustic emission, the position of the acoustic
emission source and the type of acoustic emission source are key aspects to
enhance the understanding of material failure. While source localization in carbon
fiber reinforced polymers (CFRP) still has to overcome the problem of anisotropic
acoustic signal propagation and geometrical complexity [1-3], unsupervised
pattern recognition techniques have already demonstrated their suitability to
identify particular source mechanisms in CFRP [4-12].
The basis for these feature based pattern recognition approaches is the concept of
feature extraction. Thus, typical features are calculated from the recorded acoustic
3
emission signal (i.e. peak frequency, see also section 3) to parameterize the signal.
Pattern recognition algorithms are then applied to search cluster structures in
subsets of these features [13]. In general, the outcome of the pattern recognition
process is a classification of acoustic emission signals based on their similarity to
each other. The correlation of one group of acoustic emission signals to a
particular source type is a separate task.
One problem common to all unsupervised pattern recognition approaches is the
evaluation of the clustering result. For the case of acoustic emission analysis, the
following two errors can occur:
1) A group of acoustic emission signals is assigned to the wrong source type
2) An acoustic emission signal is assigned to the wrong group
In literature, various methods are established to assign a group of AE signals to a
particular source type [4-12]. In our previous publications [11,12,14-16] we use
finite element modeling of acoustic emission signals for various source
mechanisms validated in a variety of experimental configurations to perform this
task.
However, a prerequisite for valid source identification is a statistically meaningful
group of signals. If no distinction between the acoustic emission signals can be
made based on their feature values, any subsequent discussion of the underlying
source type is disputable. Since the exact assignment of one particular signal to
one mechanism is by definition unknown, it is useful to express the error of the
classification procedure as uncertainty of the assignment.
Within the present investigation we present a numerical method that is capable to
quantify this uncertainty of assignment. We demonstrate how this method is used
to calculate the uncertainty of assignment for an experimental dataset. We
consider the influence of dispersion and attenuation effects during propagation of
Lamb-waves on the extracted acoustic emission features and elaborate the
experimental factors that cause an increase in the uncertainty of classification.
2 Pattern recognition
Since a comprehensive description of the pattern recognition method used in this
investigation was previously reported in [11], we only give a brief summary in the
following.
The presented method was inspired by the work of Anastassopoulos et al. and
Günter et al. [17, 18] and is based on an exhaustive screening taking into account
all combinations of signal features extracted from the recorded acoustic emission
signals. For each possible combination of signal features an investigation of the
classification performance of the k-means algorithm is evaluated ranging from
two to ten classes. The numerical degree of cluster separation of each partition is
calculated utilizing the Davies–Bouldin () and Tou ( ) indices,
Rousseeuw’s silhouette validation method () and Hubert’s Gamma statistics ()
[19-22]. Since the various cluster validation methods are comprehensively
described in the authors’ original work, their definition is not repeated in the
following.
4
In the spirit of [17, 18] the individual rating of each cluster validation technique is
cumulated based on a voting scheme and is evaluated for the number of clusters
with best performance. This is defined as the best partitioning for the given
feature combination. This methodology can be used as an automated evaluation of
the number of natural clusters and their partitions without previous knowledge
about the cluster structure of acoustic emission signals.
The assignment of a group of acoustic emission signals to a source mechanism is
achieved by a comparison to acoustic emission signals calculated by a finite
element modeling approach. Since this is beyond the scope of the current
investigation, we summarize the correlation between particular microscopic
failure mechanisms and the respective acoustic emission source configuration
briefly. The following correlation between micromechanical failure modes and
acoustic emission source properties (notated in brackets) for fiber reinforced
polymers is used:
• Interfiber fracture (matrix cracking or interfacial failure, in-plane)
• Fiber-Matrix Debonding (interfacial failure, in-plane and out-of-plane)
• Fiber-Matrix Pull-Out (interfacial failure, in-plane and out-of-plane)
• Interply delamination (matrix cracking or interfacial failure, out-of-plane)
• Fiber fracture (fiber breakage, in-plane)
A precise description of the implementation of the particular source types in finite
element models is found in Ref. [15, 16]. The description of mesoscopic failure
modes (e.g. fiber bridging) is beyond the scope of the proposed acoustic emission
source models. The suitability of the proposed method for a variety of specimen
geometries and loading conditions has been demonstrated using artificial, as well
as experimental datasets [11, 12, 14, 15].
2.1 Definition of the uncertainty of assignment
As discussed in [11] the purpose of the pattern recognition technique is the
detection of the natural clusters, which are defined as numerically best separation
of the dataset investigated. However, the detection of natural clusters does not
imply a classification error suitable for statistically significant identification of
particular failure mechanisms in a material. Naturally, clusters will always have
some overlap relative to each other, which causes ambiguous assignment of
signals at the border between to clusters. Thus, a measure for the uncertainty of
assignment during the classification process is required. In the following, the
values of cluster validity measures , , and shall be used for this
purpose.
In order to establish an analytical correlation between the cluster validity
measures and the uncertainty of assignment we investigate artificially generated
datasets. Following the approach of Milligan [23] and the refinement by Qiu and
Joe [24] we generate datasets according to the implementation within the software
package “R” by Qiu et al. [25]. We use the measure of the degree of separation
as introduced by [26]. The measure is based on the separation of two clusters
generated from two univariate normal distributions N(0, 1) and N(0,A) [26]. For
the values of A = 8, A = 6 and A = 4 the measure of the degree of separation
ranges from “well-separated” ( = 0.342), “separated” ( = 0.213) to “close” ( =
0.010) as shown in the scatter plots in figure 1.
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The proposed pattern recognition method can identify the number of clusters
accurately down to J = 0.010 [11]. Since the internal structure of the artificial
datasets are initially known this allows a direct quantification of the misclassified
cluster members of the test data sets.
Figure 1: Visualization of cluster structure for degree of separation of J = 0.342 (a), J=0.213 (b)
and J=0.010 (c).
As measure of the mismatch between two partitions the Rand index is applied
[27]. In statistics, the Rand index is used as a direct measure of the percentage of
decisions that are correct and thus is a direct measure of the classification error.
Next we consider the correlation between the calculated cluster validity measures
, , and and the Rand index.
To this end, we investigated a number of artificial datasets with varying degree of
separation between -0.45 and 0.45. The number of objects in each cluster was
randomly chosen within the range [50, 200] which reflects reasonable variation of
the cluster sizes. Table 1 summarizes the remaining parameters used in the study.
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-15
-10
-5
0
5
10
15
(a)
J = 0.342
Class 1
Class 2
Class 3
Feature 2
Feature 1
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-5
0
5
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(b)
Feature 2
Feature 1
Class 1
Class 2
Class 3
J = 0.213
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Feature 2
Feature 1
(c)
J = 0.010
Class 1
Class 2
Class 3
