Nearest Neighbor and Learning Vector Quantization classification for
damage detection using time series analysis
Oliver R. de Lautour
Department of Civil and Environmental Engineering
The University of Auckland
Private Bag 92019
Auckland Mail Centre
Auckland 1142
New Zealand
Email: oliver.delautour@beca.com
Piotr Omenzetter (corresponding author)
Department of Civil and Environmental Engineering
The University of Auckland
Private Bag 92019
Auckland Mail Centre
Auckland 1142
New Zealand
Email: p.omenzetter@auckland.ac.nz
Tel.: +64 9 3737599 ext. 88138
Fax: +64 9 3737462
Abstract
The application of time series analysis methods to Structural Health Monitoring (SHM) is a
relatively new but promising approach. This study focuses on the use of statistical pattern
recognition techniques to classify damage based on analysis of the time series model
coefficients. Autoregressive (AR) models were used to fit the acceleration time histories
obtained from two experimental structures; a 3-storey laboratory bookshelf structure and the
ASCE Phase II Experimental SHM Benchmark Structure in undamaged and various damaged
states. The coefficients of the AR models were used as damage sensitive features. Principal
Component Analysis and Sammon mapping were used to firstly obtain two-dimensional
projections for quick visualization of clusters amongst the AR coefficients corresponding to
the various damage states, and later for dimensionality reduction of data for automatic
damage classifications. Data reduction based on the selection of sensors and AR coefficients
was also studied. Two supervised learning algorithms, Nearest Neighbor Classification and
Learning Vector Quantization were applied in order to systematically classify damage into
states. The results showed both classification techniques were able to successfully classify
damage.
Keywords: Structural Health Monitoring, Damage detection, Time series analysis,
Autoregressive models, Nearest neighbor classification, Learning Vector Quantization.
1
1. Introduction
Over the two past decades considerable research efforts have focused on developing robust
and reliable methods for Structural Health Monitoring (SHM), amongst which vibration
based techniques are the most widely used and appear to be the most promising. Extensive
literature reviews of such methods can be found elsewhere [1, 2] and the focus here is
restricted to techniques relevant to this study, namely, the application of time series methods
for damage detection in civil infrastructure and the use of statistical pattern recognition
techniques for damage classification.
Time series analysis techniques, originally developed for analyzing long sequences of
regularly sampled data are inherently suited to SHM applications. However, applications of
such techniques for SHM can still be considered as an emerging and a relatively unexplored
approach. An interesting and promising idea is to use time series coefficients as damage
sensitive features. A pioneering study by Sohn et al. [3] used Autoregressive (AR) models to
fit the dynamic response of a concrete bridge pier. By performing statistical analysis on the
coefficients of the AR models the authors were able to distinguish between the healthy and
damaged structure. In a later study, Sohn et al. [4] applied a similar methodology to health
monitoring of a surface-effect fast patrol boat. However, these studies focused only on
damage detection and the authors did not attempt to classify damage into states
corresponding to its severity. Gul et al. [5] presented a study in which AR coefficients from a
laboratory steel beam with varying support conditions were classified using a clustering
algorithm or multivariate statistics. Omenzetter and Brownjohn [6] used a vector Seasonal
Autoregressive Integrated Moving Average model to detect abrupt changes in strain data
collected from the continuous monitoring of a major bridge structure. The seasonal model
was used because of the strong diurnal component in the data caused by temperature
variations. Nair et al. [7] used an Autoregressive Moving Average time series to model
vibration signals from the ASCE Phase II Experimental SHM Benchmark Structure. The
authors defined a damage sensitive feature used to discriminate between the damaged and
undamaged states of the structure based on the first three AR coefficients. The localization of
the damage was achieved by introducing another feature, also based on the AR coefficients,
found to increase from a baseline value when damage was near. Nair and Kiremidjian [8]
investigated Gaussian Mixture Modeling, an unsupervised pattern recognition technique to
model the first AR coefficients obtained from fitting ARMA time series to an analytical, 12DOF model of the ASCE Phase II Experimental SHM Benchmark Structure. The extent of
damage was shown to correlate well with the Mahalanobis distance between undamaged and
damaged feature clusters.
In order to differentiate between undamaged and damaged systems and locate and quantify
damage analytical techniques are required to interpret patterns in the damage sensitive
features. Such methods can broadly be divided into supervised techniques when data from
damaged structures is available for training of pattern recognition algorithms, and
unsupervised when it is not. Widely employed supervised techniques include Artificial
Neural Networks [9-11], Genetic Algorithms [12], Support Vector Machines [13, 14] and
discriminant analysis [15]. A number of unsupervised methods have also been investigated
and these include outlier detection [16-18], control chart analysis [3, 19] and clustering
analysis [5, 20, 21]. The present study utilizes the techniques of Nearest Neighbor
Classification (NNC) and Learning Vector Quantization (LVQ). These algorithms have been
applied in the past in a number of damage detection studies mostly concerned with faults in
composite materials, mechanical systems and aerospace structures. Philippidis et al. [22] and
2
Godin et al. [23] classified the waveforms of acoustic emission signals recorded during
destructive tensile tests on coupons of composite materials. Raju Damarla et al. [24] and
Doyle and Fernando [25] analyzed high frequency vibration data from damaged composite
panels and classified damage using frequency and time domain features. Tse et al. [26]
studied a tapping machine using statistical features of dynamic forces and torques measured
in the various machine components and identified several states in which the machine was
operating. Abu-Mahfouz [27] considered classification of damage in a gearbox system
combining frequency response data and basic statistical measures of time-domain dynamic
signals. Trendafilova et al. [28] considered detection of damage by examining frequency
response functions of the FEM model of an aircraft wing. Despite the interest these
techniques drew amongst the composite materials, machine and aircraft monitoring
community, the application of NNC and LVQ for classification of damage to civil
infrastructure has not yet been studied.
This study develops a method for structural damage detection that integrates the use of AR
models to establish damage sensitive features and application of the NNC and LVQ
techniques for classification of the AR coefficients depending on damage type and location.
The proposed method firstly fits the structural acceleration time histories in the undamaged
and various damaged states of the system with AR models. The coefficients of these AR
models are then chosen as damage sensitive features. In order to enable visualization of the
data and check the presence of clusters amongst the AR coefficients corresponding to the
different damage states they are initially projected onto two dimensions using Principal
Component Analysis (PCA) and Sammon mapping. The vectors of damage sensitive features,
with dimensions reduced via PCA or selection of subsets of sensors and AR coefficients,
were later analyzed using NNC and LVQ algorithms in order to classify damage into states.
The method has been applied to experimental data obtained from a simple 3-storey laboratory
bookshelf structure excited on a shake table and the more complex ASCE Phase II
Experimental SHM Benchmark Structure excited by a small shaker.
2. Theory
2.1 Autoregressive models
In this study, AR time series models are used to describe the acceleration time histories. AR
models are a staple of time series analysis [29] and are used in the analysis of stationary time
series processes. A stationary process is a stochastic process, one that obeys probabilistic
laws, in which the mean, variance and higher order moments are time invariant. AR models
attempt to account for the correlations of the current observation in time series with its
predecessors. A univariate AR model of order p, or AR(p), for the time series {xt} (t =
1,2,…n) can be written as:
xt  1 xt 1  2 xt 2  ...   p xt  p  at
(1)
where xt,…xt-p are the current and previous values of the series and {at} is a Gaussian white
noise error time series with a zero mean. The AR coefficients 1,…, p can be evaluated
using a variety of methods [29]. In this study, the coefficients were calculated using a least
squares solution. While using the least squares method for small samples may introduce bias
in the AR coefficient identification, for larger samples, as in our case, this bias is usually
negligible [30, 31].
3
2.2 Nearest Neighbor Classification
NNC is a simple supervised pattern recognition technique [32]. Given a set of pre-selected
and fixed reference or codebook vectors mi (i = 1, …, k) corresponding to known classes, an
unknown input vector x is assigned to the class which the nearest mi belongs. Several
distance measures can be used including Manhattan, Euclidean, Correlation and
Mahalanobis. In this research the Euclidean and Mahalanobis distance measures were used.
The Euclidean distance DE(x,y) between two vectors x and y can be calculated using:
DE x, y  
x  y T x  y 
(2)
The Mahalanobis distance DM(x,y) between two vectors of the same multivariate distribution
with a covariance matrix  can be calculated from:
DM x, y  
x  y T  1 x  y 
(3)
The Mahalanobis distance accounts explicitly for the different scales and correlations
amongst vector entries and can be more useful in cases where these are significant.
2.3 Learning Vector Quantization
LVQ is a supervised machine learning technique designed for classification or pattern
recognition by defining class borders [32]. It is similar to NNC in that it uses a set of
codebook vectors and seeks for the minimum of distances of an unknown vector to these
codebook vectors as the criterion for classification. However, unlike in NNC where codebook
vectors are fixed, an iterative procedure is adopted in which the position of the codebook
vectors is adjusted to minimize the number of misclassifications. Learning of the optimal
codebook vector positions can be performed using several algorithms. In this study, the
Optimized-Learning-Rate LVQ1 algorithm was used. This algorithm has an individual
learning rate for each codebook vector, resulting in faster training.
Given a set of initial codebook vectors mi (i = 1, …, k) which have been linked to each class
region, the input vector x is assigned to the class which the nearest mi belongs, i.e. an NNC
task is performed. Let c define the index of the nearest codebook vector, i.e. mc. Learning is
an iterative procedure in which the position of the codebook vectors is adjusted to minimize
the number of misclassifications. At iteration step t let x(t) and mi(t) be the input vector and
codebook vectors respectively. The mi(t) are adjusted according to the following learning
rule:
 
 
     

m c t  1  1  s t  c t  m c t  s t  c t x t
m i t  1  m i t for i  c
(4)
where s(t) equals +1 or -1 if x(t) has been respectively classified correctly or incorrectly, and
c(t) is the variable learning rate for codebook vector mc:
4
 c t  
 c t  1
1  s t  c t  1
(5)
The learning rate must be constrained such that c(t) < 1.
2.4 Principal Component Analysis
PCA is a popular multivariate statistical technique often used to reduce multidimensional
data sets to lower dimensions [33]. Given a set of p-dimensional vectors xi (i = 1, …, n)
drawn from a statistical distribution with mean x and covariance matrix , PCA seeks to
project the data onto a new p-dimensional space with orthogonal coordinates via a linear
transformation.
Decomposition of the covariance matrix by singular value decomposition leads to:
  VVT
(6)
where  = diag[  12 , …,  2p ] is a diagonal matrix containing the eigenvalues of  ranked in
the descending order  12  …   2p , and V is a matrix containing the corresponding
eigenvectors or principal components. The transformation of a data point xi into principal
components is:
z i  VT x i  x 
(7)
The new coordinates zi are uncorrelated and have a diagonal covariance matrix . Therefore,
the entries of zi are linear combinations of the entries of xi which explain variances  12 ,
…,  2p . To reduce the dimensionality, a selection q < p of principal components can be used
that retains those components that contribute most to the data variance, thus reducing the
dimension of the data to q.
2.5 Sammon mapping
Sammon mapping [34] is a nonlinear transformation used for mapping a high dimensional
space to a lower dimensional space in which local geometric relations are approximated.
Consider a set of vectors xi (n = 1, …, n) in a p-dimensional space and a corresponding set of
vectors yi in a q- dimensional space, where q < p. For visualization purposes q is usually
chosen to be two or three. The distance between vectors xi and xj in p-dimensional space is
given by:

Dij*  D x i , x j

(8)
and the distance between the corresponding vectors yi and yj in q-dimensional space is:

Dij  D y i , y j
5

(9)
where D is a distance measure, usually the Euclidean distance. Mapping is achieved by
adjusting the vectors yi to minimize the following error function by steepest descent [35]:
E
n
1
i 1  j i Dij*
n

i 1 j i
D
ij
 Dij*

2
Dij*
(10)
3. Application to a 3-storey bookshelf structure
The 3-storey experimental bookshelf structure used in this study was approximately 2.1m
high and constructed from equal angle aluminum column sections and stainless steel floor
plates bolted together with aluminum brackets as shown in Figure 1. The stainless steel plates
were 4mm thick and 650mm  650mm square. The column sections were 30mm  30mm
equal angles. Two section thicknesses were used for the columns, either 3.0mm or 4.5mm for
the damaged and undamaged states respectively. Each column was made of 3  0.7m high
segments, rather than one long angle, in order to make them easily replaceable for simulation
of localized damage at different stories. The column sections were fastened at each end with
two M6 bolts to aluminum brackets (Fig. 1b). Additional brackets were installed at the base
of the structure to minimize torsional motion. The whole structure was mounted on a 20mm
plywood sheet fixed with M10 bolts to the shake table.
The structure was instrumented with four uniaxial accelerometers, one for measuring the
table acceleration and one for each storey. Accelerations were measured in the direction of
ground motion at 400Hz using a computer fitted with a data-logging card. Afterwards the
data was decimated by a factor of four for modal analysis and eight for time series modeling.
Damage was introduced into the structure by replacing the original 4.5mm thick columns of a
particular storey with 3.0mm angles. Four damage states were considered; these were labeled
D0, D1, D2 and D3 corresponded to no damage (healthy structure), 1st storey damage, 2nd
storey damage and simultaneous 1st and 2nd story damage. Figure 2 schematically shows
these four damage states together with the percentage of remaining stiffness obtained via
model updating described below.
In order to understand and quantify the dynamic properties of the structure modal analysis
was conducted in each damage state and modal parameters were estimated from the
identification of a discrete state-space model using the Prediction Error Method [36]. For
modal analysis the input into the shake table was white noise. Table 1 shows the estimated
natural frequencies fe and percentage changes of the frequencies fe in states D1, D2, and D3
in relation to D0. Modal damping was approximately 1% for all modes. The modal analysis
results were used to update simple lumped mass-spring models of the structure. The updating
was performed in order to assess the structural stiffness in different damage states. The
masses lumped at each storey were calculated from known section sizes and initial estimates
of lateral storey stiffness were provided from simple hand calculations. Modal frequencies
were used to update the mass-spring models using a sensitivity matrix approach and the
analytical and experimental mode shapes were checked using the Modal Assurance Criterion
(MAC) [37]. From these models the lateral stiffness of each storey in each damage state was
estimated. The results from model updating are given in Table 2. Reductions in lateral
stiffness between 7-10% were observed (see also Figure 2).
6
The objective of the damage detection study was to classify seismic responses measured in
the four damage states and to that end eight scaled earthquake records were used to excite the
structure. Table 3 lists the earthquakes used, the Peak Ground Acceleration (PGA) of the
original and scaled records, the duration of the record and the frequency at which the
earthquake was sampled. The earthquakes were scaled so that a range of response amplitudes
was obtained, while ensuring no yielding of the structure occurred. The acceleration time
history of each storey was modeled using a univariate AR model. Although the structural
response due to earthquake motion would normally be considered as nonstationary, analysis
of small segments only can satisfactorily preserve stationarity [38].
In this study, the optimal AR model was selected using the Akaike Information Criterion
(AIC) [39]. The AR coefficients were estimated from a 500-point window advancing 100
points until the end of the record was reached. Averaging AICs over the 3 stories showed a
minimum at order 24. Model diagnostic checking, comprising testing the stationarity and
normality of the residuals and significance of sample residual Autocorrelation Function
(ACF) and Partial Autocorrelation Function (PACF), was conducted [29]. The KolmogorovSmirnov test [40] was used to test for normality of the residuals and the Ljung-Box statistic
[29] was used to test for correlation. The tests indicated that the residuals conformed to a
Gaussian white noise process. A data set of 388 points (vectors of AR coefficients)
containing 97 points for each damage state was obtained.
3.1 Damage classification in the 3-storey bookshelf structure
Using three univariate AR(24) models, one for each floor, resulted in 72-dimensional vectors
of damage sensitive features. As a preliminary investigation, to visualize and check the
presence of clusters in the data, PCA and Sammon mapping were used to create two and
three-dimensional projections of the vectors of AR coefficients. The results of the twodimensional projections are shown in Figures 3 and 4 for PCA and Sammon mapping,
respectively. Projection of the data onto the first two principal components showed no clearly
defined clusters with data points from all four damage states scattered amongst one another.
In contrast, the Sammon map showed some organization of the data into overlapping bands,
although again, no distinct clusters could be drawn. Using three-dimensional mappings did
not provide a better separation. These preliminary insights indicate that higher dimensional
data need to be investigated to achieve separation of the AR coefficients from different
damage states. Simple visual techniques will then be inadequate and more advanced
approaches such as NNC and LVQ are required.
The two previously described pattern recognition techniques, NNC and LVQ were used to
classify damage into the states D0-D3. Two data reduction techniques were investigated: (i)
projection of the data onto a lower dimensional space using PCA, and (ii) selection of subsets
AR coefficients. The reason for studying the two data reduction techniques is that PCA
systematically picks up feature components that contribute most to its variance, which may
on the other hand be difficult to achieve through selecting AR coefficients by a trial and error
approach. Due to the computational effort required for Sammon mapping this technique was
not used at this stage.
The feature dimension was reduced by projecting the AR coefficients onto the first 60, 40,
30, 20 or 10 principal components using PCA. The 388-point data set, consisting of 97 points
from each damage state was randomly divided into 300 codebook vectors and 88 testing
7
points respectively. Five different random sets of codebook vectors were considered. Using
NNC and averaging the results from five runs, the obtained number of misclassifications and
percentage errors is given in Table 4 for both the Euclidean and Mahalanobis distance
measures. The Mahalanobis distance measure outperformed the Euclidean by a considerable
margin and adequate results with 6% misclassifications were obtained using 20 principal
components. Good classification results, 5% or less misclassifications, were achieved using
more than 30 principal components while excellent classification, 1% or less
misclassifications, required 60 principal components. Using the Euclidean distance measure a
large number of misclassifications were obtained, in excess of 34% misclassifications. The
difference in performance between the Mahalanobis and Euclidean distance measures could
be explained by the fact that the Mahalanobis distance accounts for the different scales of
each principal component.
Four subsets of AR coefficient were investigated with (i) first five coefficients, (ii) first ten
coefficients, (iii) first fifteen and (iv) first twenty coefficients selected from all three models.
Using NNC and the Mahalanobis distance, the obtained number of misclassifications and
percentage errors is given in Table 5 as an average of five runs with different codebook
vectors. Good classification, less than 5% errors, was obtained using 10 AR coefficients or
more. Similar performance between the two data reduction techniques was observed for
smaller feature dimensions. For example, using 10 AR coefficients, equating to a feature
dimension of 30, 4 misclassifications were recorded. When using 30 principal components, 4
misclassifications were also recorded. However, for larger feature dimensions PCA
performed better than selection of subsets of AR coefficients. Using 20 AR coefficients and
60 principal components, 3 and 1 misclassifications were obtained respectively.
Although NNC performed well, performance could be improved by using a more advanced
classification technique such as LVQ. The LVQ classification was used with the Mahalanobis
distance measure and the PCA dimensionality reduction technique. The 388-point data set
was divided into 300 points for training and 88 points for testing. The number of codebook
vectors was chosen to be 30, 50 or 100. These were initialized by random selection from the
training data set. The results averaged from five runs with different initialized codebook
vectors are shown in Table 6. The table shows that increasing the number of codebook
vectors reduced the number of misclassifications. Good classifications were obtained when
20 or more principal components were used while excellent classification was achieved using
30 principal components. Overall, LVQ performed better than NNC with excellent
classifications obtained using 30 principal components and much smaller numbers of
codebook vectors compared to 60 principal components using NNC.
4. Application to ASCE Phase II Experimental SHM Benchmark Structure
The ASCE Phase II Experimental SHM Benchmark Structure [41] is a 4-storey 2-bay by 2bay steel frame with a 2.5m  2.5m floor plan and a height of 3.6m (Figure 5a). The columns
were B1009 sections and the floor beams were S7511 sections, all sections were Grade
300 steel. The beams and columns were bolted together. Bracing was added in all bays with
two 12.7mm diameter threaded steel rods, see Figure 5b. Additional mass was distributed
around the structure to make it more realistic. Four 1000kg floor slabs were placed on the 1st,
2nd and 3rd floors, one per bay. On the 4th floor, four 750kg slabs were used. Two of the slabs
per floor were placed off-centre to increase the coupling between translational and torsional
motion.
8
In the following discussions the locations in the structure are referred to using their respective
geographical directions of north (N), south (S), east (E) and west (W).
A total of 9 damage scenarios were simulated on the structure, these involved the removal of
bracing and the loosening of bolts in the floor beam connections. Table 7 lists the damage
states and gives a description of damage. The different configurations give a mixture of
minor and extensive damage cases.
A series of ambient and forced vibration tests were carried out on the structure. Of primary
interest in this study were the forced random vibration tests conducted using an electrodynamic shaker mounted on the SW bay of the 4th floor on the diagonal. Input into the shaker
was band-limited 5-50Hz white noise. The structure was instrumented with 15
accelerometers; three accelerometers each for the base, 1st, 2nd, 3rd and 4th floors. These were
located on the E and W frames to measure motion in the N-S direction and in the centre to
measure E-W motion. Acceleration data was recorded at 200Hz using a data acquisition
system and filtered with anti-aliasing filters.
The optimal model order was selected using AIC and the residual error series was checked
for confirmation of a Gaussian white noise process following the same process as described
earlier. Univariate AR(30) models were adopted and fitted to the acceleration data from each
of the 15 accelerometers. This resulted in the dimension of feature (AR coefficients) vectors
of 300. The AR coefficients were estimated using least squares from 1000-point segments
advancing 200 points until the end of the record was reached. A data set of 1035 feature
vectors was obtained, 115 from each configuration.
4.1 Damage classification in the ASCE Phase II Experimental SHM Benchmark Structure
Preliminary investigations showed that projection of the data onto the first two principal
components allows much clearer clustering in the data to be seen than in the case of the 3storey bookshelf structure (Figure 6). Two larger clusters were apparent that consisted of the
configurations 1-6 and configurations 7-9. Some overlapping existed between certain damage
configurations, in particular configurations 1, 5 and 6. Using Sammon mapping, see Figure 7,
six large-scale clusters could be seen. Two of the clusters consisted of configurations 1, 5,
and 6 and configurations 3 and 4, while the remaining clusters were solely formed by a single
configuration.
Two data reduction techniques were once again investigated (i) projection of the data onto a
lower dimensional space using PCA and (ii) selection of subsets of AR coefficients and/or
accelerometers. The reason for considering the selection of sensors as a data reduction
method was that in practical situations one will not have the comfort of having a large
number of accelerometers to perform PCA but will rather have to decide on their practically
feasible numbers and locations before deploying a monitoring system.
For the purpose of damage classification using NNC, the feature dimension was reduced by
projection of the data onto the first 20, 10, 5 or 3 principal components. The 1035-point data
set was randomly divided into 700 codebook vectors and 335 testing points. Averaging the
results from five runs with different selection of codebook vectors the number of
misclassifications and percentage errors is given in Table 8. In this case, similar performance
was obtained using both the Euclidian and Mahalanobis distance measures. This can be
9
attributed to the fact that the data from different damage configuration seems to be well
separated. Excellent performance (1% or less misclassifications) was obtained using only 10
principal components. These results represent significant reduction in dimensionally while
good accuracy was maintained.
Reducing the number of AR coefficients and/or accelerometers may adversely affect
information contained in the data and degrade the performance of the damage detection
method. Therefore several combinations of reduced AR coefficients and accelerometers were
systematically investigated to ascertain their practical minimum numbers. The number of AR
coefficients was reduced by selecting the first few coefficients only. This was chosen to be
either (i) the first coefficient, (ii) the first two coefficients, (iii) the first three coefficients, (iv)
the first four coefficients, or (v) the first six coefficients. Similarly, the number of
accelerometers and their location was either (i) the full set (15 accelerometers), (ii) omitted
accelerometers located on the base and those of the W face measuring N-S motion (8
accelerometers), or (iii) same as case (ii) but with all remaining accelerometers on stories 1
and 3 omitted (4 accelerometers). The numbers of misclassifications using NNC and the
Euclidean distance averaged for five runs with different codebook vectors and for different
subsets of AR coefficients and/or accelerometers are shown in Figure 8. The boundaries of
5% and 1% percent misclassifications are also shown in the figure. It can be seen that good
classification was obtainable using at least 2 AR coefficients and/or 4 accelerometers (feature
dimension 8), while excellent performance required at least 3 AR coefficients and/or 4
accelerometers (feature dimension 12). These results are only slightly inferior compared to
PCA.
LVQ classification was applied to PCA-reduced data with the same number of principal
components as previously and the same sized training and testing data sets were used. The
results from NNC showed that performance was similar for both distance measures, and
hence only the Euclidean distance was chosen for LVQ. The number of codebook vectors
was either 50, 100 or 200. These were initialized by random selection from the training set.
The results obtained from averaging the number of misclassifications from five runs are
shown in Table 9. The table shows that increasing the number of codebook vectors generally
resulted in better performance. Excellent performance was obtained using 20 principal
components for 100 and 200 codebook vectors. Good classification was still achieved using
10 principal components, however, errors became significant once fewer principal
components were used. Overall, performance was slightly worse than NNC.
5. Conclusions
SHM and damage detection methods can benefit from the applications of time series analysis
techniques, which were developed for understanding long, regularly sampled sequences of
data. In this study, a supervised damage detection and classification method using AR models
and NNC and LVQ statistical pattern recognition algorithms has been developed and applied
to a simple 3-storey laboratory bookshelf structure and more complex ASCE Phase II
Experimental SHM Benchmark Structure. Acceleration time histories of the structures in
different simulated damage cases and under dynamic excitation were recorded and fitted
using AR models. The coefficients of these AR models were chosen as damage sensitive
features. Dimensionality reduction of the damage sensitive feature was achieved via PCA and
Sammons mapping and served the purpose of visualization of clusters amongst the AR
coefficients and lessening computational burden of the pattern recognition techniques.
Selection of subsets the AR coefficients and accelerometers was also investigated as a means
10
for data reduction. Systematic damage classification was studied using the NNC and LVQ
supervised learning algorithms with either Euclidian or Mahalanobis distance measures.
The studies on the 3-storey laboratory structure demonstrated that for localized stiffness
reduction of 7% to 10% the AR coefficient corresponding to different damage states, when
projected on two dimensions, do not form clearly separable clusters. However, when the
number of principal components used was increased to 60 and 30 respectively, both NNC and
LVQ with the Mahalanobis distance measure were able to classify damage with no more than
approximately 1% of misclassifications. The Euclidian measure, on the other hand produced
approximately 35% misclassifications. Using selection of AR coefficient subsets as an
alternative feature reduction technique, NNC produced approximately 3% misclassifications
with a feature dimension of 60, indicating the systematic approach using PCA offered some
benefits. For the data from the ASCE Phase II Experimental SHM Benchmark Structure
much more clearly delineated clusters of AR coefficients were seen in two dimensions. In
this case the performance of both distance measures was comparable. The results using NNC
and LVQ showed that approximately 1% misclassifications could be achieved using 20
principal components. The selection of AR coefficient subsets showed similar
misclassifications to PCA for equivalent feature dimensions when analyzed using NNC.
Overall, the results showed that AR coefficients perform well as damage sensitive features,
NNC and LVQ are reliable tools for damage classification, and significant reductions in the
data dimensionality could be achieved whilst maintaining good performance.
In its current form outlined in the paper the proposed damage detection method faces some
challenges. Firstly, the supervised nature of the method requires data from damaged states to
be available. This represents a practical challenge to the applicability of the method to real
life structures. However, in a real-world application data from damaged states could be
obtained from an analytical model of the structure. Secondly, in their basic forms used in the
paper both NNC and LVQ use a predefined number of clusters or damage states and if
presented with a feature from another damage state they will classify it incorrectly as
belonging to one of those states. This problem could be addressed by accounting for
probabilistic or possibilistic nature of cluster classification. A mapping between the physical
changes in the structure due to damage and resulting positions of corresponding feature
clusters may be established, in practice using an analytical structural model. Then,
probabilities or membership functions can be constructed based on distances from a damage
feature to the given predefined clusters. Samples with probabilities or membership function
values close to one will be assigned to the respective clusters and those with smaller values
will not. For samples rejected from the predefined clusters their damage could be inferred by
interpolating the mapping between the physical changes in the structure due to damage and
the resulting positions of feature clusters. The extensions of the proposed method outlined is
this paragraph will be studied in future.
Acknowledgement
The authors would like to express their gratitude for the Earthquake Commission Research
Foundation of New Zealand for their financial support of this research (Project No.
UNI/535).
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14
Table 1. Natural frequencies and percentage changes at different damage states for 3-storey
bookshelf structure.
fe (Hz)
Mode
D0
st
1
1.928
2nd
5.52
3rd
8.55
a
Based on D0.
D1
1.879
5.43
8.30
D2
1.837
5.46
8.09
D3
1.840
5.42
8.15
D1
-2.5%
-1.6%
-2.9%
fe (%)a
D2
-4.7%
-1.0%
-5.3%
D3
-4.6%
-1.8%
-4.6%
de Lautour and Omenzetter
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Table 2. Updated stiffnesses from analytical models for 3-storey bookshelf structure.
Stiffness of 1st storey (N/m)
Stiffness of 2nd storey (N/m)
Stiffness of 3rd storey (N/m)
D0
3.77104
0.60104
0.77104
D1
3.49104
0.60104
0.77104
D2
3.77104
0.54104
0.77104
D3
3.49104
0.54104
0.77104
de Lautour and Omenzetter
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Table 3. Earthquake records used to excite 3-storey bookshelf structure.
Earthquake
Duzce 12/11/1999
Erzincan 13/3/1992
Gazli 17/5/1976
Helena 31/10/1935
Imperial Valley 19/5/1940
Kobe 17/1/1995
Loma Prieta 18/10/1989
Northridge 17/1/1994
PGA (g)
0.535
0.496
0.718
0.173
0.313
0.345
0.472
0.568
Scaled PGA (g)
0.027
0.033
0.048
0.035
0.031
0.035
0.047
0.038
Duration (sec)
25.885
20.780
16.265
40.000
40.000
40.960
39.945
40.000
Sampling frequency (Hz)
200
200
200
100
100
100
200
50
de Lautour and Omenzetter
17
Table 4. Number and percentage of misclassifications using NNC and PCA reduced data for
3-storey bookshelf structure.
Number of principal components
60
40
30
20
10
Euclidean distance
31 (35%)
34 (39%)
30 (34%)
34 (39%)
34 (39%)
Mahalanobis distance
1 (1%)
3 (3%)
4 (5%)
5 (6%)
10 (11%)
de Lautour and Omenzetter
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Table 5. Number and percentage of misclassifications using NNC and subsets of AR
coefficients for 3-storey bookshelf structure.
Number of AR coefficients selected per model
20
15
10
5
Mahalanobis distance
3 (3%)
4 (5%)
4 (5%)
10 (11%)
de Lautour and Omenzetter
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Table 6. Number and percentage of misclassifications using LVQ for 3-storey bookshelf
structure.
Number of principal components
30
20
10
30
0 (0%)
4 (5%)
17 (19%)
Number of codebook vectors
50
100
0 (0%)
0 (0%)
3 (3%)
3 (3%)
15 (17%)
14 (16%)
de Lautour and Omenzetter
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Table 7. Damage configurations for ASCE Phase II Experimental SHM Benchmark
Structure.
Configuration
1
2
3
4
5
6
7
8
9
Damage
No damage
All bracing removed on the E face
Bracing removed on floors 1-4 on a bay on the SE corner
Bracing removed on floors 1 and 4 on a bay on the SE corner
Bracing removed on floors 1 on a bay on the SE corner
All bracing removed on E face and floor 2 on N face
All bracing removed
Configuration 7 + loosened bolts on floors 1-4 on E face N bay
Configuration 7 + loosened bolts on floors 1 and 2 on E face N bay
de Lautour and Omenzetter
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Table 8. Number and percentage of misclassifications using NNC for ASCE Phase II
Experimental SHM Benchmark Structure.
Number of principal components
20
10
5
3
Euclidean distance
1 (0.3%)
5 (1%)
23 (7%)
62 (19%)
Mahalanobis distance
1 (0.3%)
5 (1%)
24 (7%)
65 (19%)
de Lautour and Omenzetter
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Table 9. Number and percentages of misclassifications using LVQ for ASCE Phase II
Experimental SHM Benchmark Structure.
Number of principal components
20
10
5
3
Number of codebook vectors
50
100
200
6 (2%)
5 (1%)
4 (1%)
13 (4%)
10 (3%)
6 (2%)
24 (7%)
29 (9%)
23 (7%)
75 (22%)
68 (20%)
67 (20%)
de Lautour and Omenzetter
23
(a)
(b)
(c)
Figure 1. 3-storey laboratory bookshelf structure: (a) general view, (b) detail of column-plate
joint, and (c) dimensions and accelerometer locations.
de Lautour and Omenzetter
24
Legend:
100%of stiffness
100%of stiffness
100%of stiffness
100%of stiffness
4.5mm thick angle
100%of stiffness
100%of stiffness
90%of stiffness
90%of stiffness
93%of stiffness
100%of stiffness
93%of stiffness
D1
D2
D3
3.0mm thick angle
100%of stiffness
D0
Figure 2. Damage states in 3-storey bookshelf structure showing percentage of lateral
stiffness at each story.
de Lautour and Omenzetter
25
Figure 3. Projection of data from 3-storey bookshelf structure onto the first two principal
components.
de Lautour and Omenzetter
26
Figure 4. Projection of data from 3-storey bookshelf structure via Sammon mapping.
de Lautour and Omenzetter
27
(a)
(b)
Figure 5. ASCE Phase II Experimental SHM Benchmark Structure: (a) general view, and (b)
beam-column joint and bracing.
de Lautour and Omenzetter
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Figure 6. Projection of data from ASCE Phase II Experimental SHM Benchmark Structure
onto the first two principal components.
de Lautour and Omenzetter
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Figure 7. Projection of data from ASCE Phase II Experimental SHM Benchmark Structure
via Sammon mapping.
de Lautour and Omenzetter
30
Figure 8. Number of misclassifications using subsets of AR coefficient and/or accelerometers
for ASCE Phase II Experimental SHM Benchmark Structure.
de Lautour and Omenzetter
31