Research Article
Relevance feature selection of modal
frequency-ambient condition pattern
recognition in structural health
assessment for reinforced concrete
buildings
Advances in Mechanical Engineering
2016, Vol. 8(8) 1–12
Ó The Author(s) 2016
DOI: 10.1177/1687814016662228
aime.sagepub.com
He-Qing Mu1,2, Ka-Veng Yuen3 and Sin-Chi Kuok4
Abstract
Modal frequency is an important indicator for structural health assessment. Previous studies have shown that this indicator is substantially affected by the fluctuation of ambient conditions, such as temperature and humidity. Therefore, recognizing the pattern between modal frequency and ambient conditions is necessary for reliable long-term structural health
assessment. In this article, a novel machine-learning algorithm is proposed to automatically select relevance features in
modal frequency-ambient condition pattern recognition based on structural dynamic response and ambient condition
measurement. In contrast to the traditional feature selection approaches by examining a large number of combinations
of extracted features, the proposed algorithm conducts continuous relevance feature selection by introducing a sophisticated hyperparameterization on the weight parameter vector controlling the relevancy of different features in the prediction model. The proposed algorithm is then utilized for structural health assessment for a reinforced concrete
building based on 1-year daily measurements. It turns out that the optimal model class including the relevance features
for each vibrational mode is capable to capture the pattern between the corresponding modal frequency and the ambient conditions.
Keywords
Bayesian inference, feature selection, maximum likelihood, model class selection, structural health monitoring
Date received: 7 April 2016; accepted: 6 July 2016
Academic Editor: Jun Li
Introduction
The goal of structural health monitoring (SHM) is to
assess the health status of a structure based on structural responses and ambient conditions measurement.1
Modal frequency, which is related to structural stiffness, is an important indicator in SHM. Previous studies have shown that this indicator is substantially
affected by the fluctuation of ambient conditions, such
as temperature and relative humidity.2–6 In order to
depict the relationship between modal parameters
and ambient conditions, a number of long-term
1
School of Civil Engineering and Transportation, South China University
of Technology, Guangzhou, P.R. China
2
State Key Laboratory of Subtropical Building Science, South China
University of Technology, Guangzhou, P.R. China
3
Faculty of Science and Technology, University of Macau, Macao, China
4
Department of Civil and Environmental Engineering, Cornell University,
Ithaca, NY, USA
Corresponding author:
He-Qing Mu, School of Civil Engineering and Transportation, South China
University of Technology, Guangzhou 510640, P.R. China.
Email: [email protected]
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License
(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without
further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/
open-access-at-sage).
2
monitoring systems were operated for different types
of structures, including bridge,7–9 reinforced concrete
buildings,10–12 and other types.13–15 In these studies, it
has been shown that the fluctuation of the structural
dynamical properties was significant due to the variation of ambient conditions, and the modal frequencies
exhibited strong correlation with temperature and
relative humidity. If the modal frequency-ambient
condition pattern is not appropriately captured, there
will be substantial bias in the structural health assessment results. Toward the goal for reliability enhancement of structural health assessment under changing
ambient conditions, pattern recognition and machinelearning approaches have received tremendous attention.16,17 A number of machine-learning algorithms
have been proposed, including principal component
analysis,18 support vector machine,19 nonlinear principal component analysis,20 support vector machine
multi-class clustering algorithm,21 and kernel-based
algorithm.22
Due to the fact that structural health assessment
exhibits significant level of uncertainty,23–26 a number
of system identification approaches have been proposed.27–32 In particular, the Bayesian inference–based
approach has attracted special attention as it provides
a rigorous solution for uncertainty quantification and
it is applicable for different problems in pattern recognition and system identification,33–35 such as modal/
model updating,36–38 robust signal processing, and sensor configuration.39–43 In this article, a novel machinelearning approach is proposed to automatically select
the relevance features in modal frequency-ambient condition pattern recognition based on structural dynamic
response and ambient condition measurement.
In the traditional feature selection approaches,44,45 a
set of model class candidates is first constructed, and
each candidate contains a subset of the components of
Figure 1. (a) Side view and (b) layout of the East Asia Hall.
Advances in Mechanical Engineering
the full extracted feature vector. Model plausibility evaluation is implemented for each candidate, and it is
obvious that the total number of candidates for evaluation is huge although the number of the components of
the extracted feature vector is moderate. For instance,
when the number of components of the full extracted
feature vector is 10, 210-1 candidates in total are
required for consideration given that at least one feature of the full extracted feature vector should be
included in a candidate. Therefore, optimal model class
determination in the traditional approaches requires
large computational effort. In contrast, the proposed
approach is capable of conducting continuous relevance feature selection, which is done by introducing a
sophisticated hyperparameterization on the weight
parameter vector controlling the relevancy of different
features in the prediction model. The proposed algorithm is then utilized for structural health assessment
for a reinforced concrete building based on 1-year daily
measurements.
The structure of this article is outlined as follows.
Dataset and feature extraction are first presented.
Then, the approach for relevance feature selection is
developed. Finally, the proposed approach is utilized
for modal frequency-ambient condition pattern recognition for the monitored structure based on
measurement.
Dataset and feature extraction
Description of dataset
The structure considered in this study is a 22-story reinforced concrete residential building, namely, the East
Asia Hall. Figure 1 shows the side view and its layout
plan. The floor plan of the building is in asymmetric Lshape, with the building height and the length of the
two spans being 64.70, 51.90, and 61.75 m, respectively.
Mu et al.
3
Figure 2. Identified squared modal frequencies of the first three modes (v21 , v22 , v23 ) and measurement of the temperature (T) and
the relative humidity (H).
The monitoring period is 1 year between 2 May
2008 and 1 May 2009. During the monitoring period,
the structure was subjected to four typhoons of 15 days.
In order to focus on the pattern between the structural
properties and ambient conditions under normal
weather conditions, the 15-day typhoon attacking
period is excluded and the rest of 350 datasets are utilized in the analysis. Two accelerometers, operated
based on standard exploration geophone mass-spring
systems with 50 V/g sensitivity, were installed on the
18th floor with two orthogonal directions (directions 1
and 2) shown in Figure 1(b).
In order to minimize the spatial temperature difference of the monitored structure, acceleration was measured at 11:00 p.m. every day. Based on the 10-min
acceleration time history of each day, the Bayesian
spectral density approach46–48 is applied to identify the
modal frequencies of the building. The Bayesian spectral density approach is a well-developed probabilistic
approach for modal identification.46–48 The modal
identification is performed by utilizing the spectral density obtained from the structural response as the data
to estimate the uncertain modal parameters. The efficiency and flexibility of the Bayesian spectral density
approach on modal identification had been demonstrated through successful applications with in-field
measurements.11,12,38 Herein, each set of the 10-min
acceleration measurement is partitioned into four segments with an equal time duration and then the averaged spectrum is obtained for modal identification.
Since the squared modal frequencies of first three
modes are considered, the range ½0, 4s2 that covers
the concerned squared modal frequencies is used in the
modal identification.
Figure 2 shows the identified squared modal frequencies of the first three modes (v21 , v22 , v23 ) and measurement of the temperature (T) and the relative
humidity (H). The ranges of temperature and relative
humidity are 138C T 35:08C and 32:0% H 95:0%, respectively. It can be seen that high temperature associates with high relative humidity, while low
temperature associates with scattering relative humidity, reflecting the weather pattern influenced by monsoon and subtropical climate.
Feature extraction
Extracted features are constructed based on the
previous research of the relationship between modal
4
Advances in Mechanical Engineering
Figure 3. Correlation coefficients between the squared modal frequency and each feature component for the three modes:
(a) upper: mode 1; (b) middle: mode 2; and (c) lower: mode 3.
frequency and ambient conditions. Watson and
Rajapakse49 correlated structural properties with temperature based on a polynomial function; Xia et al.5
proposed a linear relationship between modal frequencies and temperature along with humidity; Rincón
et al.50 attempted to investigate the influence of structure due to the changing of relative humidity; Yuen
and Kuok11,12 proposed a second-order polynomial
function for modal frequencies with respect to both
temperature along with relative humidity; Moser and
Moaveni51 and Moaveni and Behmanesh52 utilized
high-order (up to fourth) polynomials for correlating
modal frequencies and temperature only. In summary,
most of the existing works considered low-order polynomials for modal frequencies with respect to temperature only or temperature and relatively humidity; on
the other hand, very few works considered high-order
polynomials (up to fourth) for frequencies with respect
to temperature only, neglecting the cross combinations
of temperature and relatively humidity. In this study,
in order to comprehensively take different possible patterns into consideration, the extracted features include
not only higher order functions of the temperature and
the relative humidity but also the cross combinations
of them. Finally, the feature vector x containing totally
15 components is given as follows
x = 1, Tn , Hn , Tn2 , Tn Hn , Hn2 , Tn3 , Tn2 Hn , Tn Hn2 , Hn3 , Tn4 ,
Tn3 Hn , Tn2 Hn2 , Tn Hn3 , Hn4 T
ð1Þ
where the normalized temperature Tn = T=Tmax and the
normalized relative humidity Hn = H=Hmax are
obtained by considering the rescale of T and H by the
maximum value Tmax = max(T ) and Hmax = max(H),
respectively. Figure 3 shows the correlation coefficients
between the squared modal frequencies and different
feature components for the three modes. It is observed
that all the modal frequencies possess positive correlation with each component of the feature vector.
Toward the goal for reliability enhancement of the prediction model charactering modal frequency-ambient
condition pattern, it is aimed to select the suitable relevance features based on the available dataset. In next
section, a novel Bayesian inference–based learning
approach is proposed. In contrast to the traditional
approach requiring selecting relevance features by
examining a large number of combinations of extracted
features,44,45 the proposed approach conducts continuous relevance feature selection by introducing a sophisticated hyperparameterization on the weight parameter
vector controlling the relevancy of different features in
the prediction model.
Mu et al.
5
Relevance feature selection
Consider the mth vibrational mode of the structure.
The modal frequency-ambient condition pattern given
the feature vector with Nb (Nb = 15) components for
the dataset with N (N = 350) points is given as follows
Ym = Xbm + em
ð2Þ
where Ym = ½v2m, 1 , . . . , v2m, N T 2 RN is the mth mode
output vector; X = ½X1 , . . . , XNb 2 RN 3 Nb is the
design/input matrix characterized by the feature vector
of equation (1); bm 2 RNb is the mth mode weight parameter vector to be trained; and em 2 RN is the mth
mode residual vector following Gaussian distribution
N (em j0, s2m IN ) with the corresponding mth mode
prediction-error variance s2m . Then, the likelihood function for Dm =fX, Ym g of the mth mode is given as
follows48
p(Dm jbm , s2m ) = N (Ym jXbm , s2m IN )
ð3Þ
In order to conduct continuous relevance feature
selection for the mth mode, a sophisticated hyperparameterization on the weight parameter vector bm is introduced. The automatic relevance determination (ARD)
prior is adopted for hyperparameterization,53–55 which
is defined as a zero-mean Gaussian distribution
p(bm jam ) = N (bm j0, A1
m (am ))
Optimization of hyperparameter vector and
prediction-error variance
According to Bayes’ theorem, the posterior PDF of the
hyperparameter vector am and the prediction-error
variance s2m is given by p(am , s2m jDm ) } p(Dm jam , s2m )
p(am , s2m ), where p(Dm jam , s2m ) is the evidence conditional on am and s2m , and p(am , s2m ) is the prior PDF of
am and s2m . As the maximum a posteriori estimation
can be well approximated by the maximum evidence
estimation, the optimal values for am and s2m can be
determined by considering the following optimization57
ð4Þ
where Am (am ) = diagfam, 1 , am, 2 , . . . , am, Nb g is the precision matrix (inverse of the covariance matrix) and
am = ½am, 1 , am, 2 , . . . , am, Nb T is the hyperparameter
vector parameterizing the prior probability density
function (PDF) of the weight parameter vector bm .
Using Bayes’ theorem, the posterior PDF for bm is
given by56
p(bm jDm , am , s2m )} p(Dm jbm , s2m )p(bm jam )
^ m)
= N (bm j^bm , S
bmle
m, i is the maximum likelihood estimate of bm, i , and its
associated feature Xi is retained in the model for modal
frequency-ambient condition pattern of the mth mode
in equation (2). On the other hand, if am, i ! ‘, then
the prior PDF p(bm, i jam, i ) is Gaussian distribution with
very high precision (i.e. very small variance), corresponding to the case that the posterior PDF is dominated by the prior PDF so bm, i ! 0 and its associated
feature Xi is irrelevant to the model in equation (2). In
the remaining of this section, aiming for searching the
optimal model class with the relevance features for each
vibrational mode of the structure, an efficient strategy
is proposed for optimization of the hyperparameter vector am along with the prediction-error variance s2m .
ð5Þ
where
^ m, s
a
^ m2 =
arg max
+
Nb
am 2R , s2m 2R
Lm am , s2m
ð8Þ
where Lm (am , s2m ) [ ln p(Dm jam , s2m ) is the logarithm
of the evidence of the mth vibrational mode, which can
be evaluated as follows57
Lm am , s2m
ð
= ln p Dm jam , s2m p(bm jam )dbm
=
1
N ln 2p+ln jCm j + YTm C1
m Ym
2
ð9Þ
with the matrix Cm 2 RN 3 N given by
^ m XT Ym
^bm = s2 S
m
^ m = s2 XT X + Am 1
S
m
ð6Þ
ð7Þ
The posterior mean vector ^bm and covariance matrix
^
Sm , as well as the relevancy of each feature in the design
matrix, continuously depends on the hyperparameter
vector am . This conclusion can be validated by considering two extreme cases for the value of the ith component of the hyperparameter vector am, i . If am, i ! 0,
then the prior PDF p(bm,i jam,i ) is Gaussian distribution
with very small precision (very large variance), corresponding to the case that the posterior PDF is dominated by the likelihood function so bm, i ! bmle
m, i where
Cm = s2m IN + XT A1
m X
ð10Þ
where IN is the N 3 N identity matrix. The optimization
in equation (8) is (Nb + 1)-dimensional, which is infeasible when the number of components of the feature vector Nb is large. Here, separation optimization strategy is
proposed for efficiency enhancement. Consider the following decomposition for the matrix Cm
Cm = s2m IN +
Nb
X
T
a1
m, k Xk Xk
k =1
T
= Cm, i + a1
m, i Xi Xi
ð11Þ
6
Advances in Mechanical Engineering
P
T
where Cm, i = s2m IN + k6¼i a1
m, k Xk Xk represents Cm
1
excluding the ith term in the sum am, i Xi XTi . Then, jCm j
58
and C1
m can be rewritten as follows
1
T
jCm j = jCm, i j1 + a1
m, i Xi Cm, i Xi
1
C1
m = Cm, i T 1
C1
m, i Xi Xi Cm, i
am, i + XTi C1
m, i Xi
ð12Þ
ð13Þ
Let am, i denote the hyperparameter vector am
excluding am, i . By plugging equations (12) and (13)
into equations (9) and (10), Lm (am , s2m ) can be decomposed into two parts
Lm am , s2m = Lm am, i , s2m + lm am, i , s2m
ð14Þ
The first term Lm (am, i , s2m ) is the log-evidence
removing am, i (and thus bm, i and Xi ) from the model of
the mth mode
i
1h
T 1
Lm am, i , s2m = N ln 2p + lnC1
m, i + Ym Cm, i Ym
2
ð15Þ
The second term l(am, i , s2m ), related to am, i and s2m ,
is given by
lm am, i , s2m
"
#
Q2m, i
1
ln am, i lnðam, i + Sm, i Þ +
=
am, i + Sm, i
2
ð16Þ
with
Sm, i = XTi C1
m, i Xi
ð17Þ
Qm, i = XTi C1
m, i Ym
ð18Þ
Figure 4. The procedures of the proposed algorithm.
As the ith hyperparameter am, i has been isolated
from the log-evidence Lm (am , s2m ), the optimal value of
am, i can be obtained by taking the partial derivative of
lm (am, i , s2m ) in equation (16) with respect to am, i
a1 S 2 (Q2m, i Sm, i )
∂lm ðam;i ; s2m Þ
= m, i m, i
∂am;i
2(am, i + Sm, i )2
ð19Þ
Thus, the optimal value of the ith hyperparameter
a
^ m, i is achieved when ∂lm (am, i , s2m )=∂am, i = 0 under the
constrain that am, i 0 (because am, i is the diagonal element of the precision matrix Am ). Thus, there are two
possible values for a
^ m, i
(
a
^ m, i =
‘,
Q2m, i Sm, i ,
2
Sm,
i=
if Q2m, i Sm, i
if Q2m, i .Sm, i
ð20Þ
Based on the updated value of am , the optimal
prediction-error variance is re-estimated by considering
the derivative of Lm (am , s2m ) with respect to s2m 53,54
s
^ m2 =
T
(Ym X^bm ) (Ym X^bm )
P b
^ m, i
N Nb + Ni =
^ m, i S
1a
ð21Þ
^ m is obtained by
where ^bm is obtained by equation (6); S
^
equation (7) and Sm, i is the ith diagonal element of the
^ m.
posterior covariance matrix S
Summary of the proposed approach
The procedures of the proposed algorithm are shown
in Figure 4 and are as follows. For pattern recognition
of the mth mode, the algorithm starts from a single feature and proceeds by adding, pertaining and/or deleting
features.
1.
Initialize s2m . The default value is 20% of the
sample variance of the mth mode output
Ym = ½v2m, 1 , . . . , v2m, N T .
Mu et al.
7
Table 1. Weight parameter identification results of the optimal models of the three modes.
Mode
bm, 1
bm, 2
bm, 3
bm, 4
bm, 5
bm, 6
bm, 7
bm, 8
m=1
m=2
m=3
1.920 (0.015)
2.582 (0.025)
3.332 (0.013)
–
0.226 (0.069)
–
0.040 (0.037)
–
–
–
–
–
–
–
20.033 (0.057)
–
–
–
–
–
–
20.931 (0.127)
20.187 (0.126)
–
Mode
bm, 9
bm, 10
bm, 11
bm, 12
bm, 13
bm, 14
bm, 15
s2m (3104 )
m=1
m=2
m=3
–
–
–
–
–
–
–
–
20.015 (0.012)
1.014 (0.144)
–
–
0.255 (0.073)
0.358 (0.087)
0.332 (0.061)
–
–
–
–
–
–
8.68
6.02
7.34
– indicates that the weight parameter is not included in the optimal model.
2.
3.
Select the initial feature Xi . The correlation coefficients between the output Ym and each feature
are calculated. The initial feature is selected as
the one which possesses the maximum absolute
correlation coefficient.
Initialize am . Since only one of the initial feature
Xi is included, each component of am, i is infinity and, hence, Cm, i = s2m IN . The hyperparameter associated with the initial feature Xi can
be estimated using equation (20)
2
Sm,
XTi Xi
i
=
am, i = 2
2
Qm, i Sm, i
(XTi Ym ) =(XTi Xi ) s2m
4.
5.
6.
7.
standard deviations (inside the parentheses), and the
sign ‘‘–’’ indicates that the weight parameter is not
included in the optimal model, representing that the
associated feature is irrelevant. It is observed that the
optimal models of the three modes possess different relevant features shown as follows
v21 = 1:92 + 0:04H0 0:931T02 H0
+ 1:014T03 H0 + 0:255T02 H02
ð23Þ
v22 = 2:582 + 0:266T0 0:187T02 H0 + 0:358T02 H02 ð24Þ
ð22Þ
2 T
^ T
^
Compute ^bm = s2
m Sm X Ym and Sm = (sm X
1
X + Am ) by equations (6) and (7) based on Am
and s2m .
and
Qm, j =
Update
Sm, j = XTj C1
m, j Xj
T 1
Xj Cm, j Ym by equations (17) and (18) for
j = 1, . . . , Nb . Select a candidate feature Xj ,
update its associated hyperparameter am, j as
follows
If Q2m, j Sm, j , then remove Xj and set am, j = ‘;
If Q2m, j .Sm, j and am, j \‘, then update am, j =
2
2
Sm,
j =(Qm, j Sm, j ) (implying that Xj has been
already included in the model in current iteration);
If Q2m, j .Sm, j and am, j = ‘, then include Xj and
2
2
update am, j = Sm,
j =(Qm, j Sm, j ).
2
Update s
^ m by equation (21) with the updated
^ m and ^
bm .
S
Repeat Steps 4–6 until convergence achieves so
the final learning results are obtained.
Modal frequency-ambient condition
pattern recognition
Table 1 shows the weight parameter identification
results of the optimal models of the three modes, the
optimal values (outside the parentheses), and the
v23 = 3:332 0:033T0 H0 0:931T02 H0
0:015T04 + 0:332T02 H02
ð25Þ
Both the temperature and relative humidity exist in
all the three optimal models, showing that these two
ambient factors influence the modal frequencies of different modes and they should be considered simultaneously in modal frequencies prediction. This conclusion
is consistent with that by Xia et al.5 and Yuen and
Kuok.11,12 On the other hand, the optimal models for
different modes possess different combinations of the
two ambient factors. The reasons are given as follows.
In practical monitoring of a target structure under
changing ambient and operational conditions, fluctuations of modal frequencies are induced by multiple environmental factors, not only temperature and relative
humidity but also other factors, such as wind load,59
rainfall, support reaction,60 and soil–structure interaction,61,62 and the degrees of participation of different
environmental factors to the modal parameters of different modes are possibly not identical. On the other
hand, it is worth noting that although only the temperature and relative humidity are utilized in the study, the
temperature and relative humidity are the primary
effects while other factors (wind load, rainfall, support
reaction, soil–structure interaction) are the secondary
effects in modal frequencies prediction. This conclusion
is supported by checking the normalized residual plot
of the three modes shown in Figure 5 that most of the
8
Figure 5. Normalized residual plot of the three modes
(e1 , e2 , and e3 are the normalized residuals of modes 1, 2, and 3,
respectively).
Advances in Mechanical Engineering
normalized residuals satisfy jem j 1:96, m = 1, 2, 3,
where the threshold ‘‘1.96’’ corresponds to that there is
95% in probability for a standard normal distributed
sample falling into the interval [21.96, 1.96]. Therefore,
it can be concluded that the optimal models given in
equations (23)–(25) are capable to properly capture the
wandering pattern of the modal frequencies under normal weather conditions.
Note that in Bayesian spectral density approach,46–
48
not only identified values but also associated uncertainties of modal frequencies are available. It is worth
noting that unequal uncertainty values for modal frequencies of different days may lead to heterogeneous
error pattern, requiring the development of the heterogeneous Bayesian inference and learning framework.63
Figures 6 and 7 show the squared modal frequencies
versus the temperature and relative humidity of the
three modes, respectively. The modal frequency–
temperature curves of Figure 6 are plotted with the
relative humidity evaluated at its sample mean (67.9%),
and the modal frequency–relative humidity curves of
Figure 7 are plotted with the temperature with its sample mean (24.7°C). The increasing trends of these curves
satisfy the correlation analysis results of Figure 3 for
different modes. Finally, the design modal frequency
contours of the optimal values and the associated
Figure 6. The squared modal frequencies versus the temperature of the three modes.
Mu et al.
9
Figure 7. The squared modal frequencies versus the relative humidity of the three modes.
uncertainties for the three modes are given in Figure 8.
The left three subplots are for the optimal values of
modal frequencies given the temperature and the relative
humidity for modes 1, 2, and 3, respectively, and the
right three subplots are for the corresponding associated
uncertainties for modes 1, 2, and 3, respectively. Note
that different modes associate with different uncertainty
contours because they possess different optimal models.
In addition, it can be observed that the uncertainty level
of the corner regions in the uncertainty contours is higher
than that of the central region. This is due to the nonuniform distribution of the input measurement that most
of the measured data points locate in the center region
while very few locate in the corner region, especially for
the top right and bottom right regions. On the other
hand, the non-uniform distribution of the input measurement also reflects the ambient condition pattern of the
city for the monitored structure that, high temperature is
with intermediate level of relative humidity rather than
high or low level of relative humidity.
Conclusion
In this article, a novel machine-learning algorithm is
proposed to automatically select relevance features in
modal frequency-ambient condition pattern recognition
based on structural dynamic response and ambient
condition measurement. In the traditional feature selection approaches, 3 3 (215-1) candidates in total are
required to obtain the optimal model classes for the
three modes based on the 15-component full extracted
feature vector, which requires large computational
effort. On the other hand, in the proposed approach,
the optimal model classes for modal frequency predictions of different vibrational modes of the structure are
determined by first introduction of the hyperparameter
in the ARD prior for the weight parameter vector,
which controls the relevancy of different features in the
prediction model, then conducting hyperparameter
optimization by maximum evidence estimation. During
the optimization process, the irrelevance components
of the weight parameter vector as well as the corresponding features are automatically pruned out while
the relevance terms are retained. The proposed algorithm is utilized for structural health assessment for a
reinforced concrete building based on 1-year daily measurements. It turns out that the optimal model classes
of different vibrational modes of the structure are capable to capture the modal frequency-ambient condition
10
Advances in Mechanical Engineering
Figure 8. Design modal frequency contours of the optimal values (left three subplots) and the associated uncertainties (right three
subplots) for the three modes.
patterns of different modes. In addition, the design
modal frequency contours of the optimal values and
the associated uncertainties are given, which provides
an efficient tool for modal frequency prediction in
future analysis.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
article.
Funding
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this
article: This work was supported by the National Natural
Science Foundation of China (51508201), the China
Postdoctoral Science Foundation, and the State Key
Laboratory of Subtropical Building Science, South China
University of Technology (2016ZB26).
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