Reconstruction to Sensor Measurements Based on

applied sciences Article

Reconstruction to Sensor Measurements Based on a Correlation Model of Monitoring Data Wei Lu 1, *, Jun Teng 1,2 , Chao Li 1 and Yan Cui 1 1

2

*

Department of Civil and Environmental Engineering, Harbin Institute of Technology Shenzhen Graduate School, Shenzhen 518055, China; [email protected] (J.T.); [email protected] (C.L.); [email protected] (Y.C.) College of Civil Engineering, Fujian University of Technology, Fuzhou 350118, China Correspondence: [email protected]; Tel: +86-755-2603-3056

Academic Editors: Gangbing Song, Chuji Wang and Bo Wang Received: 19 January 2017; Accepted: 28 February 2017; Published: 3 March 2017

Abstract: A sensor failure will lead to sensor measurement distortion, and may reduce the reliability of the whole structure analysis. This paper studies the method of monitoring information reconstruction based on the correlation degree. For the faulty sensor, the correlation degree of the normal response of this sensor and the measurements of the other sensors is calculated, which is also called the correlation degree of reconstructed variables and response variables. By comparing the correlation degrees, the response variables, which are needed to establish the correlation model, are determined. The correlation model between the reconstructed variables and the response variables is established by the partial least square method. The value of the correlation degrees between the reconstructed variables and the response variables, the amount of the monitoring data which is used to determine the coefficients of the correlation model, and the number of the response variables are used to discuss the influence factors of the reconstruction error. The stress measurements of structural health monitoring system of Shenzhen Bay Stadium is taken as an example, and the effectiveness of the method is verified and the practicability of the method is illustrated. Keywords: structural health monitoring; responses reconstruction; correlation model; stress measurements; Shenzhen Bay Stadium

1. Introduction The sensors are the basic instruments of the structural health monitoring system, while the measurements of sensors are the necessary information for estimating the structural safety. However, the operation duration of a sensor is limited and the sensor measurements may be abnormal because of sensor failure or environmental influences [1]. The technique for reconstructing the sensor information should be developed as a result, which can ensure saving the complete monitoring information for structural safety estimation. The monitoring information is diverse in the different measured locations and the different types of structural responses, which are correlated with each other to some extent [2]. That is to say, it is necessary to reconstruct the information for the failure sensor using the effective correlation model between the failure sensor and the other normal sensors. For the sensor information reconstruction, the researchers proposed several methods to solve the question. For the problem of data distortion in the structural health monitoring system of a bridge, Huang [3] analyzed the relationship of the monitoring data by using Granger causality testing [4] in which the sensor data with a closer relationship is selected to be the input vector for an extreme learning machine to recover the missing sensor signal data. However, the individual prediction value could be a large error, and the training results need to be further improved because the tendency of the predicted data is not consistent with the theory value. The radial basis function (RBF) neural Appl. Sci. 2017, 7, 243; doi:10.3390/app7030243

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network model is another method to recover the unreliable data, which is also easy to fall into a local optimum [5]. The Kalman filtering method was proposed to build the model between structural displacements and structural accelerations [6], in which the structural model parameters and external excitation should be known in advance. The method of data compression sampling is used for data recovery of wireless sensors, while the recovery data error would be larger for the accelerations when the sparse of the original data becomes worse [7]. Feng [8] proposed a data recovery method based on a least square support vector machine. A forecasting model based on the method is established, and signal recovery is accomplished by on-line learning diagnosis. However, this method mainly aims at the data reconstruction of the sensor within short duration, and the effect is not good for a high sampling rate and for long-term failure prediction. This paper, thus, focuses on the structural responses and the correlation model between the reconstructed variable and the response variable is established, which can be used for reconstructing the sensor information. In this paper, the correlation degree and the partial least square method are applied to the data reconstruction of the fault sensors. As the strain sensors are placed on the roof steel structure of Shenzhen Bay Stadium, the strain measurements represent the distributed responses of the structure and have an inherent association, which is suitable to verify the effectiveness of the proposed method in the paper as well. 2. Correlation Model of Monitoring Data 2.1. Correlation Analysis of Monitoring Data The monitoring structural responses from different locations and different types of sensors are in the correlated relationship. The Pearson correlation coefficient is an index to measure the linear correlation degree of structural responses. The structural response matrix of m measuring points under the load in n time steps is denoted as X:   x11 x12 · · · x1m  h i   x21 x22 · · · x2m  X = x1 x2 ... xm =  (1) .. .. ..    .. . . .   . xn1 xi =

h

xi1

xi2

xn2 · · · iT

xnm

· · · xin

(2)

In the Equations (1) and (2), xi represents the measurements of the measuring point i under the load of n time steps. When the fault sensor works in normal, the measurement of the measuring point where the fault sensor located is denoted as y. The Pearson product-moment correlation coefficient [9] of the measurement from the arbitrary measuring point and the measuring point where the faulty sensor located is denoted as ρi ρi =

E(xi − µxi )(y − µy ) cov(xi , y) = σxi σy σxi σy σxi =


1 n x i ( k ) − µx i ∑ n − 1 k =1

σy =

 1 n  y ( k ) − µy ∑ n − 1 k =1

cov(xi , y) =

(3)

2

(4)

2


 1 n xi (k) − µxi y(k) − µy ∑ n − 1 k =1

(5)

(6)

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In Equation (3), σxi and σy represent the response variance of the measurement point i and the fault sensor, respectively; cov(xi , y) is covariance, xi (k) and y(k) represent the measurement of point i and the fault sensor in the kth time step, respectively; µxi and µy are the mean value of the measurements xi and y, respectively. 2.2. Correlation Modeling Using Partial Least Squares and Multiple Regression 2.2.1. Reconstructed Variable and Response Variables The correlation coefficients of the normal measurement of fault sensor and the measurements of the other sensors are calculated, which are also called the correlation coefficients of the reconstructed variable and response variables. The correlation matrix can be denoted as P: P=

h ρ1

· · · ρm

ρ2

i

(7)

The correlation coefficients are sorted from larger value to small value based on their absolute values, while the updated correlation coefficient’s sorting matrix can be denoted as DP :

· · · ( ρ )i

DP = {(ρ)1

· · · (ρ)m

o

(8)

(ρ)i represents the ith correlation coefficient absolute value, while (ρ)1 is the largest correlation coefficient absolute value and (ρ)m is the smallest correlation coefficient absolute value, respectively. The corresponding measurement to (ρ)i is denoted as (x)i , while the updated structural response sorting matrix can be denoted as DX :

· · · (x) i

DX = {(x)1

· · · (x) m

o

(9)

The faulty sensor information is considered as the reconstructed variable YP , and p measurements are selected as response variables XP : XP = {(x)1

· · · (x) p

(x)2

o

Y P = {y}

(10) (11)

2.2.2. Establishment of Correlation Model Using the Partial Least Square Method The reconstructed variables and response variables are standardized as:

(x)i∗ =

( x ) i − µ(x) i σ(x) i

y∗ =

(12)

y − µy

(13)

σy

The response variables XP and the reconstructed variable YP are standardized into E0 and F0 , respectively: o  E0 = (x)1∗ (x)2∗ · · · (x)∗p (14) F0 = { y ∗ }

(15)

The main components set of E0 and F0 can be obtained by using principal component analysis, which are denoted as T and U, respectively: T=

n

t1

t2

...

th

...

tA

o

(16)

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U=

n

u1

u2

...

uh

...

uA

o

(17)

The regression equations regarding E0 and F0 using partial least squares [10] and first principal components are: E0 = t1 p1 T + E1 (18) F0 = t1 r1 T + F1

(19)

where the regression coefficient vector is: E0T t1

p1 =

k t1 k2 F0T t1

r1 =

k t1 k2

(20)

(21)

The regression equations for the hth principal component are: E h −1 = t h p h T + E h

(22)

F h −1 = t h r h T + F h

(23)

where the regression coefficient vector is: EhT−1 th

ph =

rh =

k t h k2 FhT−1 th

k t h k2

(24)

(25)

The relationship between the hth principal component th and the hth axis of E0 is: t h = E h −1 w h

(26)

where wh is the unit eigenvector corresponding to the maximum eigenvalue θh 2 of matrix E h −1 T F h −1 F h −1 T E h −1 . According to the Equations (18), (19), (22), and (23), E0 and F0 can be expressed as: E0 = t1 p1 T + t2 p2 T + · · · + th ph T + Eh

(27)

F0 = t1 r1 T + t2 r2 T + · · · + th rh T + Fh

(28)

Substituting Equation (26) into Equation (22) yields: E h = E h −1 (I − w h p h T )

(29)

Substituting Equations (26) and (29) into Equation (27) yields: h

Eh = E0 ∏ (I − w j p j T )

(30)

j =1

Substituting Equation (30) into Equation (26) yields: h −1

th = Eh−1 wh = E0 ∏ (I − w j p j T )wh = E0 wh ∗ j =1

(31)

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h −1

wh ∗ =

∏ (I − w j p j T )wh

(32)

j =1

Substituting Equation (31) into Equation (28) yields: F0 = E0 w1 ∗ r1 T + · · · + E0 wh ∗ rh T + Fh

(33)

Assuming the number of the principal components is h when the principal components satisfy the cross-validation principle. Meanwhile, Fh approaches 0, which can be ignored. Equation (33) is written as: F0 = E0 (w1 ∗ r1 T + · · · + wh ∗ rh T ) (34) Substituting Equations (14) and (15) into Equation (34) yields: 

h

∗  ∑ r j w j1 j = 1    ...  h ∗ ∗ ∗  r j w ji ∗ y∗ = [(x)1 . . . (x)i . . . (x) p ]  j∑  =1  ...   h  ∑ r j w jp ∗

             

(35)

j =1

If: αi ∗ =

h

∑ rj wji ∗

(36)

j =1

then: y∗ = α1 ∗ (x)1 ∗ + · · · + α p ∗ (x) p ∗

(37)

Substituting Equations (12) and (13) into Equation (37), the correlation model between the reconstructed variable and the response variables is: y(i ) = α1 (x(i ))1 + · · · + α p (x(i )) p + α0

(38)

2.2.3. The Reconstruction Error Evaluation Index The average absolute error between the reconstructed value and actual value ea is: ea =

1 n |y(k) − y0 (k)| n i∑ =k

(39)

The average relative error between the reconstructed value and actual value eb is: 1 n y(k) − y0 (k) eb = ∑ × 100% n k =1 y0 (k)

(40)

where y(k) represents the reconstructed response of kth time step of the supposed fault sensor, and y0 (k ) represents the actual response of kth time step of the supposed fault sensor, respectively.

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3. Appl. Sci. 2017, 7, 243 to 3. Reconstruction Reconstruction to Stress Stress Measurements Measurements of of Shenzhen Shenzhen Bay Bay Stadium Stadium

6 of 12

3. Reconstruction to Stress Measurements of Shenzhen Bay Stadium

3.1. 3.1. Project Project Description Description 3. Reconstruction to Stress Measurements of Shenzhen Bay Stadium 3.1. Project Description The The structural structural health health monitoring monitoring system system of of Shenzhen Shenzhen Bay Bay Stadium Stadium began began to to be be implemented implemented in in The structural health monitoring system of Shenzhen Bay Stadium began to be implemented in 3.1. 2010, while the 2010,Project whileDescription the continuous continuous monitoring monitoring of of the the steel steel roof roof in in both both the the construction construction phase phase and and service service 2010, while the continuous monitoring of the steel roof in both the construction phase and service phase began to out time [11]. The measurements of the sensors phaseThe began to be be carried carried out at at the the same same time The stress stress measurements sensors located located structural health monitoring system of[11]. Shenzhen Bay Stadium began of to the be implemented in phase began to be carried out at the same time [11]. The stress measurements of the sensors located at the support members are chosen as the basic analysis data for example. The layout of sensors at the support members are chosen as the basic analysis data for example. The layout of sensors 2010, while the continuous monitoring of the steel roof in both the construction phase and service at the support members are chosen as the in basic analysis data for example. The layouttwo of rods sensors located at members are shown Figures 1 and 2. locatedbegan at the thetosupport support members aresame shown in[11]. Figures and measurements 2. There There are are six sixofsupports, supports, two rods of of phase be carried out at the time The 1stress the sensors located at located at the support members are shown in Figures 1 and 2. There are six supports, two rods of which are being monitored and four stress monitoring points are arranged on each rod; the sensor which are being monitored and four stress monitoring points are arranged on each rod; the sensor the support members are chosen as the basic analysis data for example. The layout of sensors located which are being monitored and four stress 3. monitoring points arranged on each rod;monitoring the sensor installation of support shown in example, theare number of stress installation of members support is isare shown ininFigure Figure 3. 1For For number of eight eight at the support shown Figures andexample, 2. There the are six supports, two stress rods ofmonitoring which are installation of support is shown in Figure 3. For example, the number of eight stress monitoring points at support 3-1 can be denoted as 3-1-1, 3-1-2, 3-1-3, 3-1-4, 3-1-5, 3-1-6, 3-1-7, and points at support 3-1 can be denoted as 3-1-1, 3-1-2, 3-1-3, 3-1-4, 3-1-5, 3-1-6, 3-1-7, and being and 3-1 fourcan stressbemonitoring are arranged on each rod;3-1-5, the sensor installation of pointsmonitored at support denoted points as 3-1-1, 3-1-2, 3-1-3, 3-1-4, 3-1-6, 3-1-7, and 3-1-8, respectively. 3-1-8, respectively. support is shown in Figure 3. For example, the number of eight stress monitoring points at support 3-1 3-1-8, respectively. can be denoted as 3-1-1, 3-1-2, 3-1-3, 3-1-4, 3-1-5, 3-1-6, 3-1-7, and 3-1-8, respectively. 3-6 3-6 3-6

3-5 3-5 3-5

3-1 3-1 3-1

3-2 3-2 3-2

3-4 3-4 3-4

3-3 3-3 3-3

Figure Figure 1. 1. Stress Stress monitoring monitoring point point of of the the support support connecting connecting rod. rod. Figure Figure 1. 1. Stress Stress monitoring monitoring point point of of the the support support connecting connecting rod. rod.

Figure 2. 2. Distribution Distribution of of the the stress stress sensor. sensor. Figure Figure 2. Distribution of the stress sensor. Figure 2. Distribution of the stress sensor.

Figure Figure 3. 3. The The sensor sensor installation installation of of two two support support connecting connecting rods. rods. Figure 3. The sensor installation of two support connecting rods.

3.2. The of the Reconstructed Variable Variable and and the the Response Variables The Correlation Correlation Model the Response Response Variables Variables 3.2. Model of the Reconstructed 3.2. The Correlation Model of the Reconstructed Variable and the Response Variables The sensor at at location location 3-1-4 is assumed to be failure on 16 May 2014. The stress monitoring location 3-1-4 3-1-4 is is assumed assumed to to be be failure failureon on16 16May May2014. 2014. The The stress stress monitoring monitoring The sensor The sensor at location 3-1-4 is assumed to be failure on 16 May 2014. The stress monitoring information information of of this this point point is is considered considered as as the the reconstructed reconstructed variable. variable. The The measurements measurements of of the the other other information of this point is considered as the reconstructed variable. The measurements of the other locations are considered as the available response variables. The stress monitoring data from 7 May 2014 locations are variables. TheThe stress monitoring datadata fromfrom 7 May 2014 locations areconsidered consideredasasthe theavailable availableresponse response variables. stress monitoring 7 May locations are considered as the available response variables. The stress monitoring data from 7 May 2014 to 2014 are selected to the correlation to 10 10toMay May 20142014 are are selected to calculate calculate thethe correlation coefficients between measurements from 2014 10 May selected to calculate correlationcoefficients coefficientsbetween betweenmeasurements measurements from to 10 May 2014 are selected to calculate the correlation coefficients between measurements from location are shown in Table 1. location 3-1-4 3-1-4 and and the the other other locations, locations, which which are are shown shown in in Table Table1. 1. location 3-1-4 and the other locations, which are shown in Table 1.

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Appl. Sci. 2017, 7, 243correlation degree of stress monitoring data between stress sensor 3-1-4 and other 7 of 12 Table 1. The

stress sensors. Table 1. The correlation degree of stress monitoring data between stress sensor 3-1-4 and other stress Sensor sensors. Ranking Location Correlation Coefficient Ranking Sensor Location Correlation Coefficient 1 Ranking Sensor 3-1-8 Location Correlation 0.6831Coefficient Ranking 18 3-1-2 0.2076 Sensor Location Correlation Coefficient 2 3-6-6 0.5437 19 3-6-1 0.1975 1 3-1-8 0.6831 18 3-1-2 0.2076 3 3-3-2 0.5194 20 3-5-1 0.1962 2 3-6-6 0.5437 19 3-6-1 0.1975 4 3-2-8 0.4656 21 3-1-3 0.1623 3 0.5194 3-5-1 0.1962 5 3-1-73-3-2 0.4212 22 20 3-2-7 0.1113 4 0.4656 3-1-3 0.1623 6 3-2-43-2-8 0.4121 23 21 3-6-8 0.0915 5 0.4212 3-2-7 0.1113 7 3-1-63-1-7 0.4006 24 22 3-3-7 0.0742 6 0.4121 3-6-8 0.0915 8 3-2-63-2-4 0.3735 25 23 3-2-2 0.0632 7 0.4006 3-3-7 0.0742 9 3-5-73-1-6 0.3688 26 24 3-3-1 0.0624 10 3-2-33-2-6 0.3677 27 25 3-3-3 0.0547 8 0.3735 3-2-2 0.0632 11 3-5-43-5-7 0.3608 28 26 3-2-5 0.0424 9 0.3688 3-3-1 0.0624 12 3-6-33-2-3 0.3512 29 27 3-5-2 0.0214 10 0.3677 3-3-3 0.0547 13 3-5-53-5-4 0.3425 30 28 3-5-8 0.0156 11 0.3608 3-2-5 0.0424 14 3-3-6 0.3289 31 3-5-3 0.0141 12 3-6-3 0.3512 29 3-5-2 0.0214 15 3-5-6 0.2745 32 3-6-4 0.0108 13 3-5-5 0.3425 30 3-5-8 0.0156 16 3-1-5 0.2608 33 3-2-1 0.0031 14 0.3289 3-5-3 0.0141 17 3-1-13-3-6 0.2487 - 31 15 3-5-6 0.2745 32 3-6-4 0.0108 16 3-1-5 0.2608 33 3-2-1 0.0031 17 3-1-1 0.2487 In Equation (38), there is no exact number demanded of the response variables, which -may depend

on the detailed examples and the special value of the correlation coefficients. Generally speaking, In Equation (38), therecan is be no used exacttonumber of the when response which may the historical measurements identifydemanded the effectiveness the variables, different number of depend on the detailed examples and the special value of the correlation coefficients. Generally responses variables selected. If four response variables are chosen to build the correlation model, speaking, that is to say,the p =historical 4, then: measurements can be used to identify the effectiveness when the different number of responses variables selected. If four response variables are chosen to build the o correlation model, that is to say, p = 4, then: (41) DP = {0.6831 −0.5437 −0.5194 0.4656 XP1

= =

n

(x)1

DP  {0.6831 0.5437 o 0.5194 0.4656}

(x)2

(x)3

(41)

(x)4

n X P1  {(x)1 (x)2 (x)3 (x)4 } o σ3−1−{σ (t()t) σσ3−3−(2t)(t)σ σ3−(t2)}−8 (t) 6 8 (3t)18 (tσ)3−σ6− 36 6 3 3 2 3 2 8 YP1 =Y{P1y=1 }= ((tt)}) y1 {=σ{σ 3−31− 1 44

(42) (42) (43) (43)

The correlation model ofof the reconstructed variable and thethe response variable is: is: The correlation model the reconstructed variable and response variable  1.01( x(i))1 +(x0.05( 0.09((xx((i)) + 0.10(x(i))4 x(20.73 y1 (i ) = 1.01(yx1 ((i)i )) (i ))x2(i))−2 0.09 i )) i ))4 − 20.73 3 3 + 0.10( 1 + 0.05

(44) (44)

Stress(MPa)

-28 Actual value Reconstructed value

-30 -32 -34 -36 -38 0:00

4:00

8:00

12:00 Time

16:00

20:00

24:00

Reconstruction error(MPa)

The stress monitoring the sensor stress 3-6-6, sensor3-3-2, 3-6-6, 3-3-2, andused 3-2-8 are used to The stress monitoring datadata of theofstress 3-1-8, and3-1-8, 3-2-8 are to reconstruct reconstruct the stress measurements of the sensor located at point 3-1-4 on 16 May 2014. the stress measurements of the sensor located at point 3-1-4 on 16 May 2014. The contrast curves The contrast curves values between reconstructed and in actual values arereconstructed shown in Figure between reconstructed and actual valuesvalues are shown Figure 4. The values 4. The reconstructed values have the same trend as the actual values, and there are small differences have the same trend as the actual values, and there are small differences between them. The average between them. The MPa average error is 0.89 MPa and the relative error is 2.71%. absolute error is 0.89 andabsolute the relative error is 2.71%.

(a)

5 4 3 2 1 0 0:00

4:00

8:00

12:00 Time

16:00

20:00

24:00

(b)

Figure Thecomparison comparisonbetween betweenthe thereconstructed reconstructedvalue valueand and the actual value:(a)(a)the the curve Figure 4. 4.The the actual value: curve ofof reconstructed value and the actual value; and (b) the curve of absolute error. reconstructed value and the actual value; and (b) the curve of absolute error.

3.3. Influence of Correlation Degree on Reconstruction Error In order to discuss the influences of the selection of response variables, two additional cases are discussed, which are:

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3.3. Influence of Correlation Degree on Reconstruction Error

-28

Actual value Reconstructed value Actual value

-30 -32

Reconstructed value

Stress(MPa) Stress(MPa)

-28 -30

Reconstruction error(MPa) Reconstruction error(MPa)

In order to discuss the influences of the selection of response variables, two additional cases are discussed, which Appl. Sci. 2017, 7, 243are: 8 of 12 n o Appl. Sci. 2017, 7, 243 8 of 12 XP2 = (x)13 (x)14 (x)15 (x)16 nX  {(x) o (45) (x)14 (x)15 (x)16 } 2 13 = X σP ( t ) σ ( t ) σ ( t ) σ ( t ) (45) 3−{(5x −)5 3−3(− 6) 3− 5−6 3−1−5 ( x ) x ( x ) }  {σ133 5 5 (t)14 σ 3 3156 (t) σ163 56 (t ) σ 31 5 (t)} P2 (45)  {σ 3 5 5 (t) σ 3 36 (t) σ 3 56 (t ) σ 31 5 (t)} and: n o and: X = ( x ) ( x ) ( x ) ( x ) P3 30 31 32 33 and: nX  {(x) o (46) (x) ( x) ( x) } = σP 33−5−8 (30t) σ331−5−3 (t32) σ3−336−4 (t) σ3−2−1 (t) (46) X P 3  {({σ x)30 ((xt))31 σ(x)32 (t)(x)σ33 } (t) σ (t)} 3 5 8 3 5 3 36  4 3  2 1 (46) (t) σ 3 5 3 (t) σ 36  4 (t) σ 3 2 1 (t)} The correlation model usingX{σ P23 is: 5 8 The correlation model using X P 2 is: The correlation model is:(x(i )) + 0.39(x(i )) + 0.02(x(i )) − 23.85 y2 (i ) = 0.02(using x(i ))13 X+P 20.01 (47) 16 y2 (i)  0.02( x(i))13 + 0.01(x14 (i))14 + 0.39(x(i))1515 + 0.02(x(i))16  23.85 (47) y2 (i)  0.02(x(i))13 + 0.01(x(i))14 + 0.39(x(i))15 + 0.02(x(i))16  23.85 (47) The Thecontrast contrastcurves curvesbetween betweenreconstructed reconstructedvalues valuesand andactual actualvalues valuesare areshown shownininFigure Figure5.5. The average absolute error is is 0.97 MPa and the relative error 2.97%. The contrast curves between reconstructed values andis actual values are shown in Figure 5. The average absolute error 0.97 MPa and the relative error is 2.97%. The average absolute error is 0.97 MPa and the relative error is 2.97%. 5

54 43

-32 -34

32

-34 -36

21

-36 -38 0:00 4:00 8:00 12:00 16:00 20:00 24:00 -38 Time 0:00 4:00 8:00 12:00 16:00 20:00 24:00 Time (a)

10 0:00 4:00 8:00 12:00 16:00 20:00 24:00 Time 0 0:00 4:00 8:00 12:00 16:00 20:00 24:00 Time (b)

(a) (b) Figure The reconstructed value and actual value using response variable X:P2(a) : (a) the curve Figure 5. 5. The reconstructed value and thethe actual value using response variable XP2 the curve ofof Figure 5. The reconstructed value and the actual value using response variable X P2 : (a) the curve of the reconstructed value and actual value; and curve reconstruction error. the reconstructed value and thethe actual value; and (b)(b) thethe curve of of reconstruction error. the reconstructed value and the actual value; and (b) the curve of reconstruction error. The correlation model using X P 3 is: The correlation model using XP3 is: The correlation model using X P 3 is:

-28 Actual value Reconstructed value Actual value

-30-32

Reconstructed value

Stress(MPa) Stress(MPa)

-28-30

-32-34

Reconstruction error(MPa) Reconstruction error(MPa)

y3 (i)  0.01(x(i))30 + 0.01(x(i))31  0.01(x(i))32  0.01(x(i))33  32.03 (48) y3 (i ) = −0.01y ((xi)(i )) (x(ix)) 0.01 (x(i ))33 − 32.03 (48) 30 + 32 − 0.01( x(i0.01 ))30 + 0.01( (i))3131 −  0.01( x((i))x32(i))0.01( x(i))0.01  32.03 (48) 3 33 The contrast curves between reconstructed values and actual values are shown in Figure 6. contrast curves between values are shown shown in Figure 6. The contrast values error and actual values TheThe average absolute error is 1.03 reconstructed MPa and the relative is 3.13%. The The average average absolute absolute error error is is 1.03 1.03 MPa MPa and andthe therelative relativeerror errorisis3.13%. 3.13%. 5

54 43 32

-34-36

21

-36-38 0:00 4:00 8:00 12:00 16:00 20:00 24:00 -38 Time 0:00 4:00 8:00 12:00 16:00 20:00 24:00 Time (a)

10 0:00 4:00 8:00 12:00 16:00 20:00 24:00 0 Time 0:00 4:00 8:00 12:00 16:00 20:00 24:00 (b) Time

(a) (b) Figure 6. The reconstructed value and the actual value using response variable XP3: (a) the curve of Figure 6. The The reconstructed reconstructed value and the the actual value using response variableXXP3 P3 (a) the the curve curve of of the reconstructed value and the actual value; and (b) the curve of reconstruction error. Figure 6. value and actual value using response variable :: (a) the reconstructed reconstructed value value and and the the actual actual value; value; and and(b) (b)the thecurve curveof ofreconstruction reconstructionerror. error. the

The errors of the reconstructed information using different correlation coefficients of the The errors of the summarized reconstructedininformation different correlation coefficients of the response variables Table 2.usingusing The errors of theare reconstructed information different correlation coefficients of the response response variables are summarized in Table 2. variables are summarized in Table 2. Table 2. The summarized errors using different response variables. Table 2. The summarized errors using different response variables. Response Variables

Response X Variables P1

ea (MPa)

eb

ea (MPa) 0.89

eb 2.71%

P2

0.89 0.97

2.71% 2.97%

X PX2 P 3

0.97 1.03

2.97% 3.13%

X PX1

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Table 2. The summarized errors using different response variables. Response Variables

ea (MPa)

eb

XP1 XP2 XP3

0.89 0.97 1.03

2.71% 2.97% 3.13%

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It can be seen from Table 2 that the errors become larger when the coefficients of response variablesItbecome smaller, which the the average error and error areofminimum can be seen frominTable 2 that errorsabsolute become larger whenrelative the coefficients response when XP1 is chosen the response It absolute indicated thatand therelative response variables with closer variables becomeas smaller, in whichvariables. the average error error are minimum when correlation the reconstructed variable should It be indicated chosen, which for thevariables reconstruction of the XP1 is to chosen as the response variables. that is thebetter response with closer correlation to the reconstructed variable should be chosen, which is better for the reconstruction of sensor measurements. the sensor measurements. 3.4. Influence of Monitoring Data Quantity on Reconstruction Error 3.4. Influence of Monitoring Data Quantity on Reconstruction Error In order to discuss the influences of the selection of response variables, two additional cases In order to discuss influences of the selection of response variables, two additional are discussed. The originalthe stress monitoring data is collected from 7 May 2014 to 10 Maycases 2014are to discussed. The original stress monitoring data is collected 7 May 2014 Maylocations. 2014 to calculate the correlation coefficients between measurements fromfrom location 3-1-4 andto the10other calculate the correlation coefficients between measurements fromtolocation 3-1-4and andfrom the 7other The stress monitoring data of two additional cases are from 7 May 2014 12 May 2014 May locations. The stress monitoring data of two additional cases are from 7 May 2014 to 12 May 2014 2014 to 14 May 2014, respectively. and 7 May 2014 to using 14 Maystress 2014,monitoring respectively.data from 7 May 2014 to 12 May 2014 is: The from correlation model The correlation model using stress monitoring data from 7 May 2014 to 12 May 2014 is: y4 (i ) = 0.89(xy(i(i)) + 0.09 (i ))x2(i− i ))+30.04( + 0.04 (i ))4 − 33.62 ) 10.89( x(i)) (+x0.09( )) 0.20  0.20((x x((i)) x(i)) (x33.62 4

1

2

3

(49) (49)

4

Stress(MPa)

-28 Actual value Reconstructed value

-30 -32 -34 -36 -38 0:00

4:00

8:00

12:00 Time

16:00

20:00

24:00

Reconstruction error(MPa)

contrast curves between reconstructedvalues valuesand andactual actual values values are are shown shown in The The contrast curves between reconstructed in Figure Figure7.7. The average absolute error is 0.65 MPa and the relative error is 1.99%. The average absolute error is 0.65 MPa and the relative error is 1.99%. 5 4 3 2 1 0 0:00

(a)

4:00

8:00

12:00 Time

16:00

20:00

24:00

(b)

Figure 7. The reconstructed value actual value usingdata datafrom from77May May 2014 2014 to to 12 12 May May 2014: Figure 7. The reconstructed value andand thethe actual value using 2014: (a) the curve of reconstructed value and the actual value; and (b) the curve of reconstruction error. (a) the curve of reconstructed value and the actual value; and (b) the curve of reconstruction error.

The correlation model using stress monitoring data from 7 May 2014 to 14 May 2014 is: The correlation model using stress monitoring data from 7 May 2014 to 14 May 2014 is:

Reconstruction error(MPa)

Stress(MPa)

y5 (i)  0.35(x(i))1  0.16(x(i))2  0.11(x(i))3 + 0.03(x(i))4  39.45 (50) y5 (i ) = 0.35(x(i ))1 − 0.16(x(i ))2 − 0.11(x(i ))3 + 0.03(x(i ))4 − 39.45 (50) The contrast curves between the reconstructed values and the actual values are shown in Figure 8. The average absolute error is 0.22 MPa and the relative error is 0.68%. The contrast curves between the reconstructed values and the actual values are shown in Figure 8. The average absolute error is 0.22 MPa and the relative error is 0.68%. 5 -28 Actual value The errors of the reconstructed information using stress monitoring data from different time 4 -30 Reconstructed value periods are summarized in Table 3. 3 -32 It can-34be seen from Table 3 that the errors become smaller when the quantity of stress monitoring 2 data become larger, in which the average absolute error and relative error are a minimum when the 1 -36 stress monitoring data from 7 May 2014 to 12 May 2014 0is chosen to build the correlation model. -38 0:00 4:00 8:00 12:00 16:00 20:00 24:00 0:00 4:00 8:00 12:00 16:00 20:00 24:00 Time This indicated that the larger be used, which is better Time quantity of stress monitoring data should for reconstruction of sensor measurements. In theory, the number of measurements larger than the (a) (b) unknown coefficients is the basic requirement for regression function identification; in fact, there are Figure 8. The reconstructed value and actual value using data from 7 May 2014 to 14 May 2014: (a) the curve of the reconstructed value and the actual value; and (b) the curve of the reconstruction error.

The errors of the reconstructed information using stress monitoring data from different time periods are summarized in Table 3. It can be seen from Table 3 that the errors become smaller when the quantity of stress

Actual value Reconstructed value

-30

Stress(MPa)

Reconstruction error(MPa

-28

-32 -34 -36

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4:00

8:00

12:00 Time

16:00

20:00

24:00

5 4 3 2 1 0 0:00

10 of 12 4:00

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24:00

(a) included in the measurements, the measurements (b) 2017, 7, 243 of 12 noisesAppl. andSci.temperature effects in at least 10 one full day are better. Figure 7. The reconstructed value and the actual value using data from 7 May 2014 to 12 May 2014:

correlation model. This indicated that the larger quantity of stress monitoring data should be used, (a) the curve of reconstructed value and the actual value; and (b) the curve of reconstruction error. which is better forsummarized reconstruction of using sensorstress measurements. theory, the number measurements Table 3. The errors monitoring In data from different timeofperiods. largerThe than the unknown coefficients is the basic requirement for regression function identification; correlation model using stress monitoring data from 7 May 2014 to 14 May 2014 is: in fact, there are noises and temperature effects included in the measurements, the measurements in Data Acquisition Time ea (MPa) eb y5 (i)  0.35(x(i))1  0.16(x(i))2  0.11(x(i))3 + 0.03(x(i))4  39.45 (50) at least one full day are better. 2014.05.07~2014.05.10 0.89 2.71% 1.99% The contrast curves 2014.05.07~2014.05.12 between the reconstructed values0.65 and the actual values are shown in Figure 8. Table 3. The summarized errors using stress monitoring data from different time periods. 2014.05.07~2014.05.14 0.22 is 0.68%. 0.68% The average absolute error is 0.22 MPa and the relative error

Stress(MPa)

-30

ea (MPa) 0.89 5 0.65 4 0.22 Reconstruction error(MPa)

Data Acquisition Time 2014.05.07~2014.05.10 Actual 2014.05.07~2014.05.12 value Reconstructed value 2014.05.07~2014.05.14

-28

-32

eb 2.71% 1.99% 0.68%

3

-34 3.5. Influence of Response Variable Number on Reconstruction2 Error -36

1

In order to discuss the influences of the different number of response variables, three 0 -38 0:00 4:00 8:00 12:00 16:00 20:00 24:00 0:00 cases 4:00 are 8:00 12:00 16:00 20:00 24:00 additional discussed, which are: Time Time (a)

X P 5  {(x)1 (x)2 (x)3 (x)4 (x)5 }

(b)

(51)

The reconstructed and actual value using from data from 7 2014 May to 2014 14 2014: May 2014: FigureFigure 8. The8.reconstructed valueXvalue and actual 14 to May (a) the {(x)1 value (x)2 (using x)3 (data x)4 (x)5 7(May x)6 } (52) P6 (a) the curve of the reconstructed value and the actual value; and (b) the curve of the reconstruction error. curve of the reconstructed value and the actual value; and (b) the curve of the reconstruction error. X P7  {(x)1 (x)2 (x)3 (x)4 (x)5 (x)6 ( x)7 } (53) The errors of the reconstructed information using stress monitoring data from different time 3.5. Influence of Response Variable Number on Reconstruction Error periods summarized in using Table 3.X P 5 is: Theare correlation model It can be seenthe from Table 3 ofthat errors number become of smaller when the quantity stress In order to discuss influences the the different response variables, three of additional y ( i )  1.33( x ( i )) + 0.10( x ( i ))  0.03( x ( i ))  0.08( x ( i ))  0.20( x ( i ))  32.82 (54) a monitoring data become larger, in which the average absolute error and relative error are 6 1 2 3 4 5 cases are discussed, which are: minimum when the stress monitoring data from 7 May 2014 to 12 May 2014 is chosen to build the o The contrast curves between reconstructed values and actual values are shown in Figure 9. (51) = {( x)1 and(xthe )2 relative (x)3 error (x)4is 2.45%. (x)5 P50.80 The average absolute errorXis MPa The correlation model using X P 6 is: o

(52) XP6 = {(x)1 (x)2 (x)3 (x)4 (x)5 (x)6 y7 (i)  1.33(x(i))1 + 0.11( x(i))2  0.04( x(i))3  0.11( x(i))4  0.16(x(i))5 + 0.08(x(i))6  30.95 (55) o (53) X = x ) ( x ) ( x ) ( x ) {( ( x ) ( x ) ( x ) P7 1 2 3 4 5 actual6 values 7 are shown in Figure 10. The contrast curves between reconstructed values and

The average absolute error is 0.80 MPa and the relative error is 2.44%. The correlation model using X P 7 is

The correlation model using XP5 is:

y6 (i ) =y1.33 (x(i )) 0.10x)(x(i0.07( ))2 − (x(xi )) 0.08 i ))4 x− (xx()i ))526.93 − 32.82  1.30( x)11++0.09( x)0.03  0.08( )4 3− 0.19( x)(5 +x(0.12( )6 +0.20 0.13( 8 2 3 7

(54)

(56)

Stress(MPa)

-28

Actual value Reconstructed value

-30 -32 -34 -36 -38 0:00

4:00

8:00

12:00 Time

16:00

20:00

(a)

24:00

Reconstruction error(MPa)

The contrast curves between actual showninin Figure The contrast curves between reconstructed the reconstructedvalues values and and the actualvalues values are shown Figure 11. 9. The average absolute is 0.77 MPa and relativeerror errorisis2.45%. 2.36%. The average absolute errorerror is 0.80 MPa and thethe relative 5 4 3 2 1 0 0:00

4:00

8:00

12:00 Time

16:00

20:00

24:00

(b)

Figure 9. The reconstructed value and actual value using response variables XP5: (a) the curve of the

Figure 9. The reconstructed value and actual value using response variables XP5 : (a) the curve of the reconstructed value and the actual value; and (b) the curve of reconstruction error. reconstructed value and the actual value; and (b) the curve of reconstruction error.

The correlation model using XP6 is: y7 (i ) = 1.33(x(i ))1 + 0.11(x(i ))2 − 0.04(x(i ))3 − 0.11(x(i ))4 − 0.16(x(i ))5 + 0.08(x(i ))6 − 30.95 (55)

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Actual value Reconstructed value

Stress(MPa)

-30 -32 -34 -36

Appl.-38 Sci. 2017, 7, 243 0:00 4:00 8:00

Stress(MPa)

-28

12:00 Time

(a)

-30

16:00

20:00

24:00

Actual value Reconstructed value

5 4 3 2 1 0 0:00

Reconstruction error(MPa) Reconstruction error(MPa)

-28

Reconstruction error(MPa)

The contrast curves between reconstructed values and actual values are shown in Figure 10. Appl. Sci. 2017,absolute 7, 243 11 of 12 The average error is 0.80 MPa and the relative error is 2.44%.

4:00

8:00

5

12:00 Time

16:00

20:00

11 of 12 24:00

(b)

4

Figure10. 10.The Thereconstructed reconstructed value and actual value using response variables X:P6(a) : (a)the thecurve curveofofthe the Figure value and actual value using response variables XP6 -32 3 reconstructed value and the actual value; and (b) the curve of reconstruction error. reconstructed value and the actual value; and (b) the curve of reconstruction error. -34

-36 -28

Stress(MPa)

Actual value The correlation model using XP7 is

-38 -30 value 0:00 4:00 8:00 12:00 Reconstructed 16:00 20:00 24:00 Time -32 y8 = 1.30(x)1 + 0.09(x)2 − 0.07(x)3 − 0.08(x)4 (a) -34

2

51

40 0:00 3 (x) − 0.19

4:00 5

8:00

12:00 Time

16:00

20:00

24:00

+ 0.12(x)6 + 0.13(x)7 − 26.93

(56)

(b)

2

The contrast curves between value the reconstructed values the actual valuesXP6 are shown in Figure Figure 10. The reconstructed and actual value using response variables : (a) the curve of the 11. -36 1and The average absolute error is 0.77 MPa and the relative error is 2.36%. reconstructed value and the actual value; and (b) the curve of reconstruction error. -38 0 4:00

8:00

16:00

20:00

24:00

(a)

0:00

4:00

8:00

12:00 Time

16:00

20:00

24:00

5 (b) Actual value 4 Reconstructed value Figure 11. The reconstructed value and actual value of the response variables XP7: (a) the curve of -32 3 of the reconstruction error. reconstructed value and the actual value; and (b) the curve -28

Stress(MPa)

12:00 Time

Reconstruction error(MPa)

0:00

-30

-34

The -36

2

errors of the reconstructed information using1 different number of the response variables are summarized in Table 4. -38 0 0:00

4:00

8:00

12:00 Time

16:00

20:00

24:00

0:00

4:00

8:00

12:00 Time

16:00

20:00

24:00

Table 4. The (a)summarized errors using different number of the response (b) variables. Response Variable ea (MPa) eb variables XP7: (a) the curve of Figure The reconstructed value and actual value response Figure 11.11. The reconstructed value and actual value of of thethe response variables XP7 : (a) the curve of reconstructed value and the actual value; and (b) the curve of the reconstruction error. X 0.89 2.71% P 1 (b) the curve of the reconstruction error. reconstructed value and the actual value; and

XP5 0.80 2.45% The errors of the reconstructed information using different number of the response variables The errors of the reconstructed information using different number XP6 0.80 2.44% of the response variables are are summarized in Table 4. summarized in Table 4. X P7 0.77 2.36% Table 4. The summarized errors using different number of the response variables. Table 4. The summarized errors using different number of the response variables.

It can be seen from Table Response 4 that the errors become Variable ea (MPa)smaller eb when the number of response variables become larger, inResponse which the average absolute error and Variable e (MPa) ebrelative error are minimum when a X P1 0.89 2.71% XP7 is chosen as the response variables. However, there is only a slight difference between these XP1 0.89 2.71% X 0.80 2.45% discussed cases, and the number with the same level of correlation XP5of theP 5 response 0.80variables 2.45% 0.80 2.44% 0.80 variables. 2.44% P6 theX P6 coefficients influencing the errorsXof reconstructed XP7 0.77 2.36% X P7 0.77 2.36% 4. Conclusions ItIn can be paper, seenseen from TableTable 4ofthat errors smaller when the number responseofvariables It can be from 4the that the become errors reconstruction become smaller when response this a method sensor information based on the theofnumber correlation degree is become larger, in which the average absolute error and relative error are minimum when X is chosen P7 variables become larger, in which the average absolute error and relative error are minimum when proposed. The monitoring information of a faulty sensor can be reconstructed by the correlation asmodel response variables. However, therenormal is onlysensors. a slight between thesehealth discussed cases, Xthe P7 is chosen the response variables. However, theredifference is only slight difference between these and theas measurements of other Based on athe structural monitoring and the number of the response variables with the same level of correlation coefficients influencing the discussed cases, and the number of the response variables with the same level of correlation system of Shenzhen Bay Stadium, the stress measurements of sensors located at support members errors of the reconstructed variables. coefficients influencing the errors of the reconstructed are used to verify the proposed method. Furthermore, variables. the influences for the errors of reconstructed monitoring information of a faulty sensor caused by three kinds of factors are discussed in the 4. Conclusions paper, which are the selection of response variables with different correlation, the quantity of the measurements froma method differentoftime periods, and thereconstruction number of response variables. The selection In this paper, sensor information based on the correlation degreeof is response and the quantity of the time periods arecorrelation two main proposed.variables The monitoring information of measurements a faulty sensorfrom can different be reconstructed by the factors, which influence the of reconstructed errors for structural the faultyhealth sensor directly. model and the measurements other normalinformation sensors. Based on the monitoring The number of the response variables influences the errors of reconstructed information a little if system of Shenzhen Bay Stadium, the stress measurements of sensors located at support members

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4. Conclusions In this paper, a method of sensor information reconstruction based on the correlation degree is proposed. The monitoring information of a faulty sensor can be reconstructed by the correlation model and the measurements of other normal sensors. Based on the structural health monitoring system of Shenzhen Bay Stadium, the stress measurements of sensors located at support members are used to verify the proposed method. Furthermore, the influences for the errors of reconstructed monitoring information of a faulty sensor caused by three kinds of factors are discussed in the paper, which are the selection of response variables with different correlation, the quantity of the measurements from different time periods, and the number of response variables. The selection of response variables and the quantity of the measurements from different time periods are two main factors, which influence the reconstructed information errors for the faulty sensor directly. The number of the response variables influences the errors of reconstructed information a little if the selected response variables are in the same correlation level to reconstructed variable. Furthermore, the proposed method is verified its effectiveness to a real world project. Acknowledgments: This research is supported by National Science Foundation of China (Grant No. 51678201, 51308162), and the Fujian National Science Foundation (2014J01171). Author Contributions: Wei Lu wrote the paper; Wei Lu and Jun Teng conceived and designed the Structural Health Monitoring (SHM) System of Shenzhen Bay Stadium; Chao Li conducted experimental data analysis; Yan Cui conducted experimental data acquisition and collation. Conflicts of Interest: The authors declare no conflict of interest

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3. 4.

5. 6. 7. 8. 9. 10. 11.

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