Prediction and Analysis of the Tunnel Arch Top Settlement Based... the Fuzzy Support Vector Regression Machine

ORIENT ACADEMIC FORUM
Prediction and Analysis of the Tunnel Arch Top Settlement Based on
the Fuzzy Support Vector Regression Machine
WU Weidong, GENG Shuai
School of Civil Engineering and Architecture, Southwest Petroleum University, Chengdu, Sichuan,
China, 610500
[email protected]
Abstract: This article use fuzzy math and support vector machine(SVM) to predict the tunnel arch top
settlement(TATS). Use the practical measured data to establish the estimate and prediction regression
model of fuzzy support vector machine(FSVM) for TATS ,and analysis the predicted result.
Keywords: regression model, fuzzy support vector machine, tunnel arch top settlement (TATS)
1
Introduction
The monitoring of the tunnel arch top settlement (TATS) is a required part of the site monitoring. But as
the length of the tunnel is increasing, the monitoring of TATS is becoming harder. So the best solving
method is the prediction of TATS. SVM follows the structural risk minimization principle and has the
high generalization ability at the small sample case, the support vector regression machine is a part of it.
But before the SVR model is esatblished, the original data must be processed by the fuzzy math which is
good at processing the fuzzy information which is uncertainty and inaccuracy, because the original data
have the fuzzy characteristics due to the bad monitoring environment, personal error etc.on the data
collection stage.
So, we may integrate fuzzy math and support vector machine applying the both superiority, and
establish fuzzy support vector regression machine (FSVR) and predict TATS.
2
The Prediction Model of TATS Based on FSVR
2.1 The prediction model of FSVR of TATS
Support the training data set{(Xi,Yi)},i=1,…,n are observed where Xi is d-vector of real numbers and in
the model of FSVM of TATS is time and each Yi is triangular fuzzy number of the TATS every day. Let
xij is element of Xi . Then ,we assume xij≥0 by simple translation of all vectors.
Let W=W=(W1,W2,…,Wd), where Wi (mWi ,αWi,βWi ),αWi,βWi,≥0,i=1,...,d and let B=(mB,αB,βB),αB,βB≥0.
We now consider the following model:
f ( X ) = B + (W , X ) = B + W1 x1 + W2 x 2 + L + Wd x d
(1)
∈
∈
∈
Where B T(R) is the bias ,W T(R)d is the weight matrix ,X Rd as a vector,T(R)d is the set of
d-dimensional of the triangularfuzzy number .
Assume that
W
2
= mw
2
+ mw − α w
2
+ mw + β w
β w = ( βW1 ,..., βWd ) ， α w = (αW1 ,..., αWd ) ,
2
,where
mw = (mW1 ,..., mWd )
，
and through the introduction of ε-insensitive loss
function, the solution of equation (1) become to solve the convex optimization problem as below:
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min
1
W
2
2
l
3
+ C ∑∑ (ξ ki + ξ kj* )
K =1 i =1
myi − ( K ( mW , X i ) + mB ) ≤ ε + ξ1i ,

*
( K ( mW , X i ) + mB ) − myi ≤ ε + ξ1i ,

(mYi − α Yi ) − ( K ( mW , X i ) + mB − K (α W , X i ) − α B ) ≤ ε + ξ 2i ,

s.t.( K ( mW , X i ) + mB − K (αW , X i ) − α B ) − ( mYi − α Yi ) ≤ ε + ξ 2*i

(mYi + βYi ) − ( K (mW , X i ) + mB + K ( βW , X i ) + β B ) ≤ ε + ξ 3i
( K ( m , X ) + m + K ( β , X ) + β ) − ( m + β ) ≤ ε + ξ *
W
i
B
W
i
B
Yi
Yi
3i

*
ξ , ξ ≥ 0, k = 1,2,3.
 ki ki
(2)
Where C is the penalty number, ξ is slack variables, K (x, y) = φ (x) φ (y) as the kernel function, its dual
form:
1 3 l
*
*
 2 ∑ ∑ (α ki − α ki )(α kj − α kj ) K ( X i , X j ) +
 k =1 i , j =1
l
 3 l
min ε ∑ ∑ (α ki + α ki* ) − ∑ mYi (α1i − α1*i ) −
i =1
 k =1 i , j =1
l
 l
∑ ( mYi − α Yi )(α 2i − α 2*i ) − ∑ (mYi + β Yi )(α 3i − α 3*i )
 i =1
i =1
(3)
 l
*
∑ (α ki − α ki ) = 0, k = 1,2,3,
s.t. i =1
α , α * ∈ [0, C ], k = 1,2,3.
 ki ki
*
Where α , α is the Lagrange multipliers, then, put the solution of equation (3) into equation(1),can
derive the equation (4):
i
l
i =1
i =1
f ( X ) = B + (∑ (α 1i − α 1*i ) K ( X i , X ),∑ [(α 1i − α 1*i ) − (α 2i − α 2*i )]K ( X i , X ),
l
∑ [(α
i =1
3i
− α ) − (α 1i − α )]K ( X i , X ))
（ α ,β ）,according to the condition of KKT, we can derive the m ，,the
The next step is to find B mB,
method is below:
B
B
B
l

(α1 j − α1*j ) K (X j , X i ) − ε
m
m
=
−
∑
Yi
 B
j =1


l
m = m − (α − α * ) K (X , X ) + ε
1j
1j
Yi
∑
j
i
 B
j =1
So we can derived the
（4）
*
1i
*
3i
α ,β
B
B
α1i ∈ (0, C ),
（5）
α ∈ (0, C ),
*
1i
by solving the optimization problem.
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l
min
α B ,βB ≥0
∑| m
Yi
i =1
l
− α Yi −∑ (α 2 j − α 2* j ) K ( X j , X i ) − m B + α B | ε
j =1
l
（6）
l
+ ∑ | m Yi + β Yi - ∑ (α 3 j − α ) K ( X j , X i ) − m B − β B |ε
*
3j
i =1
At last ,put B
3
j =1
（m ,α ,β ）into equation (4),we can derive the model of FSVR of TATS.
B
B
B
Applications
3.1 Project profile
Some tunnel is located at Liannan County in Guangdong province. The tunnel is single-hole tunnel, the
length is 1330m.The method of the monitoring of TATS is shown in the fig.1,one monitoring section
includes three monitoring points as D,C,E.the monitoring sections are placed approximately every 30
40m along the tunnel. The equipment is Steel Ruler, tower ruler and level. The time intervals of the
monitoring can be viewed in the table 1.
～
D
C
E
8
m
12m
Figure.1 method of monitoring of TATS
Table 1 The interval of the monitoring
1st to 15th
16th to 30th
30th to 90th
Time
Interval time of
monitoring
Once or twice/day
Once/2day
Once or
twice/week
More than 90th
Once or third /month
Fuzzy processing of original data
Because the original data has fuzzy characteristics due to the bad environment and person factors, we
have to fuzzy the original data before establish model. According to the characteristics and the reason of
the errors, so the fuzzy method in this article is that select the data of a monitoring point as the central
values, and then, adds a percent as the right spread and reduces a percent as the left spread. The original
data which can be seen in the table.2 is the data of the D point at the 3rd monitoring section at tunnel
exit.
3.2
Table.2 Original data
Time(d)
Aug.1
Aug.2
Aug.3
Aug.4
Aug.5
Aug.6
Data(mm)
-14.49
-13.24
7.44
5.98
2.25
7.42
Aug.8
Aug.9
Aug.10
Aug.11
Aug.12
Aug.13
Aug.8
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13.48
13.69
4.06
10.63
2.14
4.86
Aug.1
According to the reason of the data errors, when the steel ruler (the average measurement is 4.2m)and
the tower ruler(the average measurement is 1.6m) have 3 ° deviation from vertical, these will cause the
maximal error 8mm.After linear regression analysis and error estimates to the data of many monitoring
points, we can estimate the relative error is 33%. So let 33% be the basic percent to fuzzy the original
data, and after make the original data fuzzy ,all number plus a positive number to ensure they are greater
than 0,so we select 20 as that positive, at last carry out normalization processing, the result can be seen
in table.3
：
3.3
Table.3 The fuzzy and normalization of the data
Time(d)
Aug.1
Aug.2
Aug.3
Aug.4
Aug.5
Aug.6
a(mm)
0.00
0.04
0.65
0.62
0.55
0.65
m(mm)
0.13
0.16
0.71
0.67
0.57
0.71
Β(mm)
0.26
0.28
0.78
0.73
0.59
0.78
Aug.7
Aug.8
Aug.9
Aug.10
Aug.11
Aug.12
Aug.13
0.75
0.76
0.59
0.7
0.55
0.6
0.6
0.87
0.88
0.62
0.8
0.57
0.64
0.64
0.99
1
0.66
0.89
0.59
0.69
0.68
Establish the estimate and prediction model of FSVR of TATS
Put the data in Table 3 into the equation (4), the data of the 1st day to the 11th day as the training set, the
data of the 12th day and 13th day as the prediction set, Gaussian function was selected as kernel
function and used genetic algorithms to find the optimal parameters C and ε, The results is C = 78.5053,
ε = 0.01, and to train the model to calculate а, а *, then put а, а * into equation (6) and equation (7) to
calculate B, finally ,we put B and а, а *into equation (5) to get the prediction model. the prediction value
can be seen in table.4 and the effect of the model of FSVR of TATS can be seen in figure.2
：
Time(d)
actual number
predicted number
Table.4 Prediction and actual number
Aug.12
a(mm)
0.60
m(mm)
0.64
β(mm)
0.69
α’(mm)
0.53
m’(mm)
0.60
β’(mm)
0.68
Aug.13
0.60
0.64
0.68
0.53
0.60
0.68
From Table 4, we can see that the predicted value and actual value are almost the same, the absolute
error is 0.04,the average relative error of the maximal and minimal border and central values is 6.9%,
the model can fit the train data of TATS very well, the nature of the prediction of the central
value(actual value) is that using SVR to predict the future data, so if the parameters is best decided by
genetic algorithms , the prediction of the central value will be very well, but the maximal and minimal
borders are not like using the model of SVR to predict, so Here we mainly discuss the effect of the
maximal and minimal borders to the monitoring work ( the fig.2):
(1)The distance from the maximal border and minimal border is decided by аYi βYi of Yi= mYi аYi
βYi in the train set. The more difference between аYi and βYi , the larger distance it has. So if we want to
improve the precision, we must low down the difference between аYi βYi from the central value first.
）
，
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ORIENT ACADEMIC FORUM
(2)The actually changes of TATS don't out the maximal and minimal border and the distance between
borders and central value is not larger which indicate that the precision of prediction is well.
(3)when the monitoring data is not accurate, the three trend lines of TATS will give more information
than one line for the monitoring person.
(4)When select the train data ,it will be better to select the new data because it will offer more new
information about the TATS.
Figure.2
4
The effectiveness of the model of FSVR of TATS
Conclusion
This article using a model of FSVR of TATS to predict the future data of TATS ,the result show that the
new model not only have the high generalization ability and good precision like the original SVR but
also offer two maximal and minimal borders to give the monitoring person the more information than
only one line does . This will make the monitoring person will be more sensitive to the change of
TATS ,especially,when the general trend is close to the alert line.
Author in brief:
WU Weidong(1969-),male,an associate professor ,engaged in engineering management teaching
research and practice.
References
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[3]. V.Vapnik. The Nature of Statistical Learning Theory, Springer, Berlin, 1995.
[4]. Nello Cristianini, John Shawe Taylor.An Introduction to Support Vector Machines and Other
Kernel-based Learning Methods, 2000, Cambridge University Press: 98-105.
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