This document proposes a new method for detecting outliers in bridge deflections using a multi-agent system. Agents measure bridge deflections using methods like fiber-optic sensors. They create clusters of similar bridges by comparing characteristics. To detect outliers, agents calculate influence function matrices to analyze measurement patterns and identify outliers. Considering data from neighboring bridges in the cluster allows for more accurate outlier detection than looking at each bridge individually. A simulation shows the multi-agent approach detects more outliers than single-agent methods.

1. Detecting Outliers in Bridge Deflections Using Multi-Agent System
Myungho Jung
University of Utah
myungho.jung@utah.edu
Detecting dangerous bridges is important issue because it is related to many people’s life. Al-
though many methods to measure bridge deflections are studied, those cannot be applied to every
case. For example, newly constructed bridges cannot be checked accurately by the existing methods.
In this paper, a new method to effectively check bridges using multi-agent systems is introduced.
Agents create clusters and detect outliers by comparing data each other to produce more informa-
tion. This method will be helpful to analyze the status of bridges.
I. INTRODUCTION
Safety of bridges should be checked as frequently as
possible. However, there are issues to implement it.
First, bridges are usually filled with vehicles. Thus, the
process must not obstruct traffic. Second, the method to
measure bridge deflections should be cost-effective and
be able to be installed in every type of bridge. More-
over, the measurement should be always reliable. Fi-
nally, although there are few data because the bridge is
newly constructed, the status of the bridge should be
checked correctly. Through measuring bridge deflections
and clustering similar bridges, the safety of bridges can
be checked exactly and efficiently.
II. METHOD
This system can be divided into three parts. First,
bridge deflections are measured. There are many meth-
ods to measure the displacement of bridges. Each of the
measurement has pros and cons and thus, a method to
measure the deflection of each bridge should be chosen
depending on surroundings. After that, we have to judge
that the change of measurement is acceptable. The mea-
surement of a bridge can be checked by comparing with
data history of itself. However, the measurement can be
changed by many environmental factors. If clusters for
similar bridges are created and the data of bridges in the
same clusters are compared with each other, the state of
the bridge can be checked more exactly. Finally, outlier
in the database should be detected. When an abnormal
value occurs, agents in the cluster will confirm that which
bridge has a problem and the error will be notified.
A. Measurement of bridge deflections
There are a lot of methods to measure displacement
of bridges. Direct measurement such as dial gages and
linear variable differential transformers are generally re-
liable and stable. However, those cannot be installed
at various type of bridges and are not accurate in some
cases. Therefore, indirect measurements like photogram-
metry, the geodetical measurement techniques and the
global positioning system (GPS) are developed. More-
over, other techniques using inclinometers, strain gages,
or accelerometers are studied. Finally, fiber-optic-based
displacement transducer is also one of the best meth-
ods to measure bridge deflections in short-term and long
term inspection. It is stable, cost-effective, and accurate.
Different methods can be used to measure the deflection
depending on the environment.
FIG. 1. measuring bridge deflections
1. Fiber-optic-based displacement transducer[8]
This method measures the difference between initial
and transformed height at the same position of bridges.
To do this, liquid leveling system will be installed at
many points of bridges. Some of them will be located
at columns and others will be located between columns.
If the bridge moves vertically, the height of the liquid
column will change. By comparing heights of systems,
displacement of bridges can be measured. When time is
ti in Figure1, relative displacement at a point of bridge
can be calculated below.
Deflection = (hA
ti − hA
t0) − (hR
ti − hR
t0) (2.1)
In other words, the measurement means the relative
change of the height. Errors in this system are less than
0.1 mm in ideal conditions. The measurement of bridge
deflections at each point will be stored in millimeters.

2. 2
However, this method has drawbacks. First, the mea-
surement can be affected by pressure caused by the wind.
This will be overcome by installing air circuit systems to
balance the air pressure at each transducer. Second, if
there are bubbles in the liquid circuit, it will have an
adverse effect on the result. To prevent this, the system
should be carefully installed. In addition, errors will be
able to occur by leakages. This error will be detected by
checking abnormal changes. Finally, extreme tempera-
tures can result in errors because of freezing or evapo-
ration. In this case, other types of liquid or antifreeze
products will be used to minimize errors. The value of
measurement can be different depending on the shape of
bridges. Thus, the variance of measurement should be
compared. When a bridge is created, the value of bridge
deflection is stored as a base value. Moreover, the agent
receives values from other agents and declare them as
base values. After that, relative values are calculated for
comparison.
2. Inclinometer[6]
There are many advantages of this method. First, this
doesn’t need fixed observation positions. Thus, it can
be used for various bridges. Second, it is hardly influ-
enced by the environment, such as temperature or pres-
sure. Third, the deflection as well as the curve can be
measured by installing many inclinometers on a bridge.
Moreover, it can be easily installed and the installation
cost is low. Finally, it is proved to be relatively accu-
rate through hundreds of tests. Thus, it can be used in
various environments although it is not one of the most
precise methods like fiber-optic-based method.
B. Clustering similar bridges
Bridge deflections can be detected by the measurement
above. However, if the bridge is newly constructed or
there is an exceptional case, it is hard to obtain reli-
able results. Moreover, bridge deflections can be influ-
enced by environments like temperature. In such a case,
we should be able to figure out the cause of the change
of bridge deflections. To overcome this problems, clus-
tering similar bridges and comparing data are required
to get more credible results. Each bridge in a cluster
should have similar characteristics. Clusters can be cre-
ated through matchmaking. Before clustering, charac-
teristics of bridges should be quantified to be stored in
the database.
1. Characteristics of bridges[1]
To compare two bridges, traits of bridges should be
represented by data. Data types for each bridges are
below.
Description Type Example
Bridge name String E-16-MU, E-16-LA, D-16-
DM
Bridge type Integer Prestressed(1), Steel I-beam
(2), Steel plate girder(3)
Number of spans Integer 1, 2, 3, 4
Length (ft) Float 112.0, 255.5, 243.2
Width (ft) Float 128.5, 46.5, 100.7
Year built Integer 1983, 1990, 1995
ADTT(trucks/day)a
Float 450, 20, 334
a
ADTT: average daily truck traffic.
TABLE I. Data types for bridges
All values except the bridge name will be used for com-
parison. Two bridges that have totally the same charac-
teristics may not exist. Thus, if the difference between
characteristics of bridges is within the limit, the bridges
will be included in a cluster.
2. Similarity measurement
To make clusters, similarity between bridges should be
measured[7]. The value of measurement is from 0 to 1. If
the measurement is 1, it means that two agents are the
same. On the other hand, when two agents are totally
different, the measurement becomes 0. Otherwise, the
value indicates how similar two bridges are. More similar
the bridges are, more the measurement increases. Let A
and B be factors of two agents. If the number of grains
of each bridge is n, A = a1..an and B = b1..bn. Then,
the measurement for similarity can be calculated below:
Sim(A, B) =
n
i=1 max(0, 1 − αi|ai − bi|)
n
(2.2)
A scaling parameter α is used. It will change depending
on the significance of the factor. Default value of α is 1.
3. Clustering algorithm by matchmaking[5]
If the system for a bridge is assumed to be an agent,
overall systems can be operated on multi-agent system.
Although agents are decentralized, they can create clus-
ters by matchmaking algorithm. Main processes are be-
low:
1. Preclustering: Each agent creates granules of
grains.
2. Bootstrapping: Find at least an agent that can
communicate with each other.
3. Similarity Metric: Measure the similarity between
two agents.
4. Clustering: Similar agents become a cluster and
repeat through referrals.

3. 3
Similar to Yenta system[5], we can call each factor of the
agent as a grain and a group of related grains as a granule.
However, only a granule is necessary in this simulation
because there are not many grains. Each agent has a
cluster cache and rumor cache. A cluster cache stores
the information on similar agents and a rumor cache of
an agent includes the information on other agents that
the agent has contacted with.
After an agent finds another agents through bootstrap-
ping, two agents compare their granules with each other.
If the similarity is above the threshold, the agent adds
the information on other agent to the cluster cache; oth-
erwise, it is added to the rumor cache. After this, the
agents exchange the information in the rumor cache and
repeat the same process. When there are three agents,
A, B and C, although A and B are in a cluster and B
and C are in a cluster, A and C may not be in a cluster.
Let granules of agents be Ai, Bi, and Ci (1 ≤ i ≤ the
number of granules). A1 is equal to B1 and B2 is equal to
C2. However, A1 can be different from C1. Through this
algorithm, agents can create clusters and communicate
with other agents in the same cluster.
function Matchmaking( )
Create granules with grains
Search other agents that can contact with
Compare granules each other
if The difference is within the limit then
Create a cluster
Store the agent in cluster cache
else
Store the information in rumor cache
end if
Compare with the rumor cache of other agent and re-
peat the process
end function
FIG. 2. Algorithm for creating clusters
C. Detection of outlier
To check the outliers, data should be analyzed. The
outliers in time series can be detected through the influ-
ence function matrix[3]. This method is also used for the
earthquake prediction[2]. Similar to earthquake detec-
tion, bridge deflections will be measured daily. A differ-
ent part is that there is an extra calculation for the influ-
ence function based on the measurements of neighbors.
With this process, more latent outliers can be detected.
1. The influence function matrix
The influence function[4] is defined as below:
I[F, T(F), x] = lim
→0
{T((1 − )F + δx) − T(F)}/ (2.3)
The x is a variable, F is a distribution function, T is
a statistic, is a positive real number, and δx is the
distribution function that has the unit probability at the
point x. Let ρ(k) be the autocorrelations of lag k in
time series. When there are n observations x1..xn, the
autocorrelation can be defined as below:
ρ(k) =
n−k
i=1 (xi − ¯x)(xi+k − ¯x)
n
i=1(xi − ¯x)2
(2.4)
If F is a bivariate distribution function, the influence
function for F can be defines as below:
I[F, ρ(k), (x1, x2)] = z1z2 − ρ(k)(z2
1 + z2
2)/2 (2.5)
The zi is the normalized form of xi. We can calculate
the influence functions of any pair of observations by
this equation. The standardized observations zi influ-
ence other estimates that are apart from k time lags.
If an influence function estimate has a large impact on
many other lagged estimates, it can be assumed as an
outlier. Let n be the number of time lags and m be the
number of observations. When t is 1..n and k is 1..m,
n × m matrix can be created from below:
I[F, ρ(k), (zt, zt+k)] = [1 − ρ2
(k)]ut,k,1ut,k,2
ut,k,1 =
1
2
·
(zt + zt+k)
1 + ρ(k)
+
(zt − zt+k)
1 − ρ(k)
ut,k,2 =
1
2
·
(zt + zt+k)
1 + ρ(k)
−
(zt − zt+k)
1 − ρ(k)
The ut,k,1 and ut,k,2 are observations from independent
distribution. The value of I is the distribution of a con-
stant times a product of standard normal random vari-
ables because µ, σ, and ρ(k) are known through the time
series. Values for the influence function will be checked
if those are abnormally large for a realization from a sta-
tionary Gaussian process through the distribution.
We can analyze the patterns of the influence function
and detect outliers by marking outliers in the matrix.
If the absolute values of influence function estimates are
more than the critical value, the position is marked as ‘+’.
As a result, an observation that has many ‘+’ symbols on
the horizontal and upper diagonal line can be considered
as an outlier. The critical value is recommended to be
selected based on the product standard normal distribu-
tion.
III. ALGORITHM
Contrary to the existing detection in time series, we
can utilize other similar neighbors’ data. Therefore, ex-
tra routines are added to the algorithm. First, detect
outlier of itself. If the data is considered as an outlier,
the agent notices others. After all agents check data of
themselves, they compare the data with others by cre-
ating the influence function matrix of the mean of other
agents except abnormal data. By this process, the agents
can more accurately inspect the status.

4. 4
Time
Lag
1 2 3 · · · n-2 n-1 n
1 +
2 +
3 +
...
...
m-3 +
m-2 +
m-1 +
m + + + · · · + + +
...
TABLE II. an outlier at time m in the influence function
matrix
function CheckDeflections( )
Measure the deflection of the bridge
Create influence function matrix of itself
if Measurement is an outlier then
Notify other agents
return
else
Receive data from other agents except outliers
Compute the mean of the observations of others
Create influence function matrix of the means
Detect outliers
return
end if
end function
FIG. 3. Algorithm for creating clusters
IV. SIMULATION
To test this algorithm, a simulation is implemented by
Matlab. In this simulation, it is assumed that there are
10 bridges in the same cluster. 100 random numbers are
created for the measurement of each agent. The random
numbers are generated to be normally distributed(σ = 2
and µ = 0). Time lag is set to 10. After creating the
influence function matrix, if the number of ’+’ or ’-’
marks are more than one, the measurement is regarded
as an outlier. There are two kinds of data. One is errors
detected by an agent and the other is the total number
of errors by applying the algorithm in this paper. A
multi-agent system is only applied the second case. Each
measurement is averaged over 100 runs. The relationship
between the number of detected outliers and the critical
value is below:
1 1.2 1.4 1.6 1.8 2
10
15
20
25
30
the critical value
thenumberoferrors
error detected by single agent
error detected by multi-agent
FIG. 4. relationship between the number of detected errors
and the critical value
As seen in the graph above, more errors are detected
when a multi-agent system is used. Moreover, we can
confirm the relationship between the critical value and
the number of detected errors. For the reliability of the
result, confidence intervals of the number of errors were
measured. These values were calculated from the results
above. Each sample size is 100 and α = 0.1 (95% confi-
dence interval)
the number of errors detected by single agent
critical value lower bounds upper bounds
1.05 25.1925 28.3475
1.10 25.9572 29.4828
1.15 22.4535 25.7265
1.20 20.6236 23.3964
1.25 21.4515 24.4485
1.30 19.6557 22.8043
1.35 18.0731 21.1469
1.40 15.7286 18.4114
1.45 14.5058 17.2542
1.50 14.3224 16.5576
1.55 14.1884 16.7716
1.60 12.8506 15.5894
1.65 12.7334 15.0466
1.70 11.4461 13.7539
1.75 10.2045 12.1555
1.80 10.6664 13.0136
1.85 10.7072 13.1728
1.90 9.5708 11.6092
1.95 8.9798 10.9202
2.00 8.6398 10.9002
TABLE III. confidence intervals for the number of errors de-
tected by single agent

5. 5
the number of total errors
critical value lower bounds upper bounds
1.05 29.0460 31.9940
1.10 29.8893 33.1307
1.15 26.5512 29.5888
1.20 24.7857 27.4543
1.25 25.2103 28.1097
1.30 23.6448 26.5152
1.35 21.9551 24.8049
1.40 19.6864 22.1936
1.45 18.2770 20.9430
1.50 18.2450 20.2950
1.55 17.7035 20.1165
1.60 16.6957 19.3843
1.65 16.5724 18.8876
1.70 15.3268 17.4732
1.75 13.3992 15.3208
1.80 14.1860 16.3940
1.85 14.2225 16.6375
1.90 12.5387 14.5413
1.95 12.1924 14.0876
2.00 11.9143 14.1057
TABLE IV. confidence intervals for the number of total errors
detected by multi-agent
V. CONCLUSION
Many methods to measure bridge deflections and to
detect outliers have been studied. However, those can be
used in limited environments. In this paper, a method us-
ing a multi-agent system is introduced. Although the real
data on bridge deflections, the simulation shows that the
algorithm can detect more errors than the existing meth-
ods. And the number of detected errors largely changed
depending on the critical value. The critical value can be
adjusted depending on the type of bridges or an environ-
ment. This method will be useful in detecting outliers
among measurements as well as in analyzing the status
of the bridge.
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