Journal of Intelligent Learning Systems and Applications, 2012, 4, 247-254
http://dx.doi.org/10.4236/jilsa.2012.44025 Published Online November 2012 (http://www.SciRP.org/journal/jilsa)
247
Comparison between Neural Network and Adaptive
Neuro-Fuzzy Inference System for Forecasting Chaotic
Traffic Volumes
Jiin-Po Yeh1, Yu-Chen Chang2
1Department of Civil and Ecological Engineering, I-Shou University, Kaohsiung City, Taiwan; 2Institute of Civil Engineering Tech-
nology, National Kaohsiung University of Applied Sciences, Kaohsiung City, Taiwan.
Received May 26th, 2012; revised June 19th, 2012; accepted June 26th, 2012
ABSTRACT
This paper applies both the neural network and adaptive neuro-fuzzy inference system for forecasting short-term chaotic
traffic volumes and compares the results. The architecture of the neural network consists of the input vector, one hidden
layer and output layer. Bayesian regularization is employed to obtain the effective number of neurons in the hidden
layer. The input variables and target of the adaptive neuro-fuzzy inference system are the same as those of the neural
network. The data clustering technique is used to group data points so that the membership functions will be more tai-
lored to the input data, which in turn greatly reduces the number of fuzzy rules. Numerical results indicate that these
two models have almost the same accuracy, while the adaptive neuro-fuzzy inference system takes more time to train. It
is also shown that although the effective number of neurons in the hidden layer is less than half the number of the input
elements, the neural network can have satisfactory performance.
Keywords: Neural Network; Adaptive Neuro-Fuzzy Inference System; Chaotic Traffic Volumes; State Space
Reconstruction
1. Introduction
It has been known for decades that chaotic behaviors
exist in traffic flow systems Gazis et al. [1] developed a
generalized car-following model, known as the GHR (Ga-
zis-Herman-Rothery) model, whose discontinuous beha-
vior and nonlinearity suggested chaotic solutions for a
certain range of input parameters. Due to the capacity di-
mension [2] of the attractor being fractal and first Lyapu-
nov exponent [3] being positive, Disbro and Frame [4]
showed the presence of chaos in this General Motors’ mo-
del without signals, bottlenecks, intersections, etc. or with
a coordinated signal network. Chaos was observed in a
platoon of vehicles described by the traditional GHR mo-
del modified by adding a nonlinear inter-car separation
dependent term [5,6], Poincaré maps of which appear as a
cloud of points without any repeat. Traffic volume col-
lected at 2-min interval on the Beijing Xizhimen highway,
China, was also found to posses chaotic behaviors [7].
Because of nonperiodic behaviors, chaotic time series
seem to be unpredictable, but a variety of short-term
forecast models have been attempted and proven to be
successful, such as models employing Kalman filtering
theory [8], the local linear model using information based
on past values [9], the polynomial model [10], neural
network-based black-box models [11-15], a model con-
sisting of a fuzzy C-means clustering and a radial-ba-
sis-function neural network [16], etc. This paper also
tries to forecast the short-term chaotic traffic volume at
the intersection. Two kinds of models are presented for
comparison. One is the neural network, where the delay
coordinates [2,17,18] of the reconstructed state space of
the traffic flow system are used as the input vector of the
neural network and the first delay coordinate of next state
as the target of the neural network. The other model is
the adaptive neuro-fuzzy inference system [19,20], where
inputs and targets are identical to the first one, but mem-
bership functions and fuzzy rules [21,22] replace neurons
in the neural network. The number and the shapes of the
membership functions are decided and tuned by a data
clustering technique and backpropagation neural network,
respectively, which is different from the Park’s model
[16] in the ways of data clustering and learning process.
2. Diagnosis of Chaos
The Poincaré map, time series, autocorrelation function,
etc., can often provide graphic evidence for chaotic be-

Comparison between Neural Network and Adaptive Neuro-Fuzzy Inference System for
Forecasting Chaotic Traffic Volumes
248
havior, while the fractal dimension and largest Lyapunov
exponent are two principal quantitative measures of chaos.
This paper selects the fractal dimension, largest Lyapu-
nov exponent and autocorrelation function to show the
existence of chaos of the traffic flow. Brief introduction
to them is as follows.
2.1. Fractal Dimension
If there is only one measurement available for a system,
delay coordinates are usually used to reconstruct its state
space [17]. Given a time series x(t) and time delay
 , an
n-dimensional state space can be reconstructed with the
delay coordinates:
   

{,,2,, 1xt xtxtxtn



  . To get
the appropriate dimension for reconstructing the state
space of a chaotic dynamical system, the first step is to
obtain the fractal dimension of the chaotic attractor in the
state space. There are a number of ways to measure the
chaotic attractor dimension [2]. Among them, this paper
chose the method of correlation dimension, because it is
much easier to implement and not time-consuming. Con-
sider an orbit discretized to a set of N points in the state
space. A sphere of radius r is poisoned at each point of
the orbit and the number of points within each sphere
with Euclidean distance less than r is counted. A correla-
tion function is then defined as [2,23]
 


1
lim
1ij
nij
CrHri j
NN


 XX (1)
where ij is the Euclidean distance between
points Xi and Xj and H is the Heaviside function (or unit
step function ). For many attractors, this function C(r)
exhibits a power law dependence on r, as ; that is
XX
0r

0
lim d
rCr ar  (2)
Hence, a correlation dimension is defined by the expres-
sion

0
ln
lim ln
cr
Cr
dr

 (3)
The chaotic attractor dimension will often approach an
asymptote d with the dimension of the reconstructed state
space gradually increasing. To embed a d-dimensional
chaotic attractor, the state space may be reconstructed
with dimension greater than or equal to 2d + 1, which ac-
cording to Takens [24] will be sufficient to have generic
delay plots.
2.2. The Largest Lyapunov Exponent
The largest Lyapunov exponent of a chaotic orbit is de-
fined by the expression [3]
2
10
1
lim log
mi
mii
d
md

 
 (4)
The calculation is initiated by locating the nearest
neighbor to the first point of a reference trajectory in the
reconstructed state space and the distance between them
is denoted by d0i. This pair of points is then propagated
through the attractor for a fixed short time  and its final
separation di is computed. After that, a replacement for
the propagated pair is attempted by the following proce-
dure: 1) The distance of each delay coordinate point in
the attractor to the propagated point of the reference tra-
jectory is determined; 2) Points closer than a given length
and away from another much smaller length (to avoid
noise) are examined to see if the angle between the
original pair and attempted pairs is less than a given
small angle (e.g. 0.3 radians); and 3) The attempted pair
with the smallest angle is used as replacement for the
next propagation. The repeating of propagating and re-
placing are carried out for m cycles.
2.3. Autocorrelation Function
To find out the resemblance of the signal


x
t with
itself as time passes, the autocorrelation function
 
0
lim d
T
T
Rxtxt 


t
 
(5)
is an often-seen tool to achieve this purpose. If


R

approaches the square of the mean of the function


x
t
as

, it means that the signal is only correlated
with its recent past [25], i.e., sensitive to the initial con-
ditions. Furthermore, the time lag

at which


R

first crosses the square of the mean of the function


x
t
is usually considered as the time delay

for recon-
structing the state space.
3. Forecasting Models
There are two models applied in this paper to forecast
short-term chaotic traffic volumes: the feedforward back-
propgation neural network and the adaptive neuro-fuzzy
inference system, which are described as follows.
3.1. Feedforward Backpropagation Neural
Network Model
The first forecasting model used in this paper is a feed-
forward neural network with the backpropagation train-
ing algorithm, as shown in Figure 1. The transfer func-
tion in the single hidden layer is the tan-sigmoid function

1e ,1,2,3,,
1e
i
i
n
ii n
afni s



 
 (6)
where ,1 1,22,12R ,,,,
ii iiRRi
nwxwxwxbxxx

 
are the elements of the input vector, s is the number of

Comparison between Neural Network and Adaptive Neuro-Fuzzy Inference System for
Forecasting Chaotic Traffic Volumes
249
Neuron 1
Neuron 2
Neuron S
x1
x2
xR
B
b2
b1
b
s
w1,1
Neuron
w1,2
w1,S
wR,S
W1,1
W1,2
W1,S
Input Hidden Layer Output Layer
Figure 1. The structure of the feed forward back propaga-
tion neural network.
neurons, ,1, 2, R
are the weights connecting the
input vector and the ith neuron, and bi is the bias of the
ith neuron. The output layer with a single neuron is given
by the linear function
,,,
ii i
ww w

afn n (7)
where
1,1 11,22,1,11,2, are
the weights connecting the neurons of the hidden layer
and the neuron of the output layer, and B is the bias of
the output neuron.
,, ,,
is sis
nWaWaWa BWWW 
There are many variations of the backpropagation al-
gorithm, aiming to minimize the network performance
function, i.e., the mean square error between the network
outputs and the targets, which is

2
1
1
Minimize MSE
m
jj
j
ta
m



(8)
where tj and aj are the jth target and network output, re-
spectively. This paper chooses the Levenberg-Marquardt
algorithm [26-28] as the training function to minimize
the network performance function. This algorithm inter-
polates between the Newton’s algorithm and the gradient
descent method. If a tentative step increases the per-
formance function, this algorithm will act like the gradi-
ent descent method, while it shifts toward Newton’s
method if the reduction of the performance function is
successful. In this way, the performance function will
always be reduced at each iteration of the algorithm. To
avoid the problem of overfitting, there are two methods
to improve the network generalization: Bayesian regu-
larization [29] and early stopping. The Bayesian regu-
larization can provide a measure of how many network
parameters (weights and biases) are being effectively
used by the network. From this effective number of pa-
rameters, the number of neurons required in the single
hidden layer of the neural network can be derived by the
following equation

 
1Rs ssP

 (9)
where R is the number of elements in the input vector, s
is the number of neurons in the hidden layer, and P is the
effective number of parameters found by the Bayesian
regularization. In the strategy of early stopping, the
available data is divided into three sets: the training set,
validation set and testing set. The training set is used for
computing the gradient and updating the network weights
and biases, while the error of the validation set is moni-
tored during the training process. When the network be-
gins to overfit the training data, the error on the valida-
tion set typically begins to rise. Once the validation error
keeps increasing for a specified number of iterations, the
training is stopped and the weights and biases at the
minimum of validation error are returned. The testing set
is not used during the training, but is used to check the
performance of the trained network. To evaluate the per-
formance of the trained network, this paper performs
linear regression analysis between the network outputs
and the corresponding targets, and computes the correla-
tion coefficient [30].
3.2. Adaptive Neuro-Fuzzy Inference System
Model
The second forecasting model used in this paper is an
adaptive neuro-fuzzy inference system, as shown in Fig-
ure 2. This model consists of two components: a fuzzy
inference system and a backpropagation algorithm. For
an ordinary fuzzy inference, the parameters in the mem-
bership functions are usually determined by experience
or the trial-and-error method. However, the adaptive neu-
ro-fuzzy inference system can overcome this disadvan-
tage through the process of learning to tailor the mem-
bership functions to the input/output data in order to ac-
count for these types of variations in the data values,
rather than arbitrarily choosing parameters associated
with a given membership function. This learning method
works similarly to that of neural networks. The fuzzy in-
ference incorporated into the adaptive neuro-fuzzy in-
ference system is the first-order Sugeno-type inference
[31], the typical rule of which, if there are only two in-
puts x and y, has the form
If input1and input 2,
then output is
x
y
zaxbyc


 (10)
The output level zi of each rule is weighted by the firing
strength wi of the rule, which is
 


12
Min ,
i
wFxFy
(11)
where


1
F
x and


2
F
y are the membership func-
tions of inputs 1 and 2, respectively. Finally the output of
the inference system yields

Comparison between Neural Network and Adaptive Neuro-Fuzzy Inference System for
Forecasting Chaotic Traffic Volumes
250
Input Input
Membership function Rule Output Membership function Output
Figure 2. The structure of the adaptive neuro-fuzzy infer-
ence system.
1
1
N
ii
iN
i
i
wz
Z
w




(12)
where N is the number of the rules. Because the number
of the input variables and data sets are large in this paper,
the “subtractive clustering” technique [32] is adopted to
cluster the data and assign every data point a membership
grade for each cluster. According to the number of
membership functions and input variables, the number of
rules is then decided. Due to the fact that membership
functions are more tailored to the input data, the fuzzy
inference system will end up having much fewer rules
than that without clustering.
4. Numerical Results
The eastbound traffic volumes at the intersection of
Jiouru Road and Ningsia Street, Kaohsiung City, Taiwan,
are taken as examples. The traffic volume data were col-
lected by the vehicle traffic counter in November, 2008,
totaling 14 days excluding weekends. Due to data being
recorded every five minutes, three time intervals are
chosen: 5-min, 10-min and 15-min. The data are divided
into three sets: training data (8 days), validation data (4
days) and testing data (2 days). As mentioned previously,
training and validation data are used to train forecasting
models, while testing data are used to examine how good
the trained models are. All forecasts are only one time
interval ahead of occurrence, i.e., 5-min, 10-min or 15-
min ahead of time. The MATLAB software [33] is ap-
plied to build the neural network and adaptive neuro-
fuzzy inference system.
To get a reasonable time delay for reconstruction of
the traffic flow system, the autocorrelation function

R

is plotted. Figures 3-5 show the autocorrelation
function for 5-min, 10-min and 15-min traffic volumes,
respectively, where the dotted horizontal line represents
the square of the mean of time series of the traffic vol-
ume. All the three curves tend to approach the dotted
horizontal line and the time lag η for the autocorrelation
to first cross the dotted horizontal line is found approxi-
mately at 300 min for all these three time intervals.
Hence the time delayτ to reconstruct the flow system is
60 for 5-min interval, 30 for 10-min interval and 20 for
15-min interval. By using the corresponding time delay
and gradually increasing the dimension n of the state
space, the correlation dimension of the chaotic attractor
will reach an asymptote as n increases. These processes
are shown in Figures 6 to 8 for 5-min, 10-min and 15-
min, respectively. These figures indicate that the correla-
tion dimension d for 5-min interval is 6.687, for 10-min
interval is 6.766 and for 15-min interval is 6.637. There-
fore, the embedding dimension (2d + 1) is 15 for these
three time intervals. Aside from the fractal dimension,
the largest Lyapunov exponent of the attractor is also
calculated to show the presence of chaos. The largest
Lyapunov exponents are all positive for different time
intervals and almost identical for each time interval with
different evolution steps, as shown in Table 1. Only after
obtaining the required embedding dimension and time
delay can the forecasting commence. The training input/
output data is a structure whose first component is a 15-
dimensional input:
Table 1. The largest Lyapunov e xponent found for different
time intervals and evolution steps.
Time interval 5-min 10-min 15-min
1 3.56E−04 3.50E−04 4.55E−04
3 3.62E−04 3.50E−04 4.55E−04
5 3.64E−04 3.49E−04 4.60E−04
7 3.62E−04 3.50E−04 4.56E−04
No. of evolution
steps
9 3.62E−04 3.50E−04 4.61E−04
05001000 1500 20002500
3200
3600
4000
4400
4800
5200
5600
 (min)
R
(  )
Figure 3. Autocorrelation function of 5-min traffic volume.

Comparison between Neural Network and Adaptive Neuro-Fuzzy Inference System for
Forecasting Chaotic Traffic Volumes
251
05001000 1500 2000 2500
12000
14000
16000
18000
20000
22000
R
(  )
 (
m
in)
Figure 4. Autocorrelation function of 10-min traffic volume.
05001000 1500 20002500
2 8000
3 2000
3 6000
4 0000
4 4000
4 8000

(
m
i
n
)
R(  )
Figure 5. Autocorrelation function of 15-min traffic volume.








,,2,, 1xi xixixi4

  , where


x
i is
the observation of the time series of traffic volume
and
 is the time delay, and whose second component is
the output:
thi


1xi. As mentioned previously, the time
delay is chosen to be 60, 30 and 20 for 5-min, 10-min,
and 15-min traffic volumes, respectively. Numerical re-
sults for the neural networks and adaptive neuro-fuzzy
inference system are discussed as follows.
4.1. Neural Networks
By using the Bayesian regularization, the effective net-
work parameters (weighs and biases) can be found and
the number of effective neurons in the hidden layer is
then calculated from Equation (9). The results for three
time interval are listed in Table 2, which shows the
number of neurons actually required in the hidden layer
is indeed less than half the number of input elements.
The performance of a trained network can be measured
to some extent by the errors on the training, validation
and test sets. One option is to perform a regression
analysis between the network response and the corre-
sponding targets. Through linear regression analysis, the
correlation coefficients between outputs and targets for
different time intervals and data sets are obtained and
shown in Table 2, ranging from 0.951 to 0.985.
4.2. Adaptive Neuro-Fuzzy Inference System
By using the “subtractive clustering” technique, the mini-
0246
-16
-12
-8
-4
0
ln
r
ln C( r )
State-space dimension
n =3-23
(a)
048 12162024
0
1
2
3
4
5
6
7
State-s
p
ace dimension n
d
c
(b)
Figure 6. (a) The curves of


lnCrvs with state space
dimension increasing from 3 to 23 (up to bottom); (b)
The curve of the correlation dimension vs for 5-
min traffic volume.

ln r
c
d
nn

Comparison between Neural Network and Adaptive Neuro-Fuzzy Inference System for
Forecasting Chaotic Traffic Volumes
252
0246
-16
-12
-8
-4
0
ln
r
ln C( r )
State-space dimension
n =3-23
(a)
04812 16 20 24
0
1
2
3
4
5
6
7
State-s
p
ace dimension n
d
c
(b)
Figure 7. (a) The curves of vs with state
space dimension increasing from 3 to 23 (up to bottom);
(b) the curve of the correlation dimension vs for
10-min traffic volume.

lnCr

ln r
c
d
nn
Table 2. The number of effective neurons in the hidden
layer of the neural network and the correlation coefficient.
Time interval 5-min 10-min 15-min
No. of neurons 6 6 5
Training data 0.951 0.978 0.985
Validation data 0.961 0.981 0.977
Correlation
coefficient
Testing data 0.953 0.977 0.981
mum inference rules are found for 5-min, 10-min and 15-
min intervals, respectively. The results are shown in Ta-
ble 3. The number of rules found by clustering tech-
nique is indeed much fewer than that without clustering.
Through the learning process, the parameters of the
membership functions in the antecedent and the con-
stants in the equation of the consequent of each rule are
decided. After simulating the fuzzy inference, the corre-
lation coefficients between outputs and targets for dif-
ferent time intervals and data sets are found, as shown in
Table 3, ranging from 0.951 to 0.990.
5. Conclusion
The phenomena of the fractal dimension, the positive
0246
-16
-12
-8
-4
0
ln
r
ln C( r )
State-space dimension
n =3-23
(a)
048 12162024
0
1
2
3
4
5
6
7
State-s
p
ace dimension n
d
c
(b)
Figure 8. (a) The curves of


lnCrvs with state
space dimension increasing from 3 to 23 (up to bottom);
(b) The curve of the correlation dimension vs for
15-min traffic volume.

ln r
c
d
nn

Comparison between Neural Network and Adaptive Neuro-Fuzzy Inference System for
Forecasting Chaotic Traffic Volumes
253
Table 3. The number of inference rules of th e ad ap tive neuro-
fuzzy inference system and the correlation coefficient.
Time interval 5-min 10-min 15-min
No. of inference rules 29 19 19
Training data 0.964 0.982 0.990
Validation data 0.951 0.973 0.969
Correlation
coefficient
Testing data 0.962 0.972 0.971
largest Lyapunov exponent and the autocorrelation ap-
proaching the square of the mean of the time series con-
firm the existence of chaos in the traffic flow system.
Two forecasting models of the chaotic traffic flow pre-
sented in this paper prove to be very successful with sat-
isfactory accuracy. The Bayesian regularization applied
to the neural network to get effective number of neurons
in the hidden layer and the subtractive clustering tech-
nique applied to the adaptive neuro-fuzzy inference sys-
tem to get the minimum number of fuzzy rules are both
quite useful and effective. The numerical results show
that the prediction accuracies of these two modes are
almost the same, as far as the correlation coefficient is
concerned, but the adaptive neuro-fuzzy inference system
requires more time to train, because more parameters
need to be determined and that the number of effective
neurons in the hidden layer is usually less than the num-
ber of elements in the input vector.
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