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sets and systems Fuzzy Sets and Systems 93 (1998) 173 183
Real-time adaptive on-line traffic incident detection H . X u a, C . M . K w a n a'*, L , H a y n e s a, J . D . P r y o r b alntelligent Automation Inc., 2 Research Place, Suite 202, Rockville, MD 20850, USA bUnited States Army, Space & Strategic Defense Command, Concepts and Applications Division, P.O. Box 1500, Huntsville, AL 35807-3801, USA
Received November 1994; revised July 1996
Abstract A new approach to traffic incident detection is proposed in this paper. The method consists of two stages. First, a real-time adaptive on-line procedure is used to extract the significant components of traffic states, namely, average velocity and density of moving vehicles. Second, we apply a new neural network called Fuzzy CMAC (Cerebellar Arithmetic Computer) to identify traffic incidents. Fuzzy CMAC is an ideal candidate for this purpose for the following reasons. First, the Fuzzy CMAC learning structure is a creative use of fuzzy logic and CMAC based neural networks. Expert knowledge in terms of linguistic rules can be incorporated into the design. Second, the learning process is well suited for real-time application since the training process is an order of magnitude faster than conventional neural nets. Third, the Fuzzy CMAC can be implemented in high speed, highly parallel hardware. The importance of this research is three-fold. One is that a good traffic incident detection system will help drivers to select an optimum route. The second one is that the system will be able to provide information for efficient dispatching of emergency services. Lastly, it will provide accurate knowledge of existing traffic conditions in order to guide effective on-line traffic controls. © 1998 Elsevier Science B.V. Keywords. Fuzzy logic; Neural networks; Principal Component Analysis; Traffic
1. Introduction Whenever there is a traffic incident such as a car accident or a stalled car in a congested highway, the traffic flow will be seriously affected. The objective of this research is to develop an efficient, real-time traffic incident detection system in order to * Corresponding author. Tel. (301)-590-3155; Fax. (301)-5909414; e-mail:
[email protected]. 1Research supported by US Army Space & Strategic Defense Command under contract DASG60-95-C-0084. 0165-0114/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved Pll S01 6 5 - 0 1 1 4 ( 9 6 ) 0 0 2 2 9 - 1
alleviate traffic jams, reduce travel time for drivers and improve safety. There are quite a n u m b e r of excellent papers on traffic incident detection [4-7, 10-12, 14, 15, 17]. In 4-7, 11, 12, 14, 15, the researchers directly utilize information available from the presence detectors. One major problem in these m e t h o d s is that they usually produce false alarms when traffic compression waves occur. They also do not perform well in light or moderate traffic conditions. In [17], modelbased techniques were used to detect freeway incidents. The basic reasoning is that the use of
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dynamic models that capture traffic characteristics should help in extracting information concerning incidents in which the direct effect on the observables may not be dramatic. Two methods were presented. However, there are many assumptions in these methods that are not true in practice. As admitted by the authors in [17], "none of these assumptions is valid". In this paper, a new nonmodel-based intelligent approach to traffic incident detection is proposed. Our method does not make any assumptions since we do not need a model. The only requirement for our system to work is that we need to place sensors on the highway to measure the traffic flows. This requirement is also necessary to implement Willsky et al. method [17]. The method consists of two stages. In the first stage, a real-time, adaptive, online procedure is used to extract the most significant components of the traffic states that consist of the average velocities and densities of the moving vehicles. The importance of a real-time algorithm is due to the fact that the traffic flow is a time-varying process and incidents occur randomly. In addition, this adaptive procedure provides data compression so that there are fewer inputs to the neural network classifier immediately following the first stage. Consequently, the classification speed will be greatly increased. In the second stage, a new neural network called Fuzzy CMAC (Cerebellar Model Arithmetic Computer) is utilized to identify the traffic incidents. Fuzzy CMAC is an ideal candidate for this purpose for the following reasons. First, the Fuzzy CMAC learning structure is a creative use of fuzzy logic and CMAC-based neural networks. Expert knowledge in terms of linguistic rules can be incorporated into the design. Second, the learning process is well suited for real-time application since the training process is an order of magnitude faster than conventional neural nets. Third, the Fuzzy CMAC can be implemented in high speed, highly parallel hardware. The paper is organized as follows. In Section 2, we present a traffic model which was developed in [10]. Unlike the approach in [10], the main purpose of the model is to generate training and testing data to verify our algorithm. A real-time on-line adaptive algorithm will be described in Section 3. The method was developed by Chen and Chang
[15] and has a very fast convergence speed. Hence it is suitable for real-time applications. We will describe the Fuzzy CMAC network in Section 4. Simulation results will be given in Section 5. Finally, a brief conclusion will be given in Section 6.
2. Traffic model Basically, a traffic model describes the dynamics of traffic flow. The model was developed by Isaksen and Payne [10]. The variables in the dynamic model are spatial mean velocities (v, in miles/h), densities (p, in cars/miles/lane), and flows (¢, in cars/h/lane) over links of the freeway between presence detector locations. This yields a spatially discretized set of coupled equations, dpi
O i - t -- ~i
dt
6xi
dvi dt -
(2.1a)
vi(vi - vi-1) v~(pi) - vl + ~i ½((SXz + 6 x i - O + T
! [.
-]
(2.1b)
where ¢~ = vlpi. Here subscripts are used to denote the link number with which each variable is associated, cSxi is the length of link i, coz represents acceleration noise, and v and T are parameters introduced to model driver response characteristics. Fig. 1 depicts a picture of traffic links. The vC(p) term represents the driver's desired equilibrium speed as a function of the density of traffic. Fig. 2 shows the vC(p) curve which contains three important parameters, vf~e¢is the equilibrium velocity under light traffic conditions; Pfree is the density at which the equilibrium velocity begins to decrease; and Phm is the maximum density of cars that the freeway can hold. The curve between Pfre~ and P jar, is logarithmic, i.e. vf~¢~In [Pjam/P] v~(P) = In [Pjam/PfreJ '
Prro¢~< P ~< Pjam.
(2.2)
Suppose we are modeling M links of the freeway (i = 1 the upstream link, i = M the downstream link), Eq. (2.1) must be modified to account for the
H. Xu et al. / Fuzzy Sets and Systems 93 (1998) 173-183 link i
1
P~
Traffic flow
Fig. 1. Definition of links.
~;free
Pfwe
/gja,n
Fig. 2. T h e v°(p) curve.
boundary conditions. On link 1 the equations are dpa
flow-
dt
vlPl
(2.3a)
6xl
dv, __ v e ( p ) - v,
v I [
dt
T p, U(~-x, ~--~-x2iJ"
T
P2 - Pt
(2.3b)
The variable f l o w in (2.3a) is assumed to be a Poisson arrival process with a specified mean value, which was used to control the overall level of traffic. For link M, a zero density gradient assumption acrossing the last boundary leads to dvM -
dt
-
VM(VM -- VM-1) _+ ve(p~) -- VM + ogM ½(~)XM- 1 -t- ~)M) T
(2.4)
3. Real-time adaptive on-line principal component analysis Principal component analysis (PCA) has been applied to many situations in signal processing, image processing, and pattern recognition. Its main function is to retain the most important characteristics of inputs. It is one of the most general purpose
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feature extraction methods. Feature extraction from complex, high-dimensional inputs is fundamentally important in many practical applications such as pattern recognition, communications, etc. It can eliminate information redundancy and disturbances and allow the most important information to be expressible in lower dimensions. As a result, information storage, processing, and transmission can become easier and efficient. Usually, PCA is done by processing the covariance matrix of the input data off-line. However, there are urgent needs to have an adaptive PCA which has several advantages. First, in practice, some input vectors may contain thousands of elements which makes the procedure of finding eigenvalues of the covariance matrix a numerically ill-conditioned and time-consuming one. Hence, off-line computation of the principal components is only feasible for a relatively small number of elements in the input vector. Second, it is better to have a recursive procedure which can process new input data whenever it is available to the system. Third, the input data may have some time-varying features. It may not be appropriate to use the old data to perform PCA. In other words, the information of the most recent data should be used to update the database. Fourth, in many situations such as military applications, real-time processing of missile sensor and guided missile seeker scenes is most critical. It is much better to have real-time capability for PCA. There are many methods for performing adaptive PCA. The basic idea behind them is to cast the process as a neural network structure [9]. A picture of the network is shown in Fig. 3. A hidden layer is used to compress the input data from a dimension of n to p. The output layer reconstructs the original input data from the compressed data. This network differs from other types of networks such as multilayer perceptron or Hopfield in that efficient unsupervised learning procedures exist to train it. The importance of this is that no desired outputs are needed. The network can automatically adjust its weights. Under certain statistical conditions on the inputs, it can be shown that the networks weights converge to the principal eigenvectors of the inputs and the outputs of the hidden layer converge to the principal components of the inputs [9].
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Inputs x~(k) ~
x,,(k)
~
.
Principal Components ~ ~
Reconstructed inputs y~(k)
w:~
YA k)
Fig. 3. A neural network analogy of the adaptive PCA procedure.
Step 1: Set weight vectors wi(0)e R" (such that I[wi(0)[I 2 ~ 1/2 2 and the estimate of eigenvalues
2i(0) = 6 > 0 for i = 1,2, ... , m. Step 2: Draw a new pattern x(k) at time k with k greater or equal to 1, and present it to the network. Step 3: Calculate the output Vi's V,(k) = w~x(k),
i = 1,2 . . . . ,p.
(3.1)
Step 4: Estimate the eigenvalues In
other
words,
by referring to Fig. 3, Wp = [Wpl Wp2 .., Wpn]T will converge to the Pth eigenvector of the covariance matrix of the input x(k). One immediate application of this network is for information transmission. At the transmitter end, one can compress the transmitted data by using PCA. Then the original signal is reconstructed from the compressed signal at the receiving end. Adaptive PCA concerns the determination of the input-to-hidden layer network weights in Fig. 3. Oja [13] introduced a linear neuron model with constrained Hebbian-type learning which is an unsupervised learning procedure. Sanger [16] extended the procedure to the multi-neuron case to compute the first m principal components. The demerit of this method is that the principal eigenvectors are learnt in a sequential manner, i.e. after the first principal component vector has converged, the procedure will start to learn the second principal component vector. This may be very time consuming and may not be suitable for real-time applications. In [2], a recursive least-squares learning procedure is proposed to perform the PCA. Variable step size is used for updating. The convergence speed is very fast. Chen and Chang [2] also proposed an adaptive learning algorithm which is insensitive to large eigenvalue spread in the covariance matrix. All the eigenvectors are converged almost at the same rate. The reason for this rapid convergence is that the learning rate parameters are selected automatically and adaptively according to the eigenvalues of the input covariance matrix that are estimated during the learning process. Their algorithm consists of seven steps:
2,(k) = L ( k - 1) + 7(k) [(w~x,(k)/[[ w,(k)[I2) - f~i(k - 1)], i = 1, 2, . . . , p.
(3.2)
where xi(k) = x ( k ) - Z~-=lt Vj(k)wj(k). The value of 7(k) is set to be smaller than one and decreased to zero as k approaches ~ . Step 5: Modify the weights w:s wi(k q- 1) = wi(k ) q- qi(k)Vi(k)
j=l where th(k) = fli(k)/2i(k). The value of fldk) is set to be smaller than 2(x/~ - 1) and decreased to zero as k approaches oo. Step 6: Check the length of wi's w,(k + 1) = 1
1
Sx//~(wi(k + 1)/Jlw,(k+ 1)[[, /f I[w,((k+ 1)l]2 > + Mk + 1) ( wdk + l) otherwise. Step 7: Increase the time k by one and go back to Step 2 for the next input pattern until all of the wi's are mutually orthonormal. The proof can be found in [3]. It can be shown that the p weight vectors can converge quickly to the first p principal component vectors at almost the same rate. Compared with Sanger's Generalized Hebbian Algorithm, the method converges quickly to the desired eigenvectors irrespective of the eigenvalue spread in the input covariance matrix while Sanger's method diverges in the large eigenvalue spread case. Similar to Sanger's method, the above learning scheme is also an unsupervised method, i.e. no desired output pattern is needed.
H. Xu et al. / Fuzzy Sets and Systems 93 (1998) 173-183
4. Fuzzy CMACneural network 4.1. Comparison of a (.'MAC network with a fuzzy logic controller (FLC) Two decades ago, a unique neural network model called a Cerebellar Model Arithmetic Computer (CMAC) was established by Albus [1] based on a model of the human memory and neuromuscular control system. The CMAC is a perceptronlike associative memory that performs nonlinear function mapping over a particular region of a function space. A CMAC network has the capability to learn an unknown nonlinear mapping by examples, and to reproduce multiple outputs in response to multiple inputs. Because of its table look-up mechanism, and its hash-code based mapping structure, CMACs are able to cope with high-dimensional input/output applications without severely deteriorating their processing speed and performance. Fuzzy set theory was initially proposed by Zadeh as a tool to model the imprecision that is inherent in human reasoning, especially when dealing with complexity. The fuzzy theory has seen its most widespread application in the area of control. Controllers using control laws specified with fuzzy set theory (or fuzzy logic) are known as fuzzy controllers. Such controllers are easier (relative to nonfuzzy controllers) to design, especially in cases where the laws are nonlinear and the systems are complex. The advantages of a CMAC over a FLC are: • There are very efficient learning laws to update the values of weights based on experience and examples. • There is a random mapping mechanism to reduce the physical memory requirement for multiple input and high-resolution applications. • There exist efficient input encoding schemes for high-dimensional input vectors. The advantages of a FLC over a CMAC are: • It is possible to interpret the implication of weight values using linguistic labels. • The membership functions and the firing strengths contain additional information as to how close the input vector is to each linguistic variable.
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Therefore, the number of input space partitions may be smaller to achieve the same generalization and output smoothness. • The fuzzy rules can take a variety of forms while only numeric values can be associated with CMAC associative memory locations. • There are many methods to construct a fuzzy control knowledge base, such as expert's experience and knowledge.
4.2. Fuzzy CMAC architecture In this section, we will present a new neural network which combines the advantages of both CMAC and FLC. We call this a Fuzzy CMAC network. Intelligent Automation Inc. has successfully applied this network to active vibration control, finger print analysis, and chaotic time-series prediction [8]. Fig. 4 illustrates the architecture of the Fuzzy CMAC. The Fuzzy CMAC inherits the preferred features of arbitrary function approximation, learning, and parallel processing from the original CMAC neural network, and the capability of acquiring and incorporating human knowledge into a system and the capability of processing information based on fuzzy inference rules from fuzzy logic. The combination of a neural network and fuzzy logic yields an advanced intelligent system architecture. At the input stage, the Fuzzy CMAC uses the fuzzification method of a FLC as its input encoding scheme. Fuzzy rules can be assigned to each associative memory cell. These rules may not necessarily have a crisp consequent part. The output generation uses a defuzzification approach which sums weighted outputs of the activated rules based on the firing strengths sl. The overall mapping function of a Fuzzy CMAC can be formalized as M
Z(U) • p~l SpWp
(4.1)
where u = [u~ u2 ..- uN]T is the input vector; wp, p = 1,2 . . . . . M, are the weights of the network; M - - j l if N =- 1 and M = ~ = 2 ( J l - 1) [ I i ; I m t - 1 + j a , f o r N > 1, i = 1,2 . . . . . Nandm~is the number of knot points (refer to Fig. 5) on the ith input dimension. Thejith knot point on the ith input dimension is denoted as ui.j,, jl = 1,2 . . . . . mi.Sp,
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Fuzzy Encodeing
Fire Slrength F u z z y Rasoning Rules
I'----'=Jl~l~'
x:
,--.
~
Associated Memory Cell Vector
S ai
Y
hted Sum
:)I
z •
U2
~
~
ustment
S~,~
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