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Stationary freeway traffic flow modelled by a linear stochastic partial differential equation
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Tnrupn. Rex-& Vol. 268. No. 2. ~9. 115-126. 1992 Prind In Great Brlrun.
STATIONARY FREEWAY TRAFFIC FLOW MODELLED BY A LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATION ELLO WEITS’ Delft University of Technology, The Netherlands (Received 24 May 1590) Abstract-A stochastic model of stationary high density freeway traffic flow is presented. The high density assumption implies the choice of a macroscopic model. The assumption of stationarity allows us to linearize the model equations, but at the same time leads us to adding an appropriate stochastic term to the equation obtained from linearizing the original equations. The result is an infinite-dimensional stochastic differential equation describing the evolution of R(f.x). R(t.x) is the deviation of the density around some fixed value at time I and location x. R(t,x) is a stationary Gaussian process, if the initial value is suitably chosen. The process R&x) is determined by three parameters. Knowledge of these parameters may enable the traffic engineer to characterize the traffic stream as homogeneous or inhomogeneous. The model is tested with data measured on a Dutch freeway.
I. INTRODUCTION
Freeway traffic flow has been studied extensively during the last three decades, mainly because of the practical problems that accompanied the increasing intensity of the traffic. Nevertheless, a unified theoretical approach has not been developed. There are several obvious reasons why there is not one particular approach in modelling freeway traffic flow that has proven itself superior to the others. First, a model can be microscopic or macroscopic; individual vehicles are observed in practice, but is it judicious to incorporate individual behaviour into the model? Second, in microscopic as well as in macroscopic models there are a lot of factors that cannot be modelled exactly, but have to be viewed as random disturbances; it is, however, by no means obvious how this should be done and in most cases introducing stochastic components greatly complicates the analysis of the model. The answer to these questions primarily depends on the class of freeway traffic flow situations one is studying. Every traffic flow model has its own limited area of application. Or, alternatively, every type of freeway traffic flow requires its own mathematical model. The model that is presented below specifically applies to stationary high density multilane freeway traffic flow. In the case of high density multilane freeway traffic flow, the analogy between traffic flow and fluid flow can fruitfully be exploited. The result is a macroscopic model (or continuum model), i.e., we will only deal with the macroscopic variables p&x) (the density of the flow) and U(M) (the flow-velocity) as the relevant dependent variables, r and x denoting time and place, respectively. One of the consequences of the assumption of stationarity is that we concentrate on situations in which the traffic flow passes a stretch of a freeway of fixed length without entrances or exists. Usually a macroscopic model consists of two equations
(1)
‘Supported by the Netherlands Foundation for Technical Research (STW). Present address: Department of Mathematics, Agricultural University of Wageningen. DreYen Laan 4. 6703 HA Wageningcn, The Netherlands.
116
E.
WEITS
The first equation states the ‘conservation of vehicles.’ The second equation is an evolution equation for the velocity. In general, J( - - - ) is a non-linear function of its arguments. The functionfis usually constructed in such a way that the model has equilibrium solutions p = pO,u = u,,, p,, and u,,being constants. See Payne (1971) and Smulders (1988) for some examples of choices of J. A further consequence of the assumption of stationarity is that we can linearize the eqns (1) and (2) around some equilibrium point (pO,vO).This yields (for common choices off)
g=,a’R at
aR ax2
-
‘0 ax’
where R is the deviation of p around po, i.e. R = p - po. K and co are positive constants depending on po. If, however, according to the stationarity-assumption large (deterministic) effects are absent, we must take into account the (smaller) stochastic disturbances. Therefore, we add a stochastic term to the equation: aR -=
at
Ka2R -a.?
g
Co ax
+ noise-term.
(3)
The interpretation of this model equation is as follows: K is a parameter that determines how strong the smoothing tendency of the process counteracting the influence of the noise term; co (approximately) equals the mean velocity of the traffic stream. The noise term accounting for the stochastic disturbances of the traffic stream must be chosen with care. A suitable choice will be presented in section 2. In section 2 we will also specify boundary and initial conditions that complete the description of the model and provide us with a (unique) solution. 2. SPECIFICATION OF THE MODEL
The default choice of the noise term is, of course, white noise in two dimensions (x and t). ‘White’ means that the noise contributions at different times or at different sites are uncorrelated. In the present application, however, we modify the white noise term in order to obtain ‘conservation of vehicles’ on a scale, the length of which will be called S. Twodimensional white noise can be seen as a sum of infinitely many noise components. Each component is characterized by a one-dimensional white noise and a certain wavelength. The modification of the (two-dimensional) white noise term consists in leaving out all components with wavelength larger than S. Thus the disturbances have a range that is bounded by S. We write this modified white noise term as ‘udB(t)/dt,’ where u is a positive parameter controlling the amplitude of the noise and B(t) is a suitable integrated noise process. (‘B stands for ‘Brownian motion’-cf. Appendix 2.) Let us now rewrite eqn (3) as dR(t) -= dt
Kd2R -dx2
c+Cl Co dx
dB(t) -z-’
(4)
We suppressed the ‘x,’ writing R(t) instead of R(t,x). because we will consider R(t) as a random element of a function space. For the same reason ‘d/dx’ replaces ‘a/ax.’ Equation (4) is called a stochastic partial differential equation (see also Appendix 1). The stretch of a freeway on which we observe the traffic flow is represented by the interval [OJ]. We call the parameter L the ‘observation length.’ Two other lengths are of importance, namely S and M. S is called ‘interaction length;’ it determines the maximum range of the disturbances (which are represented by the noise term ‘udB(t)/dt’). We
117
Stationary freeway traffic flow
assume that S 4 M and furthermore that M = mS for m large. M might be called ‘world length;’ we may think of it as the length of a very long circular freeway, from which the vehicles cannot escape. Of course, L has to be somewhere halfway between S and M, we suppose that S 4 L < M. The time interval [0, q is the interval during which we observe (partially) the process. From Fig. 1 we can ‘deduce’ the following boundary conditions:
R(tS’) = R(t,M)
$
(f,O) =
z
and
(f,M).
Furthermore R, is a suitable initial value. If the noise term were absent from eqn (4) the solution would be immediate using semi-group theory, namely R(t) = U,R,, R. E L2[0,M]. Here U, denotes the semi-group generated by the operator K d2/dx2 - c,,d/dx on the Hilbert-space L2[0,M], if we assume the boundary conditions mentioned above. We recall that a semigroup is a family of bounded linear operators, {U,: t z 0)) (acting on, for example, a Hilbert space H such as L2[0,M]) which satisfy 1. u,u, = u,,,, 2. u, = I (I is the identity operator) and 3. the map I 4 U,f is continuous for each f E H. A concise introduction 25). The usefulness of (deterministic) process For (deterministic)
to the theory of semigroups is given in Goldstein (1985, pp. 3a semigroup lies in the fact that it describes the evolution of the R(I) on the basis of the value of R,,. evolution equations of the type
df(t) _
K
dr
d2
zfU)
+ h(O.
where h is a continuous function from [O,oo] into L2[0,M] and f. E L2[0,M], the unique solution reads
s I
f(f)
=
Utfo
o U,_,h(s)ds.
+
(cf. Goldstein 1985, pp. 84 & 85). By analogy we define as the (weak) solution of eqn (4) the solution of the following integral equation:
A
0
t
Fig. 1. The vehicles are driving on a circular road with circumference M. We observe the traffic on a small portion of the circle of length L, L l M. The maximum length of the disturbance S is in turn much smaller than L.
118
E.
WEITS I
R(f)
= U,Ro + Q s o U,-,dB,,
(5)
where R. is suitably chosen. The integral appearing in eqn (5) is a stochastic integral, whose definition is explained in Appendix 2. We make a further step in building the model and assume that R. is stochastic and distributed according to the invariant measure associated with the process. Then R(r) or R&x) is a stationary Gaussian process, whose covariance function is given by
r(A,d)
a2M m 1 = exp( -Xi KlAl) c 4$K is..,, ?
cos 2ri
(6)
for A = (I - s) and d = (x - y); (f,x) and (s,y) are two points in the plane. This formula shows that the covariance function of the density-fluctuation R(f) arises from noise contributions, which have a wavelength less than S. The fact that the counter i starts at m indicates that only the high frequency components of the original white noise have been retained, i.e. the large wavelengths components of the white noise have been omitted. Finally, we let M (and m) tend to infinity. Provide the solution of eqn (4) with the subscript m. If we let m tend to infinity, the process R&,x) converges to a limit process, which we denote by R&x). This limit process is a strictly stationary centered Gaussian process having covariance function
-$
K/Al)
cos(l~, 59dL
It is this limit process that is the model of the density-fluctuations to be used in the next sections. The process R(r.x) has continuous, but non-differentiable sample paths with respect to both I and x. The process is strictly stationary in both coordinates, I and x.
3. INTERPRETATION
OF THE MODEL
The model essentially consists of a description of the fluctuations of the density of a freeway traffic stream as a stationary Gaussian process, denoted by R(r,x); t and x are the time and space variable, respectively. The process is completely specified by its covariance function r(A,t).
r(A,z)
s
“1 = A , 7 exp( -PIAl/2)
cos
where A = a2S/( 4r’K) and /3 = 4r2K/S2. K, Q and S are parameters occurring in the stochastic partial differential equation having the Gaussian process as its stationary limit solution. Alternatively, A, 4, and S may be chosen as the relevant parameters. Indeed, they are the ones that are estimated. The interpretation of these parameters is as follows. A is the squared amplitude of the fluctuations, S is the typical length or range of a fluctuation and P is a damping constant that determines how fast or slowly a certain configuration of fluctuations changes; 6 = 0 would mean that a configuration would travel along the freeway (almost) unchanged. The damping is most easily observed, if one moves along with the traffic stream. From eqn (7) we see that r(A&A)
s-
1 = A , 7 exp( -PlAlI’)dl.
Stationaryfreewaytraffic flow
119
of S is eliminated from the covariance function. In a sense c,, is also an unknown parameter. It is possible, however, to measure it directly, since it equals the mean velocity of the traffic stream. Large values of A and of /3 as well as small values of S characterize a wildly changing process. A large value of A means that the amplitude of the fluctuations is large. A large value of fl indicates that the fluctuations are rapidly changing in time, i.e. the fluctuations occur rather spontaneously; a small value of S says that the fluctuations are rapidly changing in the space direction. It is clear that an increasing value of A implies that the traffic becomes more inhomogeneous. But inhomogeneity also increases, if, for constant A, the parameter 0 grows large or if the parameter S grows small, because in that case on a stretch of a freeway of fixed length and during a fixed time interval more and more fluctuations of the same average amplitude occur. As the fluctuations become more numerous, the probability of a particularly large positive fluctuation (an extreme value) increases. We claim that the probability of an extreme value is a very relevant notion as far as the occurrence of congestion is concerned. Thus we propose to interpret ‘homogeneity’ not only in terms of the amplitude of the fluctuations, but also in terms of the rapidity of change of the fluctuations. Before ending this section we briefly explain why we switch from the parameters K, u and S to the parameters A, 8, and S. The reason is that it is convenient to carry out the estimation procedure in two steps. First we obtain estimated covariances at a number of points in the time-space rectangle [O,T] x [O,t] and, secondly, we apply to these data a nonlinear regression procedure for which eqn (7) is the regression function. This procedure has two advantages. One advantage is that in an early stage of the analysis we obtain a very significant compression of the amount of measured data. The other one is the fact that the parameter S shows up in the regression function, whereas in a regression based on the original equation eqn (4) this parameter is located in the error term. Once we have decided to estimate the parameters via the estimated covariances of the process it is natural to consider A, 0, and S instead of K, u, and S. The change of point of view also has some disadvantages. In the first place we must separate the density fluctuations from the mean density. Note that in practice the sum of these two quantities, the total density, is observed. Further, the error terms appearing in the regression problem are strongly correlated. It turns out to be difficult to deal with these facts in an adequate way. We will come back to these problems in the next section. The influence
4. APPLICATION
TO TRAFFIC
DATA
4.1. Description of the data The raw data consist of 104,936 records, each record containing the time a vehicle passes a detector station, its speed and the number of the detector. Four detectors make up one detector station. There is one detector for each lane; three lanes are for regular use and the fourth is the emergency lane. The detector stations are situated on the western carriageway of the freeway Al3 near the city of Delft (at the locations A13W9.0 up to A13W16.5). They are numbered from 1 up to and including 16. The observations were recorded on September 27, 1989 in connection with on-ramp metering experiments. The Transportation and Traffic Engineering Division (DVK) of the institute ‘Rijkswaterstaat’ of the Dutch Ministry of Transport has granted the use of the data on behalf of this research. The observations used here were made between 15.30 h and 16.40 h. (The clocktime ran from 8559.493 to 12758.709 seconds.) During this period of the day the traffic on the freeway can usually be described as high density (and more or less) stationary freeway traffic. In order to eliminate as much as possible the influence of the on- and off-ramps only observations coming from the detector stations 8 up to and including 15 (which cover the freeway between Delft-Zuid and Zestienhoven-locations A13W12.5 up to A13W16.0) are considered. Thus about half of the records are discarded. It is clear that the data only permit estimation of covariances of the form r(A.0). Unfortunately, we lack measurements of the density along the direction L = cd. lR(B)26:2-C
I20
E.
WEITS
4.2. Preparation of the data for the analysis Once we have decided on these preliminary questions, some more delicate decisions have to be made. namely (i) how is the density of the traffic defined in terms of the observations? (ii) how do we extract the mean density and the fluctuations from the observed sum? (iii) what must be the length of the time interval on which the estimates of the parameters are based and? (iv) how do we determine the constant c,? We have chosen to define the density at a fixed x-value (the position of a detector station) as follows. At each instant we count the number of vehicles within a 100 metres distance, upstream or downstream. This number can be non-integer as each vehicle is thought to be spread out, backwards and forwards, over half of the following distances. Denote at fixed t the position of some vehicle of interest by xJt) and its velocity by v,,,(t). The index k stands for the lane of the vehicle; the index i is now 1 for a vehicle that is about to enter the 200 metres interval, it is n, for the vehicle that last left the interval (on lane k). Thus the density (vehicles per km per lane) equals #($
_ 3 + &.I + 100 + 100 - “‘Q-J. =I
xk,2
-
xk.l
xk.n,
-
xk,“k-I
Of course, we do not know exactly the positions of the vehicles. They must be estimated assuming that the vehicles do not change their velocities during the time that they are near the detector station, so that the position of a particular vehicle relative to the detector station equals the time interval between time t and the passing time multiplied by its velocity. Clearly, the constant velocity assumption limits the length of the interval. The length of 100 metres is chosen as it corresponds to a driving time of about 4 seconds. In the absence of incidents a period of 4 seconds seems short enough to guarantee constant velocities. Further, a length of 200 meters seems a reasonable choice of an increment of the space variable. The next question concerns the extraction of the mean density and the fluctuations from the observed sum. To get an idea of how the density process behaves, the observed process at detector station no. 8 is shown in Fig. 2 for a period of half an hour. Apart from the short range fluctuations a medium range fluctuation of the mean density can be discerned with a typical time of, say, 80 seconds. We have chosen, therefore, to calculate the mean density at a particular detector station as the moving average of the observed process using a window of 80 seconds. This choice immediately implied another decision, namely not to combine the data of adjacent detector stations as far as the mean density is concerned. The mean density is changing so fast that we cannot assume it to be constant over the length of the freeway covered by the detector stations. But then the question arises, whether or not we also should try to estimate the parameters that determine the fluctuation process for each detector station separately. As the answer to this question is not clear a priori, both possibilities can be pursued. In order to keep the length of this article within reasonable limits we will below only show the results of the analysis for the combined data. More results can be found in Weits (1990). Further, we have to decide on the length of the time interval on which the estimates of the parameters are based. Some experimenting with the data coming from detector station no. 8 shows that a time interval of 15 minutes is appropriate if we consider only one detector station at a time. For the case of combined data a time interval of 10 minutes seems to contain sufficient information. Finally, c0 is determined by taking every second of the ‘estimation interval’ the mean of the velocities of all vehicles present in the 200 metres interval. Now, we do not consider the vehicles as spread out, instead they are treated as ‘point vehicles.’ The constant is set equal to the mean of these mean velocities over the appropriate time interval (and, possibly, the relevant set of detector stations). Note that the slower vehicles are counted more often than the faster ones. This feature compensates for the fact that simply taking the mean of all passing velocities overestimates the mean velocity co, as a fast car is more likely to be detected than a slow one.
121
Stationary freeway traffic flow 40
detector station I
35
30 25 . 20 ’ 16
10 5 .
:650
I750
:I50
1950
9050
9150
9250
9350
9450
9550
lime
0
i
9550
9650
9760
9860
9950
10060
10150 10150
10350 10460
iims Fig. 2. The evolution of the density (vehicles per km per lane) at detector station no. 8 situated on the western carriageway of freeway Al3. The time is given in seconds.
4.3. Estimation of the parameters for combined data
Table 1 and Fig. 3 present the results of the estimation of the parameters on the basis of the combined data (from detector stations nos. 8-15). Although the theoretical covariance matrix of the error terms appearing in the regression problem can be calculated to a good approximation, it was not used simply because
Table I. Regression results from 6 data sets (combined data from all detector stations). c, (km/s)
:: 3. 4. 5. 6.
0.0257 0.0279 0.028 I 0.0284 0.0274 0.0278
The abbreviation
A (km-*)
B(s-‘)
S
s.d. of A
23.801 34.666 36.792 30.458 23.091 27.060
0.0072 0.0045 0 0.0126 0.0018 0.0081
2.357 1.812 2.054 2.077 2.689 1.745
I1.524 .453 1.438
s.d. stands for standard
deviation
1.262 1.191 1.712
s.d. of j3
0.0025 0.0014 0 0.0019 0.0014 0.0027
s.d. of S
0.054 0.044 0.034 0.049 0.062 0.049
E.
122
WEITS
40
Irn-““’
30 20 Cm
10 .-a ..".._ X...
0 '.. -10
-200
.** . ...a
-20 ~ 20
100
40
0
20
40
60
60
100
time
lime
,w,,ill,, ru,,
40 30 It
_l;iJ-b
CO”
Co” 10
-20+
0
100
20
40
60
60
too
lime
40
,101, ,101 r.r,.
30
!. 20 *
r..n,.
20 ....
. *.
...
. ..d
..
.
0
20
40 uma
60
60
100
-204 0
...a ..*
: . ...*
4o#L -10
-10 -20 k 0
11111
.cov *o
c0v IO 0
11‘11
30.
20
40
60
60
100
time
Fig. 3. The estimated covarianccs of the density-fluctuations (9 for all detector stations together with ordinary least squares fits (solid line). The time is given in seconds.
the regression algorithm (the Gauss-Newton method) failed to converge when it used this matrix. With a view to the fact that this weighted regression algorithm performed well on data obtained from simulations, we conclude that the theoretical covariance matrix does not completely catch all features of the covariances of the errors terms. Apparently the range of the correlations of the errors is smaller than suggested by the theoretical covariante matrix. Also the weighted regression procedure is computer-time consuming and therefore expensive. For these reasons we have chosen to use the ordinary least squares criterion. We will now make some remarks concerning the results. 1. We see that the estimates of the parameter fl are (very) small. (In one case the parameter is virtually zero.) When the value of /3 is too small (say below O.OOl), problems arise in computing accurately the values of the regression function and its derivatives. Therefore, if /3 is too small, we set it equal to zero and estimate the remaining parameters using an alternative regression procedure from which B is absent. 2. The value of S is closely related to the value of co. If a wrong value of co is taken, this only affects the estimation of S. 3. In the neighborhood of I = 0 the values of P tend to be reduced. The effect is certainly caused by the way the density is calculated. The assumption that the vehicles pass through the 200 metres interval with constant velocity causes some smoothing of the values of the density. 4. The pictures of the estimated covariances of the process suggest that there are two dips
stali0Mt-y
frawoy traffic flow
123
for r * 30 seconds and two peaks for r I 60 seconds. One might suggest that this is due to different behaviour of the left and the middle lane and the right lane (which is characterized by much heavy traffic). 5. A rather delicate question is the estimation of the standard deviations of the estimates of the parameters. The estimates given in Table 1 are based on a result on the asymp totic distribution of the parameters under the assumption that the errors are uncorrelated. These values are too small, because the correlation of the errors makes it possible that a large part of each error is absorbed by the estimated regression function, so that the uncertainty of the estimates of the parameters increases. Experiments with artificial data sets suggest that parameters A and S can be estimated with an accuracy of about 10%. Estimation of parameter /3 is more difficult. If the parameter is very small, little more than that can be said. If it is somewhat larger, say about 0.01 (for S = 2.5), an accuracy of 30% is likely. The larger /.Iis, the more accurately it can be determined. 6. The normality of the process R has been tested by sampling once every hundred seconds from the data at detector stations nos. 11 and 14. It is assumed that these values are (almost) independent. Normal probability plots suggest that these data are approximately normally distributed. 5. CONCLUSIONS
AND REMARKS
On the basis of the data (from detector station no. 8) the model has been adapted in two ways. First we allowed for medium-range variations of the mean density and, secondly, we used the ordinary least squares criterion in the estimation of the parameters. The adapted model describes reasonably well the observed covariances of the density. We note that the fact that the regression procedures almost always converged indicates that the amount of damping present in the observed covariances is covered by the model. In the one exceptional case, putting 0 = 0 yields acceptable results. We conclude that the observed covariances obey the constraint that B L 0. It is probably the most successful feature of the model that it reveals the importance of the parameter @and, at the same time, provides a means to determine the value of this ‘hidden’ parameter. Whereas the squared amplitude of the fluctuations A as well as the length S are readily deduced from the data, 6 is not easily determined. If we would have observed the density along with the traffic stream, so that estimates of r(A,c&) could be obtained, the influence of the parameter /3 would be less obscured, since in that case there is no interference with the parameter S. Therefore, the model’s usefulness lies in the fact that it is able to determine the approximate values of the parameter /3, especially in the case of the rather limited observations at detector stations. Perhaps the most important open question is the range of values of the parameter /3. As the data used here were gathered in pre-peak-hour period, we conjecture that in the middle of the peak-hour much larger values of the parameter occur implying that the values found here can all be said to be (very) close to zero. If this guess is confirmed, one of the consequences will be that the parameter B can be estimated with sufficient precision when every it is important to know its value, i.e. when it is large. The following application is proposed. We interpret the phenomenon of large total density- which corresponds to a local concentration of vehicles-as a situation in which there is a large probability of congestion. Clearly. this phenomenon depends on the value of the mean density and on the value of the parameters. Of these the parameters A and fl are the most important ones. Large values of the mean density and the parameters A and B imply a large probability of an extreme value above some crucial level. A small value of S has the same effect, but it seems that this parameter is not likely to change very much. If the total density exceeds the crucial level the flow is likely to break down. Of course, the value of this crucial level will depend on the mean density. The parameters can be determined as indicated above. It does not make much difference whether data from one or several detector stations are used for the estimation of the parameters.
E.
124
WEITS
This sketch does not lead to quantitative statements. Therefore, the critical values as well as the range of the parameters under various circumstances have to be determined from studying a large amount of data. Once the critical values of the parameters have been found, the state of the traffic flow can be assessed (in an on-line configuration) and appropriate measures to ‘regularize’ the stream be taken, for example with the help of a signalling system. Acknowledgemenls--I thank Pict Grocneboom for his support in developing the model of freeway traffic flow presented in this article. Also I thank the editor and the referee for their valuable comments.
REFERENCES Curtain R. F. (1977) Stochastic evolution equations with general white noise disturbance. J. Murh. Anul. Appl.. 40.570-595.
Funaki T. (1983) Random motions of strings and related stochastic evolution quations. Nugoyu Muth. JournaL 89. 129-193. Goldstein J. A. (1985) Semigroups 01 Lineur Operufors & Applicufions. Oxford University Press. New York, pp. 3-25 lkeda N. and Watanabe S. (1981) Stochastic Dif/erential Equations und Diffusion Processes. North-Holland, Elscvicr/Kodansha. Tokyo. It6 K. (1984) Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia. Payne, Harold J. (1971) Models of freeway traffic and control. In Marhemuficul Models of Public Systems (Simulation Council Proc.). 1. 51-61. Smuldcrs S. A. (1988) Confrol o/f+eewuy TrufJc Flow. Thesis, Twente University of Technology, Enschcde. Weits E. A. G. (1990) A Stochustic Heuf Equafion for Freeway Trq/jic Ffow. Thesis, Delft University of Technology, Delft, The Netherlands.
APPENDICES
There are two appendices to the article. The first one gives some extra (mathematical) details of the specification of the model (see Section 2). The second appendix supplies a more or less rigorous probabilistic background for the expositions of Section 2 as well as Appendix I. Appendix
I
In this appendix we discuss some details concerning the semigroup U,. white noise in two dimensions and its modifications used in the present model and, thirdly, the solution of eqn (4). The semi-group U, generated by the operator A I K d’/dx’ - c&/dx on the Hilbcrt-space L’[O.M], with domain ‘D(A) = cfe L*[O.M]:/‘E
L’(O.M],f(O)
= f(M),f’(O)
=f’(M)).
can be represented as M U,f = s D q(r.x.y)f(y)dy q(rJ*Y)
=
IM
+
2 [6,(X -
forf e L’[O,M]
and
Cd)&(Y) + 6(x - corNdY)l exp( -VW.
,.I
where 6,(x) = asin(Zzix/M) and G,(x) = ~cos(Z~ix/M),h, = 4r’i’/M’. Set $.,0(x) = I/&. Note that (0,: i z 1; $,: i L 0) is an orthonormal basis of L’[O,M]. Secondly we consider the noise term. White noise on [O,T] x [O.M] can be represented as the time derivative of the cylindrical Brownian motion
where (b,‘:i z I; b,‘:i L 0) is a family of independent standard one-dimensional Brownian motions (see Curtain, 1977 and Appendix 2 for more information). In order to guarantee conservation of vehicles on a scale of length S, we modify the white noise term and set B(r) = c
b,‘(r)&, + b,‘(W,.
I2.W
We leave out all components with i c m. This procedure retains the white noise property for the timecoordinate, whereas the property is only locally retained for the space-coordinate. The (weak) solution of cqn (4) is defined as the solution of the integral qn (5). where R0 is chosen such that R0 E UE L’[O.M]: < f,b, > = 0. c f.& > = 0. vi s m - 1) . Equation (4) is called a stochastic partial
Stationary freeway traffic flow
125
differential equation. It is a parriol differential quation because of the appearance of partial derivatives and it is srochasric differential equation. because the differential equation is interpreted in terms of a stochastic integral. The stochastic integral used here is discussed in Appendix 2. If we assume that R,, is stochastic and distributed according to the invariant measure associated with the process we can represent the solution of qn (4) as
R(M)
= 2 Ia:(rM,(x 1-m
- cd) + a:(r)d,(x
- cot)].
where {a,‘. a,‘: i L m} is a family of independent Gaussian processeshaving covariance function
for A = It - ~1. Of course, the ‘c’ and the ‘r’ refer to ‘cosine’ and ‘sine’, respectively. From this representation it follows that R(r.x) is a ccntred Gaussian process and also that its covariance function is given by eqn (6). Appendir 2
We give a short account of the definition of a stochastic integral on a real separable Hubert space H with inner product (a;) and norm 11.11.In this article we have always set H = L’[O,M]. The one-dimensional stochastic integral (with respect to Brownian motion) corresponds to the case H = R . This so-called Ito-integral is the basis of the following survey. For information on the one-dimensional stochastic integral and the corresponding stochastic (ordinary) differential equation 1 refer to the book by Ikeda and Watanabe (1981). Let (ft.5.P) be a complete probability space with a right-continuous increasing family of sub-o-algebra’s (3, : I L 0). each containing all P-null sets. We say that the set-up (tM,( g,}.P) satisfies the usual conditions. H denotes a real, separable Hilbert space with inner product (* ;) and norm 11.I(. Lo(fi.5.P) is the space of all real random variables on (tI.S.P). Identifying X and Y E f.,,(n) if P(X = Y) = 1. WChave that endowed with metric d(X,Y) w &( IX - Y] I\ 1) L,(Q) is a FrCchet space. The next three definitions have been taken from an article by Funaki (1983). Definirion 1. A linear random funcrional on H is a linear map from H IO Ldll.5.P). We call a family of linear random functionals {B,:r 2 0) on H a cylindrical Brownian morion on H if ir sarisjies rhe following condirion: for every x a H (x + 0). r - B,(x)/]]xl] is a standard Brownian morion (adapted ro rhe filrrarion of /T;‘he case that H = L’[O.M] the time derivative of the cylindrical Brownian motion can be identified with white noise in IWO dimensions. Specifically, for 0 5 I, S rz. x, s x,. 0 s s, s sz and y, s yr it holds that
which allows us to identify two-dimensional white noise integrated over a small rectangle (r,.r,] x (x,.xJ with the random variable B, (I,, ,,,*,) - B,,( 1,,,+t ) and, equivalently, white noise integrated over (s,.s,] x (y,yJ withB~,tt,,J21) - B,,(~,,A). We will now use the s&arability of H; consider a complete orthonormal system in H: (e,: i z 1). Let f be an arbitrary element of H:f = Z, up,. As a consequence of the orthonormality (B,(e,) :i z I) is a family of independent standard one-dimensional Brownian motions, so that we can rewrite B,(f) = Z:., aB,(e,) as B,o-I = r;:., (f.e,) b,(r). where (6,(r) : i 2 I } is a family of independent standard one-dimensional Brownian motions. Formally, we write
4 = ,t e,b,(O.
(9)
Definirion 2. Ler A’([O. T] x Cl.H) be rhe Hitbert space of all H-valued, I,-odapred and measurable functions f(r.4 satisfying E ]: ]If(r)]]’ dr < 0. For every f B A’([O, TJ x D.H) we define
s r
o (lW.dB,) - ,t
5:(lW.e,)dB,k).
where {e,: i t I } is a complere orrhonormal sysrem in H. Definition 3. Ler &:I (H) denote rhe Hilberr space of Hilbert-Schmidr operarors on H wirh norm I]* llHsand let A’([O. Tj x O.CJH)) be rhe Hilberr space of all C/H)-valued. B,-adapred and measurable funcrions F(r.w) sarisfying E 1: ]]F(r)]]ns’dr < w. For every F E Al@. TJ x D, L/H)) an H-valued srochasric inregral ]: F(r)dB, is defined by the following equality:
where F*(r) is rhe adjoinr of F(r). By lineariry ir is enough lo let f run rhrough a orrhonormal basis of H. Within the framework established by these definitions the stochastic integral appearing in eqn (5) is interpreted as
126
E.
WEITS
s I
o
W-Ad&
where Q is the projection operator on the subspace of L’[Om given by cf c L*[O.M]: < f,e, > = 0, vi s m - 1) . The symbol B, here denotes the cylindrical Brownian motion as defined above. The modification of the noise term is taken care of by the operator Q. We notice that U,_,Q is a Hlbcrt-Schmidt operator for all s c I
IlLQll~s’ Recall that A, = 4di*/M’.
= 2 c exp( -tA,K(r ram
- 5)).
Define for c. > 0 r-e” I(f - c,) - s o U,-,QdB,.
Suppose that e. 1 0 as n tends to infinity. We see that, for I > n. EMr
- e.] - Itr -
f,)IIZ
t-e,
=
I
2 ,_,~,~exp(-2A,Ktr
5 c+ #2rn
,
- exP(-Wf,
- s]]ds -
f,]))
= c ; (expf-2&c,) #*aI , s
C(f.
-
- e~~(-2h,~.)]
f,)e.
We conclude that the sequence I( I - e.) is converging in the space f.*(O,S.P)
as z. 1 0.
Tnrupn. Rex-& Vol. 268. No. 2. ~9. 115-126. 1992 Prind In Great Brlrun.
STATIONARY FREEWAY TRAFFIC FLOW MODELLED BY A LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATION ELLO WEITS’ Delft University of Technology, The Netherlands (Received 24 May 1590) Abstract-A stochastic model of stationary high density freeway traffic flow is presented. The high density assumption implies the choice of a macroscopic model. The assumption of stationarity allows us to linearize the model equations, but at the same time leads us to adding an appropriate stochastic term to the equation obtained from linearizing the original equations. The result is an infinite-dimensional stochastic differential equation describing the evolution of R(f.x). R(t.x) is the deviation of the density around some fixed value at time I and location x. R(t,x) is a stationary Gaussian process, if the initial value is suitably chosen. The process R&x) is determined by three parameters. Knowledge of these parameters may enable the traffic engineer to characterize the traffic stream as homogeneous or inhomogeneous. The model is tested with data measured on a Dutch freeway.
I. INTRODUCTION
Freeway traffic flow has been studied extensively during the last three decades, mainly because of the practical problems that accompanied the increasing intensity of the traffic. Nevertheless, a unified theoretical approach has not been developed. There are several obvious reasons why there is not one particular approach in modelling freeway traffic flow that has proven itself superior to the others. First, a model can be microscopic or macroscopic; individual vehicles are observed in practice, but is it judicious to incorporate individual behaviour into the model? Second, in microscopic as well as in macroscopic models there are a lot of factors that cannot be modelled exactly, but have to be viewed as random disturbances; it is, however, by no means obvious how this should be done and in most cases introducing stochastic components greatly complicates the analysis of the model. The answer to these questions primarily depends on the class of freeway traffic flow situations one is studying. Every traffic flow model has its own limited area of application. Or, alternatively, every type of freeway traffic flow requires its own mathematical model. The model that is presented below specifically applies to stationary high density multilane freeway traffic flow. In the case of high density multilane freeway traffic flow, the analogy between traffic flow and fluid flow can fruitfully be exploited. The result is a macroscopic model (or continuum model), i.e., we will only deal with the macroscopic variables p&x) (the density of the flow) and U(M) (the flow-velocity) as the relevant dependent variables, r and x denoting time and place, respectively. One of the consequences of the assumption of stationarity is that we concentrate on situations in which the traffic flow passes a stretch of a freeway of fixed length without entrances or exists. Usually a macroscopic model consists of two equations
(1)
‘Supported by the Netherlands Foundation for Technical Research (STW). Present address: Department of Mathematics, Agricultural University of Wageningen. DreYen Laan 4. 6703 HA Wageningcn, The Netherlands.
116
E.
WEITS
The first equation states the ‘conservation of vehicles.’ The second equation is an evolution equation for the velocity. In general, J( - - - ) is a non-linear function of its arguments. The functionfis usually constructed in such a way that the model has equilibrium solutions p = pO,u = u,,, p,, and u,,being constants. See Payne (1971) and Smulders (1988) for some examples of choices of J. A further consequence of the assumption of stationarity is that we can linearize the eqns (1) and (2) around some equilibrium point (pO,vO).This yields (for common choices off)
g=,a’R at
aR ax2
-
‘0 ax’
where R is the deviation of p around po, i.e. R = p - po. K and co are positive constants depending on po. If, however, according to the stationarity-assumption large (deterministic) effects are absent, we must take into account the (smaller) stochastic disturbances. Therefore, we add a stochastic term to the equation: aR -=
at
Ka2R -a.?
g
Co ax
+ noise-term.
(3)
The interpretation of this model equation is as follows: K is a parameter that determines how strong the smoothing tendency of the process counteracting the influence of the noise term; co (approximately) equals the mean velocity of the traffic stream. The noise term accounting for the stochastic disturbances of the traffic stream must be chosen with care. A suitable choice will be presented in section 2. In section 2 we will also specify boundary and initial conditions that complete the description of the model and provide us with a (unique) solution. 2. SPECIFICATION OF THE MODEL
The default choice of the noise term is, of course, white noise in two dimensions (x and t). ‘White’ means that the noise contributions at different times or at different sites are uncorrelated. In the present application, however, we modify the white noise term in order to obtain ‘conservation of vehicles’ on a scale, the length of which will be called S. Twodimensional white noise can be seen as a sum of infinitely many noise components. Each component is characterized by a one-dimensional white noise and a certain wavelength. The modification of the (two-dimensional) white noise term consists in leaving out all components with wavelength larger than S. Thus the disturbances have a range that is bounded by S. We write this modified white noise term as ‘udB(t)/dt,’ where u is a positive parameter controlling the amplitude of the noise and B(t) is a suitable integrated noise process. (‘B stands for ‘Brownian motion’-cf. Appendix 2.) Let us now rewrite eqn (3) as dR(t) -= dt
Kd2R -dx2
c+Cl Co dx
dB(t) -z-’
(4)
We suppressed the ‘x,’ writing R(t) instead of R(t,x). because we will consider R(t) as a random element of a function space. For the same reason ‘d/dx’ replaces ‘a/ax.’ Equation (4) is called a stochastic partial differential equation (see also Appendix 1). The stretch of a freeway on which we observe the traffic flow is represented by the interval [OJ]. We call the parameter L the ‘observation length.’ Two other lengths are of importance, namely S and M. S is called ‘interaction length;’ it determines the maximum range of the disturbances (which are represented by the noise term ‘udB(t)/dt’). We
117
Stationary freeway traffic flow
assume that S 4 M and furthermore that M = mS for m large. M might be called ‘world length;’ we may think of it as the length of a very long circular freeway, from which the vehicles cannot escape. Of course, L has to be somewhere halfway between S and M, we suppose that S 4 L < M. The time interval [0, q is the interval during which we observe (partially) the process. From Fig. 1 we can ‘deduce’ the following boundary conditions:
R(tS’) = R(t,M)
$
(f,O) =
z
and
(f,M).
Furthermore R, is a suitable initial value. If the noise term were absent from eqn (4) the solution would be immediate using semi-group theory, namely R(t) = U,R,, R. E L2[0,M]. Here U, denotes the semi-group generated by the operator K d2/dx2 - c,,d/dx on the Hilbert-space L2[0,M], if we assume the boundary conditions mentioned above. We recall that a semigroup is a family of bounded linear operators, {U,: t z 0)) (acting on, for example, a Hilbert space H such as L2[0,M]) which satisfy 1. u,u, = u,,,, 2. u, = I (I is the identity operator) and 3. the map I 4 U,f is continuous for each f E H. A concise introduction 25). The usefulness of (deterministic) process For (deterministic)
to the theory of semigroups is given in Goldstein (1985, pp. 3a semigroup lies in the fact that it describes the evolution of the R(I) on the basis of the value of R,,. evolution equations of the type
df(t) _
K
dr
d2
zfU)
+ h(O.
where h is a continuous function from [O,oo] into L2[0,M] and f. E L2[0,M], the unique solution reads
s I
f(f)
=
Utfo
o U,_,h(s)ds.
+
(cf. Goldstein 1985, pp. 84 & 85). By analogy we define as the (weak) solution of eqn (4) the solution of the following integral equation:
A
0
t
Fig. 1. The vehicles are driving on a circular road with circumference M. We observe the traffic on a small portion of the circle of length L, L l M. The maximum length of the disturbance S is in turn much smaller than L.
118
E.
WEITS I
R(f)
= U,Ro + Q s o U,-,dB,,
(5)
where R. is suitably chosen. The integral appearing in eqn (5) is a stochastic integral, whose definition is explained in Appendix 2. We make a further step in building the model and assume that R. is stochastic and distributed according to the invariant measure associated with the process. Then R(r) or R&x) is a stationary Gaussian process, whose covariance function is given by
r(A,d)
a2M m 1 = exp( -Xi KlAl) c 4$K is..,, ?
cos 2ri
(6)
for A = (I - s) and d = (x - y); (f,x) and (s,y) are two points in the plane. This formula shows that the covariance function of the density-fluctuation R(f) arises from noise contributions, which have a wavelength less than S. The fact that the counter i starts at m indicates that only the high frequency components of the original white noise have been retained, i.e. the large wavelengths components of the white noise have been omitted. Finally, we let M (and m) tend to infinity. Provide the solution of eqn (4) with the subscript m. If we let m tend to infinity, the process R&,x) converges to a limit process, which we denote by R&x). This limit process is a strictly stationary centered Gaussian process having covariance function
-$
K/Al)
cos(l~, 59dL
It is this limit process that is the model of the density-fluctuations to be used in the next sections. The process R(r.x) has continuous, but non-differentiable sample paths with respect to both I and x. The process is strictly stationary in both coordinates, I and x.
3. INTERPRETATION
OF THE MODEL
The model essentially consists of a description of the fluctuations of the density of a freeway traffic stream as a stationary Gaussian process, denoted by R(r,x); t and x are the time and space variable, respectively. The process is completely specified by its covariance function r(A,t).
r(A,z)
s
“1 = A , 7 exp( -PIAl/2)
cos
where A = a2S/( 4r’K) and /3 = 4r2K/S2. K, Q and S are parameters occurring in the stochastic partial differential equation having the Gaussian process as its stationary limit solution. Alternatively, A, 4, and S may be chosen as the relevant parameters. Indeed, they are the ones that are estimated. The interpretation of these parameters is as follows. A is the squared amplitude of the fluctuations, S is the typical length or range of a fluctuation and P is a damping constant that determines how fast or slowly a certain configuration of fluctuations changes; 6 = 0 would mean that a configuration would travel along the freeway (almost) unchanged. The damping is most easily observed, if one moves along with the traffic stream. From eqn (7) we see that r(A&A)
s-
1 = A , 7 exp( -PlAlI’)dl.
Stationaryfreewaytraffic flow
119
of S is eliminated from the covariance function. In a sense c,, is also an unknown parameter. It is possible, however, to measure it directly, since it equals the mean velocity of the traffic stream. Large values of A and of /3 as well as small values of S characterize a wildly changing process. A large value of A means that the amplitude of the fluctuations is large. A large value of fl indicates that the fluctuations are rapidly changing in time, i.e. the fluctuations occur rather spontaneously; a small value of S says that the fluctuations are rapidly changing in the space direction. It is clear that an increasing value of A implies that the traffic becomes more inhomogeneous. But inhomogeneity also increases, if, for constant A, the parameter 0 grows large or if the parameter S grows small, because in that case on a stretch of a freeway of fixed length and during a fixed time interval more and more fluctuations of the same average amplitude occur. As the fluctuations become more numerous, the probability of a particularly large positive fluctuation (an extreme value) increases. We claim that the probability of an extreme value is a very relevant notion as far as the occurrence of congestion is concerned. Thus we propose to interpret ‘homogeneity’ not only in terms of the amplitude of the fluctuations, but also in terms of the rapidity of change of the fluctuations. Before ending this section we briefly explain why we switch from the parameters K, u and S to the parameters A, 8, and S. The reason is that it is convenient to carry out the estimation procedure in two steps. First we obtain estimated covariances at a number of points in the time-space rectangle [O,T] x [O,t] and, secondly, we apply to these data a nonlinear regression procedure for which eqn (7) is the regression function. This procedure has two advantages. One advantage is that in an early stage of the analysis we obtain a very significant compression of the amount of measured data. The other one is the fact that the parameter S shows up in the regression function, whereas in a regression based on the original equation eqn (4) this parameter is located in the error term. Once we have decided to estimate the parameters via the estimated covariances of the process it is natural to consider A, 0, and S instead of K, u, and S. The change of point of view also has some disadvantages. In the first place we must separate the density fluctuations from the mean density. Note that in practice the sum of these two quantities, the total density, is observed. Further, the error terms appearing in the regression problem are strongly correlated. It turns out to be difficult to deal with these facts in an adequate way. We will come back to these problems in the next section. The influence
4. APPLICATION
TO TRAFFIC
DATA
4.1. Description of the data The raw data consist of 104,936 records, each record containing the time a vehicle passes a detector station, its speed and the number of the detector. Four detectors make up one detector station. There is one detector for each lane; three lanes are for regular use and the fourth is the emergency lane. The detector stations are situated on the western carriageway of the freeway Al3 near the city of Delft (at the locations A13W9.0 up to A13W16.5). They are numbered from 1 up to and including 16. The observations were recorded on September 27, 1989 in connection with on-ramp metering experiments. The Transportation and Traffic Engineering Division (DVK) of the institute ‘Rijkswaterstaat’ of the Dutch Ministry of Transport has granted the use of the data on behalf of this research. The observations used here were made between 15.30 h and 16.40 h. (The clocktime ran from 8559.493 to 12758.709 seconds.) During this period of the day the traffic on the freeway can usually be described as high density (and more or less) stationary freeway traffic. In order to eliminate as much as possible the influence of the on- and off-ramps only observations coming from the detector stations 8 up to and including 15 (which cover the freeway between Delft-Zuid and Zestienhoven-locations A13W12.5 up to A13W16.0) are considered. Thus about half of the records are discarded. It is clear that the data only permit estimation of covariances of the form r(A.0). Unfortunately, we lack measurements of the density along the direction L = cd. lR(B)26:2-C
I20
E.
WEITS
4.2. Preparation of the data for the analysis Once we have decided on these preliminary questions, some more delicate decisions have to be made. namely (i) how is the density of the traffic defined in terms of the observations? (ii) how do we extract the mean density and the fluctuations from the observed sum? (iii) what must be the length of the time interval on which the estimates of the parameters are based and? (iv) how do we determine the constant c,? We have chosen to define the density at a fixed x-value (the position of a detector station) as follows. At each instant we count the number of vehicles within a 100 metres distance, upstream or downstream. This number can be non-integer as each vehicle is thought to be spread out, backwards and forwards, over half of the following distances. Denote at fixed t the position of some vehicle of interest by xJt) and its velocity by v,,,(t). The index k stands for the lane of the vehicle; the index i is now 1 for a vehicle that is about to enter the 200 metres interval, it is n, for the vehicle that last left the interval (on lane k). Thus the density (vehicles per km per lane) equals #($
_ 3 + &.I + 100 + 100 - “‘Q-J. =I
xk,2
-
xk.l
xk.n,
-
xk,“k-I
Of course, we do not know exactly the positions of the vehicles. They must be estimated assuming that the vehicles do not change their velocities during the time that they are near the detector station, so that the position of a particular vehicle relative to the detector station equals the time interval between time t and the passing time multiplied by its velocity. Clearly, the constant velocity assumption limits the length of the interval. The length of 100 metres is chosen as it corresponds to a driving time of about 4 seconds. In the absence of incidents a period of 4 seconds seems short enough to guarantee constant velocities. Further, a length of 200 meters seems a reasonable choice of an increment of the space variable. The next question concerns the extraction of the mean density and the fluctuations from the observed sum. To get an idea of how the density process behaves, the observed process at detector station no. 8 is shown in Fig. 2 for a period of half an hour. Apart from the short range fluctuations a medium range fluctuation of the mean density can be discerned with a typical time of, say, 80 seconds. We have chosen, therefore, to calculate the mean density at a particular detector station as the moving average of the observed process using a window of 80 seconds. This choice immediately implied another decision, namely not to combine the data of adjacent detector stations as far as the mean density is concerned. The mean density is changing so fast that we cannot assume it to be constant over the length of the freeway covered by the detector stations. But then the question arises, whether or not we also should try to estimate the parameters that determine the fluctuation process for each detector station separately. As the answer to this question is not clear a priori, both possibilities can be pursued. In order to keep the length of this article within reasonable limits we will below only show the results of the analysis for the combined data. More results can be found in Weits (1990). Further, we have to decide on the length of the time interval on which the estimates of the parameters are based. Some experimenting with the data coming from detector station no. 8 shows that a time interval of 15 minutes is appropriate if we consider only one detector station at a time. For the case of combined data a time interval of 10 minutes seems to contain sufficient information. Finally, c0 is determined by taking every second of the ‘estimation interval’ the mean of the velocities of all vehicles present in the 200 metres interval. Now, we do not consider the vehicles as spread out, instead they are treated as ‘point vehicles.’ The constant is set equal to the mean of these mean velocities over the appropriate time interval (and, possibly, the relevant set of detector stations). Note that the slower vehicles are counted more often than the faster ones. This feature compensates for the fact that simply taking the mean of all passing velocities overestimates the mean velocity co, as a fast car is more likely to be detected than a slow one.
121
Stationary freeway traffic flow 40
detector station I
35
30 25 . 20 ’ 16
10 5 .
:650
I750
:I50
1950
9050
9150
9250
9350
9450
9550
lime
0
i
9550
9650
9760
9860
9950
10060
10150 10150
10350 10460
iims Fig. 2. The evolution of the density (vehicles per km per lane) at detector station no. 8 situated on the western carriageway of freeway Al3. The time is given in seconds.
4.3. Estimation of the parameters for combined data
Table 1 and Fig. 3 present the results of the estimation of the parameters on the basis of the combined data (from detector stations nos. 8-15). Although the theoretical covariance matrix of the error terms appearing in the regression problem can be calculated to a good approximation, it was not used simply because
Table I. Regression results from 6 data sets (combined data from all detector stations). c, (km/s)
:: 3. 4. 5. 6.
0.0257 0.0279 0.028 I 0.0284 0.0274 0.0278
The abbreviation
A (km-*)
B(s-‘)
S
s.d. of A
23.801 34.666 36.792 30.458 23.091 27.060
0.0072 0.0045 0 0.0126 0.0018 0.0081
2.357 1.812 2.054 2.077 2.689 1.745
I1.524 .453 1.438
s.d. stands for standard
deviation
1.262 1.191 1.712
s.d. of j3
0.0025 0.0014 0 0.0019 0.0014 0.0027
s.d. of S
0.054 0.044 0.034 0.049 0.062 0.049
E.
122
WEITS
40
Irn-““’
30 20 Cm
10 .-a ..".._ X...
0 '.. -10
-200
.** . ...a
-20 ~ 20
100
40
0
20
40
60
60
100
time
lime
,w,,ill,, ru,,
40 30 It
_l;iJ-b
CO”
Co” 10
-20+
0
100
20
40
60
60
too
lime
40
,101, ,101 r.r,.
30
!. 20 *
r..n,.
20 ....
. *.
...
. ..d
..
.
0
20
40 uma
60
60
100
-204 0
...a ..*
: . ...*
4o#L -10
-10 -20 k 0
11111
.cov *o
c0v IO 0
11‘11
30.
20
40
60
60
100
time
Fig. 3. The estimated covarianccs of the density-fluctuations (9 for all detector stations together with ordinary least squares fits (solid line). The time is given in seconds.
the regression algorithm (the Gauss-Newton method) failed to converge when it used this matrix. With a view to the fact that this weighted regression algorithm performed well on data obtained from simulations, we conclude that the theoretical covariance matrix does not completely catch all features of the covariances of the errors terms. Apparently the range of the correlations of the errors is smaller than suggested by the theoretical covariante matrix. Also the weighted regression procedure is computer-time consuming and therefore expensive. For these reasons we have chosen to use the ordinary least squares criterion. We will now make some remarks concerning the results. 1. We see that the estimates of the parameter fl are (very) small. (In one case the parameter is virtually zero.) When the value of /3 is too small (say below O.OOl), problems arise in computing accurately the values of the regression function and its derivatives. Therefore, if /3 is too small, we set it equal to zero and estimate the remaining parameters using an alternative regression procedure from which B is absent. 2. The value of S is closely related to the value of co. If a wrong value of co is taken, this only affects the estimation of S. 3. In the neighborhood of I = 0 the values of P tend to be reduced. The effect is certainly caused by the way the density is calculated. The assumption that the vehicles pass through the 200 metres interval with constant velocity causes some smoothing of the values of the density. 4. The pictures of the estimated covariances of the process suggest that there are two dips
stali0Mt-y
frawoy traffic flow
123
for r * 30 seconds and two peaks for r I 60 seconds. One might suggest that this is due to different behaviour of the left and the middle lane and the right lane (which is characterized by much heavy traffic). 5. A rather delicate question is the estimation of the standard deviations of the estimates of the parameters. The estimates given in Table 1 are based on a result on the asymp totic distribution of the parameters under the assumption that the errors are uncorrelated. These values are too small, because the correlation of the errors makes it possible that a large part of each error is absorbed by the estimated regression function, so that the uncertainty of the estimates of the parameters increases. Experiments with artificial data sets suggest that parameters A and S can be estimated with an accuracy of about 10%. Estimation of parameter /3 is more difficult. If the parameter is very small, little more than that can be said. If it is somewhat larger, say about 0.01 (for S = 2.5), an accuracy of 30% is likely. The larger /.Iis, the more accurately it can be determined. 6. The normality of the process R has been tested by sampling once every hundred seconds from the data at detector stations nos. 11 and 14. It is assumed that these values are (almost) independent. Normal probability plots suggest that these data are approximately normally distributed. 5. CONCLUSIONS
AND REMARKS
On the basis of the data (from detector station no. 8) the model has been adapted in two ways. First we allowed for medium-range variations of the mean density and, secondly, we used the ordinary least squares criterion in the estimation of the parameters. The adapted model describes reasonably well the observed covariances of the density. We note that the fact that the regression procedures almost always converged indicates that the amount of damping present in the observed covariances is covered by the model. In the one exceptional case, putting 0 = 0 yields acceptable results. We conclude that the observed covariances obey the constraint that B L 0. It is probably the most successful feature of the model that it reveals the importance of the parameter @and, at the same time, provides a means to determine the value of this ‘hidden’ parameter. Whereas the squared amplitude of the fluctuations A as well as the length S are readily deduced from the data, 6 is not easily determined. If we would have observed the density along with the traffic stream, so that estimates of r(A,c&) could be obtained, the influence of the parameter /3 would be less obscured, since in that case there is no interference with the parameter S. Therefore, the model’s usefulness lies in the fact that it is able to determine the approximate values of the parameter /3, especially in the case of the rather limited observations at detector stations. Perhaps the most important open question is the range of values of the parameter /3. As the data used here were gathered in pre-peak-hour period, we conjecture that in the middle of the peak-hour much larger values of the parameter occur implying that the values found here can all be said to be (very) close to zero. If this guess is confirmed, one of the consequences will be that the parameter B can be estimated with sufficient precision when every it is important to know its value, i.e. when it is large. The following application is proposed. We interpret the phenomenon of large total density- which corresponds to a local concentration of vehicles-as a situation in which there is a large probability of congestion. Clearly. this phenomenon depends on the value of the mean density and on the value of the parameters. Of these the parameters A and fl are the most important ones. Large values of the mean density and the parameters A and B imply a large probability of an extreme value above some crucial level. A small value of S has the same effect, but it seems that this parameter is not likely to change very much. If the total density exceeds the crucial level the flow is likely to break down. Of course, the value of this crucial level will depend on the mean density. The parameters can be determined as indicated above. It does not make much difference whether data from one or several detector stations are used for the estimation of the parameters.
E.
124
WEITS
This sketch does not lead to quantitative statements. Therefore, the critical values as well as the range of the parameters under various circumstances have to be determined from studying a large amount of data. Once the critical values of the parameters have been found, the state of the traffic flow can be assessed (in an on-line configuration) and appropriate measures to ‘regularize’ the stream be taken, for example with the help of a signalling system. Acknowledgemenls--I thank Pict Grocneboom for his support in developing the model of freeway traffic flow presented in this article. Also I thank the editor and the referee for their valuable comments.
REFERENCES Curtain R. F. (1977) Stochastic evolution equations with general white noise disturbance. J. Murh. Anul. Appl.. 40.570-595.
Funaki T. (1983) Random motions of strings and related stochastic evolution quations. Nugoyu Muth. JournaL 89. 129-193. Goldstein J. A. (1985) Semigroups 01 Lineur Operufors & Applicufions. Oxford University Press. New York, pp. 3-25 lkeda N. and Watanabe S. (1981) Stochastic Dif/erential Equations und Diffusion Processes. North-Holland, Elscvicr/Kodansha. Tokyo. It6 K. (1984) Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia. Payne, Harold J. (1971) Models of freeway traffic and control. In Marhemuficul Models of Public Systems (Simulation Council Proc.). 1. 51-61. Smuldcrs S. A. (1988) Confrol o/f+eewuy TrufJc Flow. Thesis, Twente University of Technology, Enschcde. Weits E. A. G. (1990) A Stochustic Heuf Equafion for Freeway Trq/jic Ffow. Thesis, Delft University of Technology, Delft, The Netherlands.
APPENDICES
There are two appendices to the article. The first one gives some extra (mathematical) details of the specification of the model (see Section 2). The second appendix supplies a more or less rigorous probabilistic background for the expositions of Section 2 as well as Appendix I. Appendix
I
In this appendix we discuss some details concerning the semigroup U,. white noise in two dimensions and its modifications used in the present model and, thirdly, the solution of eqn (4). The semi-group U, generated by the operator A I K d’/dx’ - c&/dx on the Hilbcrt-space L’[O.M], with domain ‘D(A) = cfe L*[O.M]:/‘E
L’(O.M],f(O)
= f(M),f’(O)
=f’(M)).
can be represented as M U,f = s D q(r.x.y)f(y)dy q(rJ*Y)
=
IM
+
2 [6,(X -
forf e L’[O,M]
and
Cd)&(Y) + 6(x - corNdY)l exp( -VW.
,.I
where 6,(x) = asin(Zzix/M) and G,(x) = ~cos(Z~ix/M),h, = 4r’i’/M’. Set $.,0(x) = I/&. Note that (0,: i z 1; $,: i L 0) is an orthonormal basis of L’[O,M]. Secondly we consider the noise term. White noise on [O,T] x [O.M] can be represented as the time derivative of the cylindrical Brownian motion
where (b,‘:i z I; b,‘:i L 0) is a family of independent standard one-dimensional Brownian motions (see Curtain, 1977 and Appendix 2 for more information). In order to guarantee conservation of vehicles on a scale of length S, we modify the white noise term and set B(r) = c
b,‘(r)&, + b,‘(W,.
I2.W
We leave out all components with i c m. This procedure retains the white noise property for the timecoordinate, whereas the property is only locally retained for the space-coordinate. The (weak) solution of cqn (4) is defined as the solution of the integral qn (5). where R0 is chosen such that R0 E UE L’[O.M]: < f,b, > = 0. c f.& > = 0. vi s m - 1) . Equation (4) is called a stochastic partial
Stationary freeway traffic flow
125
differential equation. It is a parriol differential quation because of the appearance of partial derivatives and it is srochasric differential equation. because the differential equation is interpreted in terms of a stochastic integral. The stochastic integral used here is discussed in Appendix 2. If we assume that R,, is stochastic and distributed according to the invariant measure associated with the process we can represent the solution of qn (4) as
R(M)
= 2 Ia:(rM,(x 1-m
- cd) + a:(r)d,(x
- cot)].
where {a,‘. a,‘: i L m} is a family of independent Gaussian processeshaving covariance function
for A = It - ~1. Of course, the ‘c’ and the ‘r’ refer to ‘cosine’ and ‘sine’, respectively. From this representation it follows that R(r.x) is a ccntred Gaussian process and also that its covariance function is given by eqn (6). Appendir 2
We give a short account of the definition of a stochastic integral on a real separable Hubert space H with inner product (a;) and norm 11.11.In this article we have always set H = L’[O,M]. The one-dimensional stochastic integral (with respect to Brownian motion) corresponds to the case H = R . This so-called Ito-integral is the basis of the following survey. For information on the one-dimensional stochastic integral and the corresponding stochastic (ordinary) differential equation 1 refer to the book by Ikeda and Watanabe (1981). Let (ft.5.P) be a complete probability space with a right-continuous increasing family of sub-o-algebra’s (3, : I L 0). each containing all P-null sets. We say that the set-up (tM,( g,}.P) satisfies the usual conditions. H denotes a real, separable Hilbert space with inner product (* ;) and norm 11.I(. Lo(fi.5.P) is the space of all real random variables on (tI.S.P). Identifying X and Y E f.,,(n) if P(X = Y) = 1. WChave that endowed with metric d(X,Y) w &( IX - Y] I\ 1) L,(Q) is a FrCchet space. The next three definitions have been taken from an article by Funaki (1983). Definirion 1. A linear random funcrional on H is a linear map from H IO Ldll.5.P). We call a family of linear random functionals {B,:r 2 0) on H a cylindrical Brownian morion on H if ir sarisjies rhe following condirion: for every x a H (x + 0). r - B,(x)/]]xl] is a standard Brownian morion (adapted ro rhe filrrarion of /T;‘he case that H = L’[O.M] the time derivative of the cylindrical Brownian motion can be identified with white noise in IWO dimensions. Specifically, for 0 5 I, S rz. x, s x,. 0 s s, s sz and y, s yr it holds that
which allows us to identify two-dimensional white noise integrated over a small rectangle (r,.r,] x (x,.xJ with the random variable B, (I,, ,,,*,) - B,,( 1,,,+t ) and, equivalently, white noise integrated over (s,.s,] x (y,yJ withB~,tt,,J21) - B,,(~,,A). We will now use the s&arability of H; consider a complete orthonormal system in H: (e,: i z 1). Let f be an arbitrary element of H:f = Z, up,. As a consequence of the orthonormality (B,(e,) :i z I) is a family of independent standard one-dimensional Brownian motions, so that we can rewrite B,(f) = Z:., aB,(e,) as B,o-I = r;:., (f.e,) b,(r). where (6,(r) : i 2 I } is a family of independent standard one-dimensional Brownian motions. Formally, we write
4 = ,t e,b,(O.
(9)
Definirion 2. Ler A’([O. T] x Cl.H) be rhe Hitbert space of all H-valued, I,-odapred and measurable functions f(r.4 satisfying E ]: ]If(r)]]’ dr < 0. For every f B A’([O, TJ x D.H) we define
s r
o (lW.dB,) - ,t
5:(lW.e,)dB,k).
where {e,: i t I } is a complere orrhonormal sysrem in H. Definition 3. Ler &:I (H) denote rhe Hilberr space of Hilbert-Schmidr operarors on H wirh norm I]* llHsand let A’([O. Tj x O.CJH)) be rhe Hilberr space of all C/H)-valued. B,-adapred and measurable funcrions F(r.w) sarisfying E 1: ]]F(r)]]ns’dr < w. For every F E Al@. TJ x D, L/H)) an H-valued srochasric inregral ]: F(r)dB, is defined by the following equality:
where F*(r) is rhe adjoinr of F(r). By lineariry ir is enough lo let f run rhrough a orrhonormal basis of H. Within the framework established by these definitions the stochastic integral appearing in eqn (5) is interpreted as
126
E.
WEITS
s I
o
W-Ad&
where Q is the projection operator on the subspace of L’[Om given by cf c L*[O.M]: < f,e, > = 0, vi s m - 1) . The symbol B, here denotes the cylindrical Brownian motion as defined above. The modification of the noise term is taken care of by the operator Q. We notice that U,_,Q is a Hlbcrt-Schmidt operator for all s c I
IlLQll~s’ Recall that A, = 4di*/M’.
= 2 c exp( -tA,K(r ram
- 5)).
Define for c. > 0 r-e” I(f - c,) - s o U,-,QdB,.
Suppose that e. 1 0 as n tends to infinity. We see that, for I > n. EMr
- e.] - Itr -
f,)IIZ
t-e,
=
I
2 ,_,~,~exp(-2A,Ktr
5 c+ #2rn
,
- exP(-Wf,
- s]]ds -
f,]))
= c ; (expf-2&c,) #*aI , s
C(f.
-
- e~~(-2h,~.)]
f,)e.
We conclude that the sequence I( I - e.) is converging in the space f.*(O,S.P)
as z. 1 0.