Using principal curves to analyse traffic patterns on freeways

TLDR: This work introduces the concept of calibration curves to determine the relationship between the latent variable (represented by the parametrisation of the principal curve) and the traffic density and applies LPCs to a variety of speed–flow diagrams from Californian freeways, including some so far unreported patterns.

Abstract: Scatterplots of traffic speed versus flow have received considerable attention over the past decades due to their characteristic half-moon shape. Modelling data of this type is difficult as both variables are actually not a function of each other in the sense of causality, but are rather jointly generated by a third latent variable, which is a monotone function of the traffic density. We propose local principal curves (LPCs) as a tool to describe and model speed–flow data, which takes this viewpoint into account. We introduce the concept of calibration curves to determine the relationship between the latent variable (represented by the parametrisation of the principal curve) and the traffic density. We apply LPCs to a variety of speed–flow diagrams from Californian freeways, including some so far unreported patterns.

Figures

Figure 3: Calibration curves for speed-flow diagram from figure 1; left: using the fundamental identity; right: using measured occupancies.

Figure 9: Road layouts corresponding to each pattern considered in Section 3.4. Traffic flows from left to right. Lane 1 is the fastest (“top”) lane in each panel. The detectors are positioned in the center of each extract. The pictures are taken from PeMS.

Figure 5: Speed-flow data from VDS 716450 with local principal curve, HS principal curve, and Greenshields curve, for six consecutive weekdays starting 9th of July 2007. 14

Figure 8: Representative speed-flow patterns collected from PeMS, with fitted local principal curves. The corresponding road layouts are provided in figure 9.

Figure 4: Road layout for freeway 1280-N around VDS 716450 (symbolized through a triangle). Lane 1 is the fastest lane, i.e. the lane in the top. Traffic flows from left to right. Picture taken from PeMS.

Table 1: Estimated capacities in veh/hour for Lane 1 of freeway I-280N on six consecutive Tuesdays, with means and standard deviations of the six estimates given underneath.

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Einbeck, Jochen and Dwyer, Jo (2011) 'Using principal curves to analyse trac patterns on freeways.',
Transportmetrica., 7 (3). pp. 229-246.
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Using principal curves to analyze traffic patterns on
freeways
Jochen Einbeck
∗
and Jo Dwyer
Durham University, Department of Mathematical Sciences,
Science Laboratories, South Road,
Durham, UK
Abstract
Scatterplots of traffic speed versus flow have caught considerable attention over
the last decades due to their characteristic half-moon like shape. Modelling data
of this type is difficult as both variables are actually not a function of each other in
the sense of causality, but are rather jointly genera ted by a third latent variable,
which is a monotone function of the traffic density. We propose local principal
curves as a tool to describe and model speed-flow data, which takes this viewpoint
into a c c ount. We introduce the concept of calibra tion curves to determine the
relationship between the latent variable (represented by the parametrization of
the principal curve) and the traffic density. We apply local principal curves to a
variety of speed-flow diagrams from Californian freeways, including some so far
unreported patterns.
Key Words: fu ndamental diagram; capacity; local principal curves; smoothing.
∗
jochen.einbeck@durham.ac.uk

1 Introduction
Scatterplots of speed versus flow have been widely analyzed and discussed in trans-
portation science, and have recently attracted new interest with th e rapid advances in
the development of Intelligent Transportation Systems. As an example, consider data
plotted in figure 1 (left), r ecorded on 10th July 2007 (00:00 to 23:59) on the Califor-
nian Freeway I280-N, Lane 1, VDS (“vehicle detector station”) number 716450. The
data show a characteristic and frequently reported half-moon like shap e. Roughly,
the upper and the lower cluster correspond to uncongested and congested operating
condition, respectively, and the few data points between them to an unstable transi-
tion region. Based on a cluster analysis, Xia & Chen (2007) argued that actually five
different operating conditions should be distinguished.
Under equilibrium conditions, i.e. stationary speed and spatially homogeneous
density, it is well known that the speed v and the flow q are related through the
fundamental identity q = k v, where k is the traffic density. The association between
speed, flow and density is often referred to as the fundamental diagram. As Wu (2002)
points out, the fundamental identity specifies the fundamental diagram only up to one
degree of freedom. In other words, one has to impose an additional constraint on
any pair of the three variables in order to specify the fundamental diagram fully.
This is usually achieved by fixing the k − v relationship. For instance, the original
model suggested by Greenshields (1935) uses k(v) = k
j
(1 − v/v
f
), where k
j
is the
jam density corresponding to v = 0 and and v
f
the free-flow speed. Having fixed the
speed-density relationship, the speed-flow relationship is determined by q(v) = k(v)v,
which in the special case of the Greenshields model (hereafter: GM) takes the shape
q(v) = k
j
v(1 − v/v
f
), i.e. a parabola without an intercept. Several other, generally
more complex, functional relationships between k and v have been proposed since
2

then, see e.g. Kockelman (2001) or Wu (2002) for an overview on this literature.
An interesting and early reference comparing different speed-density models from a
statistical point of view is Drake, Schoefer & May (1967). More recently, Wu (2002)
proposed to avoid the usually applied “trial and error” model selection strategy by
relating the parameters of the fundamental diagram to microscopic road parameters.
The reason why the k − v relationship is preferred to any other pair of variables is
simply that th is is th e only one which is monotonic. This is illustrated in figu re 1
(right), using here occupancy, the quantity returned by default by PeMS, which is
roughly linearly related to dens ity (Hall, 2002).
50 100 150 200
20 40 60 80 100 120
Flow (veh/5 min)
Average Speed (km/h)
0.0 0.1 0.2 0.3 0.4
20 40 60 80 100 120
Occupancy
Average Speed (km/h)
Figure 1: Fundamental diagram recorded on Freeway I280-N; left: speed-flow; right:
speed-occupancy.
There have hardly been made any attempts at modelling the q − v relationship
directly (rather than through the k − v relationsh ip); Li (2008) mentions one instance
used in the Highway capacity manual 2000. One reason for this reluctance may be
that any functional form between q and v is hard to justify. Obviously, v cannot be
3

seen as a function of q as we have potentially two different outp uts for the same input.
But also the other way round, q = q(v), seems somewhat contrived: speed v is quite
difficult to measure while the flow q is very easy to measure — realistically, nobody
would be interested in predicting flow from speed. Also, traffic flow is not a function
of speed in the sense of causality, it is rather that drivers have to obey the constraints
set by the current road conditions, and this will affect both speed and flow. As a
consequence, it seems more natural to consider both variables as the two-dimensional
output of a function


q
v


(t) of some (latent) variable, say t. This also does the job
of fixing the remaining degree of freedom in the fundamental diagram, but it implies a
symmetric view on the variables; the resulting model is invariant w.r.t. interchanging
the coordinate axes for q and v.
The statistical concept corresponding to this viewpoint is a principal curve: a
smooth curve passing through the “middle of the data cloud”. Principal curves were
introduced by Hastie & Stuetzle (1989) (hereafter: HS) as a nonparametric extension
to linear principal component analysis. Chen, Zhang, Tang & Wang (2004) have ap-
plied HS principal curves to speed-flow d ata and showed that this leads generally to
better fits than the Greenshields-type parametric m odels of flow given speed. We will
take thin gs on from here and illustrate the benefits and relevance of principal curves
in the context of the fundamental diagram. The methodology that we will be using for
the actual curve fitting is that of local principal cu rves (Einbeck, Tutz & Evers, 2005b,
hereafter LPC). In Section 2, we explain briefly how LPCs work, and we demonstrate
that the curve parametrization, representing the latent variable, is a monotonic func-
tion of the traffic density, which was singled out as “the primary factor to define the
level of service on a freeway” by Xia & Chen (2007). Specifically, we introduce the
novel concept of a calibration curve, which relates the curve parametrization to den-
4

Frequently asked questions

Q1. What are the contributions in "Using principal curves to analyze traffic patterns on freeways" ?

The authors propose local principal curves as a tool to describe and model speed-flow data, which takes this viewpoint into account. The authors introduce the concept of calibration curves to determine the relationship between the latent variable ( represented by the parametrization of the principal curve ) and the traffic density. The authors apply local principal curves to a variety of speed-flow diagrams from Californian freeways, including some so far unreported patterns. 

Q2. What are the future works in "Using principal curves to analyze traffic patterns on freeways" ?

For instance, even if one manages to calibrate a curve as in figure 8 ( d ), it does not seem very likely that this provides a reliable reference for the behavior of the road in the future. 

Q3. What is the effect of a reduced speed?

The reduced speed implies a reduced stopping distance, allowing each vehicle to become closer to the one in front, which in turn leads to increasing flow. 

Q4. What is the statistical concept corresponding to this viewpoint?

The statistical concept corresponding to this viewpoint is a principal curve: asmooth curve passing through the “middle of the data cloud”. 

Q5. What is the common reason for the monotonic calibration curve?

For almost all speed-flow diagrams which can be represented by a single-branchedprincipal curve, the calibration curve will be monotonic irrespective of the parametrization used. 

Q6. What is the reason for the speed-flow relationship?

An explanation for this is that free flow at a very low flow rate is likely to occur very late at night when the freeways are at their least busy, therefore many road users will choose to drive at a slightly slower speed than they might do at such a flow rate during the day. 

Q7. What is the reason why traffic flow is preferred to any other pair of variables?

traffic flow is not a function of speed in the sense of causality, it is rather that drivers have to obey the constraints set by the current road conditions, and this will affect both speed and flow. 

Q8. How can one quantify the relationship between parameter and density?

The relationship between parameter and density can be quantified through a calibration curve, an approximate version of which can be generated even without knowledge of the traffic density via the fundamental identity of traffic flow (in principle, also an external “reference” calibration curve, which would form a characteristic of the road under certain default conditions, could be used instead, if one standardizes the “0” value of the parametrization). 

Q9. What is the way to model the speed-flow data?

Modelling speed-flow data through principal curves would only make really senseif the fitted curve is repeatable; i.e. if two principal curves fitted at the same location at different days are similar (under otherwise similar conditions). 

Q10. What is the method used to measure the flow of traffic?

Their method is to measure the flow, the number of vehicles that go over a “loop” per unit time, and occupancy, the amount of time each vehicle takes to drive over a loop, of traffic every 30 second period. 

Q11. What is the purpose of the principal curves?

Principal curves were introduced by Hastie & Stuetzle (1989) (hereafter: HS) as a nonparametric extension to linear principal component analysis. 

Q12. What is the main reason for the monotonicity of the principal curve?

Minor perturbations from this monotonicity will occur if (and only if) the speed-flow data cloud is so strongly skewed that there exists a line through the origin cutting the principal curve twice. 

Q13. Why is the k v relationship preferred to any other pair of variables?

The reason why the k − v relationship is preferred to any other pair of variables is simply that this is the only one which is monotonic.