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Short-term Highway Traffic State Prediction Using Structural
State Space Models
Chung-Cheng Lu, Ph.D.
(Corresponding author)
Institute of Information and Logistics Management, National Taipei Univ. of Technology
1 Section 3 Chung-Hsiao East Road, Taipei, 10608, Taiwan
Tel: +886-2-27712171 ext 2306; E-Mail: jasoncclu@gmail.com
Xuesong Zhou, Ph.D.
School of Sustainable Engineering and Built Environment, Arizona State University
Tempe, Arizona, 85287, USA. E-Mail: xzhou@asu.edu
Abstract
This research proposes a short-term highway traffic state prediction method based on a
structural state space model, with the intention to provide a robust approach for obtaining
accurate forecasts of traffic state under both recurring and non-recurring conditions. True
traffic state is decomposed to three components, namely, regular traffic pattern, structural
deviation and random fluctuation. Particularly, the structural deviation term reflects the
change of true traffic state from regular (historical) pattern, due to unexpected capacity
reduction and/or demand variations. A polynomial trend is adopted to describe the temporal
evolution of structural deviations across different time intervals. We derive an analytical form
of structural deviations in a single bottleneck case based on cumulative flow count diagrams.
The proposed model is incorporated in a Kalman filtering-based algorithmic framework,
together with an adaptive scheme for determining the variances of random errors. A set of
numerical experiments were conducted on two test beds in the northern Taiwan highway
network. Experimental results show that the proposed approach is particularly superior to an
ordinary Kalman filtering method presented in the literature under non-recurring conditions,
highlighting the advantage of combining both the polynomial trend model and historical
pattern into the proposed short-term traffic state prediction approach.
Keywords: traffic state estimation and prediction; structural state space model; Kalman filter.
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1. Introduction
Recent advances in route guidance systems (e.g., Mahmassani and Peeta, 1995;
Ben-Akiva, 1997; Bottom, 2000; Mahmassani et al., 2005), coordinated ramp metering (e.g.,
Hegyi et al., 2005; Papamichail et al., 2010), and state-dependent congestion and value
pricing (e.g., Dong et al., 2011; Lou et al., 2011) have exemplified that traffic control
measures and travel demand management strategies based on anticipatory traffic state
information produce better system performances than those based on prevailing traffic state
measurements. These research findings underscore the importance of traffic state estimation
and prediction to Advanced Traveler Information Systems (ATIS) and Advanced Traffic
Management Systems (ATMS) functionalities.
Existing traffic state estimation and prediction approaches can be classified into three
categories: (i) approaches based on statistical methods or other techniques for predicting link
traffic states or path travel times, (ii) approaches based on macroscopic models for estimating
traffic flows on freeway corridors, (iii) approaches based on dynamic traffic assignment for
estimating and predicting network-wide origin-destination (OD) trip demands and route
choices so as to obtain network flow patterns on links with and without observations (e.g.,
Zhou and Mahmassani, 2007). This research focuses on the first category approaches, in
particular, for predicting short-term highway link traffic states with point measurements. A
rich body of the literature has been devoted to traffic state prediction approaches in the past
decades, most of which focused on travel time prediction using, for instance, linear regression
(e.g., Zhang and Rice, 2003; van Hinsbergen and van Lint, 2008), Kalman filtering (e.g.,
Chen and Chien, 2001; Nanthawichit et al., 2003; Xia et al., 2011; Anand et al., 2013) neural
networks (e.g., Rilett and Park, 2001; van Lint et al., 2002; Abdulhai et al., 2002), Bayesian
forecasting (e.g., van Hinsbergen et al., 2009; Fei et al., 2011; Lu, 2012), support vector
regression (e.g., Lam and Toan, 2008; Chandra and Al-Deek, 2009), and Fuzzy methods (e.g.,
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Zhang and Ye, 2008).
This research proposes a Kalman filtering-based algorithm for traffic state (e.g., link
density, speed, flow, or travel time) estimation and prediction on freeway corridors. Kalman
filter is a state estimation and prediction process applied to a dynamic system that involves
time-dependent system state transition and random perturbations. In this process, real-time
observations are obtained during each prediction period, and then processed through a
measurement equation to achieve a posteriori estimation of each period’s state variables. The
dynamic characteristics of state variables from one period to another are captured by a
transition equation, relating state variables across different periods. If the dynamic system
and measurement noise distributions are Gaussian, Kalman filter is an optimal filter with
minimum mean square error. This approach has been shown to be especially suitable for
real-time system applications (e.g., Chui and Chen, 1999). It has the capability of processing
real-time observations sequentially so that the computational burden and memory
requirements for large systems can be dramatically decreased. For instance, Chen and Chien
(2001) and Chien and Kuchipudi (2003) developed travel time prediction models using
Kalman filtering algorithm with simulated synthetic probe data. Nanthawichit et al. (2003)
used a macroscopic traffic flow model along with Kalman filtering technique to predict travel
time with synthetic detector data and probe vehicle data. Yang et al. (2004) presented a
Kalman filter method for online traffic speed prediction. Chu et al. (2005) developed an
adaptive Kalman filtering-based travel time prediction method that fuses both point detector
data and probe vehicle data. van Lint (2008) proposed censored extended Kalman filter (EKF)
and delayed EKF algorithms for online travel time prediction. Wang et al. (2005, 2006) also
developed a real-time freeway traffic estimation approach using an EKF algorithm.
The aforementioned literature demonstrates a wide spectrum of Kalman filtering
applications and constitutes the state-of-the-art techniques for traffic state or travel time
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estimation and prediction. Nevertheless, most of them focused on predicting traffic states or
travel times under recurring conditions (in which traffic flow follows its regular daily pattern),
and may require special (or ad-hoc) treatments under non-recurring conditions (in which
traffic flow is impacted by irregular or unforeseen events, such as incidents, accidents, or bad
weather). Under recurring conditions, traffic states can be estimated and predicted accurately
using the existing methods with sensor measurements. However, under non-recurring traffic
conditions, because traffic states dramatically deviates from regular traffic patterns, traffic
state estimates and forecasts obtained by the existing methods may be significantly biased. A
robust highway traffic state estimation and prediction method should provide a unified
framework for obtaining accurate forecasts under both recurring and non-recurring traffic
conditions.
With a particular emphasis on highway traffic state estimation and prediction under both
recurring and non-recurring conditions, our work adapts a structural state space model (e.g.,
Chui and Chen, 1999) that decomposes true traffic state into three components, namely
regular traffic pattern, structural deviation and random fluctuation. Specifically, regular traffic
pattern is estimated using historical data (e.g., mean or median of historical data), and
structural deviation reflects the difference between true traffic state and regular traffic pattern,
due to unexpected capacity reduction and/or demand fluctuations (e.g., Lan et al., 2008). In
this modeling framework, structural deviation vanishes under recurring conditions, while
forecasting actual traffic states under non-recurring conditions focuses mainly on the
prediction of structural deviations based on real-time sensor measurements of traffic states. In
addition, random fluctuation (or error) is included to account for the effect of other
unobserved factors and the inherent stochastic nature of time-varying traffic states.
The structural deviation term captures the influence of combined external factors that
cause true traffic state to change from average pattern. This research adapts a polynomial
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trend model to describe the temporal evolution of structural deviations across different time
intervals. Very few previous studies incorporated a similar term in modeling the impact of
various exogenous factors on regular traffic pattern. For instance, Zhou and Mahmassani
(2007) developed a structural state-space model for estimating and predicting dynamic OD
travel demands, and detrended structural deviations from the regular pattern of dynamic OD
travel demands represented by historical averages. Moreover, an analytical derivation of the
structural deviation term in a single bottleneck case is also proposed in this paper, based on
cumulative flow count diagrams.
The idea of the structural state space model with a linear (or polynomial) trend to
describe the structural deviations is illustrated in Fig. 1. In the figure, solid lines represent
true traffic state; dotted lines represent estimated historical pattern (i.e., regular pattern); red
arrows denote predicted pattern. The plot on the left hand side depicts a case of recurring
conditions in which the realized true state is similar to the regular pattern, while the plot on
the right hand side presents a case of non-recurring conditions in which the true state is
significantly different from the regular pattern. In the former case, the historical pattern and
linear trend model successfully captures the true state series and obtains accurate predictions.
On the other hand, in the latter case, the structural deviations between the true state and
regular pattern are significant due to an unexpected event. Our approach utilizes the
polynomial filter to absorb possible structural deviations, aiming to produce robust prediction
results under both recurring and non-recurring conditions.
[Insert Fig. 1 about here]
The next section describes the proposed structural state space model for traffic state
estimation and prediction, and the polynomial trend model based on a multiple-order Taylor’s
series which approximates a smooth function of structural deviation. Section 3 presents the
Kalman filtering-based solution algorithm, which integrates short-term traffic state estimation
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and prediction equations under recurring and non-recurring conditions. An adaptive scheme
for updating measurement and system errors variances is also illustrated. Section 4 reports a
set of numerical experiments conducted on two test beds in northern Taiwan. Concluding
remarks are given in Section 5.
2. Structural state space model for traffic state estimation and prediction
Notation
a index for links with traffic measurements, a = 1,…, N
t index for prediction time intervals, t = 1,…, T
length of a prediction time interval
current time stamp
s, p, m order indices in a polynomial trend model
traffic state on link a at time interval t
Kt vector of traffic states at time interval t
a priori estimate (i.e., regular pattern) of traffic state based on historical data on link a
at time interval t, where the superscript h stands for historical data
vector of a priori state estimates based on historical data at time interval t
structural deviation from the a priori estimate
on link a at time interval t
,
′,
′′ the first-, second-, and third-order derivatives of structural deviation ,
respectively
,
,
the pth-, mth-, and sth-order of derivatives of structural deviation ,
respectively
Zt vector of structural deviations from the a priori estimate at time interval t
observation on link a at time interval t
Yt vector of observations at time interval t
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system process error of traffic state on link a at time interval t
vector of system process errors at time interval t
measurement error on link a at time interval t
vector of measurement errors at time interval t
state transition matrix from time interval t1 to time interval t
H measurement matrix
Qt vector of process error variances at time interval t
Rt vector of measurement error variances at time interval t
vector of a priori traffic state predictions at time interval t
vector of posteriori traffic state estimates at time interval t
a priori prediction of state variance-covariance matrix at time interval t
posteriori estimate of state variance-covariance matrix at time interval t
Kalman gain matrix at interval t for state estimation under recurring conditions
vector of a priori structural deviation predictions at time interval t
vector of posteriori structural deviation estimates at time interval t
a priori prediction of structural deviation variance of link a at time interval t
a priori prediction of structural deviation variance matrix at time interval t
posteriori estimate of structural deviation variance of link a at time interval t
posteriori estimate of structural deviation variance matrix at time interval t
Kalman gain matrix at interval t for state estimation under non-recurring conditions
system process error (or noise) on link a at time interval t
process error for pth-order derivative of structural deviation
Wt vector of system process errors at time interval t under non-recurring conditions
Vt vector of measurement errors at time interval t under non-recurring conditions
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vector of process error variances at time interval t under non-recurring conditions
vector of measurement error variances at interval t under non-recurring conditions
var( ) the variance function
cov( ) the covariance function
2.1 Structural state space model
This research aims to estimate and predict traffic states, , for a
near future (short-term) time interval t on a set of N highway links with real-time
observations available from traffic surveillance systems. As mentioned previously, this work
considers that true traffic state is composed of three components:
True traffic state = regular traffic pattern + structural deviations + random fluctuations.
Realistically, only the a priori estimates of the regular pattern,
based
on historical data, are available before forecasting real-time traffic states. These a priori
estimates, such as means or medians of historical data, are used to reflect regular traffic
pattern. Thus, the true traffic state on link a at time interval t, , is modeled as a linear
combination of the a priori estimate, structural deviation, and random disturbance, as shown
in Eq.(1).
. (1)
The proposed structural state space model consists of the following two equations.
State transition equation:
, where
t ~ N(0, Qt). (2)
Measurement equation:
, where
t ~ N(0, Rt). (3)
The state transition equation describes the temporal change in traffic state from time interval
t1 to time interval t, while the measurement equation models the relationship between the
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state predictions, Kt, and real-world measurements at time t, Yt. In Eqs.(2) and (3),
t and
t
are white noise terms uncorrelated with initial state K0 and with each other.
In this modeling framework, for traffic state prediction under recurring conditions, the
structural deviation, , is considered as zero, and the regular pattern,
, can be
constructed based on historical data under recurring traffic conditions. On the other hand, for
traffic state prediction under non-recurring conditions, a polynomial trend model can be
adopted to provide a close approximation for the short-term prediction of the structural
deviation at time interval t.
2.2 Traffic state estimation and prediction under recurring conditions
In the proposed approach, under recurring conditions, the a priori prediction of traffic
states at time t, , is determined as:
, (4)
and the a priori prediction of variances, , is determined as:
. (5)
Then, the posteriori traffic state estimates, , is updated through a linear function of the a
priori prediction, , and a weighted difference, .
. (6)
By assuming that the measurement error covariance Rt is uncorrelated with Gt and Yt, a
general formulation for the variance-covariance matrix of the posteriori estimate error can be
derived as follows:
. (7)
where I is an identity matrix with appropriate dimensions and Tr denotes the transpose of a
matrix. In Eqs.(6) and (7), Gt is known as the Kalman gain factor used to obtain the posteriori
estimate and its covariance . The optimal form of Gt can be derived as:
. (8)
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Under the optimal formulation of the Kalman gain matrix, a simplified expression of the
estimation error covariance is derived as follows:
. (9)
2.3 Traffic state estimation and prediction under non-recurring conditions
Under non-recurring conditions, structural deviations, Zt, may result from various
demand and/or capacity changes. In the proposed structural state space model, the transition
equation of structural deviations is as follows:
, where Wt ~ N(0, ). (10)
The measurement equation is as follows:
, where Vt ~ N(0, ). (11)
The prediction equations for the (a priori) structural deviation vector, , and its
variance-covariance matrix, , are presented in Eqs.(12) and (13), respectively.
, (12)
. (13)
Accordingly, the predicted traffic state vector and its variance-covariance matrix can be
obtained as follows:
. (14)
. (15)
The estimation equations for the (a posteriori) non-recurring structural deviation, , and its
variance-covariance matrix, , are given in Eqs.(16) and (17), respectively.
. (16)
. (17)
where
. (18)
2.4 Polynomial trend model
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Let
be the current time. Consider the state prediction on link a under non-recurring
conditions, an m-order Taylor’s series can be used to approximate a smooth function of
structural deviation, z(a,
+
), expanded about the point z(a,
); that is,
. (19)
The above representation assumes that structural deviation at time (
+
) can be captured
locally by an m-order polynomial function as Eq.(19) near time
for a small value of
, while
derivatives of higher orders are assumed to be zero. A general form for the pth-order
derivative of this polynomial trend model is given as follows (Chui and Chen, 1999):
. (20)
Based on the above representation of structural deviation, the temporal change of derivative
from time
to (
+
) can be described as follows:
. (21)
As an example, the structural deviation transition equation based on the polynomial trend
model with order m = 3 for link a is given in Eq.(22), where the state vector consists of the
zeroth to third-order derivatives of structural deviation from the a priori regular pattern.
′
′′
′′′
′
′′
′′′
′
′′
′′′
. (22)
Specifically, the transition equations with zeroth-order (p = 0), first-order (p =1), second-order
(p =2), and third-order (p =3) derivatives can be expanded as Eqs.(23)-(26), respectively.
′
′′
′′′ . (23)
′
′′
′′′
′. (24)
′′
′′′
′′ . (25)
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′′′
′′′ . (26)
Eq.(22) can be extended to include the derivatives of the structure deviations of the N
(independent) links with real-time measurements, and hence the complete transition equation
of structural deviations in Eq.(10) can be obtained by defining the state vector as:
, (27)
the state transition matrix as:
, (28)
where
, for a = 1,…, N. (29)
and the evolution noise vector as:
. (30)
Although a sophisticated higher-order polynomial describes structural deviations in
greater detail than a lower-order model, it might result in potential large prediction errors. As
shown in the following equation, prediction errors of the proposed model depends on the
order m of the model and the length of prediction interval,
.
′
′
′
′
. (31)
Moreover, the computational complexity of Kalman filter is on the order of O(D3), where D =
N m. A lower-order model can decrease the size of the state vector and hence improve
computational efficiency. However, if traffic state is drastically different from the historical
regular pattern, then a higher-order polynomial trend model will be necessary to approximate
the structural deviation.
2.5 Analytically-derived state transition matrix for a single bottleneck case
This section aims to offer an alternative approach to derive state transition matrix,
, for a particular single bottleneck case with a reduced capacity due to an incident (i.e.,
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non-recurring congestion). For the sake of simplicity, the link subscript, a, is ignored in the
following discussion. As shown in Fig. 2, under recurrent congestion, the bottleneck has a
constant queue discharging rate C. An incident that causes a reduced capacity CR happens at
time s and ends at time e. The capacity restores back to C after time e and the queue vanishes
at time f.
[Insert Fig. 2 about here]
Consider the traffic state of interest as link density or number of vehicles on this link. To
derive the state transition matrix for this case, it is necessary to analyze the additional number
of vehicles in queue due to the incident. Depending on the relative position of the prediction
horizon, [
,
+
], to the incident time window, [s, e] there are three scenarios under
consideration. First, if the prediction horizon is inside the incident duration (i.e., s <
< e
),
then the structural deviations at time
: z
= (C CR) (
s) and at time
+
: z
+
= (C
CR) (
+
s). Thus, the corresponding state transition matrix (or factor) can be derived as:
. (32)
Second, if e
<
< e, then the structural deviations at time
: z
= (C CR)(
s) and at
time
+
: z
+
= C (
+
s) CR (e s) C (
+
e) = (C CR)(e s). Hence,
. (33)
Lastly, if e <
< f, then the structural deviations at time
: z
= (C CR)(e s) and at time
+
: z
+
= (C CR)(e s). Therefore,
. (34)
The same approach could be applied to derive the state transition matrix for other traffic
states of interest, such as link travel time. Interestingly, the derived state transition matrices
for additional link travel time are the same as those for link density under the three scenarios
discussed above, as the link density and additional travel time are directly connected through
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the same slope of output capacity. Similar matrices could also be derived for severs weather
and work zone cases. For a real-world corridor with multiple coupling bottlenecks, the matrix
is time-dependent and situation-dependent. The derivation in this section also provides
strong evidences that the transition matrix under non-recurring conditions can be reasonably
approximated as a time-invariant constant transition coefficient of 1 shown in Eq.(34), or a
time-dependent transition coefficient shown in Eq.(33).
The above state transition matrix relies on the information of incident duration, so an
incident duration (or effect) prediction model needs to be incorporated in the traffic state
estimation and prediction framework to apply the derived method online. It is important to
note that predicting incident duration is not in the scope of our current paper. A variety of
models exist in the literature for predicting effects of incidents. Linear regression models with
independent variables, such as incident type, weather condition, and number of vehicles and
lanes involved, were commonly used in this regard. For example, Khattak et al. (1995), Garid
et al. (1997), Ozbay and Kachroo (1999) adopted variations of this approach. Sheu et al.
(2001) and Sheu et al. (2004) presented a stochastic modeling approach based on Kalman
filtering to real-time prediction of incident effects on freeway and surface street traffic
congestion, respectively. Li and Cheng (2011) presented an incident duration prediction
method based on latent Gaussian Naïve Bayesian classifier.
3. Solution Algorithm
3.1 Kalman filtering-based approach
The traffic state prediction process can be theoretically modeled as a probabilistic
combination of recurring and non-recurring conditions, but it is difficult to estimate or predict
the probability of non-recurring conditions (because supply reduction events and demand
fluctuations that cause non-recurrent traffic states are generally unexpected). To
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accommodate both recurring and non-recurring traffic state predictions, the proposed
estimation and prediction equations under recurring and non-recurring conditions are
incorporated in a Kalman filtering-based algorithm. Essentially, for each link a and time
interval t, a threshold that consists of an upper bound (
) and a lower bound (
) is used
to define the range of regular traffic states. If an observation on link a at time t, ya,t, falls in
the range, then the corresponding traffic state ka,t is considered as under recurring conditions;
otherwise, ka,t is under non-recurring conditions. The bounds can be determined according to
the experiences of Traffic Control Center (TCC) or using historical data. For instance, the
lower bound and upper bound may be set as 30th and 70th percentiles, respectively, when the
regular pattern is defined as the median of historical data.
Algorithm: Kalman filtering-based algorithm (KF)
Step 0: (Initialization) Obtain the a priori estimate of regular traffic pattern,
, and set up
the threshold,
, for each link a and each time interval t = 1,…, T. For
each link a, set
,
,
, and
. Let t = 1.
Step 1: (State Prediction) Propagate a priori traffic state predictions (
) and its variances
() from interval t1 to t.
;
.
Step 2: (State Estimation) For each link a with real-time measurement, compute the Kalman
gain factor ( or
) and update posteriori traffic state estimate (
) and its
variance (
) based on real-time link observation at current time interval t, ya,t.
Specifically,
If
(under recurring conditions) then
;
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;
;
;
.
If
or
(under non-recurring conditions) then
;
;
;
;
;
;
.
Step 3: If t < T, then t = t + 1, and go to Step 1; otherwise (t = T), stop.
3.2 Adaptive scheme for updating measurement and system errors variances
Specifying the values of evolution error variances Qt and measurement error variances
Rt is an important task for operating the proposed recursive algorithm. This research adapts
the adaptive Kalman filtering approach (e.g., Chui and Chen, 1999) which utilizes statistical
properties of measurement residual sequence to dynamically estimate measurement error
variances Rt. Let
. (35)
By taking the variance on both sides, we can obtain the following
. (36)
Then,
, where
(37)
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Thus,
. (38)
Next, a constant signal-to-noise ratio , reflecting the relative weight of inherent
system error variance with respect to observational error variances, is assumed to obtain
evolution error variances Qt. The decision-maker at TCC can select an appropriate value of
so as minimize average prediction error in training data sets. According to West and Harrison
(1997), if a constant ratio
is assumed, the limiting behavior for the Kalman gain factor can
be derived as the following:
∞
. (39)
For instance,
= 0.5 (i.e.,
∞) indicates that the a priori estimate and the real-time
(posteriori) estimate are equally important. The KF approach with the above adaptive scheme
for updating measurement and system error variances is denoted as Adaptive KF or AKF.
4. Numerical experiments
The proposed KF and AKF approaches were evaluated on two test beds in northern
Taiwan (see Fig. 3). One of them is a highway stretch between Nangang Interchange (0-km)
and Pinglin TCC (14-km) of National Highway No.5 (Jiang Wei-Shui Highway) eastbound.
Another is a highway segment between Taishan toll booth (35-km) and Taoyuan Interchange
(49-km) of National Highway No.1 (Sun Yat-Sen Highway) northbound. Taiwan’s National
Freeway Bureau has installed a dense set of detectors in these two major freeways to collect
traffic data, such as volume, speed and occupancy data (the spacing between two detectors is
less than 2-km). The traffic state of interest in the conducted experiments is density, defined
as number of vehicles per kilometer per lane. Traffic density forecasts have been used in
recently proposed coordinated ramp metering (e.g., Papamichail et al., 2010) and
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state-dependent congestion pricing approaches based on anticipatory traffic state information
(e.g., Dong et al., 2011). The observed density data on the weekdays (Tuesday, Wednesday,
and Thursday) from February to June (five months), 2011, were collected and processed by
the TCCs of Highway No.5 and Highway No.1 using estimated average vehicle length.
[Insert Fig. 3 about here]
To guide against testing hypotheses suggested by the data, a cross-validation method
was used in the numerical experiments. Specifically, the weekday (Tuesday, Wednesday, and
Thursday) density data from February to May, 2011 were used as the historical data set to
estimate regular traffic patterns and the upper and lower bounds for recurring traffic
conditions for each (5-minute) prediction time interval. Because mean density may be
affected by the presence of extreme values in a statistical sample, median density was used as
the a priori estimate of regular density pattern. In addition, according to the experiences of
TCCs of Highway No.5 and Highway No.1, the upper and lower bounds of recurring
conditions were set as the 25th percentile and the 75th percentile of historical data in each time
interval. The weekday density data in June (fourteen days) were then employed as the test set
to evaluate the prediction results. For each weekday, there are two prediction periods under
consideration, i.e., 6:00-12:00 AM and 2:00-8:00 PM.
The proposed KF and AKF were implemented with the polynomial (or linear) trend
model with order m = 1 in the state transition equation. As discussed previously, a higher
order trend model was not considered, due to the tradeoff between better approximation of
structural deviations and prediction errors (see Eq.(31)). In the proposed approach, the state
vector in the model is (,
), where the estimate of was obtained as the median of
historical structural deviations in density of link a at time t = 0 (or the beginning of the
planning horizon) and
was approximated as the change of structural deviation per unit
time (
) using historical data. The initial state vector was then input to
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the polynomial trend model with the state transition matrix
to determine the state
vector in subsequent time stamps: , , and so on.
In addition to evaluating the KF and AKF methods, this research also implements and
tests an ordinary Kalman filtering (OKF) approach (e.g., Chui and Chen, 1999; Chen and
Chien, 2001) that does not include historical data and the polynomial trend model. The state
transition and measurement equations of OKF are as follows.
, (40)
. (41)
where denotes the state transition parameters from time t1 to t, which are externally
determined. Hence, there are three predictors under comparison: KF, AKF, and OKF.
The mean absolute percentage error (MAPE) was used to measure the effectiveness of
the three traffic state prediction methods. The MAPE is a commonly used measure of
accuracy of methods for constructing fitted time series values in statistics, particularly in
trend estimation and prediction (e.g., Xia et al., 2011). It usually expresses prediction
accuracy as a percentage, and is defined by the formula:
(42)
where M is the number of observations. As shown in Eq.(42), the definition of the MAPE
involves the absolute operator which implicitly gives the same weight to the errors of
over-estimation/ prediction and under-estimation/ prediction, respectively. Typically, in
estimating and predicting a time series, both under-estimation and over-estimation errors
have to be taken into account when evaluating the performance of a method under
consideration (e.g., Lewis, 1982; Delurgio, 1998). The error of under-estimation cannot be
offset by the error of over-estimation, and vice versa. This principle is also found in several
other measures (indices) of prediction accuracy that are commonly-used in prediction, such
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as mean absolute error (MAE) and root mean squared error (RMSE).
(43)
(44)
Furthermore, one of the advantages over these other measures of prediction accuracy is that
the MAPE expresses prediction accuracy (or inaccuracy) using percentages, so that our
judgment would not be affected by the magnitude of prediction and observation values.
Typically, a prediction is considered as accurate if the MAPE is less than 20% and highly
accurate when the MAPE is lower than 10% (e.g., Lewis, 1982; Delurgio, 1998).
Table 1 and Table 2 present the MAPEs of the prediction results obtained by the three
predictors on test bed #1 (on Highway No. 5) and test bed #2 (on Highway No. 1),
respectively. As shown in the tables, the MAPEs of the three predictors are all smaller than
20%. On test bed #1, the average MAPE of KF is about 5% less than that of OKF and near
3% more than AKF. On test bed #2, the average MAPE of KF is about 6% less than that of
OKF and near 2.5% more than AKF. Furthermore, the maximum absolute percentage errors
(Max APE) for the three prediction methods (OKF, KF and AKF) under comparison in the
two test beds are reported in Table 1 and Table 2 (see the last row of the two tables),
respectively. The Max APEs of KF and AKF are much smaller than those of OKF, which
further highlights the advantage of the proposed approaches over the conventional KF
method in the literature. On both test beds, KF and AKF significantly outperform OKF in
terms of MAPE. This indicates the advantage of adopting the structural state space model
which incorporates historical data and considers structural deviations from regular traffic
patterns. Moreover, the performance of AKF is slightly better than KF, due to the adaptive
scheme, described in Section 3.2, for updating system and measurement error variances.
[Insert Table 1 and Table 2 about here]
21
To evaluate the performance of the proposed approach under non-recurring conditions,
this research selected three incident cases that occurred on National Highway No.5 in June,
2011, and investigated the prediction results in the immediate upstream segments of the
incident locations. The incident data, as shown in Table 3, were provided by Pinglin TCC,
which is responsible for the operation of National Highway No.5; the incident data on
National Highway No.1 were not available when the experiment was conducted. All the three
incidents involved multiple cars and caused one- or two-lane closure in the incident durations,
so the capacities were greatly reduced by 33% to 66% (there are three lanes in test bed #1).
[Insert Table 3 about here]
The MAPEs of OKF and AKF for these three incident cases are provided in the
rightmost two columns in Table 3, which show that AKF significantly outperforms OKF. In
the three cases, the MAPE of AKF is at least 8% lower than that of OKF. Figs. 4-1, 5-1, and
6-1 depict the prediction results of OKF, and Figs. 4-2, 5-2, and 6-2 plot the prediction results
of AKF for the three incident cases. As shown in the figures, the observed densities in the
incident durations (i.e., non-recurring conditions) deviate considerably from the historical
pattern and exceed the upper bounds (the 75th percentiles) of recurring conditions, while the
observed densities before and after incident durations follow closely with the historical
pattern. This observation lays a foundation for the proposed structural state space modeling
framework and the KF algorithm for traffic state prediction and estimation. Particularly, for
non-recurring conditions, the structural changes from the regular pattern are captured in the
proposed approach, while the forecasts for recurring conditions are based on the historical
pattern. The numerical results demonstrate that the prediction performance of AKF is better
than that of OKF under both recurring and non-recurring conditions. The superiority in
prediction performance of AKF over OKF is especially apparent under non-recurring
conditions, highlighting the advantage of incorporating the structural deviation term in the
22
model.
Essentially, under recurring conditions, the proposed approach based on the estimate of
regular pattern is capable of describing real-time traffic state time series and produces
accurate forecasts. However, in the presence of structural changes, deviations between actual
traffic states and regular pattern have non-zero means. The KF then combines the polynomial
trend filter with historical pattern to capture possible structural deviations, producing robust
forecasts under both recurring and non-recurring conditions. Therefore, a robust traffic state
predictor should incorporate both regular pattern and structural deviation components to
ensure prediction quality under a wide range of conditions.
[Insert Fig. 4-1, Fig. 4-2, Fig. 5-1, Fig. 5-2, Fig. 6-1, and Fig. 6-2 about here]
5. Concluding remarks
With a particular aim at developing a robust short-term traffic state prediction approach
that can produce accurate forecasts under both regular and irregular traffic conditions, this
research proposes the prediction model which integrates both the historical pattern and the
polynomial trend component. The proposed model decomposes actual freeway traffic state
into three components, namely, regular pattern, structural deviation, and random fluctuation,
where the median of historical data is used as the a priori estimate of regular pattern and the
polynomial trend model is adopted to describe structural changes due to unexpected capacity
reduction and demand fluctuations. In this modeling framework, freeway traffic states are
predicted by the proposed approach using a mixture of historical (median) data that capture
regular traffic patterns and real-time observations obtained from detectors that contain
intrinsic (structural deviation) information of current traffic states. We also show that the
structural deviation in a single bottleneck case can be analytically derived based on
cumulative flow counts, which provide theoretical supports on the use of polynomial trends
model under non-recurring congestion conditions .
23
The model is incorporated in the Kalman filtering-based algorithm. If a real-time
measurement falls within the range of recurring conditions, structural deviation vanishes and
the forecast of traffic state depends on historical pattern. Otherwise, structural change is
predicted and corrected based on that real-time observation. The proposed KF and AKF
algorithms were evaluated on the two test beds in the northern Taiwan highway network, and
compared with the OKF algorithm presented in the literature. The prediction results show that
the proposed approaches significantly outperform OKF in terms of MAPE on both test beds.
Furthermore, the forecasting results under incident (non-recurring) situations were
investigated. The prediction performance of AKF is particularly better than that of OKF
under non-recurring traffic conditions, highlighting the benefit of incorporating the structural
deviation term in the model.
Our future research is to compare the prediction performance of the proposed approach
with that of other state-of-the-art traffic state (or travel time) prediction methods, such as
Bayesian forecasting and neural networks, on the two test beds and some other test beds. A
set of sensitivity tests will also be conducted to investigate impacts and benefits of using
higher order polynomial trend models. Additionally, as a heterogeneous sensor network may
include a set of point, point-to-point and probe sensors, extending the proposed traffic state
predictor to fully utilize heterogeneous sources of measurement data will be an important
future research task for the deployment of AKF in traffic networks. With real-time
measurement data inevitably tagged with some degree of imprecision and uncertainty, a
robust traffic state prediction method should also explicitly address the issue of representing
and dealing with imprecise and uncertain data.
Acknowledgements
This paper is based on a project (NSC 102-2410-H-027-011-MY3) sponsored by the National
Science Council, Taiwan. The authors are grateful to three anonymous reviewers for many
24
insightful and constructive comments and suggestions which help improve the quality of this
paper. The authors are solely responsible for the content of this paper.
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Fig. 1 Illustrative examples of the structural state space model
Fig. 2 Cumulative flow count curves for a single bottleneck with reduced capacity due to an
incident starting at time s
Time
Cumulative flow count
C
C
Departure curve
with incident (D')
Incident duration
Additional number
of vehicles in queue
(structural deviation)
Departure curve (D)
Arrival curve
f
e
s
Reduced capacity (CR)
Time
True state
Historical pattern
Deviation
29
Fig. 3 Two test beds in northern Taiwan area
(Source: National Freeway Bureau, Taiwan http://www.freeway.gov.tw/english/way_net.aspx)
Table 1 MAPEs (%) of the prediction results on test bed #1 (Highway No.5)
Methods
OKF
KF
AKF
Date
AM
PM
AM
PM
AM
PM
2011/6/1
17.04
15.78
11.51
9.11
7.36
7.66
2011/6/2
17.64
17.96
13.64
11.01
10.31
8.41
2011/6/3
16.65
16.44
13.13
11.95
9.84
8.95
2011/6/8
17.16
15.37
9.24
9.11
7.55
8.52
2011/6/9
16.26
13.31
9.06
10.38
8.31
7.47
2011/6/10
17.16
17.06
11.15
10.87
7.98
7.95
2011/6/15
16.78
15.99
12.42
13.36
8.25
9.68
2011/6/16
16.58
17.32
8.73
13.66
4.83
9.63
2011/6/17
14.75
15.15
8.36
11.68
4.97
8.31
2011/6/22
15.81
15.87
10.12
9.23
8.34
7.85
2011/6/23
15.97
15.27
10.52
8.87
7.63
5.76
2011/6/24
16.32
17.89
12.43
10.72
9.32
6.33
2011/6/29
15.69
15.11
13.36
9.03
8.70
6.62
2011/6/30
15.34
14.63
10.55
12.29
8.02
8.09
Average
16.37
15.74
11.02
10.61
8.04
7.95
Max. APE
28.21
30.95
18.88
18.47
14.13
13.89
Table 2 MAPEs of the prediction results on test bed #2 (Highway No.1)
Methods
OKF
KF
AKF
Date
AM
PM
AM
PM
AM
PM
2011/6/1
18.32
16.28
12.16
9.67
9.35
8.08
2011/6/2
17.50
15.84
9.44
9.48
9.32
8.27
2011/6/3
18.84
16.90
12.58
12.82
7.19
9.11
2011/6/8
14.91
16.72
9.63
8.82
8.94
8.14
30
2011/6/9
17.79
14.60
8.34
8.73
8.31
7.77
2011/6/10
17.51
14.59
10.77
8.24
8.35
6.62
2011/6/15
14.13
14.62
9.55
9.94
7.40
7.41
2011/6/16
14.80
15.39
8.19
11.22
7.76
9.22
2011/6/17
15.54
15.73
10.03
9.78
6.46
9.57
2011/6/22
14.74
15.03
9.38
10.38
7.16
9.37
2011/6/23
15.27
17.40
8.42
6.44
7.40
5.94
2011/6/24
16.33
15.71
12.97
8.51
6.18
7.02
2011/6/29
17.44
13.90
11.51
8.35
8.33
6.27
2011/6/30
18.29
14.24
11.19
11.37
7.23
8.77
Average
16.52
15.49
10.30
9.75
7.82
7.97
Max. APE
25.05
24.13
17.75
17.39
13.91
13.85
31
Table 3 Prediction performances for three selected incidents on Highway No.5 in June, 2011
No.
Location
Date
Start
Time
End
Time
# of Blocked
Lanes
OKF
(MAPE)
AKF
(MAPE)
1
11.2-km Eastbound
2011/6/1
16:50
17:45
1
17.87%
9.84%
2
7.1-km Eastbound
2011/6/9
9:10
10:05
1
18.12%
9.91%
3
4.5-km Eastbound
2011/6/15
11:10
12:15
2
18.15%
9.23%
Fig. 4-1 Prediction results of OKF for Incident No. 1
Fig. 4-2 Prediction results of AKF for Incident No. 1
0
20
40
60
80
100
120
140
Density (number of vehicles/lane-km)
Observations
Predictions
Historical median
0
20
40
60
80
100
120
140
Density (number of vehicles/lane-km)
Observations
Predictions
Historical median
Incident
Incident
32
Fig. 5-1 Prediction results of OKF for Incident No. 2
Fig. 5-2 Prediction results of AKF for Incident No. 2
Fig. 6-1 Prediction results of OKF for Incident No. 3
0
20
40
60
80
100
120
07:30
07:35
07:40
07:45
07:50
07:55
08:00
08:05
08:10
08:15
08:20
08:25
08:30
08:35
08:40
08:45
08:50
08:55
09:00
09:05
09:10
09:15
09:20
09:25
09:30
09:35
09:40
09:45
09:50
09:55
10:00
10:05
10:10
10:15
10:20
10:25
10:30
10:35
10:40
10:45
10:50
10:55
11:00
11:05
11:10
11:15
11:20
11:25
11:30
11:35
11:40
Density (number of vehicles/lane-km)
Observations
Predictions
Historical median
0
20
40
60
80
100
120
07:30
07:35
07:40
07:45
07:50
07:55
08:00
08:05
08:10
08:15
08:20
08:25
08:30
08:35
08:40
08:45
08:50
08:55
09:00
09:05
09:10
09:15
09:20
09:25
09:30
09:35
09:40
09:45
09:50
09:55
10:00
10:05
10:10
10:15
10:20
10:25
10:30
10:35
10:40
10:45
10:50
10:55
11:00
11:05
11:10
11:15
11:20
11:25
11:30
11:35
11:40
Density (number of vehicles/lane-km)
Observations
Predictions
Historical median
0
20
40
60
80
100
120
140
Density (number of vehicles/lane-km)
Observations
Predictions
Historical median
Incident
Incident
Incident
33
Fig. 6-2 Prediction results of AKF for Incident No. 3
0
20
40
60
80
100
120
140
Density (number of vehicles/lane-km)
Observations
Predictions
Historical median
Incident
