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Mining spatiotemporal features of city traffic
Panagiotis Fafoutellis1*, Emmanouil Kampitakis1, Eleni I. Vlahogianni1, Nectarios
Koziris2, George Yannis1, John C. Golias1
1School of Civil Engineering, National Technical University of Athens
2School of Electrical and Computer Engineering, National Technical University of Athens
*Email: panfaf@mail.ntua.gr
Abstract
Short-term traffic forecasting is a field of research that has always attracted significant attention. The
recent introduction of Machine Learning techniques in traffic forecasting has broadened the researchers’
horizons, making fresher approaches possible. However, researchers should not disregard the
importance of spatiotemporal relations of a road network and classic statistical modeling, which also
provide better interpretation. In this paper, we detect the spatiotemporal relationships of the extended
2nd ring road network of Xi’an, China using Pearson’s Correlation, Mutual Information and Dynamic
Time Warping on the network’s speed time series. The first two give an indication of the spatial
dependency between road sections by comparing their speeds’ contributions, while Dynamic Time
Warping takes also into account the temporal evolution of the phenomenon. Results show that, although
the first approach leads to an accurate Bayesian Network prediction model, the second one leads to an
improved accuracy using the same modeling structure.
Keywords: spatiotemporal relations, time series, trajectories, mutual information, dynamic time warping
1 Introduction
Short-term traffic forecasting has always been a field of high research interest due to its
significant importance to traffic flow management and the development of intelligent
transportation systems and user-friendly information providing applications (Vlahogianni,
Karlaftis & Golias, 2014). Accurate traffic forecasting is also essential to efficient traffic control
and sustainable road network conditions as it reduces the levels of uncertainty of the decision-
making process.
Nowadays, the extended use of smart devices and systems (GPS, smartphone, in-vehicle
telematics etc.), which are able to track a huge amount of real-time mobility data, gives
researchers the opportunity to develop prediction models that are more accurate and constantly
updated, as well as of high temporal resolution. This has been the turning point that moved
researchers’ attention from classical statistic approach to Machine Learning data-driven models
with the assistance of Data Mining and Big Data algorithms (Vlahogianni, Karlaftis & Golias,
2014). The aspect that made the development of such models possible is the massive growth of
the computational power of modern computers the previous two decades, which are able to
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cope with the high computational complexity calculations required within some seconds but
can also handle large amounts of data efficiently.
Α very popular approach for short term traffic forecasting is to identify the relations between
traffic flow variables, such as speed, of different road sections of a road network over time,
using statistical metrics, which have very solid mathematical foundations (Karlaftis &
Vlahogianni, 2010). Nevertheless, with the overwhelming use of Deep Learning, researchers
seem to disregard the importance of demystifying the spatiotemporal dynamics of traffic to the
prediction and efficient management of traffic conditions. Neural networks produce very
accurate predictions of traffic parameters, as they can approximate almost any function,
regardless of its degree of nonlinearity and without prior knowledge of its functional form
(Vlahogianni, Karlaftis & Golias, 2005). In contrast to traditional Neural Networks and
Machine Learning techniques, models that also involve the spatiotemporal relations of a road
network provide better interpretation and insight of the mechanisms creating the predictions.
The general understanding so far on how spatial data may improve traffic forecasting has been
limited by the lack of network wide traffic information (Vlahogianni et al., 2014). Interestingly,
most approaches so far either capture spatial dependency of adjacent upstream and downstream
links with a study link using correlation analysis or develop forecasting methods in a corridor
test sample, where all links are connected sequentially together, assume a similarity between
the behaviour of both parallel and adjacent links, and overlook the competitive nature of traffic
links (Ermagun & Levinson, 2018).
Moreover, the temporal-spatial features in forecasting are usually addressed internally in
advanced deep learning structures (Laña et al., 2019) or by resorting to more sophisticated
approaches, namely by constructing useful inputs for traffic flow predictors through the
extraction of correlations in the data (Stathopoulos & Karlaftis, 2002), (Vlahogianni et al.,
2005), (Sun, Huang & Gao, 2012), (Vlahogianni, 2015), (Zhao et al., 2017); considering each
location as a module in a modular network (Vlahogianni et al., 2007), (Vlahogianni, 2009);
considering each location as a task in a multitask DBN model (Huang et al., 2014). Evidently,
methods that increase the explanatory power of the forecasting models are usually preferred
against the so called “black box approaches”, in case we want to gain managerial knowledge
not only what traffic conditions are expected, but also on why these conditions are most likely
to occur.
The present paper attempts to introduce a much more sober analysis of spatiotemporal
dependencies disengaged from deep learning with the aim to increase the understanding on the
following research questions:
• Do spatial and temporal traffic dependencies exist in a road network?
• What are the impacts of spatiotemporal dependencies in short-term traffic forecasting?
To this end, this paper implements concepts spanning from classical correlation analysis, to
Information Theory, Time Sequence Analysis and Bayesian Networks. The proposed
methodological approach is implemented on the road network of the city of Xi’an, China using
trajectory data provided by Didi Chuxing Technology Co, a Chinese taxi and private car-hailing
company.
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2 Identifying Spatiotemporal Patterns
2.1 Linear Correlation and Mutual Information
To answer the question on whether there are road sections that are correlated in terms of travel
speed, we apply the concept of Mutual Information (MI) and compare the results with the
classical correlation analysis (Pearson’s correlation). Based on information theory, MI of two
random variables is a metric that quantifies the amount of information obtained for the first
random variable when observing the other random variable. Unlike the classical correlation
analysis, the mutual information takes into account nonlinear correlations as well, because the
computed measure is not connected to the linear or non-linear evolution rules of the quantities
involved, but to Shannon Entropy (Abarbanel, 1996), (Kantz & Schreiber, 1997). Let xn and yn
two equally spaced sets of random variables with joint probability density p(xn, yn) and
individual probability densities p(xn) and p(yn). The MI I(xn, yn), which quantifies the expected
information gained about xn when observing
n
y
is given by:
(1)
This approach has two main limitations: first, it is static in a sense that time is not introduced
in the analysis of travel speed interrelations between different network locations. Second,
interrelations are assessed in a pairwise manner without letting understanding on how
information from multiple locations may interact with each other and affect predictability.
2.2 Distance Based Time Series Similarity
To address the first point of criticism mentioned above, the present work implements the fast
dynamic time warping (Fast DTW). Dynamic time warping (DTW) is a dynamic programming
technique to find an optimal alignment between two given time series with the objective to
minimize a specific distance measure (Berndt & Clifford, 1994). For the time series X =
x1,x2,…,xn and Y = y1,y2,…,yn, DTW distance is given by the following recurrent equation to
the matrix γ(i…n, j…n) using dynamic programming (Lee et al., 2017):
(2)
The path that provides the optimum, namely minimum, distance is the warping path. The DTW
distance
( , ) ( , )DTW X Y n n
=
is the Euclidean distance along the warping path. DTW has a
quadratic time and space complexity that limits its use to only small size time series data sets.
To alleviate this limitation, an extension on classical DTW may be used, which first transforms
high dimensional time series to low dimensional time series and then obtain DTW distances on
the low dimensional time series. This extension known as Fast DTW operates on three steps
(Salvador & Chan, 2007): coarsening to reduce the dimensionality, projection to calculate DTW
distance in the lowest time series resolution, and refinement to project the warping path to an
incrementally higher resolution. The last two steps repeat until the path is projected to the full
time series resolution.
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2.3 Bayesian Network Classifier
Finally, to address the limitation of pairwise time series comparison, we develop a Bayesian
Network, which presents the relations between all the road sections and is based on the
calculation of conditional probability between their speeds’ contributions. A Bayesian Network
(BN) is a directed acyclic graph whose nodes represent variables. The weights of the
connections of the nodes are proportional to the relationship between the variables of the
corresponding nodes. With the above model, it is possible to calculate the conditional
probability of a variable getting a certain value when knowing the values of all the variables
that are connected to it (child nodes) (Pearl, 2000).
The BN for a set of variables Xi = {X1, …, Xn} also consists of a set Pi = {P1, …, Pn} of local
conditional probability distributions associated with each node and its parents. BN’s causal
interpretation is as follows: a directed edge from one variable to another Y, represents the claim
that X is a direct cause of Y with respect to other variables in the DAG (Friedman et al., 1997).
The joint distribution p can be factorized as a product of conditional probabilities, by specifying
the distribution of each node conditional on its parents. For a given structure B of a BN, the
joint probability distribution
()PX
for X can be written as:
1
( ) ( )
n
i i i
i
P X P X pa
=
=
(1)
where pai denotes the set of parents for
i
X
. The BN can be used as a classifier of
i
X
inputs to
a set of classes, in our case, the travel speed classes (C), by the rule (Friedman et al., 1997):
( )
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( ,..., ) argmax ( ) |
n
n n i i
i
classify x x p R p X x C
=
==
(2)
By the BN classification task, the influence of each variable (in our case the lagged information
of volume and occupancy from both the upstream and downstream location and the location of
interest) can be determined with respect to the prevailing speed class C. The selection of
influential spatio-temporal patterns of travel speeds will be based on the mutual information
criterion. Mutual information quantifies the amount of information flow between a node Xi and
the knowledge of traffic speed levels C. The mutual information I(X,C) between a variable X
and a class C measures the expected information gained about C, after observing the value of
the variable X:
( ) ( ) ( )
( ) ( )
,
|
, | ( )log |
XC cC
P X C
I X C P X C P C P X C P c
=
(3)
3. Implementation and Findings
3.1 Data Preprocessing
Data preprocessing is an essential procedure when conducting statistical analysis or applying
machine learning techniques. Well-prepared input data lead to better performing and easier
trained and tuned prediction models. The dataset used in this paper consists of the about 110000
trajectories per day of Didi’s vehicles in Xi’an for 2nd to 30th of November of 2016. Each
trajectory corresponds to the exact position of the vehicle per 2-4 seconds. More specifically,
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the attributes of the data are the latitude and the longitude of each point of the trajectory, an ID
number specifying the route, a second ID number specifying the driver and the timestamp of
the moment that the vehicle was at the particular position.
First, the coordinates of the points from the Chinese State Bureau of Surveying and Mapping
coordinate system (GCJ-02) were transformed to the World Geodetic System (WGS 84) in
order to depict properly on some of the most well-known web maps, such as Open Street Map,
and to make any further processing possible using the “eviltransform” python library. Second,
we apply map-matching to the points of our dataset using a nearest neighbor relationships
(Tveite, 2014). In general, map matching is the procedure of matching recorded GPS traces to
real-world road network edges, while at the same time correcting the system’s error at the time
of recording. In our case, each of the recorded points was matched with a road section of Xi’an’s
road network, as downloaded from Open Street Map. Last, we calculate the speed of the
vehicle’s movement from each point to the next of the same route, as the Euclidean distance of
the two points to the difference of their timestamps. This way, it is possible to generate the time
series of the speed of each one of the road network’s sections.
In order to calculate the time series of speeds of each road section, our data were grouped by
the road section ID and by the chosen time-step. For the needs of this paper, a time-step of 1
hour was used, resulting in two time series for each road section. The value of the speed of each
time-step of each road section was calculated as the average speed of all vehicles that passed
from the road section the specific time period.
It is worth mentioning that road sections that did not have any record on any of the twenty-nine
chosen days were excluded from further analysis, as it is clear that they do not play an important
role in Xi’an’s transportation system. The same applies to road sections that do not have any
record for more than an hour on any of the twenty-nine days. Figures 1 and 2 show the available
full-length time series and one day time series of a specific road section; although a daily
cyclicity is evident, there seem to be some short term features that may significantly affect the
magnitude and evolution of speed and, consequently, the prediction accuracy.
Figure 1: Sample 30-days time series
0
5
10
15
20
25
30
35
40
45
50
2/11/2016 00:00 8/11/2016 00:00 14/11/2016 00:00 20/11/2016 00:00 26/11/2016 00:00
Travel Speed (km/h)
Date and Time
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Figure 2: Sample 1-day time series
3.2 Do spatial and temporal traffic dependencies exist in a road network?
To address this research question, the Pearson’s correlation coefficient between each pair of
time series were calculated and are presented as a heatmap in Figure 3. This heatmap gives a
clear indication of which sections are related to each other. In addition, mutual information of
each pair of the series was calculated. The results are presented below as a heatmap in Figure
4. In both heatmaps, the lighter the color bands the stronger the relationship between road
sections. In the two heatmaps, there are some common patterns that are clearly noticeable.
However, MI criterion seems to produce lower values in terms of the strength of the identified
spatial patterns. By examining each column (or row) separately, one can detect which sections
are most related to the one of the columns.
0
5
10
15
20
25
30
35
40
45
22/11/2016 00:00 22/11/2016 06:00 22/11/2016 12:00 22/11/2016 18:00 23/11/2016 00:00
Travel Speed (km/h)
Date and Time
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Figure 3: Correlation heatmap of time series
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Figure 4: Mutual Information heatmap of time series
In order to compare the two metrics to each other, but also to the ones introduced later in this
paper, the road section with ID number 28258922 on Open Street Map was selected to be
presented in detail. The above road section is one of the most crowded road sections in Xi’an
at the centre of the city. The exact position of the road section is shown on Figure 5.
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Figure 5: Selected road section (green) on Xi’an City map.
Figure 6 depicts the 20 most related road sections (red) to the selected section, in terms of
Pearson’s Correlation (left) and Mutual Information (right). It seems that the two approaches
capture different spatial patterns on the same dataset. The impacts of these differences should
be further investigated in terms of prediction accuracy.
Figure 6: The 20 most related road sections (red) to the selected section (blue), in terms of Pearson’s
Correlation (left) and Mutual Information (right)
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Further, the implementation results of the Fast DTW algorithm on the available 1-h times series,
which gives as an indication of which road sections’ speed are related to each other- in terms
of temporal evolution- is seen in Figure 7. Smaller value (darker color) means higher
correlation. Figure 8 shows the 20 most related road sections to the chosen one, in terms of Fast
DTW distance. Compared to the detected patterns in Figure 5, there are clear differences
between the spatio-temporal correlations and the spatial correlation detected using MI or linear
correlation.
Figure 7: Dynamic Time Warping heatmap of 1-hour time step time series
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Figure 8: The 20 most related road sections (red) to the selected section, in terms of Dynamic Time
Warping Distance
3.3 Comparison between Mutual Information and Dynamic Time Warping
In order to compare each method’s results, we proceed to developing two prediction models of
the speed of the target road section. Both models are Bayes Network Classifiers that assign the
section’s speed to three balanced classes: <20, 20-26, >26 km/h, which is a reasonable choice
for signalized road sections, especially when we refer to travel speeds, including possible stops
(e.g. upstream of a signalized intersection), as done in our case.
The first model was “forced” to create the Bayesian Network and predict using only data from
the twenty road sections with the highest Mutual Information value, while the second using
only the twenty road sections with the smaller values of DTW distance.
As it can be clearly seen in Table 1, the model using Mutual Information as the metric to choose
which road sections to include outperforms the second one. The accuracy of the two models is
89% and 85.6% respectively. Hence, one can assume that using Mutual Information is a more
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accurate choice for the present application. This can be explained by considering again the
definitions of the two metrics. Dynamic Time Warping is a measure of similarity between two
time series, which is highly affected by the absolute size of the section’s speed and only slightly
by the time-series’ pattern. On the other hand, the estimation of Mutual Information takes into
account the trend of the timeseries and the proportional variation of speeds rather than their
absolute prices.
Table 1: Classification metrics of the two models developed
Metrics
Model 1 (Mutual Info)
Model 2 (DTW)
Accuracy
89%
86%
Recall (Sensitivity)
89%
86%
Precision
89%
86%
F1 - score
89%
86%
3.4 What are the impacts of spatiotemporal dependencies in short-term traffic forecasting?
In order to identify if the analysis conducted in the previous chapters is relevant and improves
the prediction task, we evaluate the findings by comparing them to the performance of a
Bayesian classifier that is “free” to make predictions using data from any road section. In this
case, each road section’s contribution to forecasting is proportionate to its probabilistic
relationship with the selected one. The classification results are summarized in Table 2. The
accuracy of this model is 84.5%.
Table 2: Classification metrics of Naïve Bayes model
Metrics
Model 3 (Naïve Bayes)
Accuracy
84%
Recall (Sensitivity)
84%
Precision
85%
F1 - score
85%
Results indicate that the models presented at the previous chapter produce obviously better
predictions. The first one, which uses the sections with the highest Mutual Information to the
selected one, is performing noticeably better, while the second only slightly but still better. This
result highlights the usefulness of performing spatiotemporal analysis.
Moreover, the above procedure decreases the dimensionality of the problem, which is a very
common issue when using Machine Learning algorithms. In the current case, we originally had
696 individuals (time periods) with 283 attributes (road sections’ speeds) each, which is a really
high value. After performing spatiotemporal analysis feature selection, we used only the 20
most related attributes. Furthermore, the above procedure reduces the computational resources
needed, which is equally important.
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4 Conclusions
Analyzing large scale spatiotemporal characteristics of a road network before proceeding to the
development of prediction models is essential in order to produce accurate predictions. In
addition, such models provide better interpretation of the spatio-temporal evolution of traffic.
General use metrics, such as Pearson’s Correlation, as well as more specialized for time series
such as Dynamic Time Warping’s distance and Mutual Information provide a clear insight into
the spatiotemporal relationships of the road sections of an urban road network. These
relationships occur from the relative position of the road sections and the traffic flow its one
serves; therefore, they provide explainable results.
In order to underline the importance of identifying spatio-temporal traffic patterns, a simple
structured Bayesian Network was developed to classify speeds based on the identified spatio-
temporal traffic patterns. Although the model that was used is not the best-suited choice for
time series data, the improved performance by introducing the network level spatiotemporal
data is noticeable and important for the standpoint of accuracy and computational efficiency.
Future research will certainly include processing of data from more days, in order to have more
generalized conclusions. The above requires a more powerful processing system in terms of
hardware, as well as software, because the vast amount of data involved lead to Big Data
Analysis approach. Second, more specialized and cutting-edge Machine Learning techniques
will be used, because there seems to be enough space for optimization to achieve accurate single
step ahead and larger time horizons forecasting.
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