# Identification of Driving Heterogeneity using Action-chains

Xue Yao*, Simeon C. Calvert and Serge P. Hoogendoorn Xue Yao*, Simeon C.
Calvert, and Serge P. Hoogendoorn are with Department of Transport & Planning,
Delft University of Technology, 2628CN. Delft, The Netherlands
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Current approaches to identifying driving heterogeneity face challenges in
capturing the diversity of driving characteristics and understanding the
fundamental patterns from a driving behaviour mechanism standpoint. This study
introduces a comprehensive framework for identifying driving heterogeneity
from an _Action-chain_ perspective. First, a rule-based segmentation technique
that considers the physical meanings of driving behaviour is proposed. Next,
an _Action phase_ Library including descriptions of various driving behaviour
patterns is created based on the segmentation findings. The _Action-chain_
concept is then introduced by implementing _Action phase_ transition
probability, followed by a method for evaluating driving heterogeneity.
Employing real-world datasets for evaluation, our approach effectively
identifies driving heterogeneity for both individual drivers and traffic flow
while providing clear interpretations. These insights can aid the development
of accurate driving behaviour theory and traffic flow models, ultimately
benefiting traffic performance, and potentially leading to aspects such as
improved road capacity and safety.

##  I Introduction

Driving behaviour plays a pivotal role in determining vehicle motion,
substantially affecting traffic flow, fuel consumption, and emission. It is
widely acknowledged that driving heterogeneity, which is defined as the
difference between driving behaviours of driver/vehicle combinations under
comparable conditions[1], does exist. Research has shown that this
heterogeneity contributes to increased traffic accidents and congestion [2].
Additionally, in mixed automated-human traffic, accurate descriptions and
predictions of human-driven vehicle (HDV) behaviour are crucial for the
decision-making and control of connected and automated vehicles (CAVs). These
have underlined the necessity of a better understanding and identification of
the heterogeneity in human driving.

It is well established that driving heterogeneity encompasses both intra-
heterogeneity, which refers to driver-independent variability, and inter-
driving heterogeneity, which involves differences in driving behaviour among
drivers [1, 3]. However, directly measuring or detecting driving heterogeneity
is challenging due to its reliance on human cognitive and physiological
processes. With the increasing availability of naturalistic driving data,
various efforts have been made to comprehensively and quantitatively analyse
driving heterogeneity. The identification of driving heterogeneity from
observed driving behaviour is typically approached in two ways [4]: 1)
Employing techniques to characterise driving behaviour by inferring driving
profiles from distinct driving events, and 2) Analysing driving behaviour
without explicitly creating driving behaviour profiles.

The former approach addressed the identification of driving heterogeneity as a
classification or clustering problem, resulting in categorical output with
discrete scales or numerical output with continuous scores. For example,
Hoogendoorn et al. [5] developed a method to categorise driver states into
low, medium, and high workload categories. Or, clustering techniques have been
employed to define a few driving style groups such as aggressive, normal, and
mild [2]. However, due to the stochastic and uncertain nature of driving
behaviour, these limited groups are insufficient for capturing the diverse
characteristics of driving behaviour. Additionally, the criteria used to
define these groups are somewhat ambiguous and subjective, posing challenges
in effectively eliminating individual biases.

In contrast to employing subjectively defined classes, some research has
focused on identifying driving heterogeneity by presenting a driving style
space containing a vast array of categories without explicitly establishing
driving behaviour profiles. For example, Qi et al. distinguished driving
styles based on a space that included over 20 different types [6]. Another
study converted car-following sequences into a comprehensive array of
primitive driving patterns, and the distributions of these patterns were then
utilised to analyse individual driving styles [7]. This approach allows for
the recognition of a greater degree of variability in driving behaviour by
encompassing various driving characteristics. However, it is essential to
acknowledge that this broader categorisation of driving heterogeneity may lead
to reduced clarity of the fundamental driving behaviours and a limited
understanding of driving heterogeneity. Consequently, further research in this
area is necessary to address these challenges.

To bridge these research gaps, a novel framework is proposed to identify
heterogeneity in longitudinal driving behaviour from an _Action-chain_
perspective. An _Action-chain_ is defined as a series of _Action phases_ over
time. The contributions of this research are two-fold: i) A rule-based
segmentation technique is presented to divide driving trajectories,
considering the clear physical meanings of driving behaviour. ii) The concept
of _Action phase_ and _Action-chain_ are first introduced to interpret driving
behaviours, based on which a method for evaluating driving heterogeneity is
proposed. The effectiveness of the framework was evaluated using real-world
datasets, and the results demonstrate that the proposed methods can
effectively identify driving heterogeneity at both individual drivers and
traffic flow levels, providing clear interpretations. This approach offers
valuable insights into understanding driving behaviour by uncovering
underlying heterogeneity, which supports the development of accurate and
robust driving behaviour and traffic flow models.

##  II Framework description

###  II-A Defining Action Phase and Action-chain

The concept of “action points”, which refers to specific moments of change in
acceleration during driving [8], serves as the foundation to introduce the
concept of _action trend_ in this study. While action points capture
acceleration or deceleration, they do not fully capture the complexity of
driving behaviours. To overcome this limitation, we further propose the
concept of _Action phase_ , which expands the scope by incorporating
additional variables to provide more comprehensive information about driving
behaviour.

By examining the univariate trajectory of driving behaviour, illustrated by
the example of velocity ($v$) in Figure 1, distinct states are obviously
observed. Some trajectories exhibit upward trends, others display downward
trends, while some maintain a relatively stable range of fluctuations that can
be considered as a keeping trend. We refer to these moments of driving
behaviour different tendencies as _action trends_ , which are segmented by
turning points. Specifically, _action trends_ are classified as “Increasing
(I)”, “Decreasing (D)”, or “Stable (S)”. To further refine the “Stable” trend,
it is categorised as “Stable in a high value (H)” or “Stable in a lower value
(L)”. Thus, the _action trend_ space can be represented as $S=\\{I,D,H,L\\}$,
and the driving trajectory shown in Figure 1 can be expressed as
$S_{v}=\\{D,L,I,D,L\\}$.

It is worth noting that while driving behaviour variables often exhibit
synchronisation, our definition of _action trends_ allows for variations in
the temporal changes of different variables. For example, when the velocity
state is “Increasing”, the acceleration state can be “Increasing”,
“Decreasing”, “Stable”, or a combination of them. Consequently, the definition
of _action trends_ can be extended to other driving behaviour variables, such
as acceleration and space headway. Thereafter, the concept of _Action phase_
is proposed by encompassing multiple variables, and each _Action phase_ label
consists of multiple _action trend_ names. These _action trend_ names are
estimated using uniform criteria derived from the group level of drivers in a
certain traffic flow.

Figure 1: Visualization of time-series driving trajectory: An example of
velocity

To account for the inherent sequential nature of driving behaviour, it is
essential to consider the temporal dependencies between _Actions phases_.
Hence, the concept of an _Action-chain_ is introduced to represent a sequence
of _Action phases_ and their relations. The behaviour of a vehicle over time
may consist of one or more _Action-chains_ , each corresponding to different
responses to the environment. With the _Action-chain_ structure, driving
behaviour over time can be characterised, which provides valuable insights
into the underlying patterns and heterogeneity of driving behaviour.

###  II-B Introducing the Novel Framework

The proposed framework for identifying driving heterogeneity aims to estimate
frame-wise driving trajectories and identify driving heterogeneity within
specific traffic flow conditions. The entire procedure is illustrated in
Figure 2, consisting of five main steps: Data Preparation, Trajectory
Segmentation, Action phase Extraction, Action-chain Establishment, and
Heterogeneity Evaluation. The extraction of _Action phase_ and the
establishment of _Action-chains_ involve the preceding steps called Driving
Behaviour Interpretation and Action-chain Implementation, respectively.

Data plays a crucial role in the identification of heterogeneity and serves as
a fundamental aspect of the analytical process. After data tracing and
preprocessing, the time-series driving behaviour data are used as input for
the segmentation algorithm (Algorithm 1). It is represented as
$x_{1},x_{2},...,x_{t}$, where $x_{t}$ denotes the driving behaviour variable
feature at the $t$-th frame. The segmentation algorithm (Algorithm 1) outputs
$l^{m}_{n}$, which represents the _action trend_ names of variable $m$ to be
recognised for the $n$-th segment, where $n=1,2,...,N$. Based on the
segmentation results, the driving behaviour of individual drivers can be
visualised using driving behaviour maps, which highlight the unique
characteristics of each driver. In the driving behaviour map, at the $t$-th
frame, the state of driving behaviour is denoted as
$S_{t}=\\{l^{1},l^{2},...,l^{m}\\}$. Subsequently, Algorithm 2 is designed to
detect driving behaviour segments in which all variables have a single _action
trend_. The output, denoted as _Action phase_ and represented as
$S_{n^{\prime}}=\\{l^{1},l^{2},...,l^{m}\\}$, signifies the _Action phase_ for
the $n^{\prime}$-th segment, where $n^{\prime}\in N^{\prime}$, $N^{\prime}$
denotes the total number of _Action phases_ for an individual driver. All the
output _Action phases_ form the _Action phase_ Library under a specific
traffic flow. The actual size of this Library is generally smaller than the
theoretical value $m^{4}$ due to the nonexistence of certain state
combinations in the real world, in accordance with fundamental driving
behaviour theories. The length of the _Action phase_ at the $n^{\prime}$-th
segment, referred to as the time label, is denoted as
$\mathcal{T}_{n^{\prime}}$.

Considering the time-series nature of driving behaviour, an _Action phase_
transition probability algorithm (Algorithm 3) is implemented to capture the
temporal dependencies between _Action phases_. An _Action phase_ and the next
_Action phase_ obtained through the maximum transition probability constitute
an _Action-chain_ , representing the most probable driving behaviour adopted
by drivers. The _Action-chain_ serves as a description of homogeneous driving
behaviour and is used to distinguish heterogeneity in driving behaviour.
Drivers who deviate more from the _Action-chains_ are considered to exhibit
greater heterogeneity (conducted by Algorithm 4).

Figure 2: A novel framework of identifying driving heterogeneity

##  III Method implementation

###  III-A Rule-based Segmentation

Traditional classification algorithms, such as the K-nearest neighbour method,
support vector machines, and Convolutional Neural Networks have been commonly
used for classifying driving styles or recognising driving patterns [9].
However, the segments obtained using these algorithms often lack clear
interpretability in terms of physical characteristics. In contrast, rule-based
segmentation is a relatively simple and interpretable method for dividing
driving behaviour trajectories into meaningful segments. Therefore, we propose
a rule-based method, referred to as Algorithm 1 within the framework, to
segment driving behaviour trajectories.

Let $V=\\{v_{1},v_{2},...,v_{m}\\}$ be a set of driving behaviour variables,
such as velocity, acceleration, distance, etc.
$P=\\{(x_{1},y_{1}),...,(x_{n},y_{n})\\}$ represents a set of turning points
for a single variable, which are calculated using calculus, specifically the
first and second derivatives. Algorithm 1 consists of the following steps:

1\. Data preparation: Load the turning points of the selected variable.
Calculate the variable changes $\Delta y$ and time intervals $\Delta x$
between neighbouring turning points.

2\. Threshold setting: Define threshold values $\theta_{1},\theta_{2}$ to
differentiate between segments with state Increasing (I), Decreasing (D), or
Stable (S). Set $\gamma$ to determine whether a segment is too short and
should be merged with its neighbouring segments.

3\. Initial categorisation: If $\Delta y>\theta_{1}$, meaning that the
variable increases to a certain extent, which cannot be ignored, then label
the segment as I. If $\Delta y<\theta_{2}$, in which case the variable
decreases to a non-negligible level, then label it as D. When
$\theta_{2}<\Delta y<\theta_{1}$, the variable keeps within a small range of
changes and is labelled as S.

4\. Merging: For each segment labelled as S, if the time interval $\Delta
x_{n}<\gamma$, and $\Delta x_{n-1}>\gamma,\Delta x_{n+1}>\gamma$, merge the
segment with its neighbouring segment $n+1$.

5\. “Stable” refinement: For the updated S segments, calculate the mean value
of the variable for each segment. Update the labels S as stable in High (H) or
Low (L) values based on the threshold $\delta$.

By implementing the above Algorithm 1, each variable in $V$ is assigned
_action trend_ labels with clear physical meanings. This rule-based method
allows for effective segmentation of driving behaviour trajectories based on
single variables.

Subsequently, Algorithm 2 is proposed to extract _Action phases_ with simple
steps including 1) segmenting the trajectory using turning points of all
considered variables, and 2) removing segments shorter than the threshold of
drivers’ reaction time $\tau$.

###  III-B Time-series Action Phase Probability Modeling

The length of an _Action phase_ can vary and is denoted by the labels “Long
(lg)” or “Short (st)” according to a threshold $\eta$. Consequently, the time
label space for _Action phases_ is represented as $\mathcal{T}=\\{lg,st\\}$.
Subsequently, _Action phase_ can be further described by a two-dimensional
label space $S^{\prime}=\\{S,\mathcal{T}\\}$, which is taken as input for the
Action phase transition probability model (Algorithm 3). Let
$S^{\prime}_{n^{\prime}}$ and $S^{\prime}_{{n^{\prime}}+1}$ represent two
adjacent _Action phases_ , where $n^{\prime}\in\\{1,2,\dots,N^{\prime}-1\\}$.
The transition probability between them can be mathematically represented as a
function
$\mathcal{R}\left(S^{\prime}_{n^{\prime}},S^{\prime}_{{n^{\prime}}+1}\right)$,
which captures the underlying characteristics or patterns between
$S^{\prime}_{n^{\prime}}$ and $S^{\prime}_{{n^{\prime}}+1}$. This transition
probability function provides insight into the relationship and dynamics
between consecutive _Action phases_ in the time-series analysis.

###  III-C Coupled Markov Chain Theory

Two main approaches are commonly used to implement the transition of driving
behaviour segments. The first approach utilises Markov models, including
Markov Chains and Hidden Markov Models (HMM) [7], which are easily
interpretable and capable of capturing underlying structures. However, when
dealing with a large number of hidden layers, HMM may become computationally
inefficient and less accurate due to increased complexity. The second approach
involves deep learning models such as Recurrent Neural Networks (RNN), Long
Short-Term Memory (LSTM), and Gated Recurrent Unit (GRU) Networks [7]. These
models can address the complexity limitation of HMM and capture complex
relationships between _Action phases_. Nevertheless, they typically require a
large amount of training data and are computationally expensive due to their
gating mechanisms.

In our case, the Markov Chain method is adopted to implement the Action
Probability (Algorithm 3). The concept of a coupled chain refers to the
collective behaviour of two independent systems, each following the principles
of a classical Markov chain [10]. Let’s consider two one-dimensional Markov
chains ($X_{i}$) and ($Y_{j}$) that operate on the state space
$\\{S_{1},S_{2},...,S_{n}\\}$, with positive transition probabilities defined
as

$\Pr(X_{i+1}=S_{k},Y_{j+1}=S_{f}|X_{i}=S_{l},Y_{j}=S_{m})=p_{lm,kf}$ (1)

here, the $(X_{i})$ chain describes the _Action phase_ state $S^{\prime}$ and
the $(Y_{i})$ chain describes the time label $\mathcal{T}$. Then the coupled
transition probability $p_{lm,kf}$ on the state space
$\\{S_{1},S_{2},...,S_{n}\\}\times\\{S_{1},S_{2},...,S_{n}\\}$ is given by

$p_{lm,kf}=p_{lk}\cdot p_{mf}$ (2)

Two coupled one-dimensional Markov chains can be utilised to construct a two-
dimensional spatial stochastic process on a lattice represented by
($Z_{i,j}$). The lattice consists of a two-dimensional domain of cells, as
depicted in Figure 3. The deep blue cells represent known boundary cells, the
light blue cells indicate known cells within the domain (past observations),
and the white cells represent unknown cells. The future state used to
determine the state of cell ($i,j$) is cell ($N_{x},j$), where each cell is
identified by its row number $i$ and column number $j$. Then the conditional
probabilities can be expressed as follows [11]:

Figure 3: Conditional Markov chain on the states of the future

$P_{lk}^{h}=\Pr(X_{i+1}=S_{k}|X_{i}=S_{l})$ (3)

$P_{mk}^{v}=\Pr(Y_{j+1}=S_{k}|Y_{j}=S_{m})$ (4)

The stochastic process $(Z_{ij})$ is obtained by coupling the Markov chains
$(X_{i})$ and $(Y_{j})$ while ensuring that these chains transition to the
same states. Therefore, we have:

$\displaystyle\Pr(Z_{i,j}=S_{k}|Z_{i-1,j}=S_{l},Z_{i,j-1}=S_{m})$ (5)
$\displaystyle=C\,\Pr(X_{i}=S_{k}|X_{i-1}=S_{l})\,\Pr(Y_{j-1}=S_{m})$

where $C$ is a normalising constant that arises from restricting transitions
in the ($X_{i}$) and ($Y_{j}$) chains to the same states. It is calculated as:

$C=\left(\sum_{f=1}^{n}p_{lf}^{h}\cdot p_{mf}^{v}\right)^{-1}$ (6)

By combining Equation 5 and Equation 6, the required probability can be
expressed as:

$\displaystyle p_{lm,k}$
$\displaystyle:=\Pr(Z_{i,j}=S_{k}|Z_{i-1,j}=S_{l},Z_{i,j-1}=S_{m})$ (7)
$\displaystyle=\frac{p_{lk}^{h}\cdot p_{mk}^{v}}{\sum_{f}p_{lf}^{h}\cdot
p_{mf}^{v}},\,k=1,...,n$

##  IV Data-based evaluation

###  IV-A Data Preporcessing

In this study, the NGSIM highway dataset, which includes data from I-80 and
US-101, was utilised to investigate the heterogeneity of longitudinal driving
behaviour based on our proposed framework. A comprehensive preprocessing of
the dataset, involving filtering and extraction, was conducted as described by
Sun et al. [2]. Especially, drivers with trajectories lasting at least 50
seconds were selected to ensure an adequate amount of data for analysing
longitudinal driving behaviour [12]. The final extracted dataset consisted of
123 drivers from the I-80 dataset and 848 drivers from the US-101 dataset.

The driving behaviour variables considered in this study were velocity ($v$),
acceleration ($a$), distance ($d$) between the preceding and following
vehicles, and their speed difference ($\Delta v$). The threshold values used
in Algorithm 1 were determined based on empirical knowledge from literature
[13], as summarised in Table I.

TABLE I: Parameter settings of Algorithm 1 $\Delta y$ /(unit) | $\theta_{1}$ | $\theta_{2}$ | $\delta$ | $\gamma$ | $\tau$ | $\eta$
---|---|---|---|---|---|---
$v/(m/s)$ | 2 | -2 | 20 | 30 | 10 | 50
$a/(m/s^{2})$ | 0.25 | -0.25 | 0.25 | 30 | 10 | 50
$d/(m)$ | 1 | -1 | 1 | 30 | 10 | 50
$\Delta v/(m/s)$ | 2 | -2 | 2 | 30 | 10 | 50

###  IV-B Visualisation and Analysis of Action Phase

The _action trend_ labels for the four driving behaviour variables are
obtained using Algorithm 1. These results are then visualised, generating
unique driving behaviour maps for each driver, as exemplified in Figure 4. In
the figure, various colours represent different driving behaviour variables,
with velocity, acceleration, distance, and speed difference represented in
that order. The varying intensity of the same colour indicates different
_action trend_ names, including Increasing, stable as High, stable as Low, and
Decreasing.

In Figure 4a, the dominant _action trend_ for acceleration is “L”, although
instances of “I” and “D” can also be observed. The distance remains relatively
stable without frequent _action trend_ changes. When comparing the driving
behaviour maps of the four drivers shown, driver ID1264 from the I-80 dataset
exhibits the fewest _action trend_ changes across the four variables.
Conversely, drivers ID3 and ID1035 from the US-101 dataset demonstrate a
higher frequency of the changes.

The driving behaviour map offers an intuitive approach to interpreting driving
behaviour by visualising the changes in driving behaviour over time. It is
also important to note that this visualisation method relies on observation
and should be complemented with further quantitative evaluation, which will be
carried out in subsequent steps.

(a) Driver ID270 from I-80 dataset

(b) Driver ID1264 from I-80 dataset

(c) Driver ID33 from US-101 dataset

(d) Driver ID1035 from US-101 dataset

Figure 4: Visualisation of _actions_ : the driving behaviour map

The _Action phase_ Library for a specific traffic flow was constructed using
Algorithm 2. The resulting Library consists of 142 _Action phases_ for the
I-80 dataset and 228 _Action phases_ for the US-101 dataset. Table II presents
the top 10 _Action phase_ along with their corresponding frequencies. Notably,
both traffic flows exhibit a significant overlap in their high-frequency
_Action phases_ , and the top three _Action phases_ are identical for both
datasets. These top _Action phases_ include “((L, L, H, H), st)”, “((L, L, H,
H), lg)”, and “((L, L, L, H), st)”, which indicate common driving behaviour
across the datasets. The reason behind this is that the two datasets were
collected during evening and morning rush hours respectively. In these
periods, the density of traffic flow is significantly high, with most vehicles
exhibiting car-following behaviour and even close to congestion. Due to this,
there is limited variability in driving behaviours, resulting in a scarcity of
“I” and “D” and a high frequency of “Stable (H and L)”. The high-density
traffic flows also provide an explanation for the highest frequency of
occurring “L”.

TABLE II: Statistics of _Action phase_ (Top 10) I-80 | US-101
---|---
_Action phase_ | Frequency | _Action phase_ | Frequency
((L,L,H,H), st) | 415 | ((L,L,H,H), st) | 2703
((L,L,H,H), lg) | 219 | ((L,L,H,H), lg) | 1661
((L,L,L,H), st) | 156 | ((L,L,L,H), st) | 965
((L,L,H,I), st) | 68 | ((L,L,L,H), lg) | 672
((L,L,L,H), lg) | 65 | ((D,L,H,H), st) | 651
((D,L,H,H), st) | 54 | ((L,L,L,L), st) | 480
((L,L,L,L), st) | 41 | ((I,L,H,H), st) | 469
((L,L,H,D), st) | 39 | ((D,L,H,H), lg) | 419
((D,L,H,I), st) | 38 | ((D,L,H,I), st) | 412
((D,I,H,I), st) | 30 | ((L,L,H,I), st) | 349

###  IV-C Analysis of Action-chain

The transition probabilities from one _Action phase_ to another within the
_Action phase_ Library were computed using Algorithm 3. Some _Action phases_
either have no transitions or exhibit very low probabilities of transition.
Conversely, other _Action phases_ tend to be transitioned to by a greater
number of _Action phases_. The results adhere to the fundamental principles of
driving behaviour. For example, the _Action phase_ “((L, L, L, H), st)” from
the US-101 dataset demonstrates higher probabilities of being transitioned.
This can be attributed to the fact that the driving data were collected during
the morning peak hour when there is typically high traffic flow density,
leading drivers to adopt more consistent driving behaviours with lower values.

Overall, each _Action phase_ was found to have a following _Action phase_ with
the highest transition probability, resulting in the formation of an _Action-
chain_ , as illustrated in the examples provided in Table III. In the I-80
dataset, for instance, the _Action phase_ “((D, I, I, H), st)” has a
probability of 0.68 to transition to “((L, I, I, H), st)”, which is higher
than any other _Action phases_.

TABLE III: _Action-chain_ composed by the highest joint transition probability (JTP) Dataset |  _Action phase_ from |  _Action phase_ to | JTP
---|---|---|---
I-80 | ((D, D, I, I), st) | ((D, L, L, I), st) | 0.68
((D, I, I, H), st) | ((L, I, I, H), st) | 0.68
((D, I, D, I), lg) | ((L, L, D, H), st) | 0.67
… | … | …
((D, D, H, D), lg) | ((L, L, H, H), st) | 0.52
US-101 | ((D, D, D, I), st) | ((L, L, L, L), st) | 0.64
((D, H, L, H), st) | ((L, L, L, H), st) | 0.64
((I, I, L, D), st) | ((L, L, L, H), st) | 0.58
… | … | …
((I, I, D, I), st) | ((L, L, D, L), st) | 0.32

###  IV-D Evaluation of Driving Heterogeneity

In this study, we assume that the maximum transition probability represents
the generally adopted _Action phase_ of drivers in a specific traffic flow,
indicating the average level of driving behaviour. However, in the real world,
drivers often deviate from this general level due to heterogeneity in their
driving behaviours. To quantify the heterogeneity, we define this
heterogeneity as the deviation between the actual _Action phase_ transition
and the _Action-chain_.

The Mean Squared Error (MSE) is a commonly used method for measuring the
average squared difference between two sets of data, and it serves as the
metric to quantify driving heterogeneity in this context, see Equation 8.

$DH=\frac{1}{N^{\prime}}\times\sum^{N^{\prime}}_{n^{\prime}=1}(P^{\prime}_{n^{\prime}}-P_{n^{\prime}})^{2}$
(8)

where $N^{\prime}$ is the total number of _Action phases_ , and
$P^{\prime}_{n^{\prime}}$ and $P_{n^{\prime}}$ represent the transition
probability of actual _Action phase_ and the maximum transition probability,
respectively. A higher value indicates a greater driving heterogeneity.

###  IV-E Numerical Results and Discussions

The heterogeneity of individual drivers in a specific traffic flow was
calculated and subjected to further statistical analysis using the normal
distribution. The $3\sigma$ rule of thumb is commonly employed in data
analysis to identify potential outliers or unusual behaviour. By applying the
$3\sigma$ principle, drivers with atypical driving behaviour were identified,
as summarised in Table IV. These drivers may serve as potential factors
contributing to increased traffic flow heterogeneity and negatively affecting
traffic performance. It is also noteworthy that the degree of driving
heterogeneity in the two traffic flows exhibits the same variance, see Figure
5; however, the drivers on US-101 (with $\mu=0.08$) display overall lower
levels of heterogeneity compared to those on I-80 (with $\mu=0.10$).

TABLE IV: Drivers with the highest heterogeneity I-80 | US-101
---|---
Driver ID | $DH$ | Driver ID | $DH$
295 | 0.2172 | 582 | 0.1379
535 | 0.2111 | 628 | 0.1360
1174 | 0.2004 | 1157 | 0.1464
- | - | 1647 | 0.1916

(a) I-80 dataset (b) US-101 dataset

Figure 5: Driving heterogeneity in a specific traffic flow

##  V Conclusion

In this study, a novel framework was proposed to address driving heterogeneity
in a comprehensive manner. By introducing the concepts of _Action phase_ and
_Action-chain_ , along with specialised algorithms, the framework effectively
quantified and explained driving heterogeneity at both the individual driver
and traffic flow levels. Real-world datasets were used for evaluation,
validating the framework’s ability to offer clear interpretations. Although
the contributions of novel insights and findings of heterogeneity of driving
behaviour, further validation and justification of the methods employed in
each step are still required, which is a focus of our ongoing research.

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