# Federated Learning with Differential Privacy

Adrien Banse1, Jan Kreischer2, Xavier Oliva i Jürgens3  1Exchange student at
EPFL, Switzerland from UCLouvain, Belgium  2Exchange student at EPFL,
Switzerland from Universität Zürich, Switzerland  3Student at EPFL,
Switzerland

###### Abstract

Federated learning (FL), as a type of distributed machine learning, is capable
of significantly preserving client’s private data from being shared among
different parties. Nevertheless, private information can still be divulged by
analyzing uploaded parameter weights from clients. In this report, we showcase
our empirical benchmark of the effect of the number of clients and the
addition of differential privacy (DP) mechanisms on the performance of the
model on different types of data. Our results show that non-i.i.d and small
datasets have the highest decrease in performance in a distributed and
differentially private setting.

##  I Introduction

The training of deep-learning models usually requires large and diverse
datasets. In many areas, gathering large datasets is difficult and requires
the collaboration of multiple institutions. Especially in medicine, patient
data is spread among multiple entities. While each party might not have enough
data to robustly train a model for a specific task, the union of their
datasets can potentially lead to successful data insights. Especially for rare
diseases, data sharing among many entities, which can be located in different
countries, is crucial. However, medical data contains much private information
and is potentially identifiable, making it especially sensitive with regard to
privacy. Legal regulations, such as GDPR in Europe, contain specific clauses
describing the privacy requirements that medical data has to comply with.

To address these important limitations, privacy-preserving techniques are
gaining momentum, as they can enable to perform machine learning (ML)
algorithms on sensitive data. Especially federated machine learning, a non-
cryptographic approach for privacy-preserving training of ML models, has been
increasingly studied in the last years [1, 2, 3]. This will be covered in
section II.

Another privacy-enhancing technology used for training ML models is
differential privacy (DP). DP algorithms aim at quantifying and setting an
upper-bound to the privacy loss of an individual when entering their private
data into a dataset. They rely on incorporating random noise to the data or
model. DP has also been used in the federated setting [4], where to
collectively train a model, multiple parties exchange or send differentially
private model updates to a central server to protect against an honest-but-
curious adversary. This will be covered in section II-B.

In this project, we will focus on cross-silo federated machine learning with
differential privacy. The three questions we want to answer are the following.

1) How does the level of data distribution affect model accuracy, i.e. the
convergence of the optimization process?

2) How does differential privacy affect model accuracy?

3) Is it applicable to small, more realistic datasets?

We will perform experiments for both i.i.d., where all of the data is
independently and identically distributed, and non-i.i.d cases. Finally, we
will apply our FL-DP algorithm on a small medical dataset.

##  II Theoretical Background

###  II-A Federated Machine Learning

In federated learning, the data remains under the control of their owners,
which we designate as clients, and a central server coordinates the training
by sending the global model directly to the clients, which then update the
model with their data. The updated models from the clients are sent back to
the central server and averaged to obtain an updated version of the global
model. This is done via the FedAvg (Federated Averaging) algorithm [1],
described in Algorithm 1 in Appendix -A.

###  II-B Differential Privacy

In this project, we focus on cross-silo federated learning, where data is
distributed between organizations with high computational resources, like
hospitals or banks. The biggest challenge in this setting usually lies on the
data security side.

The adversarial model in this setting includes both the central server and any
of the clients. The central server observes the updated parameters of all
clients, while a client observes its own updates and the new global model
parameters after every round. The problem lies in the fact that the model
parameters might leak information about the training data. The adversary’s
goal is to infer whether a given record was in the client’s training data
(membership inference) or learn properties about the client’s training data
(property inference).

($\epsilon$, $\delta$)-DP provides a strong criterion for privacy preservation
of distributed data processing systems. Here, $\epsilon>0$ is the
distinguishable bound of all outputs on two neighboring datasets (pairs of
databases $(\mathcal{D},\mathcal{D}_{-r})$ differing only in one row $r$. In
other words, the removal or addition of a single record in the database should
not substantially affect the values of the computed function/statistics.

$\log{\frac{\text{P}[A(\mathcal{D})=O]}{\text{P}[A(\mathcal{D}_{-r})=O]}}<\epsilon\text{
with probability }1-\delta$

Thus, $\delta$ represents the probability that the ratio of the probabilities
for two neighboring datasets cannot be bounded by $e^{\epsilon}$). Typically,
values of $\delta$ that are less than the inverse of any polynomial in the
size of the database are used.

There are three ways to apply differential privacy to Machine Learning:
objective perturbation, gradient perturbation and output perturbation. For
deep learning applications, we can not derive sensitivity bounds for the
objective and output and have to use gradient perturbation. We use PyTorch’s
module Opacus, that uses gradient perturbation with the advanced composition
method. It injects noise at every iteration by using the gradient clipping
technique [5] (see Algorithm 2 in Appendix -A).

##  III Models and Methods

###  III-A Datasets and models

In this section, we shortly describe the datasets used for the numerical
experiments, as well as the model used to classify them.

####  III-A1 MNIST

is a widely used database comprising thousands of $28\times 28$ pixels images
of 10 different handwritten digits (which means there are 10 different
classes). The samples are randomly and equally distributed among clients, we
say that the data is i.i.d.. We use the Convolution Neural Network (CNN)
defined in the PyTorch’s examples GitHub repository [6]. 60 000 data points
were used for training, and 10 000 for testing.

####  III-A2 FEMNIST

(from the LEAF benchmark [7]) is also a database consisting of $28\times 28$
images of 10 different handwritten digits (letters are discarded). The
difference between MNIST and FEMNIST datasets is that the partitioning of the
data is now based on the writer of the digit. It means that the underlying
distribution of data for each user is now consistent with the raw data,
yielding a non-i.i.d. dataset. We use the same CNN, and same training/testing
dataset sizes as for MNIST.

####  III-A3 Medical dataset

In order to tackle more realistic scenarios as explained in Section I, we use
a database of medical records of 120 patients [8]. The aim is to predict
whether patients suffer from inflammation of urinary bladder and/or nephritis
of renal pelvis origin. In this work, we focus on the pathology. The medical
data is randomly split onto the clients, representing hospitals. There are 6
attributes, namely temperature of the patient, occurrence of nausea, lumbar
pain, urine pushing, micturition pains, and burning of urethra, itch, swelling
of urethra outlet. We used logistic regression to tackle this classification
problem. 96 data points were used for training, and 24 for testing.

###  III-B Experimental setup

The parameters for the experiments are set as follows: The learning rate of
SGD is set to $0.01$. The number of client epochs per round is set to 1 (to
avoid the clients falling in different local minima) for the MNIST and FEMNIST
database, 100 for the medical dataset. The number of global rounds is set to
30. We set the batch size of all clients to 128 for MNIST and FEMNIST, and 8
for the medical dataset.

####  III-B1 First Experiment

In order to see the impact of federated learning, we will leave all training
parameters fixed and perform federated learning for
$\texttt{nr\\_clients}\in\\{1,5,10\\}$ Note that the amount of data points
remains constant but more distributed.

####  III-B2 Second Experiment

We reuse the previous training setting, but fix $\texttt{nr\\_clients}=10$.
Now, we want to analyze the effect of differential privacy on performance and
convergence. We set the clipping threshold of the gradient to $1.2$. We set
the privacy parameter $\delta$ to $\frac{1}{2n}$. We compare the training loss
and final accuracy with various protection levels for 10 clients, using
$\epsilon\in\\{10,50,100\\}$. Every round, every client uses a privacy budget
of $\frac{\epsilon}{\texttt{nr\\_rounds}}$.

We report the testing accuracy of the global model, as well as its loss for
every training round.

##  IV Results

###  IV-A MNIST

Results of the first experiment are shown in Figure 1a, and results of the
second experiment are shown in Figure 1b.

(a) Accuracy and loss for different numbers of clients.

(b) Accuracy and loss for different values of $\epsilon$, with 10 clients.

Figure 1: MNIST database

Concerning the first experiment, as expected, one can clearly observe that the
more clients there are, the slower the learning process.

Regarding the second experiment, one can see that the learning is faster when
we go from a privacy budget of $\epsilon=10$ to $\epsilon=50$. The same
observation can be made for the transition from $\epsilon=50$ to
$\epsilon=100$ by looking at the testing loss graph. However, we can observe
that the improvement is not proportional to $\epsilon$ since we observe a less
important change for the second transition. The three privacy setups lead to
the same final accuracy though.

Finally, one can observe on Figure 1a and Figure 1b the difference of
convergence of SGD with and without privacy in a federated learning context.
Without privacy, SGD achieves more than $95\text{\,}\mathrm{\char 37\relax}$
of testing accuracy, while only $75\text{\,}\mathrm{\char 37\relax}$ with
privacy, even with a large privacy budget such as $\epsilon=100$.

###  IV-B FEMNIST

Results can be founds in Figure 2a and Figure 2b.

(a) Accuracy and loss for different numbers of clients for the FEMNIST
database.

(b) Accuracy and loss for different values of $\epsilon$, with 10 clients for
the FEMNIST database.

Figure 2: FEMNIST database

We enlighten two main differences compared to Section IV-A:

1) While a single client achieves similar accuracy to the regular MNIST
dataset, the algorithm takes more global training rounds to converge with an
increasing number of clients. After 100 rounds the test accuracy of all models
reaches around 98%.

2) When using DP the training process does not really converge anymore, not
leading to a viable model for FEMNIST.

###  IV-C Medical dataset

Results can be found on Figure 3a and Figure 3b.

(a) Accuracy and loss for different numbers of clients for the medical
database.

(b) Accuracy and loss for different values of $\epsilon$, with 10 clients for
the medical database.

Figure 3: Medical database

Again, we enlighten some differences compared to Section IV-A.

1) Training shows higher variance with more clients, but all manage to
converge.

2) Adding differential privacy, the model accuracy does not get any better.
Only for $\epsilon=10$ the loss decreases, indicating that the global model is
improving.

##  V Discussion

In this section we will discuss the results in Section IV and mention further
improvements.

During the experiments, we clearly observed the trade-off between privacy and
utility. Since, participating clients only optimize using their own data, they
can end up in different local minima, making the global model sub-optimal.
This leads to a generally slower convergence with a higher number of clients.

The experiments for the MNIST dataset show that adding noise and gradient
clipping makes the maximal accuracy decrease around $20\text{\,}\mathrm{\char
37\relax}$ for the same number of training rounds, even on i.i.d data. More
clients might be needed to average out the noisy updates [9].

When it comes to non-i.i.d data, distribution of data in silos makes the
global model perform worse. A possible improvement would be to use a d-clique
topology (decentralized learning) instead of classical centralized federated
learning.

Regarding small datasets, i.e. more realistic datasets, the results are more
worrying: training is more sensitive to DP mechanisms, making it very
difficult to learn while effectively preserving privacy. Additionally, they
have a higher variance during training. We had to use bigger values than the
standard $\epsilon=0.1$, because such a bound was too restrictive to compute a
single mini-batch SGD iteration.

When it comes to limitations of our work, we have to note that we did not
focus on hyperparameter tuning and used fix parameters that might not be
optimal for the task. Additionally, there are other ways like fully
homomorphic encryption (FHE) that protects the data from attacks by encrypting
the model.

##  VI Summary

We were able to show that models trained using federated machine learning are
able to achieve similar accuracy as models trained on a centralized dataset.
However, adding DP mechanisms to preserve the privacy of the data showed a
rapid decrease of performance, especially for non-i.i.d or small datasets,
which are much more realistic datasets.

## References

  * [1] H. B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y Arcas, “Communication-efficient learning of deep networks from decentralized data,” 2017.
  * [2] R. C. Geyer, T. Klein, and M. Nabi, “Differentially private federated learning: A client level perspective,” 2018.
  * [3] F. Tramèr and D. Boneh, “Differentially private learning needs better features (or much more data),” 2021.
  * [4] K. Wei, J. Li, M. Ding, C. Ma, H. H. Yang, F. Farhad, S. Jin, T. Q. S. Quek, and H. V. Poor, “Federated learning with differential privacy: Algorithms and performance analysis,” 2019.
  * [5] M. Abadi, A. Chu, I. Goodfellow, H. B. McMahan, I. Mironov, K. Talwar, and L. Zhang, “Deep learning with differential privacy,” _Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security_ , Oct 2016\. [Online]. Available: http://dx.doi.org/10.1145/2976749.2978318
  * [6] “PyTorch’s MNIST example,” https://github.com/pytorch/examples/tree/master/mnist, (Accessed on 06/09/2021).
  * [7] S. Caldas, S. M. K. Duddu, P. Wu, T. Li, J. Konečný, H. B. McMahan, V. Smith, and A. Talwalkar, “Leaf: A benchmark for federated settings,” 2019.
  * [8] “Uci machine learning repository: Acute inflammations data set,” https://archive.ics.uci.edu/ml/datasets/Acute+Inflammations, (Accessed on 06/09/2021).
  * [9] T. Yu, E. Bagdasaryan, and V. Shmatikov, “Salvaging federated learning by local adaptation,” 2020.

###  -A Algorithms

This section groups algorithms used to train global and client models, using
differential privacy (see Section II and Section II-B).

Result: Global model trained

Initialize:  $\boldsymbol{w^{0}}$ ;

for  _each global round $t=0,1,\dots$_ do

for  _each client $k\in\\{1,\dots K\\}$_ do

$\boldsymbol{w_{k}^{t+1}}\leftarrow$ ClientUpdate($k,\boldsymbol{w^{t}}$)

end for

$\boldsymbol{w^{t+1}}=\sum_{k=1}^{K}\frac{n_{k}}{n}\boldsymbol{w^{t+1}_{k}}$;

end for

Algorithm 1 FedAvg (Federated Averaging)

In Algorithm 1, $K$ is the number of clients (each client denoted by $k$),
$\alpha$ is the learning rate, $n_{k}$ is the size of the dataset of client
$k$, $n=\sum_{k=1}^{K}n_{k}$ is the size of the entire dataset. Moreover,
$\boldsymbol{w^{t}_{k}}\in\mathbb{R}^{P}$ is the vector of $P$ parameters of
the local model of client $k$, at global round $t$, and
$\boldsymbol{w^{t}}\in\mathbb{R}^{P}$ is the vector of $P$ parameters of the
global model.

The function ClientUpdate involes privacy (see Section II-B), and is described
in Algorithm 2.

Result: $\boldsymbol{w_{k}^{t+1}}$ and privacy cost ($\epsilon$, $\delta$)
using a privacy accounting method

Input: Parameters of the global model at time $t$ $\boldsymbol{w^{t}}$

for  _$\text{epoch }e=1,...,E$_ do

Sample $\lfloor\frac{n}{B}\rfloor$ batches at random

for  _$j=1,\dots,\lfloor\frac{n}{B}\rfloor$_ do

for  _$i=1,\dots,B$_ do

Compute gradient

$\boldsymbol{g_{j}(x_{i}^{j})}\leftarrow\nabla\mathcal{L}(\boldsymbol{w_{k}},\boldsymbol{x_{i}^{j}})$

Clip gradient

$\boldsymbol{\bar{g}_{j}(x_{i}^{j})}\leftarrow\boldsymbol{g_{j}(x_{i}^{j})}/\text{max}\left(1,\frac{\mathinner{\\!\left\lVert\boldsymbol{g_{j}(x_{i}^{j})}\right\rVert}_{2}}{C}\right)$

end for

Add noise

$\boldsymbol{\tilde{g}_{j}}\leftarrow\frac{1}{B}\left(\sum_{i=0}^{B}\boldsymbol{\bar{g}_{j}(x_{i}^{j})}+\mathcal{N}(0,\sigma^{2}C^{2}\boldsymbol{I})\right)$

Descent

$\boldsymbol{w_{k}}\leftarrow\boldsymbol{w_{k}}-\eta\boldsymbol{\tilde{g}_{j}}$

end for

end for

Algorithm 2 ClientUpdate for client $k$

In Algorithm 2, $\eta$ is the learning rate, $\sigma$ is the noise scale, $C$
is the gradient norm bound, $B$ is the batch-size, $E$ is the total number of
epochs, $n$ is the size of the dataset and $\boldsymbol{x}_{i}^{j}$ is the
$i$-th datapoint in the $j$-th batch.
$\mathcal{L}(\boldsymbol{w},\boldsymbol{x})$ is the loss corresponding to the
datapoint $\boldsymbol{x}$, depending on the parameters $\boldsymbol{w}$.


# Deconstructing Pedestrian Crossing Decision-making in Interactions with
Continuous Traffic: an Anthropomorphic Model

Kai Tian, Gustav Markkula, Chongfeng Wei , Yee Mun Lee, Ruth Madigan, Toshiya
Hirose, Natasha Merat and Richard Romano Manuscript received. Corresponding
author: Kai TianKai Tian, Gustav Markkula, Yee Mun Lee, Ruth Madigan, Natasha
Merat and Richard Romano are with Institute for Transport Studies, University
of Leeds, Leeds, United Kingdom, LS2 9JT (E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>R.Romano@leeds.ac.uk).Chongfeng Wei is with School of
Mechanical and Aerospace Engineering, Queen’s University Belfast, Belfast,
United Kingdom, BT7 1NN (E-mail: C.Wei@qub.ac.uk).Toshiya Hirose is with
Department of Engineering Science and Mechanics, Shibaura Institute of
Technology, Tokyo, Japan (E-mail: hiroset@shibaura-it.ac.jp). This work was
supported by the UK Engineering and Physical Sciences Research Council under
grant EP/S005056/1.

###### Abstract

As safe and comfortable interactions with pedestrians could contribute to
automated vehicles’ (AVs) social acceptance and scale, increasing attention
has been drawn to computational pedestrian behavior models. However, very
limited studies characterize pedestrian crossing behavior based on specific
behavioral mechanisms, as those mechanisms underpinning pedestrian road
behavior are not yet clear. Here, we reinterpret pedestrian crossing behavior
based on a deconstructed crossing decision process at uncontrolled
intersections with continuous traffic. Notably, we explain and model
pedestrian crossing behavior as they wait for crossing opportunities,
optimizing crossing decisions by comparing the visual collision risk of
approaching vehicles around them. A collision risk-based crossing initiation
model is proposed to characterize the time-dynamic nature of pedestrian
crossing decisions. A simulation tool is established to reproduce pedestrian
behavior by employing the proposed model and a social force model. Two
datasets collected in a CAVE-based immersive pedestrian simulator are applied
to calibrate and validate the model. The model predicts pedestrian crossing
decisions across all traffic scenarios well. In particular, by considering the
decision strategy that pedestrians compare the collision risk of surrounding
traffic gaps, model performance is significantly improved. Moreover, the
collision risk-based crossing initiation model accurately captures the timing
of pedestrian crossing initiations within each gap. This work concisely
demonstrates how pedestrians dynamically adapt their crossings in continuous
traffic based on perceived collision risk, potentially providing insights into
modeling coupled human-AV interactions or serving as a tool to realize human-
like pedestrian road behavior in virtual AVs test platforms.

###### Index Terms:

Pedestrian-AV interaction, Pedestrian road crossing, Decision-making model,
Traffic flow, Simulation.

##  I Introduction

Continued advances in vehicle automation have brought us great anticipation
that society will adopt highly automated vehicles (AVs) in the near future.
However, this vision faces many unresolved challenges. One of them is to
achieve smooth interaction between AVs and other road users. The consensus
suggests that in the transition from manual to fully automated driving, there
will be mixed traffic with AVs and other road users on the road[1]. A typical
case is the expansion of the deployment of AVs from a few confined areas of
low risk to other road users to a range of operational design domains, which
could inevitably increase conflicts with other road users[2]. Failures in
interactions between AVs and other road users may hinder the large-scale
adoption and social acceptance of AVs [3, 4]. This, therefore, leads to the
research context of this study, which is to promote safe and smooth
communication and interaction in traffic [1, 3, 4]. Pedestrians are generally
regarded as the most vulnerable road users in modern transport systems, due to
the lack of protective equipment and slow movement compared to other road
users [5]. Given that pedestrians’ actions and intentions are
nondeterministic, and the diversity and dynamism of their behavior, moving
through this complicated environment is a challenge for AVs[6]. Moreover, AVs’
own behavior can also affect pedestrian road behavior, which introduces
further uncertainties into interactions. In particular, the issues mentioned
above become more pronounced at uncontrolled intersections where pedestrian
behavior is more unpredictable, and safety problems are more common than on
other controlled road sections, as there are no traffic signals to coordinate
the interaction process [7]. Additionally, most existing automated driving
systems regard the driving task as a pure collision-free motion planning
problem and view pedestrians in some contexts as rigid road obstacles, instead
of social beings [5, 8].

Against the above background, if AVs cannot properly understand the behavior
of pedestrians, they may not improve traffic efficiency and safety as
expected, but rather increase traffic dilemmas and additional issues[9].
Accordingly, much attention has been drawn to one pressing issue, namely
computational models for pedestrian road behavior, [10, 11, 6, 12, 13], which
may help AVs to better anticipate pedestrian intentions or serve as a tool to
implement realistic pedestrian behavior in simulated scenarios, and thus be
used in the validation and development of AVs[3, 14]. Existing computational
models for pedestrian behavior, particularly for pedestrian road-crossing
decisions have been developed based on a wide range of theories and
hypotheses, such as the cognitive models [15, 10], data-driven approaches
[16], discrete choice models[12], as well as game theoretical models [17].
However, those approaches have not yet bridged several gaps, as identified and
discussed below.

Firstly, most of these approaches are rarely based on specific behavioral or
psychological theories, such as pedestrian visual perception. Instead,
external physical factors, like time to collision (TTC), have been often used.
For example, [18, 19] developed a pedestrian crossing decision-making model
based on the vehicle deceleration distance. [20, 14] applied a minimum TTC as
the threshold for pedestrian crossing decisions. Although TTC or distance from
the vehicle has become the most used decision cue in crossing decision
models[18], growing evidence has shown that the impacts of vehicle kinematics
on pedestrians are multi-dimensional. For instance, at the same TTC condition,
a higher vehicle speed induces more pedestrians to cross the street compared
to a lower one[21]. Therefore, the TTC or distance may not properly carry the
risk information that pedestrians may perceive. As our previous research has
shown, pedestrian crossing behavior is highly correlated with their perceived
visual cues [22]. Hence, existing models lack effort in characterising
pedestrian perceived information, e.g., anthropomorphic visual cues[10, 1].

Moreover, few computational models specifically characterize pedestrian
decisions in the traffic flow scenario. In real situations, pedestrians
usually face a fleet of vehicles and accept one traffic gap after rejecting
some gaps. Thus, the decision-making in continuous traffic may not only be
based on the collision risk, but also involve many trade-offs between safety
and time efficiency[23]. Several previous studies indicated that with the
increased waiting time, pedestrians tended to accept crossing opportunities
with higher risk[7]. [14] developed a model which hypothesized that
pedestrians would change their obedience to the law when they waited a long
time. However, there is much evidence that pedestrians who tended to wait were
more cautious and less likely to accept risky gaps[24, 25, 26]. A meta-study
uncovered these conflicting results and noted that there was insufficient
evidence to support a linear relationship between waiting times and
pedestrians risking crossing the street[27]. On the one hand, the available
findings support that pedestrians may dynamically adjust their crossing
decision-making strategies in continuous traffic. On the other hand, it is
unreasonable to assume that pedestrians always tend to accept more dangerous
crossing opportunities as waiting time increases. Instead, we should treat
each case on its own merits. Therefore, it is necessary to look into the
details of pedestrian crossing behavior when interacting with traffic flow.

Finally, very limited models pay attention to the time dynamic of pedestrian
crossing decision-making. According to the cognitive decision-making theory,
pedestrian crossing initiation time (or onset time) is a variable due to the
noisy evidence in the human cognitive system [28]. In addition, it has been
shown that pedestrian crossing initiation time can be affected by many
factors. For instance, pedestrians may initiate quickly when facing a vehicle
with a low speed[21] or with a small time gap from the approaching
vehicle[29]. Accordingly, existing empirical observations highlight the time-
dynamic nature of pedestrian crossing decision-making. Recently, a class of
emerging models[15, 11, 10], namely the evidence accumulation model, detailed
model pedestrian crossing decisions and their timing by simulating the
cognitive process underlying crossing decision-making. However, given the
complexity of those models, they focused more on the details of the cognitive
process, and it is unclear whether it would be feasible to extend them to
cover additional factors, such as vehicle kinematics.

Regarding the above discussion, several research questions in existing
computational models of pedestrian crossing behavior can be summarised:

  * •

There is a lack of computational models that characterize pedestrian crossing
decisions based on anthropomorphic behavioral theory.

  * •

The decision pattern of pedestrians crossing the road when interacting with
the traffic flow remains unclear.

  * •

There is a lack of computational models that concisely consider the time-
dynamic nature of road crossing decisions and relate them to vehicle
kinematics.

In this study, a decision-making model for pedestrians interacting with
continuous traffic at uncontrolled intersections is proposed to solve the
above questions. The main contributions of this paper are as follows:

  * •

We formally apply our findings [22] and extend it to a relatively complex
traffic scenario, demonstrating that pedestrian crossing decisions are dynamic
and intrinsically linked to their perceived collision risk. Specifically, a
visual collision risk model is introduced as the main decision cue accounting
for pedestrian crossing decisions. Moreover, a novel decision strategy is
proposed to interpret pedestrian crossing decisions in continuous traffic
flow. In addition, a crossing initiation time model is developed and
associated with the collision cue model to account for the pedestrian dynamic
crossing initiation time.

  * •

Two different datasets collected in a highly immersive pedestrian simulator
are applied to calibrate and validate the model.

  * •

A simulation tool is established to reproduce pedestrian crossing decisions in
a customized traffic scenario based on the proposed model.

##  II Methodology

Figure 1: A simplified framework for pedestrians road-crossing decision-making
process. Figure 2: (a) Visual collision cue model in road crossing scenario.
Collision cues are as a (b) function of distance from and speed of the vehicle
or (c) TTC from and speed of the vehicle.

###  II-A Deconstructing the crossing decision-making process

During the decision-making process for road-crossing, several cognitive stages
may be involved to establish pedestrian situation awareness[1, 30]. Normally,
pedestrian perceived collision cues are the basis of their decisions, which
contain vehicle distance, speed, TTC, and more. Based on those visual cues,
pedestrians comprehend traffic situations and decide whether to cross the road
or not by combining some prior knowledge and strategies. Finally, there is a
reaction process before pedestrians start to move. Therefore, according to the
deconstructed three-stage cognitive process, we propose a collision cue-based
framework for road-crossing decision-making tasks (Fig. 1), assuming that the
crossing decision-making model consists of three constituent parts: visual
collision cue, decision, and crossing initiation.

###  II-B Visual collision cue model

Modeling pedestrian-vehicle interaction is challenging, partly because
existing pedestrian models lack psychological underpinnings. According to
psychological theory, when moving through the environment, people rely on
their visual perception of the space around them [31, 32]. The road crossing
task is a typical case that highly demands pedestrians to use visual cues to
evaluate the collision risk from approaching vehicles and guide their
movements. Relevant behavioral research has shown that the human visual system
is sensitive to changes in some visual cues, which may be the source of
collision perception. Specifically, one group of cues may provide reliable
collision time information, such as Tau[33]. Other cues, like visual angle and
its first temporal derivative[32], effectively translate motion information
into visual cues through images that expand on the retina. Although most daily
naturalistic road crossings involve all of the above visual cues (and possibly
others), Delucia[32] has suggested that humans may rely on collision time-
related cues when the scenarios include robust optical information or occur at
a near distance. Conversely, when the optical information in the task is
impoverished or occurs at a long distance, the visual angle and its first
temporal derivative may play a dominant role. In light of this conceptual
framework, we have previously identified that the first temporal derivative of
visual angle, $\dot{\theta}$, is a critical collision cue for making crossing
decisions at uncontrolled intersections. We have demonstrated that
$\dot{\theta}$ not only well explains pedestrian crossing decisions across a
wide range of traffic scenarios from two different datasets, but also
reasonably characterizes the impacts of vehicle speed and traffic gap on
pedestrians [22]. Therefore, in this study, we formalized the pedestrian
crossing decision model based on our previous findings. Typically,
$\dot{\theta}$ refers to the change rate of the visual angle subtended by an
approaching vehicle, $\theta$, (Fig. 2a)[31]. The following equations specify
its physical model:

$\theta=2\tan^{-1}\frac{w}{2Z}\Rightarrow\dot{\theta}\left(Z,v,w\right)=\frac{wv}{(Z)^{2}+w^{2}/4}$
(1)

where $v$ denotes the vehicle speed, $Z$ and $w$ are the distance to and width
of the vehicle. To better interpret the collision cue model, an example is
shown in Fig. 2. Suppose that a vehicle ( $w=1.95$ m) approaches the
pedestrian with two different constant speeds (30 km/h and 60 km/h) from 100
m. $\dot{\theta}$ is an approximately inversely exponential function of
distance and TTC from the approaching vehicle (Fig. 2b, c), showing that
$\dot{\theta}$ increases slowly at long distances and rapidly at close
distances, which agrees qualitatively with the observation that pedestrians
usually feel safe to cross for long distance or big time gap conditions but
not when the vehicle is close [21]. Further, it can be noticed that speed
effects vary across distance (Fig. 2b) and TTC dimensions (Fig. 2c). When
$\dot{\theta}$ is a function of distance and speed, it increases with speed,
which is opposite to the results in Fig. 2c, suggesting that pedestrians may
perceive a higher collision threat from the vehicle with higher speed at the
same distance. However, the approaching vehicle with a slower speed gives
pedestrians a bigger collision threat under the same TTC. The results tie well
with the previous experimental observations on pedestrian crossing
behavior[21, 34, 35].

Figure 3: Illustration of the initiation model. (a) Initiation time $t_{int}$
is the duration between $t_{pass}$ and the time when the pedestrian start
crossing. $t_{sg}$ denotes the actual gap to the approaching vehicle when
pedestrians initiate. (b) The shapes of the initiation model by changing
$\gamma$. (c) The positions of the initiation model by changing $\tau$.

###  II-C Decision model

Regarding crossing decisions at uncontrolled intersections, pedestrians
typically make crossing decisions by judging and selecting the appropriate
gaps between two consecutive vehicles, called gap acceptance behavior[7]. Our
previous study has proven that $\dot{\theta}$ is significantly negatively
correlated with pedestrian gap acceptance behavior, and a collision cue-based
binary choice logit model predicts pedestrian gap acceptance well across
different vehicle speeds and traffic gap experimental scenarios[22].
Furthermore, evidence from experimental observations indicated that
individuals’ judgments toward traffic gaps are not necessarily entirely static
over time, especially in traffic streams[36, 24, 25]. Due to certain learning
or comparison strategies, pedestrians may estimate different utilities for the
approaching vehicles with the same collision cues, thus adjusting their
crossing decision to balance safety and efficiency. We, therefore, propose the
following assumptions for the crossing decision-making in the traffic flow:

(i) Pedestrians make decisions mainly based on collision cues, i.e.,
$\dot{\theta}$, provided by approaching vehicles.

(ii) Pedestrians are unwilling to accept the current gap with a collision cue
equal to or greater than the maximum collision cue previously rejected. For
example, if pedestrians reject a $0.02$ rad/s cue, they would be more likely
to reject the same or bigger one upstream of traffic. The rule is given by:

${X}_{1}=\left\\{\begin{array}[]{l}1,\quad\dot{\theta}_{c}\geq\dot{\theta}_{mr}\\\
0,\quad\dot{\theta}_{c}<\dot{\theta}_{mr}\end{array}\right.$ (2)

where $X_{1}$ is the dummy variable for the rule. $\dot{\theta}_{c}$ and
$\dot{\theta}_{mr}$ represent collision cues for the current gap and maximum
rejected gap, respectively.

(iii) If pedestrians find that a gap next to the current gap has a smaller
collision cue than the current gap, they may prefer to wait for this gap
rather than accept a current gap with a greater collision threat, given the
rule:

${X}_{2}=\left\\{\begin{array}[]{l}1,\quad\dot{\theta}_{c}\geq\dot{\theta}_{f}\\\
0,\quad\dot{\theta}_{c}<\dot{\theta}_{f}\end{array}\right.$ (3)

where $X_{2}$ is the dummy variable for the decision rule. $\dot{\theta}_{f}$
represents a collision cue of the gap following the current one. Therefore,
the utility function of the decision model is formulated as:

$V=\rho_{0}\ln(\dot{\theta})+\rho_{1}X_{1}+\rho_{2}X_{2}+\rho_{3}$ (4)

where $\rho_{0}$ to $\rho_{3}$ are estimated coefficients. In this study,
every $\dot{\theta}$ only refers to the $\dot{\theta}$ value of the
approaching vehicle at the time when the rear end of the previous vehicle just
past the pedestrian (Fig. 3a). Regarding the $\ln$ transformation, we have
previously proven that it can efficiently increase the accuracy of model
fitting [22]. Since crossing decisions at uncontrolled intersections are
assumed to be a binary choice task, a logistic function is applied [7]. Then,
a decision model for crossing tasks in the traffic flow is given by:

$p(\dot{\theta},X_{1},X_{2})=\frac{1}{1+\exp\left(-V\right)}$ (5)

where $p$ is the probability of the gap acceptance. The (5) without the terms
$X_{1}$ and $X_{2}$ degenerates to the model we proposed in [22].

###  II-D Crossing initiation model

In real traffic, the time at which pedestrians start to cross the road is a
variable[28]. As illustrated in Fig. 3a, crossing initiation time, $t_{int}$,
is typically defined as the duration between the time when the rear end of the
previous car passes the pedestrians’ position, $t_{pass}$, and the time when
pedestrians start their movements[21]. Emerging cognitive models[28, 11, 10]
have shown that the crossing initiation time distribution may arise from an
underlying evidence accumulation process, but of a form that requires costly
stochastic simulation of to estimate the distribution. However, the skewed,
lognormal-like shape of the distribution is similar to those arising from
simpler evidence accumulation processes, which can be written in a closed
mathematical form, such as Ex-Gaussian, Shifted Wald (SW), and Weibull[37].
Considering the similarities of those methods, we only apply the SW
distribution instead of trying all of them. The SW distribution is a simple
and concise distribution modeling tool, which can fully qualify the crossing
initiation time distribution with three parameters: $b$ (deviation around the
mode), $\gamma$ (tail magnitude) and $\tau$ (onset of the distribution). Its
equation is defined as:

$\begin{gathered}x\sim\operatorname{SW}(b,\gamma,\tau)\\\
\Rightarrow\frac{b}{\sqrt{2\pi(x-\tau)^{3}}}\cdot\exp\left(\frac{-[b-\gamma(x-\tau)]^{2}}{2(x-\tau)}\right)\end{gathered}$
(6)

An illustration of the distributional effect that occurs by changing each of
the $\gamma$ and $\tau$ parameters are shown in Fig. 3 b and c. The tail
becomes heavier as $\gamma$ decreases, (Fig. 3b). Changes in $\tau$ control
the position of the distribution (Fig. 3c)[37].

According to our assumptions in Fig. 1, the crossing initiation time model is
affected by collision cues, so we define the initiation time model as follows:

$\begin{gathered}t_{int}\sim\operatorname{SW}(b,\gamma,\tau)\\\ \text{ with
}\gamma=\beta_{1}\ln(\dot{\theta})+\beta_{2};\tau=\beta_{3}\ln(\dot{\theta})+\beta_{4}\end{gathered}$
(7)

where $t_{int}$ is the crossing initiation time. $\beta_{1}$ to $\beta_{4}$
are estimated coefficients. The idea behind these equations is that the
strength of collision cues could affect the distribution pattern of pedestrian
initiation time. For a more intensive collision threat, if pedestrians choose
to cross, they tend to do so more quickly, so the distribution is concentrated
and has a short tail. In contrast, when the collision threat is small,
pedestrians tend to start crossing slowly, so the distribution is more likely
to have a long tail[38]. Accordingly, the SW model is not only a practical
distribution model but also provides notable psychological significance for
our decision model. In addition, $b$ is assumed to be a coefficient not
influenced by collision cues. Furthermore, since response time data are
routinely assumed to be normally distributed in many studies [21, 39], another
crossing initiation time model based on the Gaussian distribution is proposed
as a comparison to the SW model, defined as the following equations:

$\begin{gathered}t_{int}\sim\mathcal{N}(\mu,\sigma),\\\ \text{ with
}\mu=\beta_{1}\ln(\dot{\theta})+\beta_{2};\sigma=\beta_{3}\ln(\dot{\theta})+\beta_{4}\end{gathered}$
(8)

where $\mu$ and $\theta$ are parameters of the Gaussian model, $\mathcal{N}$.

###  II-E Pedestrian road-crossing decision-making model in traffic flow

Finally, a pedestrian road-crossing decision-making model based on the SW
distribution in the traffic flow (SW-PRD) is then established by employing (5)
and (7):

$\displaystyle f_{SW}(t_{\text{int
}})=\sum_{n=1}^{N}P_{n}\cdot\operatorname{SW}\left(b,\gamma\left(\dot{\theta}_{n}\right),\tau\left(\dot{\theta}_{n}\right)\right)$
(9) $\displaystyle
P_{n}=p\left(\dot{\theta}_{n},X_{1,n},X_{2,n}\right)\cdot\left(1-P_{n-1}\right)$
$\displaystyle P_{0}=0$

where $n$ is the position number of the gap in the traffic flow.
${\dot{\theta}}_{n}$, $X_{1,n}$ and $X_{2,n}$ represent the decision variables
for the $n$th traffic gap. $P_{n}$ means the recursive probability that
pedestrians accept the $n$th gap, which is calculated based on $p$ and
$P_{n-1}$. Similarly, a road-crossing decision model based on Gaussian
distribution (G-PRD) is given by:

$\displaystyle f_{G}(t_{\text{int
}})=\sum_{n=1}^{N}P_{n}\cdot\mathcal{N}\left(\mu\left(\dot{\theta}_{n}\right),\sigma\left(\dot{\theta}_{n}\right)\right)$
(10)

###  II-F Simulation tool

In this subsection, an agent-based simulation tool is proposed using the
established models to reproduce pedestrian crossing behavior at uncontrolled
intersections with traffic flow. The framework mainly includes three parts:
the decision model, environment model, and pedestrian kinematics model.
Regarding the traffic environment, as the intersections on multi-lanes are
often separated by refuges[40], pedestrians actually cross one lane at a time.
Therefore, a single-lane road with an uncontrolled intersection is considered.
On the other hand, the model is possibly extended to a multi-lane situation,
but the impacts of refuges should be further considered [41]. A fleet of
vehicles travels on the lane at a constant speed, wherein the vehicle
quantity, speed, and traffic gaps can be customized. Afterward, a basic social
force model is applied as a pedestrian kinematics model[42], which considers
the driving force towards the destination and repulsive force from the
boundary of the crosswalk. Finally, according to the information provided by
the traffic environment and kinematics model, each pedestrian’s road crossing
decision is generated through PRD models. The detailed process of the
simulation tool is provided in the supplementary file (Appendix. A-A). A
demonstration video of the simulation tool is also provided. Please see the
attachment.

##  III Model calibration and validation

In this study, two empirical datasets collected in a simulated environment,
i.e., a CAVE-based highly immersive pedestrian simulator, were applied to
calibrate and validate the PRD models. The following sections provide detailed
information on the two datasets, calibration, and validation methods.

Figure 4: Schematic diagrams and photos of traffic scenarios in simulated
experiments. The crossing scenarios and traffic of the (a) first dataset and
(b) second dataset.

###  III-A Empirical data

Dataset one. A virtual road scene with a 3.5 m wide single lane and 1.85 m
wide pavement was created in the simulator. Two consecutive vehicles of 1.95 m
in width were driven in the middle of the road at the same constant speed.
Three vehicle speeds were selected, namely, 25 mph, 30 mph, or 35 mph. The
first vehicle came into view 96 m away from the pedestrian, and the second
vehicle maintained a specific time gap behind the first vehicle, i.e. 2 s, 3
s, 4 s, or 5 s (Fig. 4a). Sixty participants were instructed to cross the road
between the two cars if they felt comfortable and safe to do so. Otherwise,
they could reject the gap. Three experimental blocks were created, and each of
the 12 scenarios (4 time gaps $\times$ 3 speeds) were presented in random
order and repeated once in each experimental block. Therefore, each
participant experienced 72 trials, and 4270 trials of data were obtained in
total.

The virtual environment and simulation process mentioned above were designed
and controlled by the Unity3D platform. Internal code automatically recorded
the positions and velocities of vehicles and participants on each time step.
Two main metrics were applied: gap acceptance, $u$, and crossing initiation
time, $t_{int}$. The gap acceptance data were the binary crossing decisions
made by participants, i.e., $u=1$ means pedestrians accepted the gap, while 0
indicated rejected the gap. The crossing initiation time was defined as
described in Section II-D and Fig.3a. For more detailed information about this
dataset, please refer to [38].

Dataset two. To explore pedestrians’ road crossing decisions in traffic flow,
pedestrians were asked to cross a one-lane road with continuous traffic in the
simulator (Fig.4b). The size of time gaps between every two consecutive
vehicles varied, which provided pedestrians with different opportunities to
make crossing decisions (Fig.4b). Four traffic scenarios with different
sequences of gap sizes (in seconds) were designed as follows:

  * •

Scenario one: 1 1 1 3 3 3 6 1 1 6;

  * •

Scenario two: 1 1 1 1 3 3 7 1 1 3 8;

  * •

Scenario three: 1 1 1 3 1 3 1 3 5 4 8;

  * •

Scenario four: 2 3 1 1 3 1 1 1 5 4 7;

Among these scenarios, the one-second and two-second time gaps between
vehicles were considered dangerous crossing opportunities that very few
pedestrians would accept. For the three-second and four-second gaps, decisions
were expected to significantly differ between participants due to their
heterogeneity (e.g., age and gender). The time gaps longer than four seconds
were considered safe gaps that most pedestrians were expected to confidently
accept. In all scenarios, a range of compact, midsize, van, and SUV vehicles
were driven at 30 mph. Since the types of the approaching vehicle were
randomly selected, in the analyses here, the width of the vehicle was
calculated by averaging the width of all vehicles in the corresponding gap in
each scenario. 60 participants completed four crossing tasks in any of the
four scenarios and repeated them once more (4 crossing tasks $\times$ 4
scenarios $\times$ 2 repetitions). We, therefore, collected data from 1920
trials. All the trials that participants experienced were in a randomized
order. Similar to the first dataset, two main metrics were used: gap
acceptance, $u$, and crossing initiation time, $t_{int}$. For more detailed
information about this dataset, please refer to[25].

###  III-B Data processing and parameter estimation

With regard to data processing, both datasets were divided into a training set
and a validation set. Regarding dataset one, as controlled experimental
variables were vehicle speed and time gap size, we separated the training and
validation sets by choosing the data from different combinations of
experimental variables (As illustrated in Section III-A, there were 12
different combinations). To have enough data in the training and validation
sets, data from 10 combinations were grouped into the training set, while the
rest of the data belonged validation set. Moreover, in order to make sure the
validation data were sufficiently different, the 2 combinations are not
adjacent to each other in terms of speed or time gap size. Accordingly, the
validation set included data in 4 s 25 mph and 5 s 35 mph conditions,
approximately accounting for $23\%$ of the initiation time data and $14\%$ of
the gap acceptance data (The data size of the two metrics was not the same as
there was no initiation time data for participants who rejected the gap). The
remaining data of all other conditions were grouped into the training set.
Similarly, with respect to dataset two, the data from traffic scenario four
were used as the validation set, accounting for $24\%$ of gap acceptance data
and $25\%$ of initiation time data.

A Maximum Likelihood Estimation (MLE) method was used to calibrate the
parameters in the models. Firstly, regarding the decision model (5), since it
assumes that crossing decisions are drawn from a Bernoulli distribution, its
likelihood function is given by:

$\begin{gathered}\mathcal{L}_{1}(\omega)=\prod_{i=1}^{n}p\left(\Theta\mid\omega\right)^{u_{i}}\left(1-p\left(\Theta\mid\omega\right)^{1-u_{i}}\right)\\\
\rho_{1},\rho_{2},\rho_{3},\rho_{4}\in\omega\\\
\dot{\theta}_{i},X_{1,i},X_{2,i}\in\Theta\end{gathered}$ (11)

where $\omega$ includes all the estimated parameters
$\rho_{1},\rho_{2},\rho_{3},\rho_{4}$. $\Theta$ denotes
$\dot{\theta}_{i},X_{1,i},X_{2,i}$ for the $i$th trial. $n$ is the size of the
dataset. With respect to the initiation models, their likelihood functions are
given by the following equations based on (7) and (8):

$\begin{gathered}\mathcal{L}_{2}(\Delta)=\prod_{j=1}^{m}\operatorname{SW}\left(t_{int,j},\dot{\theta_{j}}\mid\Delta\right)\\\
\beta_{1},\beta_{2},\beta_{3},\beta_{4},b\in\Delta\end{gathered}$ (12)

$\begin{gathered}\mathcal{L}_{3}(\Delta)=\prod_{j=1}^{m}\mathcal{N}\left(t_{int,j},\dot{\theta_{j}}\mid\Delta\right)\end{gathered}$
(13)

where $\Delta$ is the summary of the estimated parameters of crossing
initiation models. $t_{int,j}$ is the $j$th crossing initiation time data. The
data size is $m$. According to the MLE method, the maximization problem is
equivalent to minimizing the negative log-likelihood. Thus, the optimal
estimations for parameters are achieved when negative log-likelihood functions
are minimised, e.g., $-\ln\left(\mathcal{L}_{1}(\omega)\right)$. We applied a
built-in ’fminuc’ function in MATLAB to find the solution to the above
minimization problems [43].

Furthermore, there were some differences in the model estimates based on the
two datasets. Firstly, since the traffic flow scenarios were not considered in
dataset one, the models based on this dataset did not include the parameters
$\rho_{1},\rho_{2}$. Regarding dataset two, for comparison purposes, we
manipulated the SW-PRD model so that it had the proposed decision rules for
traffic flow, whereas the G-PRD model did not. The estimated parameters based
on the two datasets are presented in Table. I and Table. II. In addition, the
parameters of the social force model are adopted from [42].

###  III-C Validation methods

After calibration, the predictions were compared with the validation set to
verify the ability of the models. Two evaluation methods were applied to
compare the performance of the proposed models, namely BIC and K-S test. The
BIC is given by:

$\text{BIC}=k\ln(n)-2\ln(L)$ (14)

where $k$ is the number of parameters in the model. $n$ is the size of the
dataset. $L$ is the maximum likelihood. The preferred model is the one with
the minimum BIC [44]. K-S test is a nonparametric test, which is used to
evaluate the goodness-of-fit of the predicted results by quantifying the
distance between empirical and predicted distributions [45]. The main equation
of K-S test is:

$D_{n,m}=\sup\left|\boldsymbol{F}_{n}(x)-\boldsymbol{F}_{m}(x)\right|$ (15)

where $\sup$ denotes the supremum function. $\boldsymbol{F}_{n}(x)$ and
$\boldsymbol{F}_{m}(x)$ are the distribution functions of the observed data
and predicted result. $n$ and $m$ represent the size of the samples. The K-S
test rejects the null hypothesis, i.e., two samples are drawn from the same
probability distribution if $D_{n,m}$ is bigger than the selected threshold.
In addition, the R-squared, $R^{2}$, and Root Mean Square Error (RMSE) are
also used in the model discussion.

TABLE I: Calibration results of models based on dataset one Parameter | SW-PRD (Without flow) | G-PRD (Without flow)
---|---|---
Estimate | 95 % C.I. | Estimate | 95 % C.I.
$\beta_{1}$ | 0.03 | [-0.19, 0.24] | -0.03* | [-0.05, -0.01]
$\beta_{2}$ | 4.48* | [3.35, 5.62] | 0.15* | [ 0.07, 0.24]
$\beta_{3}$ | -0.20* | [-0.26, -1.78] | -0.21* | [-0.24, -0.18]
$\beta_{4}$ | -2.11* | [-2.43, 1.22] | -0.76* | [-0.91, -0.62]
b | 6.06* | [4.43, 7.68] | - | -
$\rho_{0}$ | -2.14* | [-2.28, -1.98] | -2.14* | [-2.28, -1.98]
$\rho_{3}$ | -9.95* | [-10.64, -9.26] | -9.95* | [-10.64, -9.26]
LL | -108.43 | -176.69
BIC | 252.37 | 381.79
Note. LL: log-likelihood of the entire model, C.I.: confidence
interval, *: significant at a 5% significance level
With/Without flow: consider/not consider decision strategies for traffic flow
TABLE II: Calibration results of models based on dataset two Parameter | SW-PRD (With flow) | G-PRD (Without flow)
---|---|---
Estimate | 95 % C.I. | Estimate | 95 % C.I.
$\beta_{1}$ | 0.47* | [0.29, 0.66] | -0.05* | [-0.06, -0.04]
$\beta_{2}$ | 7.36* | [6.15, 8.57] | 0.01 | [-0.05, 0.07]
$\beta_{3}$ | 0.04 | [-0.02, 0.10] | -0.10* | [-0.13, -0.09]
$\beta_{4}$ | -1.41* | [-1.70, -1.13] | -0.59* | [-0.68, -0.50]
b | 7.76* | [5.6, 9.90] | - | -
$\rho_{0}$ | -2.92* | [-3.16, -2.68] | -3.31* | [-3.55, -3.07]
$\rho_{1}$ | -1.29* | [-1.56, -1.02] | - | -
$\rho_{2}$ | -0.50* | [-0.84, -0.15] | - | -
$\rho_{3}$ | -13.23* | [-14.30, -12.16] | -15.50* | [-16.56, -14.46]
LL(Decision model) | -1536.40 | -1672.50
LL(CIT model) | -36.35 | -104.03
BIC | 3218.40 | 3600.40
Note. LL(Decision model/CIT model): log-likelihoods of decision models
/crossing initiation time models
Figure 5:  Validation results. Probability density functions and data based on
datasets (a) one and (b) two. The vertical dash-dotted lines in (b) indicate
the time when the rear end of the vehicle passes the pedestrian’s position.
The size of the time gap (in seconds) between every two vehicles is indicated
at the top of the diagram.

##  IV Results and Analysis

In this Section, we first discuss the calibration results of the SW-PRD and
G-PRD models. Afterward, the validation results of the two models were
compared using the BIC and K-S test. Finally, the model with better
performance is compared to two entire datasets, and the reproduced crossing
behavior patterns are discussed in detail. Additionally, regarding the first
dataset, as it does not include the traffic flow scenario, we focus on the
impacts of speed and time gap on pedestrian crossing behavior, while the
effect of traffic is discussed using the results based on the second dataset.

TABLE III: Validation results of models based on dataset one Condition | Model | LL | BIC | K-S test score | P value
---|---|---|---|---|---
25 mph 4 s | SW-PRD | -23.08 | 71.47 | 0.06 | 0.56
| G-PRD | -27.28 | 74.82 | 0.10 | 0.08
35 mph 5 s | SW-PRD | -13.19 | 54.81 | 0.05 | 0.31
| G-PRD | -24.83 | 72.41 | 0.09 | 0.02*

###  IV-A Calibration results

Dataset one. The parameters of the SW-PRD and G-PRD models were calibrated
using the first dataset. One thing to note is that as the first dataset did
not include traffic flow scenarios, these two models thus did not implement
decision strategies in traffic, which means $\rho_{1}$ and $\rho_{2}$ were not
included in the models, and two decision models in the SW-PRD and G-PRD models
were the same. The calibration results are shown in Table. I, where the
maximum log-likelihood and BIC of the SW-PRD model based on the training set
are -108.43 and 252.37, which are significantly better than those of the G-PRD
model, i.e., -176.69 and 381.79, indicating that the SW-PRD model can better
describe pedestrian crossing initiation time than the G-PRD model on the
calibration set. Moreover, it can be found that the effect of $\rho_{0}$ is
significantly negatively correlated with $\dot{\theta}$
($\text{Est}.=-2.14,\text{C.I.}=[-2.28,-1.98]$), showing that pedestrian
crossing gap acceptance decreases as the risk of collision increases.
Additionally, the estimated effect of $\beta_{3}$ in the SW-PRD model is
significantly correlated with $\dot{\theta}$ (Table. I), suggesting that
pedestrian crossing initiation time is negatively related to the collision
risk.

Dataset two. The calibration results based on the second dataset are shown in
Table. II. As the SW-PRD model implemented the decision strategies in traffic
flow, it included $\rho_{1}$ and $\rho_{2}$. However, the G-PRD model did not.
Meanwhile, as both the decision model and initiation time model in the SW-PRD
model and the SW-PRD model were different, we calculated the respective log-
likelihood of the decision and initiation time models to facilitate the
comparison of the results. Again, the SW-PRD model fits data better than the
G-PRD model, where the SW-PRD model has larger log likelihoods for both the
decision and crossing initiation time models, and its BIC is smaller than that
of the G-PRD model. In particular, concerning the SW-PRD model, except for the
significant effect of $\rho_{0}$
($\text{Est}.=-2.92,\text{C.I.}=[-3.16,-2.68]$), $\rho_{1}$ and $\rho_{2}$
also significantly affect the pedestrian gap acceptance
($\text{Est}.=-1.29,\text{C.I.}=[-1.56,-1.02];\text{Est.}=-0.50,\text{C.I.}=[-0.84,-0.15]$),
consistent with our assumed crossing decision strategies in traffic flow. In
addition, although the effect of $\beta_{3}$ in the SW-PRD model is not
significant, the positive effect of $\beta_{1}$ reduces the tail magnitude of
the distribution of crossing initiation time as $\dot{\theta}$ increases and
thus can reduce pedestrians crossing initiation time.

###  IV-B Validation results

The calibration results indicate that the SW-PRD model fits the training sets
better than the G-PRD model. In this section, the validation sets of two
datasets are compared with the predicted results of two models.

Dataset one. Regarding the validation results, as shown in Table. III, the SW-
PRD model has better BIC values and K-S scores for all conditions.
Specifically, in the 35 mph 5 s condition, the K-S test rejects the null
hypothesis and indicates that the results of the G-PRD model are different
from the observed data at a $5\%$ significance level. As shown in Fig. 5a, it
can be found that the G-PRD model tends to overestimate the initial parts of
the data, but the SW-PRD model does not.

Dataset two. The predicted results are compared to the validation set of the
second dataset. The log-likelihood of crossing initiation time models of SW-
PRD and G-PRD are presented separately for reasons explained previously
(Table. IV). Both SW-PRD and G-PRD models accurately capture the timing of
pedestrian crossing decisions in the traffic flow, i.e., the peak location of
the initiation time distribution ( Fig. 5b). The predicted peak shapes of both
models are close to the data. However, the SW-PRD model has a relatively
better performance than the G-PRD model because the log-likelihood of the
crossing initiation time model for SW-PRD is bigger than the value for G-PRD
in Table. IV. The overall predictions of the SW-PRD model are closer to the
data than these of the G-PRD model. Specifically, the SW-PRD model has a
better BIC value and log-likelihood than the G-PRD model (Table. IV). Also,
the K-S test supports that the predicted density function of the SW-PRD model
is similar to the empirical distribution. In contrast, the predicted result of
the G-PRD model is rejected by the K-S test at a $5\%$ significance level
(Table. IV). As shown in Fig. 5b, it can be found that consistent with the
empirical data, the SW-PRD model predicts a decrease in the gap acceptance
from the first 3 s gap (at $t_{pass_{2}}$) to the second 3 s gap (at
$t_{pass_{5}}$). By contrast, the G-PRD model calculates a constant value for
both 3 s gaps, resulting in a significant underestimation of gap acceptance in
the first 3 s gap. In general, the SW-PRD model has better performance than
the G-PRD model on the validation set of dataset two.

TABLE IV: Validation results of models based on dataset two Model | LL | LL(CIT model) | BIC | K-S test score |  $p$ value
---|---|---|---|---|---
SW-PRD | -578.37 | -11.23 | 1193.10 | 0.08 | 0.10
G-PRD | -707.53 | -52.76 | 1444.10 | 0.16 | 0.001*

In the following sections, we discuss the predicted pedestrian crossing
behavior patterns in detail by comparing predicted results with the two full
datasets to provide a complete understanding of the proposed crossing
decision-making model. Since SW-PRD performs better on all datasets than
G-PRD, the SW-PRD model generates our results in the following sections.

###  IV-C Dataset one: Speed and time gap effects

The SW-PRD model predictions of crossing gap acceptance for each speed and
time gap condition are compared with the observed data in Fig. 6a. According
to the empirical data, crossing gap acceptance increased with vehicle speed
and traffic gap, aligning well with previous studies [21, 35]. The SW-PRD
model reproduces these behavioral patterns very well ($R^{2}=0.890$,
$RMSE=0.050$), suggesting that pedestrians might adapt their crossing
decisions based on the changes in collision cues.

Figure 6: Predicted gap acceptance of the SW-PRD model for both datasets. The
data and the predicted results are represented in black and blue respectively.
(a) For dataset one, the proportion of gap acceptance is plotted as a function
of vehicle speed and gap size (Gap sizes are indicated by different line
styles). (b) For dataset two, the proportion of gap acceptance for each gap of
each traffic scenario is presented.

Fig. 7a shows a comparison between the predicted crossing initiation time and
observed data. In line with the literature, [34], the empirical data showed
that pedestrian crossing initiation time correlated with vehicle kinematics,
i.e., it decreased as traffic gaps and vehicle speeds decreased. This
behavioral pattern can be understood as a distance-dependent phenomenon
whereby a reduction in vehicle speed and time gap leads to a reduction in
spatial distance, resulting in an increase in the perceived risk of collision
[22]. Hence, if pedestrians choose to cross, they tend to do so more quickly.
Based on our modeling results, the proposed SW-PRD model captures this pattern
with a good fit ($R^{2}=0.890$, $RMSE=0.050$), again indicating that visual
collision cues are associated with pedestrian crossing behavior.

Moreover, a more detailed comparison between predictions and data is shown in
Fig. A.2 in Appendix A-B. It can be noticed that the SW-PRD model predicts
pedestrian crossing behavior qualitatively and quantitatively. It not only
describes the distributions of pedestrian crossing initiation along the time
axis but also captures the variation in the mean crossing initiation time.

Figure 7: Predicted crossing initiation time of the SW-PRD model for both
datasets. Error bars and the edge of blue areas indicate the $2.5\%$ and
$97.5\%$ percentiles of the data and predicted results. (a) For dataset one,
the crossing initiation time is plotted as a function of vehicle speed and gap
size. (b) For dataset two, the crossing initiation time is a function of gap
size.

###  IV-D Dataset two: Impacts of traffic flow

Predicted gap acceptances of the SW-PRD model in the traffic flow are compared
to the observed data in Fig. 6b. Firstly, it can be noticed that pedestrians
in the traffic flow did not accept gaps of the same size equally. For
instance, regarding the $4$th gap and the $5$th gap in traffic scenario one
(The size of both traffic gaps is 3 s), the probability of crossing gap
acceptance dropped significantly from $27.9\%$ to $10.5\%$. When pedestrians
faced the $6$th gap, the decreasing trend became even stronger. The
probability of crossing gap acceptance was $8.1\%$, more than three times
smaller than the value of the $4$th gap. Further looking at the predictions,
the SW-PRD model reproduces this behavioral pattern across all traffic
scenarios with reasonable goodness-of-fit Fig 6b).

Fig.7b plots the predicted crossing initiation time as a function of the time
gap and compares it with the observed data. The SW-PRD model fits the crossing
initiation time data well ($R^{2}=0.850$, $RMSE=0.038$). Consistent with
empirical observations and similar to the first dataset [29], the SW-PRD model
predicts a smaller initiation time as the time gap decreases, again suggesting
that pedestrians attempted to compensate for crossing risk in unsafe traffic
gaps by initiating faster.

Furthermore, as shown in Fig. A.3 in Appendix A-B, detailed model predictions
are compared with the observed data. Across all traffic scenarios, the SW-PRD
model accurately predicts the level, shape and location of peaks of the
crossing initiation time distribution, showing that the model has a good
ability to characterize pedestrian crossing decisions in a continuous flow of
traffic.

##  V Discussion and conclusion

This study demonstrates a novel approach to characterize pedestrian crossing
decision-making at uncontrolled intersections with continuous traffic. We
hypothesized that the crossing behavior could be understood as depending on
three stages of information processing (perceive, decide, execute), and thus
proposed a model with three corresponding constituent parts: visual collision
cue, crossing decision, and crossing initiation. Following is a summary of the
detailed research results.

In our previous study[22], we showed that the visual collision cue,
$\dot{\theta}$, could capture the effects of vehicle kinematics on pedestrian
crossing decisions in single gaps and explain why pedestrians tended to rely
on distance from vehicles to make crossing decisions [21, 35]. In this study,
this finding is formally applied to model crossing decisions and extended to a
more complicated traffic scenario, i.e., a continuous flow of traffic. The
modeling results support that $\dot{\theta}$ is capable of characterizing the
risk perceived by pedestrians, at least at uncontrolled intersections with
constant speed traffic.

Moreover, regarding our third hypothesis, i.e., pedestrian crossing initiation
is time-dynamic and influenced by vehicle kinematics, we relate the proposed
crossing initiation time model to $\dot{\theta}$. The modeling results support
our hypothesis and show that pedestrians dynamically adjust their initiation
time based on vehicle kinematics. Both the SW and Gaussian distributions can
reasonably describe pedestrian initiation time, whilst the SW distribution has
relatively better goodness-of-fit than the Gaussian distribution, which
further indicates that the distribution of crossing initiation time is right-
skewed.

Notably, to accurately reproduce pedestrian crossing behavior in continuous
traffic flow, we further hypothesize that pedestrians compare the risks of the
gaps around them before making decisions, which is supported by the fact that
the proposed crossing decision strategy for continuous traffic scenarios
significantly improves the performance of the model. The study thus concludes
with the following findings. Firstly, pedestrians may have a reduced tendency
to accept a gap if they see an upcoming larger gap. Secondly, pedestrians may
have a greater tendency to reject a gap if they have already rejected a gap of
that size or larger. Although no other studies have yet found these patterns
of crossing behavior, some empirical observations provide indirect support.
[46] showed that drivers who rejected the bigger traffic gap tended to incur a
longer delay. [26] indicated that pedestrians who tended to reject the
crossing opportunities would be more cautious and tend to accept longer gaps.
Moreover, [24] found that pedestrians who missed the first opportunity to
cross the road would not compensate for their loss by accepting a shorter
second opportunity to cross the road. The above studies reinforce our
hypothesis that pedestrians who tend to wait for safer crossing opportunities
are more cautious and more likely to optimize their crossing strategies by
comparing crossing opportunities. Unlike several previous studies, which
simply assumed pedestrians tend to accept smaller gaps with the increase in
waiting time [7, 14], the novelty is that we show that there may be other
patterns in pedestrian crossing behavior in terms of waiting for the crossing
opportunity, which may provide an explanation for the non-significant effect
of waiting time on pedestrian crossing decisions found in the meta-study [27].
Furthermore, this finding is interesting in that it reminds us that there may
be a complex changing pattern in pedestrians’ strategy toward waiting for
crossing opportunities. Future research can further attempt to disentangle the
effects of waiting time and traffic flow.

Overall, this work provides a new concept that pedestrian crossing decisions
are dynamic and intrinsically closely linked to their perceived collision
risk, and can be reinterpreted through a three-stage crossing decision-making
process. The proposed model shows good predictive performance in different
simulator datasets, and it could therefore be interesting to test the model on
naturalistic traffic datasets as a next step. Furthermore, the idea of the
deconstructed process may drive further study to involve more complicated
perceptual, decision, and initiation models.

Regarding the practical implications of this study, there are many possible
ways to extend these concepts and models to further improve research in
pedestrian-AV interactions. First, as an increasing number of studies have
been keen on using pedestrian behavior models to promote safe and efficient
interactions[47], the proposed decision model may provide predictive
information to help automated driving systems to better anticipate pedestrian
crossing intentions and initiations. Early work is emerging where researchers
are attempting to plan and coordinate the actions of AVs and pedestrians
toward common goals by considering the visual collision risk of
pedestrians[6]. Another possible application case is future traffic scenarios
involving AV platoons and pedestrians, where AV platoons may need to take into
account the dynamic pedestrian crossing decisions along the length of the
platoon and adopt the decision strategy of each AV. Moreover, there is an
urgent need to train and evaluate AVs to perform well also in safety-critical
interactions with human road users. However, due to the low frequency of
critical traffic scenarios in real life, i.e., the corner case, and safety
reasons, both academia, and industry have agreed on using simulation methods
as a complementary way to validate AVs. Reliable simulation results rely on
the behavioral authenticity of simulated road users [14]. Hence, another
practical significance of this study is that the model can serve as a module
in the microscopic transport simulation tools or virtual testing platforms to
realize naturalistic pedestrian road crossing decisions.

However, several limitations of this study need to be addressed in the future.
Since the results and model cover only scenarios with single-lane, constant-
speed traffic, the model cannot be directly generalized to other scenarios
without further development. For example, in situations with yielding
vehicles, the collision cue model used in this study alone may not provide
sufficient information to model crossing decisions. In addition, compared to
the crossing behavior in pedestrian simulators, in real traffic, pedestrians
can flexibly adjust their behaviors and be affected by many potential factors.
The pedestrian simulator allows exact experimental control of conditions but,
therefore, naturally provides a less variable environment, and the virtual
nature of the task may also affect the observed behavior. Hence, an important
future work should apply the model to a reliable naturalistic dataset.
Furthermore, the model is developed based on current theories of human
collision and does not assert that pedestrians exactly use the applied visual
cues and perception strategy. As collision perception theory is further
developed, the model can be improved accordingly.

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|  Kai Tian received the M.Sc. degree in automotive engineering from Chongqing
University, China, 2019. He is now a PhD student at the Institute of Transport
Studies, University of Leeds, UK, working from 2019. His main research
interests include pedestrian-automated vehicle interaction, human factors and
safety, and decision-making modelling.
---|---
|  Gustav Markkula received the M.Sc. degree in engineering physics and
complex adaptive systems and the Ph.D. degree in machine and vehicle systems
from Chalmers University of Technology, Gothenburg, Sweden, in 2004 and 2015,
respectively. After having worked in the automotive industry for more than a
decade, he is now Chair in Applied Behaviour Modelling at the Institute for
Transport Studies, University of Leeds, U.K. His main research interests
include quantitative, cognitive modeling of road user behavior and
interaction, and virtual testing of vehicle technology and automation.
---|---
|  Chongfeng Wei received the PhD degree in mechanical engineering from the
University of Birmingham, UK, in 2015. He is now an assistant professor at
Queen’s University Belfast, UK. His current research interests include
decision making and control of intelligent vehicles, human-centric autonomous
driving, cooperative automation, and dynamics and control of mechanical
systems. He is also serving as an Associate Editor of IEEE Open Journal of
Intelligent Transportation Systems, IEEE Transactions on Intelligent Vehicles,
and IEEE Transactions on Intelligent Transportation Systems.
---|---
|  YeeMun Lee is currently a senior research fellow at the Institute for
Transport Studies, University of Leeds. She obtained her BSc (Hons) in
Psychology and her PhD degree in driving cognition from The University of
Nottingham Malaysia in 2012 and 2016 respectively. Her current research
interests include investigating the interaction between automated vehicles and
other road users using various methods, especially virtual reality
experimental designs. Yee Mun is involved in multiple EU-funded projects and
is actively involved in the International Organisation for Standardisation
(ISO).
---|---
|  Toshiya Hirose received the master’s degrees and the Ph.D. degree from
Shibaura Institute of Technology, Tokyo, Japan in 2002 and 2005. He is
currently an Associate Professor with the Department of Engineering Science
and Mechanics, Shibaura Institute of Technology. Before joining Shibaura
Institute of Technology, he has worked with the National Traffic Safety and
Environment Laboratory in Japan, and he was in charge of developing safety
regulations for vehicles. He belonged to the Intelligent Transport Studies,
the University of Leeds as the visiting researcher from 2019 to 2020. His
active research interests include autonomous vehicles, driver assistance
systems, active safety, driving simulators and human behavior models.
---|---
|  Natasha Merat is a Professor of Human Factors and Transport Systems at ITS,
University of Leeds. She is leader of the multidisciplinary Human Factors and
Safety Group and academic lead of Virtuocity at Leeds. She has a PhD in
Psychology from Leeds, and her research interests are in understanding user
interaction with new technologies in transport.
---|---
|  Richard Romano has over thirty years of experience developing and testing
AVs and ADAS concepts and systems which began with the Automated Highway
Systems (AHS) project while he directed the Iowa Driving Simulator in the
early 1990’s. He received his BASc and MASc in Engineering Science and
Aerospace Engineering respectively from the University of Toronto, Canada and
a PhD in Motion Drive Algorithms for Large Excursion Motion Bases, Industrial
Engineering from the University of Iowa, USA. In addition to a distinguished
career in industry he has supervised numerous research projects and authored
many journal papers. In 2015 he was appointed Leadership Chair in Driving
Simulation at the Institute for Transport Studies, University of Leeds, UK.
His research interests include the development, validation and application of
transport simulation to support the human-centered design of vehicles and
infrastructure.
---|---

##  Appendix A Supplementary file

###  A-A Simulation tool

Figure A.1: Structure of the simulation tool. The traffic environment contains
a single lane (60 m long and 4.2 m wide) and a fleet of vehicles (colored
rectangles).

In this study, an agent-based simulation tool is proposed using the
established PRD models for reproducing pedestrian crossing behavior at
uncontrolled intersections with traffic flow. The framework mainly includes
three parts: PRD model, environment model, and pedestrian kinematics model
(Fig.A.1). The detailed process of the simulation tool is as follows:

(i) Generate the traffic environment using the given traffic and pedestrian
parameters.

(ii) Generate a pedestrian agent at a random location on the pavement near the
crosswalk. After that, the pedestrian walks to the edge of the pavement. Since
this study focuses on the crossing decisions in the traffic flow, the
pedestrian performs the PRD model after the first vehicle has passed him/her
(Algorithm. 1).

(iii) The PRD model generates each pedestrian’s decision and initiation time
through a Monte Carlo sampling method (Algorithm. 2).

(iv) Pedestrians cross the road and walk to the opposite side of the road. The
simulation model stops when the traffic scenario ends or all pedestrians cross
the road.

A demonstration video of the simulation tool is also provided. Please see the
attachment.

Algorithm 1 Simulation model based on the model

Input: Model parameters
$\rho_{0},\rho_{1},\rho_{2},\rho_{3},\beta_{1},\beta_{2},\beta_{3},\beta_{4},b$
Output: $\boldsymbol{u},\boldsymbol{t}_{int}$

1: $I_{r}=I$ // Number of remaining participants $I_{r}$ and total number
participants $I$

2: for $n$th gap in traffic $N$ do

3: $\dot{\theta}_{n}\leftarrow\text{Eq}.\ref{1}$

4: $X_{1,n},X_{1,n}\leftarrow\text{Eq}.\ref{2}$ and $\text{Eq}.\ref{3}$

5: $p_{n}\leftarrow\text{Eq}.\ref{5}$

6: $P_{n}=p_{n}\cdot(1-P_{n-1})\leftarrow\text{Eq}.\ref{9}$

7: for $i$th pedestrian in $I_{r}$ do

8: $u_{i}$ $\leftarrow$ $Binomial(1,P_{n,i})$ // Sampling: crossing decision

9: if $u_{i}==1$ then

10: $f(t_{int})$ $\leftarrow$ $\text{Eq}.\ref{9}$ or $\text{Eq}.\ref{10}$ //
Caulculate probability density function of crossing decision

11: $t_{int,i}$ $\leftarrow$ Algorithm. 2 // Sampling: crossing initiation
time

12: else

13: Continue

14: end if

15: end for

16: $I_{r}=I_{r}-\text{length}(\boldsymbol{t}_{int})$ // Update remaining
participants

17: end for

Algorithm 2 Monte Carlo sampling of the model

Input: $f(t_{int})$
Output: $t_{int,i}$

1: Initialise $s=1$

2: while $s\neq 2$ do

3: $\pi(x)=f(x)$

4: $s\leftarrow\text{Uniform}(0,1)$;

5: $y\leftarrow Q(x|y)$ // Arbitrary probability density

6: if $u\leq\text{min}(\frac{pi(y)Q(x|y)}{pi(x)Q(y|x)},1)$ then

7: $t_{int,i}=y$

8: $s=s+1$

9: else

10: $s=1$

11: end if

12: end while

###  A-B Detailed modeling results

Detailed comparisons between modeling results and observations are shown in
Fig.A.2 and Fig.A.3. In Fig.A.2, the probability density functions of crossing
initiation time are plotted against time gaps and vehicle speeds. While, In
Fig.A.3, the probability density functions of crossing initiation time are
plotted as a function of traffic scenarios and crossing initiation time.

Figure A.2: Predicted density function of crossing initiation time of the SW-
PRD model based on dataset one. The predicted results, including density
function, samplings and mean values of crossing initiation time, are compared
with the observed data in terms of vehicle speed and traffic gap size. Figure
A.3: Predicted density function of crossing initiation time of the SW-PRD
model based on dataset two. The predicted density functions and samplings are
compared with the observed data. Regarding each traffic scenario, the order of
traffic gaps is indicated above each sub-figure. The vertical lines represent
the time when the rear end of the related vehicle passes the pedestrian’s
position, i.e., $t_{pass}$.