# Context-Aware Target Classification with Hybrid Gaussian Process prediction
for Cooperative Vehicle Safety systems

Rodolfo Valiente∗, Arash Raftari∗, Hossein Nourkhiz Mahjoub∗
Mahdi Razzaghpour∗, Syed K. Mahmud∗∗, Yaser P. Fallah∗
∗Connected and Autonomous Vehicle Research Lab (CAVREL)
Department of Electrical and Computer Engineering, University of Central
Florida, Orlando, FL, USA
∗∗Hyundai America Technical Center, Inc. (HATCI)
Superior Township, MI, USA This material is based upon work partially
supported by the National Science Foundation under Grant No. CNS-1932037, NSF
CAREER Grant 1664968 and in part by the Hyundai America Technical Center, Inc.
(HATCI).

###### Abstract

Vehicle-to-Everything (V2X) communication has been proposed as a potential
solution to improve the robustness and safety of autonomous vehicles by
improving coordination and removing the barrier of non-line-of-sight sensing.
Cooperative Vehicle Safety (CVS) applications are tightly dependent on the
reliability of the underneath data system, which can suffer from loss of
information due to the inherent issues of their different components, such as
sensors’ failures or the poor performance of V2X technologies under dense
communication channel load. Particularly, information loss affects the target
classification module and, subsequently, the safety application performance.
To enable reliable and robust CVS systems that mitigate the effect of
information loss, we proposed a Context-Aware Target Classification (CA-TC)
module coupled with a hybrid learning-based predictive modeling technique for
CVS systems. The CA-TC consists of two modules: A Context-Aware Map (CAM), and
a Hybrid Gaussian Process (HGP) prediction system. Consequently, the vehicle
safety applications use the information from the CA-TC, making them more
robust and reliable. The CAM leverages vehicles’ path history, road geometry,
tracking, and prediction; and the HGP is utilized to provide accurate
vehicles’ trajectory predictions to compensate for data loss (due to
communication congestion) or sensor measurements’ inaccuracies. Based on
offline real-world data, we learn a finite bank of driver models that
represent the joint dynamics of the vehicle and the drivers’ behavior. We
combine offline training and online model updates with on-the-fly forecasting
to account for new possible driver behaviors. Finally, our framework is
validated using simulation and realistic driving scenarios to confirm its
potential in enhancing the robustness and reliability of CVS systems.

###### Index Terms:

Context-Aware Target Classification, Cooperative Vehicle Safety Systems,
Gaussian Process, Scalable V2X communication.

##  I Introduction

In a Cooperative Vehicle Safety (CVS) system, Connected and Autonomous
Vehicles (CAV) take advantage of the data acquired from communication to
extend their situational awareness and improve safety [1, 2, 3]. In CVS
systems, Remote Vehicles (RV) frequently broadcast their state information to
the neighboring vehicles over a wireless channel in the form of an information
message, e.g, Basic Safety Messages (BSM) [4, 5] using Vehicle-to-Everything
(V2X) communication. V2X communication allows the information from RVs to be
available at the Host Vehicle (HV) over the wireless network extending the
situational awareness of CAVs and alleviating the limitations of the sensor-
based systems that rely on line-of-sight sensing [6, 7, 8]. At the HV, the
information messages are processed to monitor the neighboring road
participants and create a real-time map of all objects in its vicinity. The
crash warning algorithms and other safety applications access the real-time
map regularly to identify potentially dangerous situations. Therefore, the V2X
communication technologies have a significant impact on the accuracy of safety
applications. In that sense, two major V2X technologies, Cellular-V2X (C-V2X)
and Dedicated Short-Range Communication (DSRC) have been developed [9, 10].
Nevertheless, they suffer from information loss and have scalability
limitations in real-time applications affecting the quality of the real-time
map and the CVS system’s performance [11, 12, 4, 5].

CVS architectures separate the design of applications from the communication
system allowing flexibility and simpler adoption of different communication
technologies. Figure 1 shows a CVS system that consists of a communication and
sensor layer, a Target Classification (TC) layer, and a safety application
layer [13, 14]. TC classifies the neighboring RVs based on their relative
locations with respect to the HV. While the safety applications have been
extensively studied [13, 14, 8, 15, 16], there are limited studies on TC [17,
18]. Current TC solutions are limited in their context-awareness and
prediction capabilities, making them vulnerable to communication losses,
subsequently affecting the performance of the safety applications [17, 18].

To address this challenge we present a Context-Aware Target Classification
(CA-TC) module for mitigating the effect of communication loss or sensor
failure. The CA-TC fits within the CVS system block and consists of two
modules: a Context-Aware Map (CAM) and a Hybrid Gaussian Process (HGP)
prediction system as shown in Figure 2. The CAM module separates application
and information/perception subsystems and leverages vehicles’ path history,
road geometry, tracking, and prediction. The prediction module is a non-
parametric Bayesian inference modeling scheme, Gaussian Process regression,
and is responsible for vehicle’s trajectory predictions when data is not
received due to communication losses or sensor failure. The prediction system
allows the CA-TC to precisely predict future behaviors and compensate for
information loss. In this paper, we focus on improving the TC layer in Figure
1. While we particularly proposed a suitable prediction method for our
architecture, the CA-TC allows the use of any prediction method to address the
aforementioned challenge.

Figure 1: Cooperative Vehicle Safety system block diagram.

Our main contributions are as follows:

  * •

We proposed a Context-Aware Target Classification module for CVS systems that
alleviate the effect of information loss on the performance of safety
applications.

  * •

A novel Hybrid-learning-based Gaussian Process prediction method is presented.
It relies on a bank of driver models learned from real data in an offline
manner to perform forecasting on-the-fly, while also allowing online updates
of the driver models, accounting for new possible driving behavior.

  * •

The proposed system is evaluated in a simulation and hardware-in-the-loop
utilizing a Remote Vehicle Emulator (RVE) based on a DSRC Wireless Safety Unit
(WSU), which facilities the analysis of the CVS applications and the
communication system performance. Additionally, we study the impact of using
different prediction techniques and demonstrate the performance gains caused
by the proposed prediction method.

##  II Literature Review

###  II-A Cooperative Vehicle Safety Systems

CVS systems are an active area of research in vehicle safety [19, 20, 18, 21].
In [22, 23] the information from V2X communication is used for increased
situational awareness and improve safety, and authors in [17, 24] present a
CVS system that continuously tracks all RVs in the vicinity of the HV. CVS
systems rely on the accuracy of the real-time map of RVs which is processed by
the safety applications to identify potential threats. The architecture must
be modular and flexible enough to allow variations in communication rate and
sensor limitations [25].

Figure 1 presents a CVS system architecture [13, 14]. The modules include the
services modules, WSU software API, sensors and wireless data handler, the TC,
and safety applications. The WSU software services modules are generic modules
supplied by the WSU that provide services and an API to enable applications to
interface to the CAN bus, GPS receiver, and wireless. Based on the data from
surrounding vehicles, the TC classifies the locations of the RVs within a
specified radius of the HV and provides relative position and velocity metrics
for the classified RVs. The application modules consist of multiple Vehicle-
to-Vehicle (V2Vs) applications and evaluate potential safety threats based on
inputs from the system framework modules [13].

###  II-B Cooperative Vehicle Safety Applications

Different from non-cooperative safety applications that depend on the line-of-
sight sensors, cooperative perception systems leverage the information
accessible through V2V communication to identify the presence and location of
the RVs increasing the perception range. In the vehicle safety communication
report from NHTSA [13, 14] the main cooperative safety applications within CVS
were described, such as Cooperative Forward Collision Warning (FCW), Control
Loss Warning (CLW), Emergency Electronic Brake Lights (EEBL), Blind Spot
Warning (BSW), Lane Change Warning (LCW) among others. In [26] a Cooperative
Vehicle-to-Pedestrian safety application is implemented to provide extended
situational awareness and threat detection, reducing crashes and improving
safety. Authors in [27] present a cooperative testing platform to evaluate the
efficacy of CVS applications under challenging conditions and a Collision
Warning Avoidance (CWA) system is presented and evaluated in [28].

###  II-C Target Classification (TC)

In order to realize the above-mentioned safety applications, the surrounding
vehicles should be tracked and classified. While the safety applications have
been widely explored, the TC layer has received less attention [17, 18].
However, it is a main component in the CVS architecture and the performance of
all the applications relies on an accurate and robust TC [17, 18]. In that
direction, a machine learning-based TC for Millimeter-Wave Radar is proposed
in [29], it uses an artificial neural network and convolutional neural network
(CNN) and shows acceptable performance in a specific domain. In [18] a TC
layer is proposed to classify the RVs with respect to the HV, which considers
road geometry, the local map, the path history, and a simple vehicle tracking
algorithm based on a Kalman filter. These approaches are limited in their
prediction capabilities making them vulnerable to communication losses and
sensor failures. In order to mitigate these issues, we propose a CA-TC layer
and a robust and suitable prediction approach for the TC layer.

###  II-D Tracking and Prediction

In order to overcome CVS performance degradation caused by non-ideal
communication, methods for tracking and predicting information have been
proposed [30, 31, 32, 33, 34]. The constant speed or acceleration model is the
most frequently accepted assumption for prediction by conventional vehicle
manufacturers. The fundamental rationale for using those models is the
standardization of the BSM communication rate to 10Hz, given the short time
between BSM the models remain valid under ideal communication channels.
Nevertheless, because of the limitations of communication technology, the
assumption is not always valid in real scenarios, particularly in heavily
congested situations [35].

Kalman filters have been also used widely for tracking position and velocity
[31, 32]. This is likely due to the relative simplicity and robust nature of
the filter itself. In [33] is presented an adaptive KF to predict vehicle
velocity, the approach is tested in a variety of driving scenarios showing a
good performance in velocity prediction.

More recent effective approaches for trajectory prediction include recurrent
neural networks [36, 37], gaussian mixture models [38, 39] and Long Short-Term
Memory NNs [40, 41]. In [40] is described an LSTM model to predict vehicle
trajectories, leveraging the LSTM network to infer the temporal relationship
from previous sequence data. Authors in [40] propose a coupling LSTM model to
effectively predict the future trajectories of vehicles. The LSTM network can
learn the temporal relation from the historical sequence data and is employed
as the basic prediction model for all vehicle trajectories with common
parameters. Following the recent developments in that field of generative
models, several works make use of autoencoder-based solutions [42, 43],
proposing solutions that include Recurrent Variational Autoencoder,
Conditional Variational Autoencoder or Generative Adversarial Network [42,
43]. Autoencoders have been used effectively to learn a better representation
and for a prediction task in previous works [44, 45, 46]. In [44] a predictive
autoencoder is proposed to produce future observation, predicting velocity map
images directly and kinematics predictions using a the predictive autoencoder.
Authors in [47] present a variational autoencoder used for prediction with
partial interpretability. The proposed model achieves good performance,
however, the expensive training cannot be done on-the-fly while new data is
available, which makes it harder to learn from current information and adapt
to new scenarios.

To address those challenges we propose a novel hybrid learning approach based
on Bayesian inference. By leveraging non-parametric Bayesian approaches we can
predict fundamental patterns of observable time series and capture coupled
driver/vehicle dynamics without enforcing any assumptions on the model
parameters. This extraordinary feature frees the learning process from the
constraints of certain function patterns. The complexity of a model created in
a non-parametric Bayesian inference framework is automatically adjusted to the
observed data, allowing it to avoid over-complex models while still capturing
unforeseen patterns in the data on-the-fly. Particularly, within the non-
parametric Bayesian inference approaches, Gaussian Process (GP) has
demonstrated a significant performance improvement in terms of packet
generation rate as well as position tracking accuracy under network congestion
[34]. GP attempts to regress observed time-series realizations by projecting a
prior distribution directly over the function space, rather than the function
parameters space, in such a way that any finite subset of draws from this
distribution represents a multivariate Gaussian random vector [48]. This
approach adjusts the model complexity to the observed data, allowing it to
capture distinct patterns as they arise during training. This idea has been
utilized in [49, 50] to enhance the performance of Cooperative Adaptive Cruise
Control (CACC) in congested traffic scenarios.

Previous works have demonstrated that within the context of Model-Based
Communication (MBC), a modeling scheme that uses GP as its inference technique
outperforms other modeling approaches [34, 51]. Different from other works
that provide a transmitter-side approach using GP for prediction via an MBC
paradigm we propose a receiver-side approach that leverages GP to forecast the
trajectories of the RV, therefore, the proposed architecture does not require
modifications in the current V2X standards.

##  III Context-Aware Target Classification (CA-TC)

In this section, we present the CA-TC. The CA-TC fits within the CVS system
block in Figure 1. The proposed CA-TC is presented in Figure 2 and consists of
the HV path prediction, the path history of all vehicles, the current received
BSMs from RVs (Local Map), a tracking module, and the HGP system, all of which
are integrated into the CAM, in which data fusion, history, and predictions
are used for lane estimation and local map reconstruction creating a real-time
context map of the surroundings. By leveraging tracking and prediction, CA-TC
provides accurate information that improves the performance of safety
applications. The CA-TC enables the applications to work based on the CAM
rather than individual BSMs or sensory data. Using the CAM, the RVs are
classified based on the relative current and predicted location with respect
to HV. Therefore, it allows the safety applications to be triggered not just
by the local map and history but also based on the predictions, either when
packages are lost or when the prediction system detects a possible warning.
For instance, as an illustration in Figure 2, the local map is showing RV1 as
behind, but the CAM is predicting RV1 will be behind right, so RV1 will be
also processed by LCW and BSW safety application to check possible future
hazards based on the CA-TC information. The proposed CA-TC allows applications
to be independent of underlying communication and sensing systems, and the
architecture is adaptable to new sensing technologies.

Figure 2: Context-Aware Target Classification module with HGP prediction for
CVS systems. The HGP prediction system is responsible for accurate vehicle
predictions and the CAM leverages vehicles’ path history, road geometry,
tracking, and prediction to improve the robustness and reliability of the
safety applications. In the figure, HV is in blue, RVs are in green, a missing
RV is in red, predictions are in shadow yellow, and the vehicle’s history is
in shadow green.

In the CA-TC (Figure 2), the map-update module (Local Map) is in charge of
refreshing the real-time map with the latest received information from
sensors, GPS, and communication module. Thus, the real-time map database keeps
the records of its surrounding entities, such as all neighboring vehicles, and
other detected objects. Every local map record consists of the latest
available information about a particular neighboring entity, such as its
position information, speed, size, etc. The local map is created and updated
based on the latest HV and RVs position. The path history (PH) module keeps
track of HV and RVs history. It keeps a buffer of the vehicle’s most recent
position and sensor data points and computes concise representations of the
vehicle’s actual PH based on the allowable position error tolerance between
the actual vehicle path and its concise representation, and updates the PH
concise representation periodically for use by the other modules. The HV path
prediction module makes use of the HGP prediction algorithm to forecast the
future path of the HV. The tracking module utilizes the HGP prediction
algorithm together with the previous and current local map information to
correct the vehicle’s position, recover current missed RVs and filter possible
outliers. The HGP prediction system uses the information from PH, tracking,
and current local map records for future path prediction points for all
vehicles. By having an accurate RV prediction, the CA-TC allows safety
applications to be triggered based on forecasting and not just current
received information.

All the aforementioned modules are used by the CAM for an extended situational
awareness subsystem that is responsible for delivering the latest and most
accurate information to the safety application to identify present and future
possible threats and generate appropriate notifications and signals. While the
CAM can leverage information from local sensors and V2X communication, in this
work, we focus on the communication side and study the impact of communication
imperfections on CVS applications.

###  III-A Target Classification

With the information provided by the CAM, the RVs are classified in different
areas based on the relative locations with respect to the HV. In addition to
the classification of RVs, the CA-TC also provides the vehicles’ predictions,
relative speeds, headings, lateral and longitudinal offsets, and tracking
positions. Figure 4 shows the classification of the RVs relative to the HV.
The CA-TC computes the RVs classifications area based on the longitudinal and
lateral distances with respect to HV [17, 18]. based on the relatives’
longitudinal distance the RVs are classified as,

$RV_{class}=\begin{cases}Xrel\geq 0,&\quad\textrm{Ahead}\\\
Xrel<0,&\quad\textrm{Behind}\\\ \end{cases}$ (1)

Next the longitudinal distance classifications are combined with the
relatives’ lateral distance ($ld$) information to classify the RVs as,

$RV_{class}=\begin{cases}ld\in(1.5w_{lane},\infty),&\quad\textrm{Far Left}\\\
ld\in(0.5w_{lane},1.5w_{lane}],&\quad\textrm{Left}\\\
ld\in[-0.5w_{lane},0.5w_{lane}],&\quad\textrm{On Centre}\\\
ld\in[-1.5w_{lane},-0.5w_{lane}),&\quad\textrm{Right}\\\
ld\in(-\infty,-1.5w_{lane}),&\quad\textrm{Far Right}\\\ \end{cases}$ (2)

in which $w_{lane}$ is the width of the lane. Finally, for the
oncoming/ongoing traffic, the lane heading angle $\phi_{lane}$ is compared
with the RVs heading angle $\phi_{RV}$ to classify the RVs as,

$RV_{class}=\begin{cases}|\phi_{RV}-\phi_{lane}|\leq\Delta\phi_{ongoing},&\quad\textrm{Ongoing}\\\
|\phi_{RV}-\phi_{lane}|\geq\Delta\phi_{oncoming},&\quad\textrm{Oncoming}\\\
\textrm{else, }&\quad\textrm{not classified}\end{cases}$ (3)

where ongoing vehicles are vehicles in the same direction, and oncoming
vehicles are vehicles in the opposite direction. $\Delta\phi_{ongoing}$ and
$\Delta\phi_{oncoming}$ are the threshold values for classification. Based on
these RV classifications the safety applications are evaluated. Figure 3
depicts the mapping of RV classification to various safety applications. Based
on the data given by the CA-TC, the safety application modules independently
analyze possible safety hazards. This demonstrates that appropriate TC is
crucial for the triggering and functioning of safety applications, as the
warnings are dependent on the RVs being classified correctly. For example, in
the event of an approaching rear-end collision with a vehicle ahead in the
same lane and direction of travel, the TC will classify the vehicle as Ahead
and the FCW application will be evaluated as indicated in Figure 3.

Figure 3: Target Classifications areas used by the safety applications. Figure
4: Target Classifications of the RVs relative to the HV.

##  IV Hybrid-learning-based Gaussian Process (HGP) prediction system

The prediction system is used at the HV (receiver side) to regress the
observed time-series realizations from PH, capturing the distinct patterns as
they emerge in the data. Each observation point is chosen from Gaussian random
variables that are not independent and have temporal relationships with their
predecessors and successors. The correlation describes the temporal
relationship between observations, making GP a useful tool for detecting
patterns in time series.

The set of $m$ observed values is represented by an m-dimensional multivariate
Gaussian random vector, which is defined by an $m\times m$ covariance matrix
and a $m$ mean vector. This covariance matrix, often known as the GP kernel,
is the foundation upon which GP detects and anticipates the underlying
behavior of time series based on their recorded history. The fundamental GP
components can be expressed mathematically as follows,

$f(t)\sim gp\big{(}m(t),k(t,t^{\prime})\big{)},\hskip 137.0pt\\\
\\{X_{i}\\}_{i=1,2,...,m}=\\{f(t_{i})\\}_{i=1,2,...,m}\sim\mathcal{N}(\mu,\,\Sigma),\hskip
10.0pt\\\ \hskip 1.0pt\mu=\big{[}m(t_{1}),...,m(t_{m})\big{]}^{T},\hskip
70.0pt\\\ \hskip 3.0pt\Sigma_{ij}=\kappa(t_{i},t_{j})\ \forall
i,j\in\\{1,2,...,m\\},\hskip 30.0pt$ (4)

where $X_{i}$, $f(t)$, $m(.)$, and $\kappa(.,.)$ are the samples of the
vehicles’ state, observed or to be predicted at the time $t_{i}$, the unknown
underlying function that the vehicles’ states are sampled from, the mean and
the covariance functions, respectively.

Instead of working directly with the position time series, the proposed
inference algorithm treats the vehicles’ heading and longitudinal speed as two
independent time series that are regressed using GPs. The proposed method is
divided into two stages: Training and Forecasting. During the offline training
procedure, the hyper-parameters of kernel functions are learned based on the
observed samples of the vehicles’ speed and heading to create a bank of
vehicle/driver short-term behavioral models. The well-known realistic data
set, Safety Pilot Model Deployment (SPMD) [52], is used for training and
selecting a group of significant models that represent the data. It was shown
through experiments that a limited-size kernel bank suffices to model joint
vehicles’ dynamic and driver behavior. Given the availability of this bank of
models, during forecasting, the most recent information of RVs and path
history is used to select the model with the highest likelihood from the bank
and use it to forecast the vehicles’ state until new information is received.

###  IV-A Training

During training, the sequences of equally-spaced samples of longitudinal speed
and heading, measured every $100ms$, with size $TW=30$ were used to learn the
GP models and subsequently construct the kernel banks. In this work, based on
the findings in [34], a compound kernel of Radial Basis Function (RBF) and a
linear kernel is used as the covariance function and the mean function is
considered to be zero. Therefore, the covariance function in (4) can be
expressed as:

$\centering\kappa(t,t^{\prime})=\alpha_{1}^{2}\exp(-\frac{||t-t^{\prime}||^{2}}{2\gamma^{2}})+\alpha_{2}^{2}tt^{\prime}.\@add@centering$
(5)

we used the Leave-One-Out cross-validation to learn a GP model and obtain the
set of parameters $\theta=\\{\gamma,\alpha_{1},\alpha_{2}\\}$ for each under
consideration sequence of speed or heading. Assuming the $i^{th}$ element of
the observed speed or heading sequence, $x_{i}$, is left out, the joint
distribution of $x_{i}$ and the rest of the sequence, $\mathcal{X}_{-i}$, can
be expressed as:

$\left[\begin{array}[]{l}{x}_{i}\\\
\mathbf{\mathcal{X}_{-i}}\end{array}\right]\sim\mathcal{N}\left(\mathbf{0},\left[\begin{array}[]{ll}\kappa({t_{i}},{t_{i}})&K_{1}^{T}\left(t_{i},\boldsymbol{t_{-i}}\right)\\\
K_{1}\left(t_{i},\boldsymbol{t_{-i}}\right)&K_{2}\left(\boldsymbol{t_{-i}},\boldsymbol{t_{-i}}\right)\end{array}\right]\right),$
(6)

where $K_{1}$ is a $(m-1)\times 1$ vector representing the covariance between
$\mathcal{X}_{-i}$ and $x_{i}$, $K_{2}$ is the $(m-1)\times(m-1)$ covariance
matrix of $\mathcal{X}_{-i}$, and $\boldsymbol{t_{-i}}$ represents the time
stamps of $\mathcal{X}_{-i}$. $K_{1}$ and $K_{2}$ can be obtained using (5).
The conditional probability of $x_{i}$ given the rest of the sequence can be
derived as

$\Big{(}\mathbf{x_{i}}\mid\mathbf{t},\mathbf{\mathcal{X}_{-i}},\mathbf{\theta}\Big{)}\sim\mathcal{N}(\mu_{i},\sigma_{i}),\\\
\mspace{-75.0mu}\mu_{i}=\mathbf{K_{1}}^{T}[\left(t_{i},\mathbf{t_{-i}}\right)|\mathbf{\theta}]\mathbf{K_{2}}^{-1}[(t_{-i},t_{-i})|\mathbf{\theta}]\mathbf{\mathcal{X}_{-i}},\\\
\mspace{6.0mu}\sigma_{i}=-\mathbf{K_{1}}^{T}[\left(t_{i},\mathbf{t_{-i}}\right)|\mathbf{\theta}]\mathbf{K_{2}}^{-1}[(\mathbf{t_{-i}},\mathbf{t_{-i}})|\mathbf{\theta}]\mathbf{K_{1}}[\left(t_{i},\mathbf{t_{-i}}\right)|\mathbf{\theta}]\\\
\mspace{18.0mu}+\kappa[({t_{i}},{t_{i}})\mid\mathbf{\theta})].$ (7)

Therefore, the $\log$ probability of observing $x_{i}$ given the rest of the
sequence, ($\mathcal{X}_{-i}$), can be calculated as

$\log
p\left(x_{i}\mid\mathbf{t},\mathcal{X}_{-i},\mathbf{\theta}\right)=-\frac{1}{2}\log\sigma_{i}^{2}-\frac{\left(x_{i}-\mu_{i}\right)^{2}}{2\sigma_{i}^{2}}-\frac{1}{2}\log
2\pi$ (8)

Defining the cross-validation objective function as the sum of the $log$
probabilities over all elements of the sequence, i.e.,
${L}(\mathbf{t},\mathcal{X},\theta)=\sum_{i=1}^{m}\log
p\left(x_{i}\mid\mathbf{t},\mathcal{X}_{-i},\theta\right)$, the optimal
parameters $\theta=\\{\gamma,\alpha_{1},\alpha_{2}\\}$, for an under
consideration sequence of vehicles’ states, can be obtained using the
conjugate gradient optimization method as proposed in [53].

In our settings, GPS information is converted into Earth-Centered, Earth-Fixed
format (ECEF). The ECEF format is offset to the center of the vehicle on the
ground and then transformed into East-North-Up (ENU) coordinates. As the
heading and longitudinal speed of the vehicle are treated as two independent
time series, we apply the GP regression technique to model and forecast the
vehicles’ speed and heading, and use the forecasted speed and heading to
predict the position, we particularly call this approach indirect prediction.
Given that the parameters of speed and heading kernel functions are obtained
by optimizing the Leave-One-Out objective function using $m-$most recent
observations of speed and heading data, the predictive distribution of future
speed and heading values, $\mathcal{S}^{\ast}$ and $\mathcal{H}^{\ast}$,
conditioned on having observed speed and heading values, $\mathcal{S}^{obs}$
and $\mathcal{H}^{obs}$, at time stamps $\boldsymbol{t}$ can be derived as

$\Big{(}\mathbf{\mathcal{S}^{*}}\mid\mathbf{t^{*}},\mathbf{t},\mathbf{\mathcal{S}^{obs}}\Big{)}\sim\mathcal{N}(\mu^{\ast}_{s},\Sigma^{\ast}_{s}),\\\
\mspace{-75.0mu}\mu^{\ast}_{s}=K_{1}[\left(t^{*},t\right)|\theta_{s}]K_{2}^{-1}[(t,t)|\theta_{s}]\mathbf{\mathcal{S}^{obs}},\\\
\mspace{6.0mu}\Sigma^{\ast}_{s}=-K_{1}[\left(t^{*},t\right)|\theta_{s}]K_{2}^{-1}[(t,t)|\theta_{s}]K_{1}^{T}[\left(t,t^{*}\right)|\theta_{s}]\\\
\mspace{18.0mu}+K_{3}[\left(t^{*},t^{*}\right)|\theta_{s}],\mspace{-206.0mu}\\\
\Big{(}\mathbf{\mathcal{H}^{*}}\mid\mathbf{t^{*}},\mathbf{t},\mathbf{\mathcal{H}^{obs}}\Big{)}\sim\mathcal{N}(\mu^{\ast}_{h},\Sigma^{\ast}_{h}),\\\
\mspace{-75.0mu}\mu^{\ast}=K_{1}[\left(t^{*},t\right)|\theta_{h}]K_{2}^{-1}[(t,t)|\theta_{h}]\mathbf{\mathcal{H}^{obs}},\\\
\mspace{6.0mu}\Sigma^{\ast}=-K_{1}[\left(t^{*},t\right)|\theta_{h}]K_{2}^{-1}[(t,t)|\theta_{h}]K_{1}^{T}[\left(t,t^{*}\right)|\theta_{h}]\\\
\mspace{18.0mu}+K_{3}[\left(t^{*},t^{*}\right)|\theta_{h}],$ (9)

where $K_{3}$, $K_{2}$ and $K_{1}$ are the covariance matrices between future
(predicted) values, observed values and observations, and future values
respectively, and can be derived using (5). Using the predictive distributions
of longitudinal speed and heading, $P(\tilde{s})$ and $P(\tilde{h})$, the
position of the vehicle in the ENU coordinate system can be predicted as
follows.

$\bar{x}\left(t_{1}\right)=x\left(t_{0}\right)+\iiint_{t_{0}}^{t_{1}}\tilde{s}\cos(\tilde{h})\operatorname{P}(\tilde{s})\operatorname{P}(\tilde{h})dtd\widetilde{s}d\widetilde{h},\\\
\bar{y}\left(t_{1}\right)=y\left(t_{0}\right)+\iiint_{t_{0}}^{t_{1}}\tilde{s}\sin(\tilde{h})\operatorname{P}(\tilde{s})\operatorname{P}(\tilde{h})dtd\widetilde{s}d\widetilde{h},\hskip
7.0pt$ (10)

Where $t_{1}$ and $t_{0}$ denote the time instance of prediction and last
received BSM respectively. In this work, since the vehicles’ states were
measured every $100ms$, we calculated the predicted positions using the
recursive piece-wise linear formulation.

$\bar{x}\left(t_{j+1}\right)=\left(t_{j+1}-t_{j}\right)\iint\tilde{s_{j}}\cos(\tilde{h_{j}})\operatorname{P}(\tilde{s_{j}})\operatorname{P}(\tilde{h_{j}})d\widetilde{s_{j}}d\widetilde{h_{j}}\\\
+\bar{x}\left(t_{j}\right)=\mathbb{E}\big{[}s_{j}\big{]}\mathbb{E}\big{[}\cos(h_{j})\big{]}+\bar{x}\left(t_{j}\right)\\\
=\mu_{s_{j}}e^{-\sigma^{2}_{h_{j}}/2}\cos(\mu_{h_{j}})+\bar{x}\left(t_{j}\right),\\\
\bar{y}\left(t_{j+1}\right)=\left(t_{j+1}-t_{j}\right)\iint\tilde{s_{j}}\sin(\tilde{h_{j}})\operatorname{P}(\tilde{s_{j}})\operatorname{P}(\tilde{h_{j}})d\widetilde{s_{j}}d\widetilde{h_{j}}\\\
+\bar{y}\left(t_{j}\right)=\mathbb{E}\big{[}s_{j}\big{]}\mathbb{E}\big{[}\sin(h_{j})\big{]}+\bar{y}\left(t_{j}\right)\\\
=\mu_{s_{j}}e^{-\sigma^{2}_{h_{j}}/2}\sin(\mu_{h_{j}})+\bar{y}\left(t_{j}\right)$
(11)

where, $t_{j}-t_{j-1}=100ms$ and $t_{0}$ is the time instance of last received
BSM. $P(\tilde{s_{j}})$ and $\mu_{s_{j}}$ are the uni-variate predictive
distribution of longitudinal speed at time instance $t_{j}$ and its
corresponding mean. $P(\tilde{h_{j}})$, $\mu_{h_{j}}$, and
$\sigma^{2}_{h_{j}}$ are the uni-variate predictive distribution of heading at
time instance $t_{j}$, its corresponding mean, and variance. These values can
be calculated using equation (9).

The pseudo-code of our model generation scheme is illustrated in Algorithm 1,
in which $PTE$ is the Position Tracking Error and is calculated as the 2D
Euclidean distance between the actual and predicted vehicle positions. Since
$PTE$ sampling is dependent on the availability of actual position updates, it
can be done at most at the sampling rate of GPS updates, which is 10 Hz for
the SPMD dataset.

1: Dataset: SPMD Trips $\mathcal{T}=\\{T_{1},...,T_{n}\\}$ ; $i=1$

2: Observe a TW history of $\mathcal{S}^{obs}$ and $\mathcal{H}^{obs}$

3: Fit first model $K_{current}$ = $HGP_{fit}$
($\mathcal{S}^{obs}$,$\mathcal{H}^{obs}$)

4: for $T_{i}\in\mathcal{T}$ do

5: Read trip ${T}_{i}$ load training $\mathcal{S}$ ; $\mathcal{H}$

6: for $t_{step}\in T_{i}$ do

7: Use $K_{current}$

8: while $(PTE<PTE_{th})$ do

9: $\mathcal{S}^{*}$, $\mathcal{H}^{*}$ =
$HGP_{predict}(\mathcal{S}^{obs},\mathcal{H}^{obs},K_{current},t_{predict})$

10: $(X^{*},Y^{*})$ = $XY_{predict}(\mathcal{S}^{*},\mathcal{H}^{*})$

11: $PTE$= $f_{PTE}(X^{*},Y^{*})$

12: end while

13: for $K_{i}\in\mathcal{K}$ do

14: $\mathcal{S}^{*}$, $\mathcal{H}^{*}$ =
$HGP_{predict}(\mathcal{S}^{obs},\mathcal{H}^{obs},K_{current},t_{predict})$

15: $(X^{*},Y^{*})$ = $XY_{predict}(\mathcal{S}^{*},\mathcal{H}^{*})$

16: $PTE_{K_{i}}$= $f_{PTE}(X^{*},Y^{*})$

17: end for

18: $PTE_{min}=\textbf{min}(PTE_{K_{i}})$

19: if $(PTE_{min}<PTE_{th})$ then

20: Load kernel,
$K_{current}=\underset{K_{i}\in\mathcal{K}}{\operatorname{argmin}}PTE(K_{i})$

21: else

22: Observe a TW history of $\mathcal{S}^{obs}$ and $\mathcal{H}^{obs}$

23: Fit new model $K_{current}$ = $HGP_{fit}$
($\mathcal{S}^{obs}$,$\mathcal{H}^{obs}$)

24: Add $K_{current}$ to $\mathcal{K}$ bank

25: end if

26: end for

27: end for

Algorithm 1 Training: HGP kernel bank generation

In the algorithm, the vehicles’ states (speed, $\mathcal{S}$ and heading,
$\mathcal{H}$) for all $N$ available trips ($T_{i}$ , $i=1,...,N$) are loaded
one after another. A kernel ( $K_{i}\in\mathcal{K}$) is selected and used to
predict the time series at each time-step, the kernel $K_{i}$ refers to the
speed and heading kernels, the speed kernel is used to predict
$\mathcal{S}^{*}$ and the heading kernel to predict $\mathcal{H}^{*}$. When a
kernel is selected, its prediction accuracy is evaluated at each time step
ahead. $HGP_{predict}$ is used to predict with the current kernel
$K_{current}$ from the bank of kernels. As long as the kernel predicts the
future positions with a $PTE$ less than the threshold ($PTE_{th}$), it remains
as the selected kernel for the model and keeps using this kernel until its
prediction error exceeds the threshold ($PTE>PTE_{th}$). The size of the time
interval, in which the latest selected model remains in use, is called Model
Persistency ($MP$). At this moment either another kernel from the kernel bank
is selected for predicting the position or, if none of the available kernels
could satisfy the $PTE_{th}$, a new one is created and added to the bank
$\mathcal{K}$, $HGP_{fit}$ is used to learn the new model. Finally, we update
$K_{current}$ and continue the prediction with the updated kernel.

After the full kernel bank is created, the models are clustered to obtain a
reduced-size kernel bank of vehicle behaviors, the limited-size kernel bank
suffices to model the joint vehicle’s dynamic and behavioral pattern of the
driver. During forecasting, the model with the highest likelihood is chosen
from the reduced kernel bank. Figure 5 shows a diagram of the training
procedure explained before.

###  IV-B Forecasting

Considering that the pre-learned reduced-size kernel bank is available at HV,
every time the HV receives a new BSM from RV, it can choose a model from this
kernel bank to forecast the trajectory (states) of the RV up until it receives
the next BSM. For this purpose, assume that during the past $T$ seconds, HV
received BSMs from RV at time instances $\mathbf{t}$ and the value of speed
and heading of the RV at these time instances (from received BSMs) are denoted
by $\mathcal{S}$ and $\mathcal{H}$. The best model from the bank, with the
highest likelihood, is selected by using the following equation.

Figure 5: Offline training and clustering.

$\hskip 5.0pt\theta^{*}_{s}=\underset{\theta_{s}\in\text{ speed bank
}}{\operatorname{argmax}}\log
p\left(\theta_{s}\mid\mathcal{S},\mathbf{t}\right)=-\frac{n}{2}\log 2\pi\\\
\hskip
30.0pt-\frac{1}{2}\mathcal{S}^{T}K_{S}^{-1}[\left(\mathbf{t},\mathbf{t}\right)\mid\theta_{s}]\mathcal{S}-\frac{1}{2}\log
det\big{(}K_{S}[(\mathbf{t},\mathbf{t})\mid\theta_{s}]\big{)}\\\
\theta^{*}_{h}=\underset{\theta_{h}\in\text{ heading bank
}}{\operatorname{argmax}}\log
p\left(\theta_{h}\mid\mathcal{H},\mathbf{t}\right)=-\frac{n}{2}\log 2\pi\\\
\hskip
30.0pt-\frac{1}{2}\mathcal{H}^{T}K_{H}^{-1}[\left(\mathbf{t},\mathbf{t}\right)\mid\theta_{h}]\mathcal{H}-\frac{1}{2}\log
det\big{(}K_{H}[(\mathbf{t},\mathbf{t})\mid\theta_{h}]\big{)}$ (12)

Where $\theta_{s}$ and $\theta_{h}$ are the hyperparameters of the speed and
heading models in the bank, n is the size of vector $\mathbf{t}$ and $K_{S}$
and $K_{h}$ are the covariance matrix of speed and heading observations given
hyperparameters $\theta_{s}$ and $\theta_{h}$ and can be constructed using
equation (5). $\theta^{*}_{s}$ and $\theta^{*}_{h}$ are hyperparameters of the
selected speed and heading models with the highest likelihood. Using the
selected model, we can predict the future values of speed and heading and
consequently predict the position by utilizing (9) and (11).

Figure 6 depicts how the final prediction is obtained from the history of
time-series observations. After the prediction model is selected, speed and
heading are predicted and used to predict X and Y and the final path. When
packets are not received, the time series history is used to select the
corresponding model. If there is no model in the reduced kernel bank that
meets the required $PTE_{th}$ a new model is created, added to the bank, and
selected for prediction. Next, the selected model is used for forecasting the
RV’s states and keeping track of the RVs even when the data is not received.

Figure 6: Path prediction example using HGP. Figure 7: On-the-fly model
Forecasting Diagram.

GP predictions are swapped with constant speed and constant acceleration
predictions if a specific criterion is met for avoiding divergence of speed
and acceleration predictions to infinity or to unfeasible values when the time
interval between two consecutive BSMs is large and we must rely on prediction
for a long period. Figure 7 shows the diagram for the hybrid prediction
procedure. Different motion models are used based on physical constraints in
order to increase the accuracy and robustness of the prediction. This hybrid
procedure probes to be advantageous, especially in situations where the
assumptions of the other models are no longer valid. For visualization, the
results of the hybrid approach against constant acceleration on a single trip
are shown in Figure 8.

Figure 8: Performance of our HGP scheme against constant acceleration (CA).
The figure (using different colors) shows the moments when CA over-performs
HGP (in terms of $PTE$) on the tracking accuracy while remaining below the
threshold. (left) shows the performance by timesteps, (right) shows the
performance in 2D ENU coordinates.

##  V Experimental Setup and Simulation Results

We study the impact of communication uncertainties on the proposed CA-TC and
compare it with the other architectures, demonstrating that the proposed
prediction system outperforms the baselines.

###  V-A Data Preprocessing

We evaluate three separate datasets to validate the proposed architecture. To
begin, we replicate realistic scenarios using SUMO simulation data. Then we
use the 100-car dataset [54] and lastly, we evaluate our approach using the
SPMD dataset.

SUMO Dataset. We first evaluate our approach using the created SUMO dataset.
We generate logs that contained information such as timestamps, vehicle IDs,
GPS coordinates, and velocities and create the BSM packets for different
timestamps based on the standard BSM transmission frequency (i.e. 10Hz). SUMO
enables the testing of any real-world road scenario without the need for
costly and time-consuming real-world data collecting. The designed scenarios
are carefully chosen from OpenStreetMap and exported into the SUMO simulator,
where we construct situations with the required vehicle density and behavior,
and then utilize the intended scenarios to evaluate the performance of our
system and CVS applications.

100-Car Dataset. We utilize the Virginia Tech Database dataset (100-car
dataset). This dataset contains trajectories for over 800 real-world
instances, some of which are near-crash or real crashes. The data is utilized
to build a lead vehicle (RV) and a following vehicle (HV) movement using a
car-following model as described in [55]. The subsequent vehicle mobility is
used to evaluate the system’s performance. As a consequence, we have
trajectories for two cars (one RV and one HV), with the RV trajectories being
based on real data and the HV trajectories being generated using the car-
following model.

SPMD Dataset. Finally, we evaluate the performance of our system using the
SPMD [52], which is a well-known data set that contains a wide range of trips
recorded on various metropolitan roads by various sorts of cars and drivers.
SPMD is a comprehensive data collection that reflects a wide variety of
driver/vehicle actions and movements. The SPMD dataset provides various in-car
data logged through CAN, such as longitudinal velocity and acceleration, yaw
rate, steering angle, turning signal status, etc., as well as vehicle GPS data
for the duration of the complete trip duration. To encourage the selection of
a diverse set of trajectories, all the trips from the SPMD dataset are first
ranked based on the experimental criteria and then 50 of the richest and most
diverse trips are chosen. All trips are ranked on the merits of trip duration,
number of successful lane changes, number of aborted lane changes, number of
times a vehicle used a turning signal in a trip, number of times a vehicle
stopped in a trip, and standard deviation of vehicles’ yaw rate, steering
angle, GPS heading and longitudinal acceleration. These 50 richest trips,
which include more than $300,000$ sample points, are plotted, and visually
inspected to confirm the proper selection.

The logs from the datasets (SUMO, 100Car, and SPMD) are used to recreate
different communication scenarios. The logs are transferred to an OBU,
replayed, and transmitted over the air by using the RVE that we have developed
jointly in collaboration with industry partners for this purpose [9]. By
leveraging hardware-in-the-loop, the RVE allows the joint study of CVS
applications and their underlying communication system in real-time. As such,
it serves the important purpose of validating the designed architecture.

###  V-B Prediction Baseline Models

In order to evaluate the performance of the Hybrid GP prediction and
demonstrate the capabilities of the architecture, different prediction schemes
are implemented. As a baseline, we compare BSM-dependent (No estimation),
prediction with Constant Speed (CS), Constant Acceleration (CA), Kalman Filter
(KF), Auto Encoder (AE), Long short-term memory (LSTM), and the proposed
hybrid GP-based scheme (HGP). In the next section, we describe the tested
baselines.

For AE, LSTM and GP prediction, the $(x,y)$ predictions can be obtained
following two different approach, i.e, direct (D) or indirect (I). In the
direct approach, $(x,y)$ are treated as two independent time series, and the
history of $(x,y)$ is used to learn models for $(x,y)$, producing direct
predictions of futures $(x,y)$. Differently, in an indirect approach, the
heading and longitudinal speed of the vehicle are treated as two independent
time series and the history of heading and longitudinal speed are used to
learn models to forecast the vehicles’ speed and heading. The predicted
heading and speed are used to indirectly predict the future position $(x,y)$.
In our experiments, when we refer to AE, LSTM, and HGP, the indirect approach
is used. When the direct approach is used we a referred to as AE-D, LSTM-D,
and HGP-D.

####  V-B1 Kinematic Models

The BSM-dependent design (i.e. no prediction) keeps the last received position
acquired from the BSM of that RV until the HV receives a new BSM from that RV.
Although it is simple to implement and reduce the computation complexity of
the system, it is not a realistic assumption and can lead to huge $PTE$ and
the consequent drastic reduction in the performance of safety applications.
The time update equation is $X(t+1)=X(t)$, where $X(t)$ is the position of
vehicle at time t.

In the CS scheme, we assume that the vehicles’ speed will remain unchanged
during each inter-packet gap (IPG). The algorithm uses the speed from RV’s
last received BSM until it receives the next packet. This assumption is
reasonable when the IPG is small and the HV is receiving BSMs at regular
intervals. The time update equations are as below.

$\displaystyle X(n)=\begin{bmatrix}x(n)\\\ s(n)\\\ a(n)\\\ \end{bmatrix}$ (13)
$\displaystyle X(n+1)=\begin{bmatrix}1&T_{s}&0\\\ 0&1&0\\\ 0&0&0\\\
\end{bmatrix}X(n)$ (14)

Where $X(t)$ is the vector of vehicle states at time t and $T_{s}$ is the
sampling time. This experiment assumes that $T_{s}=100ms$, which is the
standard message transmission rate according to the IEEE J2735 standard. In an
event of the reception of a new packet, the measurement update takes place.

The CA scheme is similar to the former approach, except that we assume
acceleration to remain constant during the IPG duration. The time update
mechanism follows the equation below:

$\displaystyle X(n+1)=\begin{bmatrix}1&T_{s}&\frac{T_{s}^{2}}{2}\\\
0&1&T_{s}\\\ 0&0&1\\\ \end{bmatrix}X(n)$ (15)

####  V-B2 Kalman Filter Model

The KF is essentially a set of mathematical equations that implement a
predictor-corrector type estimator that is optimal in the sense that it
minimizes the estimated error covariance when some presumed conditions are
met. Rarely do the conditions necessary for optimality actually exist, and yet
the filter works well for many applications. The KF addresses the general
problem of trying to estimate the state of a discrete-time controlled process,
it assumes that the state at time $k$ evolves from state $k-1$ according to
the state update equation as,

$X_{k}=AX_{k-1}+Bu_{k-1}+w_{k-1}$ (16)

And at time $k$ a measurement(observation) is made according to,

$y_{k}=H_{k}X_{k}+v_{k}$ (17)

Where $X$ and $y$ represent the state vector and measurement vector; $w_{k}$
and $v_{k}$ represent the process and measurement noise respectively. The
process and measurement noises are assumed to be independent white noises with
normal probability distribution with covariances $Q_{k}$ and $R_{k}$
respectively. The matrix $A$ ($nxn$) is the state transition matrix which is
applied to the previous state $X_{k-1}$, the matrix $B$ ($nx1$) is the control
input matrix which is applied to the control vector $u_{k}$, and the matrix
$H$ ($mxx$) is the observation model, which maps the state space into the
observed space, it relates the state $X_{k}$ to the measurement $y_{k}$.

The KF computation is divided into two parts: time update and measurement
update. The time update is responsible for projecting forward the current
state and error covariance estimates in time to obtain a priori estimates for
the next time step. The measurement update is responsible for incorporating a
new measurement into a priori estimate to obtain an improved posteriori
estimate. The final prediction algorithm resembles that of a predictor-
corrector algorithm. The time and measurement equations are as below:

Time update equations:

$X_{k}=AX_{k-1}+Bu_{k-1}+w_{k-1}$ (18) $P_{k}=AP_{k-1}A^{T}+Q$ (19)

Measurement update:

$K_{k}=P_{k}H^{T}(HP_{k}H^{T}+R)^{-1}$ (20) $X_{k}=X_{k}+K_{k}(y_{k}-HX_{k})$
(21) $P_{k}=(I-K_{k}H)P_{k}$ (22)

Particularly, as a baseline, we consider a Kalman Filter with a constant
acceleration dynamic, and describe the state transition equations as follows,

$\displaystyle X_{k}=\begin{bmatrix}x_{k}\\\ s_{k}\\\ a_{k}\end{bmatrix}$ (23)
$\displaystyle\hat{X}_{k}=\begin{bmatrix}1&T_{s}&\frac{T_{s}^{2}}{2}\\\
0&1&T_{s}\\\ 0&0&1\end{bmatrix}\hat{X}_{k-1}+w_{k-1}$ (24) $\displaystyle
y_{k}=\begin{bmatrix}1&1&1\\\ \end{bmatrix}\hat{X}_{k}+v_{k}$ (25)

When a new packet is received, both measurement and time updates are running,
and when a packet loss happens the prediction is done based on only the time
update. Based on a statistical analysis of the dataset the variance of the
measurement noise, the R matrix, is specified as,

$R=\begin{bmatrix}10&0&0\\\ 0&1&0\\\ 0&0&0.5\end{bmatrix}$

####  V-B3 Autoencoder Model

Neural network architectures such as AE use unsupervised learning or self-
supervised learning. The AE architecture can vary, but in general it consists
of an encoder, which reduces the input’s dimension, it maps the input
$\boldsymbol{x}$ to a latent feature representation $\boldsymbol{z}$ denoted
by $z=f_{w_{e}}(x)$, and a decoder, which attempts to recreate the original
input from the lower-dimensional representation, using the latent
representation $\boldsymbol{z}$ to obtain a reconstruction $\boldsymbol{y}$ of
the input $\boldsymbol{x}$ , denoted by $y=f_{w_{d}}(z)$. The goal of training
these algorithms is to be able to recreate the original input with the fewest
information losses possible, by minimizing the reconstruction error between
$\boldsymbol{x}$ and $\boldsymbol{y}$. As a generative model, an AE can be
trained for a prediction task using a similar methodology, by training the AE
to reproduce the state at time $t+1$ given the state at time $t$. In our
baseline, we design an encoder that consists of 3 fully connected (FC) layers
with 256, 128, and 64 units respectively. The encoder takes as input the time-
series observations and outputs the internal representation that is passed to
the decoder, the decoder consists of a symmetric version of the encoder. For
multiple-step predictions, the predictions are used in a chain to generate
subsequent predictions.

####  V-B4 LSTM Model

An LSTM is a kind of Recurrent Neural Network. LSTMs are particularly helpful
when using time series or sequential data due to their capacity to learn long-
term dependencies. The LSTM network is able to learn a function that maps the
time series history to feature predictions. For the LSTM baseline in this
paper, we use a model similar to [41] with two layers of 256 LSTM cells, ReLu
activation, and dropout. The model is fit using Adam and MSE loss function. It
is mainly a regression problem solved by the LSTM model with the goal to
predict future positions $(x,y)$ for the neighboring vehicles.

###  V-C Controlled Variables

We conducted a set of experiments to measure the performance of our
architecture and investigate the existence of a reduced kernel bank size that
is capable of accurately modeling driving behaviors. Then, the feasibility of
the real-time implementation of the proposed system is also analyzed in terms
of computation time. Finally, the overall system performance is evaluated
against the previous baselines. For that purpose, we choose appropriate
controlled variables, i.e, Training Window ($\mathbf{TW}$), which is defined
as the number of the latest equally spaced received samples (e.g., most recent
history of heading and speed time series) utilized as the training data to
generate each model, Cluster Size ($\mathbf{C_{size}}$), Packet Error Rate
($\mathbf{PER}$) and Transmission Rate ($\mathbf{Rate}$). We investigate how
the ${TW}$ and ${C_{size}}$ affect the model performance. Finally, In order to
study the impact of communication losses, we use ${PER}$ and ${Rate}$ and
measure the performance of the different prediction schemes and safety
applications under different ${PER}$ and ${Rate}$. When measuring the impact
of ${PER}$, the ${Rate}$ is fixed to 10Hz and when measuring the impact of
${Rate}$, the ${PER}$ is fixed to $0\%$.

###  V-D Performance Metrics

To assess the influence of the controlled variables we select suitable
performance metrics, i.e, Model Persistency ($\mathbf{MP}$), computation time
($\mathbf{t_{c}}$), Warning Accuracy ($\mathbf{WA}$) and Position Tracking
Error ($\mathbf{PTE}$). In order to gauge the impact of $TW$, we study how
$MP$ and $t_{c}$ change with $TW$ size. We evaluate the influence of the
$C_{size}$ by measuring the performance of the system in terms of $MP$ for
different $C_{size}$. Finally, to measure the overall system performance, we
select two performance metrics, $WA$ and $PTE$, that while correlated, provide
different insights into the advantages of our approach. $PTE$ is more related
to the quantitative estimation performance and $WA$ is related to the
performance of the safety applications. $WA$ and $PTE$ have been used
successfully as good indicators of the performance of the CVS applications
[11, 12].

Position Tracking Error ($PTE$). In all the experiments, $PTE$ describes the
95th percentiles of the error in tracking the position of RV. While it
measures the quantitative performance of the prediction method, it isn’t the
best indicator of the application performance. For this purpose, we use $WA$,
which indicates how well the safety application is performing.

Warning Accuracy ($WA$). For the purpose of validation in this paper, we focus
on the FCW application, and we study the impact of communication losses on FCW
accuracy($FCW_{accuracy}$). While there are many different implementations of
FCW, we use the CAMP Linear FCW algorithm [17, 24]. The used CAMP Linear FCW
algorithm is briefly described in the next section. In that sense, the
warnings are based on the CAMP Linear algorithm and ground truth data is the
output of the CAMP Linear algorithm when PER=0 $\%$ and $Rate$=10Hz. For
$FCW_{accuracy}$, we compare the ground truth and FCW application output for
different values of $PER$ and $Rates$ while different prediction schemes are
used. More precisely, $FCW_{accuracy}$ is the ratio of true negatives plus
true positives across all execution instances and is defined as follows:

$FCW_{accuracy}=\frac{T_{p}+T_{n}}{F_{p}+F_{n}+T_{p}+T_{n}}$ (26)

where $T_{p}$ (True positive) and $T_{n}$ (True negative) are the numbers of
correctly predicted hazards and safe indications; $F_{n}$ (False negative) is
the number of incorrectly predicted safe indications (misidentified actual
hazards), and $F_{p}$ (False positive) is the number of incorrectly predicted
hazard indications (misidentified actual safe situations).

We compare the $PTE$ and $FCW_{accuracy}$ (Performance metrics) of the CA-TC-
based architecture using different prediction schemes, while $PER$ and $Rate$
(Controlled variables) change from 0 to 95 $\%$ and from 10Hz to 1Hz
respectively.

Although our architecture is agnostic to the V2X communication technology, in
our setup, we used the V2X device (DSRC-based) DENSO WSU-5900A and the RVE to
evaluate the performance of the system [9]. We show the results of the
analyses for the SUMO, 100-Car, and SPMD datasets by using the mobility logs
together with the RVE. The RVE can imitate network congestion scenarios with
different $PER$ or $Rates$.

###  V-E Cooperative Forward Collision Warning (FCW)

In this section, we briefly present the FCW CAMP Linear algorithm
implementation used in the experiments [17, 24]. FCW algorithms are intended
to notify the driver when the present movement pattern of cars indicates that
a collision is imminent. The warning should be timed such that it takes into
account the driver’s reaction time, but not so quickly that it causes false
alarms. FCW is run repeatedly (e.g., 100 ms intervals) and at each interval is
evaluated if a threat exists. The FCW CAMP Linear algorithm evaluates a
“warning range” ($r_{w}$) using RV and HV data. A warning is issued if the
distance between the HV and the RV is less than the $r_{w}$ [17, 24, 11].

CAMP Linear FCW algorithm. In the CAMP Linear FCW algorithm, the $r_{w}$ is
computed as:

$\displaystyle
r_{w}=\mathrm{BOR}+(\mathrm{v}_{\mathrm{HV}}-\mathrm{v}_{\mathrm{RV}})t_{d}+\frac{1}{2}*(\mathrm{a}_{\mathrm{HV}}-\mathrm{a}_{\mathrm{RV}})t_{d}^{2}$
(27)

Where $r_{w}$ is the warning range, $BOR$ is the Brake Onset Range;
$v_{HV},v_{RV},a_{HV},a_{RV}$ are the speed and acceleration of HV and RV and
$t_{d}$ is the driver and brake system reaction delay. The $BOR$ is computed
for three different scenarios, for stationary or moving RV, as follows.

  * •

Case 1: RV stationary.

  * •

Case 2: RV moving at the beginning and end of the scenario.

  * •

Case 3: RV moving at the beginning but stopping at the end of the scenario.

$\displaystyle\textbf{Case 1:
}\mathrm{BOR}=\frac{\mathrm{v}_{\mathrm{HVp}}^{2}}{-2*a_{req}}$ (28)
$\displaystyle\textbf{Case 2:
}\mathrm{BOR}=\frac{(\mathrm{v}_{\mathrm{HVp}}-\mathrm{v}_{\mathrm{RVp}})^{2}}{-2*(a_{req}-a_{RV})}$
$\displaystyle\textbf{Case 3:
}\mathrm{BOR}=\frac{\mathrm{v}_{\mathrm{HVp}}^{2}}{-2*a_{req}}-\frac{\mathrm{v}_{\mathrm{RVp}}^{2}}{-2*a_{RV}}$

in witch, $v_{HVp}$ and $v_{RVp}$ are the predicted velocity of HV and RV,
i.e, $v_{HVp}=v_{HV}+a_{HV}t_{d}$ and $v_{RVp}=v_{RV}+a_{RV}t_{d}$. $a_{RV}$
is the acceleration (deceleration) of RV and $a_{req}$ is the deceleration
that is required at the HV for avoiding a crash and is modeled in [24] using
real data.

###  V-F Hypotheses

Based on the defined controlled variables and performance metrics we examine
the following hypotheses:

  * •

H1. _An suitable $\mathbf{TW}$ needs to be selected in order to allow an
acceptable model persistency ($\mathbf{MP}$) and meet the real-time
requirements of the CVS applications measured by computation time
($\mathbf{t_{c}}$)_.

  * •

H2. _We anticipate the existence of a limited size of kernel banks (
$\mathbf{C_{size}}$) which is capable of predicting driving behavior while
keeping an acceptable $\mathbf{MP}$ horizon. The kernel bank can be learned
offline and still be useful during on-the-fly forecasting_.

  * •

H3. _The higher the $\mathbf{PER}$ or the lower the $\mathbf{Rate}$ of
communication, the greater the impact and benefit of using our CA-TC system,
measured by $\mathbf{PTE}$ and $\mathbf{FCW_{accuracy}}$. Thus, we expect a
higher $\mathbf{FCW_{accuracy}}$ when using our HGP prediction scheme when
compared to the baselines_.

The hypotheses are investigated through the experiments in the following
sections.

###  V-G GP Prediction Model Hyper Parameter Tuning and Implementation Details

As described, in the Hybrid GP modeling scheme instead of working directly on
the X-ENU and Y-ENU time series, the heading, and speed of the vehicle are
treated as two independent time series. The GP models, which are learned from
speed and heading histories are used to forecast their future value, and then
the predicted values of these two variables are used to predict the position
in ENU coordinate system. In this section, we present the study of $TW$ and
$C_{size}$ and obtain suitable values for those variables to be used for the
HGP prediction system.

####  V-G1 Computational Complexity

To study the hypothesis H1 we investigate the effect $TW$ in computation time
and experimentally compute the computation time of our algorithm for $TW$ size
from 1 to 1000. The HGP training (new kernel generation) and forecasting
complexity is a function of the training window. In our framework, this factor
appears whenever we need to generate a new kernel while we are running the
system in real-time. The order of this complexity is heavily dependent on the
implementation strategy of GP kernel training algorithms. In the simulations,
we have used the ”Gaussian Process for Machine Learning” MATLAB toolbox,
provided by Cambridge University.

Figure 9 shows the result of the experimental estimation, in the right plot
the blue curve shows the actual spent time and the red one is the quadratic
approximations, as shown for a large number of training windows, the
complexity of our algorithm is quadratic ($\mathcal{O}(n^{2})$). The Root Mean
Square Error (RMSE) values for the ﬁtted curve are also presented in the
ﬁgure. In the left plot, a plot of the computation time for training windows
size up to 60 is presented. As shown we can train up to $TW$ size 40 and still
be under the 100ms threshold required for safety applications. The simulations
have been executed on a PC with an Intel Core i7 6700 3.4 GHz CPU with 4 cores
and 32 GB of RAM. The results in Figure 9 validate the H1 and allows us to
select a suitable $TW$ that meets the real-time requirements of the CVS
applications. Additionally, we want to mention that our approach can benefit
from parallel computing during the bank kennel creation and during
forecasting, as the model predictions can be computed independently in
parallel, therefore the use of GPU or a CPU with additional cores will further
reduce the computation time. Based on our experiments and previous work [51],
we choose a $TW$ size of 30 time-steps (3s) and $PTE_{th}=50cm$ for the
training, model generation, and forecasting. The selected training parameters
consider an acceptable path history while also meeting the computational
requirement of the safety applications.

Figure 9: Empirical time complexity of GP kernel training as a function of
training window size.

####  V-G2 Model Generation

To examine the hypothesis H2, we obtain the kernel bank and experiment with
different cluster sizes. Following the HGP training and model generation
schemes in Algorithm 1, we obtained 350 models, and reduce the kernel bank by
clustering the kernels. For each $C_{size}$ we compute the model persistency
($MP$) when predicting future trajectories while keeping the error within the
defined $PTE_{th}$. Figure 10 shows how $MP$ changes for different $C_{size}$.
The results support the hypothesis H2 and confirm the existence of a reduced-
size kernel bank capable of predicting driving behavior patterns. From those
results, we choose a cluster size of 16 that has a reduced size while keeping
an adequate $MP$, balancing complexity and performance. Therefore the models
from the original kernel bank are clustered in the 16 most distinctive models
to form the reduced kernel bank.

Figure 10: Model persistency for different cluster sizes.

During on-the-fly mode, the most recent information is used to learn the
parameters of the Gaussian process’s Kernel. In the experiments, we use BSMs
received during the last $TW$ to learn the new model. This model will be used
to forecast the state of the vehicle until new information is received or the
forecasted speed or acceleration of the vehicle triggers the model switching
based on the algorithm described in Figure 7.

Finally, to test the capability of the reduced kernel bank, we apply the
trained kernel bank to a new set of 50 randomly selected trajectories.
Therefore, they could be considered as a set of generic driving data. The
ratio of generating new models, which has been represented in Figure 11, has
an increasing trend at the beginning of test time and raises to around 10% at
its maximum level, and then it converges around 4%. This observation shows
that the learned kernel bank is meaningful during the testing phase (on-the-
fly forecasting), even for unforeseen data, further verifying the hypothesis
H2.

Figure 11: Assessment of trained kernel bank performance in terms of required
new kernel generation rate for an unforeseen set of trajectories.

###  V-H Overall system performance

Finally, to investigate the H3, we use the parameters obtained for our HGP
prediction system, i.e. $TW=3s$, $PTE_{th}=50cm$, and $C_{size}=16$, and study
the impact of communication uncertainties on the performance of a selected
safety application with proposed CA-TC and compare it with the baselines
across multiple datasets.

Experiments with SUMO simulated data We first use the data generated from the
SUMO simulator and compare the baseline models against our approach. Figure 12
demonstrates $FCW_{accuracy}$ and $PTE$ for different values of $PER$ and
$Rates$ using the SUMO data. As expected, the $FCW_{accuracy}$ of the BSM-
dependent baseline is the lowest, and the $PTE$ is the highest. For the SUMO
simulated data, the KF model and GP perform similarly which could be a
consequence of the simplistic car-following model utilized in SUMO.

Figure 12: $PTE$ and $FCW_{accuracy}$ vs $PER$ and $Rate$ for the SUMO
dataset.

Experiments with the 100-Car Dataset We also evaluated the performances using
the 100-Car dataset. Figure 13 shows how the $PTE$ and $FCW_{accuracy}$ change
for the different $PER$ and $Rate$ values. Similarly, the BSM-dependent model
shows the highest $PTE$ and lowest $FCW_{accuracy}$. HGP has a similar
performance to KF or CA models. We speculate that the reason for having
similar performances is the car-following model used to create the HV states
in the 100-Car dataset.

Figure 13: PTE and $FCW_{accuracy}$ vs $PER$ and $Rate$ for the 100-Car
dataset

Experiments with the SPMD dataset Finally, using SPMD dataset [52], we compare
the proposed HGP method with the baselines CA, KF, AE, and LSTM. We want to
mention that the AE and LSTM models require a larger amount of data and are
not suitable for the SUMO and 100-Car datasets. Figure 14 shows the
performance of all the models in the SPMD dataset. As expected, the KF model
performed similarly to the CA model due to the chosen kinematic equation to
model the vehicle motion in the KF method. However, the slight over-
performance of KF is due to its ability to reduce measurement noise, in
particular, the AE model shows poor performance for vehicle trajectory
prediction. As $PER$ increases over $60\%$, the HGP approach shows its
superior prediction, as it can forecast for longer prediction horizons, and
capture both the dynamics of the vehicle and the driver’s behavior. The
difference in position tracking is notable as $PER$ reaches 90$\%$. The CA and
KF approaches had $60\%$ larger $PTE$ compared to HGP. The LSTM model has a
similar performance to our HPG model however LSTM models are data-hungry
models and are not suitable for online on-the-fly training while new data is
available, making it difficult to extract new knowledge from new situations,
differently from the HGP approach that can learn new models on the fly, with a
limited amount of data. The results demonstrate the superiority of our HGP
approach and the benefits as $PER$ increases, confirming the hypothesis H3.

Figure 14: Comparison of CA, KF, AE, LSTM and our HGP approach performance in
the SPMD dataset, measured by $FCW_{accuracy}$ and $PTE$ Figure 15: Number of
time which $PTE$ threshold has been exceeded vs $PER$ for different values of
threshold ($PTE_{th}$)

Following in Table I we present the tabular results for the performance of the
baselines and our HGP approach in the SPMD dataset, measured by the $PTE$ at
different $PER$ values. For completeness, we include both direct (AE-D,
LSTM-D, HGP-D) and indirect (AE, LSTM, HGP) approaches. The baseline methods
are CA, KF, AE-D, LSTM-D, HGP-D, AE, LSTM, and HGP. The results from Table I
show the improved performance when using the HGP in terms of $PTE$, which are
more noticeable as communication losses increase measured by $PER$, verifying
our H3.

TABLE I: Tabular results for the performance of the baselines and our HGP approach in the SPMD dataset, measured by the $PTE[m]$ at different $PER[\%]$ values. For both direct (AE-D, LSTM-D, HGP-D) and indirect (AE, LSTM, HGP) approaches. | PER [%]
---|---
Methods | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 95
CA | 0.000 | 0.047 | 0.077 | 0.091 | 0.110 | 0.303 | 0.605 | 1.059 | 1.392 | 3.026 | 5.296
KF | 0.000 | 0.059 | 0.093 | 0.098 | 0.126 | 0.293 | 0.514 | 0.895 | 1.189 | 2.778 | 4.687
AE-D | 0.000 | 0.127 | 0.322 | 0.541 | 0.774 | 1.154 | 1.813 | 2.200 | 2.640 | 5.926 | 9.558
LSTM-D | 0.000 | 0.110 | 0.292 | 0.356 | 0.559 | 0.683 | 0.865 | 1.417 | 1.781 | 4.231 | 6.918
HGP-D | 0.000 | 0.108 | 0.299 | 0.322 | 0.413 | 0.546 | 0.824 | 1.106 | 1.599 | 4.167 | 6.473
AE | 0.000 | 0.065 | 0.103 | 0.108 | 0.138 | 0.523 | 0.866 | 1.185 | 1.508 | 4.156 | 6.156
LSTM | 0.000 | 0.051 | 0.085 | 0.100 | 0.122 | 0.243 | 0.366 | 0.595 | 0.731 | 2.329 | 3.825
HGP | 0.000 | 0.047 | 0.076 | 0.089 | 0.111 | 0.216 | 0.281 | 0.476 | 0.649 | 2.163 | 3.245

We perform a sensitivity analysis and compare the performance of CA, KF, AE,
and LSTM against the proposed method based on $PTE$ and the number of times in
which a specific $PTE$ threshold ($PTE_{th}$) has been exceeded. We
investigate different $PTE_{th}$s, i.e. from 0.2 m to 1.6m. In general,
vehicular safety applications are sensitive to position-tracking error in a
hard-thresholding manner. The value of this threshold is dependent on the
application and the driving scenario. The position error exceeding the
$PTE_{th}$ can cause false positives and negatives in the safety applications
and directly influence their performances.

In this test, in the case of a packet loss, the HV performs prediction and if
the prediction error is larger than the threshold, it is counted as an
inadmissible prediction, possibly causing a hazardous situation (PTE higher
that the admissible threshold $PTE_{th}$). In Figure 15, the total number of
inadmissible predictions during the trajectories for different $PTE_{th}$s are
plotted. For example, the plot in the top left shows the number of times that
the $PTE$ is bigger than $PTE_{th}=0.2m$, which gives a sense of how many
times an application that requires a $PTE$ less than $PTE_{th}=0.2m$ will
fail. The bottom right shows the number of times that the $PTE$ is larger than
$PTE_{th}=1.6m$ for different PERs. it shows that for a low $PER$ the $PTE$
never exceeds 1.6m, and for higher values of $PER$, the HGP performs better in
all the cases. For instance, at $PER=90\%$ and $PTE_{th}=1.6m$, CA fails more
than $200\%$ times compared to the proposed HGP, similarly, the proposed HGP
outperforms the other baselines. These results confirm the notable superiority
of our approach as $PER$ increases and further validate hypothesis H3.

##  VI Conclusion and Future Work

In this study, we proposed a Context-Aware Target Classification module with a
hybrid Gaussian Process regression as the prediction module to improve the
robustness and flexibility of CVS systems. Unlike previous works that are
strictly information message dependent, the proposed CA-TC allows safety
applications to work based on the context-aware map rather than individual
information messages, alleviating the effect of information loss on the
performance of safety applications. The HGP relies on a reduced-size bank of
driving models learned offline from real data to perform forecasting on-the-
fly, while also allowing the online adding of new driving models to the bank,
accounting for unforeseen driving behaviors. We studied the impact of
communication uncertainties, in terms of $PER$ and $Rate$, on the performance
of the safety applications using the proposed CA-TC and compare it to the
baseline. The performance is measured both in terms of $PTE$ and $WA$. The
architecture is validated by simulation and real driving scenarios using three
different datasets and a RV emulator based on a DSRC WSU. A notable
performance improvement is observed using the proposed framework against
traditional approaches.

Limitations and Future Work.  While we proposed a novel prediction method to
be used in our architecture, the CA-TC can be further improved by the
integration of other prediction methods. Data fusion and cooperative
perceptions are among the possible methods to be incorporated with the
flexible CA-TC architecture to further enhance the safety application’s
performance. In future works, we will examine the integration of error-driven
model-based communication into our system. We plan to investigate the
existence of natural and meaningful driving patterns for long-term maneuver
and driver’s intention prediction. We also plan to experiment with more CVS
applications and other baselines. In a parallel research direction, we plan to
extend our work to non-cooperative applications and test our predictive model
in different tasks from commonly used driving datasets such as KITTI and
Waymo, among others.

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