# Augmented Labeled Random Finite Sets and Its Application to Group Target
Tracking

Chaoqun Yang<EMAIL_ADDRESS>Mengdie Xu<EMAIL_ADDRESS>Xiaowei Liang
<EMAIL_ADDRESS>Heng Zhang<EMAIL_ADDRESS>Xianghui Cao<EMAIL_ADDRESS>School of Automation, Southeast University, Nanjing 210096, China  School of
Computer Engineering, Jiangsu Ocean University, Lianyungang 222000, China

###### Abstract

This paper addresses the problem of group target tracking (GTT), wherein
multiple closely spaced targets within a group pose a coordinated motion. To
improve the tracking performance, the labeled random finite sets (LRFSs)
theory is adopted, and this paper develops a new kind of LRFSs, i.e.,
augmented LRFSs, which introduces group information into the definition of
LRFSs. Specifically, for each element in an LRFS, the kinetic states, track
label, and the corresponding group information of its represented target are
incorporated. Furthermore, by means of the labeled multi-Bernoulli (LMB)
filter with the proposed augmented LRFSs, the group structure is iteratively
propagated and updated during the tracking process, which achieves the
simultaneously estimation of the kinetic states, track label, and the
corresponding group information of multiple group targets, and further
improves the GTT tracking performance. Finally, simulation experiments are
provided, which well demonstrates the effectiveness of the labeled multi-
Bernoulli filter with the proposed augmented LRFSs for GTT tracking.

###### keywords:

Random finite sets; Group target tracking; Labeled multi-Bernoulli filter;
Resolvable group target; State estimiation.

††thanks: This paper was not presented at any IFAC meeting. Corresponding
author: Xianghui Cao.

, , , ,

##  1 Introduction

Nowadays, the issue of group target tracking (GTT) has attracted increasing
interest due to its wide applications over civil and military fields, such as
drone swarms (Guerlin et al., 2020), vehicle formations (Wang and Ayalew,
2018), and group of robots (Senanayake et al., 2016), etc. In these
applications, multiple targets such as drones or vehicles, are usually closely
spaced and evolve in a coordinated manner with the same or similar dynamics
models, forming one or many groups. Moreover, these groups can split, merge,
die and rebuild (Zhang et al., 2023), performing complex dynamics of group
structure and frequency interaction with the targets within or without groups.
Thus, the GTT issue not only suffers from the difficulties encountered by the
issue of multi-target tracking (MTT) including unknown number of targets,
missed detections, clutters, and measurement origin uncertainty, but also
encounters the indeterminacy of group structure cuased by group formation,
death, merging or splitting (Zhang et al., 2022).

Generally speaking, most of the current GTT methods can be devided into two
categories: Extended target tracking (ETT) based methods and resolvable group
target tracking (RGTT) based methods. The first takes each group as an
extended target, and pays more attention to the estimation of each group’s
center and shape, while the other focuses on the estimation of both the states
of multiple targets and group structures (Mihaylova et al., 2014; Zhang et
al., 2023).

Aimming at the estimation of the time-varying number of targets and groups,
multiple targets, as well as group structure, in this paper, the RGTT methods
are considered. Depending on whether random finite set (RFS) theory is used or
not, the main RGTT methods can be devided into two kinds, data association
(DP) based methods and RFS-based methods.

For the DP-based GTT methods, Zhang et al considered the GTT problem within
the multiple hypothesis tracking (MHT) framework, and proposed a MHT-based
method to efficiently deal with the problems of group association, group
partition and the state estimation of the individual targets within groups
(Zhang et al., 2023). Gning et al combined evolving graph network, sequential
Monte Carlo method, and joint probabilistic data association methods to cope
with the GTT issue (Gning et al., 2022). Zhang et al utilized the belief
propagation method, also known as the message passing method, and addressed a
belief propagation method to jointly estimate both target states and group
structure (Zhang et al., 2022).

Due to the advantages of coping with the uncertainty of both the number and
states of multiple targets, RFS-based RGTT methods, such as labeled multi-
Bernoulli (LMB) filter, multi-Bernoulli filter, Poisson multi-Bernoulli
mixture (PMBM) filter, have received much more attention than the DP-based
methods. For example, Li et al proposed the leader follower labeled multi-
Bernoulli filter which combines the LMB filter with leader follower (LF) model
(Li, Li and He, 2023). Thereinto, the LF model was used to formulate the
dynamics of group structure, and the LMB filter was regarded as the filtering
tool. Liu et al classified the GTT issue into two steps: target state
estimation and group state including group structure, group size, etc,
estimation, of which the former utilized the multi-Bernoulli filter, and the
latter was based on graph theory (Liu et al, 2018). Similarly, Zhou et al also
adopted graph theory to describe group structure, and used virtual leader
follower model to predict the evolution of groups, then, the PMBM filter is
used to track multiple resolvable group targets (Zhou et al, 2024). Li et al
proposed the single target state transtition function based multi-Bernoulli
filter and studied its application to the sensor control problem in the RGTT
scenarios (Li, Li and He, 2022).

For the RFS-based methods, although the above existing remarkable works have
presented diverse methods on how to integrate the group structure into the
RFS-based filters to furthe improve the tracking performance, it is worth
noting that in the processing of these methods, the multi-target states and
group structures are still treated independently. They are processed
separately rather than considered as a unified whole. Specifically, these
methods formulate the multi-target state as an RFS without incorporating group
structure information into the RFS. Consequently, the estimation of multi-
target states and the estimation of group structures are two independent
processes. In fact, the estimation of group structures often serves as a post-
processing step after target state estimation. Therefore, a natural question
arises: Can the RFS theory naturally integrate group structure information and
achieve a holistic estimation of both target states and group structures? To
the best of our knowledge, there is currently no work in this direction.

Inspired by the advance of labeled RFS (LRFS), which is a special kind of RFS
that jointly integrates target states and target labels, in this paper, we
propose a new kind of LRFS, named augmented labeled LRFS, that simultaneously
integrates target states, target labels, and group structure information.
Subsequently, the statistical characteristic of the proposed augmented LRFS is
analyzed in details. Furthermore, based on the proposed augmented LRFS, we
formulate the GTT problem within the framework of the augmented LRFS based
multi-target Bayes filter, and propose the LMB filter with the augmented LRFS
to achieve the accurate tracking of group targets. By means of the proposed
augmented LRFS, both the states and labels of multiple targets, along with
group structures are simultaneously estimated. In summary, the main
contributions of this paper can be listed as follows.

  * •

We propose a new kind of RFS, named augmented LRFSs, which integrates group
information into the existing LRFSs, and we further analyze the statistical
characteristic of the proposed augmented LRFS in details.

  * •

We propose a new formulation of the GTT issue based on the proposed augmented
LRFSs, in which the dynamics of both multiple targets and groups are
integrated into the augmented LRFS based multi-target Bayes filter.

  * •

We propose a new method to cope with the GTT issue, i.e., the LMB filter with
the augmented LRFSs, and prove its feasibility to solve the GTT issue. As far
as we know, it is the first RFS-based method that naturally integrates group
structure information and achieves a holistic estimation of both target states
and group structures.

The rest of this paper is organized as follows: Section 2 presents the
proposed augmented LRFSs theory. Section 3 states the considered GTT problem,
and presents the system model. Section 4 presents the LMB filter with the
augmented LRFSs that cope with the GTT problem, Numerical experiments are
provided in section 5. Finally, section 6 concludes this paper.

##  2 Proposed Augmented LRFSs

This section will present the definition and statistical properties of the
proposed augmented labeled random finite sets, after a brief review on the
concept of both random finite sets (RFSs) and labeled random finite sets
(LRFSs).

###  2.1 Notation

In the rest of this paper, small English letters (e.g., $x$) are used to
denote random variables. Capital English letters (e.g., $X$) are used to
denote RFSs (see section 2.2). Bold face English letters (e.g., $\mathbf{X}$)
are used to represent LRFSs (see section 2.3). Capital English letters written
with scans serif font (e.g., $\mathsf{X}$) are used to represent augmented
LRFSs (see section 2.4). Additionally, spaces are denoted by blackboard bold
English letter, for example, $\mathbb{X}$ denotes the state space,
$\mathbb{L}$ denotes the label space, $\mathbb{N}$ denotes the integer space,
$\mathbb{Z}$ denotes the natural number space.

The following operators are defined. $||\cdot||_{2}$ denotes the 2-norm,
$<f,g>=\int f(x)g(x)dx$ denotes the inner product. For an RFS $X$, we define
the following exponential operator,

$h^{X}=\prod_{x\in X}h(x).$ (1)

In addition, similiar to (Vo, Vo and Phung, 2014), we also use

$\displaystyle\delta_{Y}(X)=\left\\{\begin{array}[]{ll}1,&\mbox{if $X=Y$},\\\
0,&\mbox{otherwise},\end{array}\right.$ (4)

and

$\displaystyle 1_{Y}(X)=\left\\{\begin{array}[]{ll}1,&\mbox{if $X\subseteq
Y$},\\\ 0,&\mbox{otherwise},\end{array}\right.$ (7)

to denote the Kronecker delta function and the inclusion fuction,
respectively.

###  2.2  RFSs

######  Definition 1.

An RFS like $X=\\{x_{1},x_{2},\cdots,x_{n}\\}$, is essentially a finite-set-
value random variable, which poses the two following unique features (Yang et
al, 2019):

  * •

Each element $x_{i}\in X$ is random and unordered;

  * •

The cardinality (i.e., the number of elements) of $X$, $|X|=n$ is random.

According to Finite Set Statistics (FISST) (Mahler, 2011), the statistical
characteristics of an RFS $X$ can be fully captured by its corresponding FISST
probability density function (PDF) $\pi(X)$. Specially, we introduce the
following two types of RFSs that will be used in the subsequent sections.

####  2.2.1 Possion RFS

A Possion RFS is an RFS $X$ whose PDF follows (Vo and Vo, 2013)

$\pi(X)=e^{-<v,1>}v^{X},$ (8)

where $v$ is a given function (also called intensity function).

####  2.2.2 Bernoulli RFS

A Bernoulli RFS is an RFS $X$ whose PDF follows (Vo and Vo, 2013)

$\displaystyle\pi(X)=\left\\{\begin{array}[]{ll}1-r,&\mbox{$X=\emptyset$},\\\
r\cdot p(x),&\mbox{$X=\\{x\\}$},\end{array}\right.$ (11)

where $p$ and $r$ are a PDF and the existence probability of $X$,
respectively.

####  2.2.3 Multi-Bernoulli RFS

By taking the union of a certain number of mutually independent Bernoulli
RFSs, i.e., $X=\cup_{i=1}^{M}{X^{i}}$, we further construct a multi-Bernoulli
RFS, whose PDF follows (Reuter et al, 2014)

$\pi(\\{x_{1},\cdots,x_{n}\\})=\prod_{j=1}^{M}(1-r^{(j)})\sum_{i\leq
i_{1}\neq\cdots\neq i_{n}\leq
M}\prod_{j=1}^{n}\frac{r^{(i_{j})}p^{(i_{j})}(x_{j})}{1-r^{(i_{j})}}.$ (12)

For simplicity, we usually use $\pi=\\{(r^{(i)},p^{(i)})\\}_{i=1}^{M}$ to
abbreviate the above PDF.

###  2.3  LRFSs

By augmenting the state $x$ with a distinct label $l\in\mathbb{L}$, we have
$\mathbf{x}=(x,l)$. In the context of target tracking, specifically, $l$ can
be represented as $l=(k,i)$, where $k$ is the target’s birth time, and $i$ is
a unique identifier used to distinguish targets borning at the same time. In
the following context, unless otherwise stated, $l=(k,i)$ holds true. Further,
we can construct a new RFS
$\mathbf{X}=\\{\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{n}\\}$. If the
labels of all of the elements in $\mathbf{X}$ are distinct, we call the RFS
$\mathbf{X}$ as an LRFS. Then we obtain the following definition of LRFSs.

######  Definition 2.

An LRFS with state space $\mathbb{X}$ and label space $\mathbb{L}$, like
$\mathbf{X}=\\{\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{n}\\}$, is an
RFS on $\mathbb{X}\times\mathbb{L}$ such that each realization has distinct
labels (Vo, Vo and Phung, 2014).

####  2.3.1 Labeled Multi-Bernoulli RFS

A labeled multi-Bernoulli (LMB) RFS $\mathbf{X}$ with parameter set
$\\{(r^{(l)},p^{(l)}):l\in\Psi\\}$ is a multi-Bernoulli RFS augmented with
labels associated with the nonempty Bernoulli components. If we use
$\alpha(l)$ to denote the label of the corresponding state, where
$\alpha:\Psi\rightarrow\mathbb{L}$ is a 1-1mapping. the PDF of the LMB RFS
$\mathbf{X}$ can be denoted as (Vo, Vo and Phung, 2014)

$\pi(\mathbf{X})=\bigtriangleup(\mathbf{X})1_{\alpha(\Psi)}(\mathcal{L}(\mathbf{X}))[\Phi(\mathbf{X};\cdot)]^{\Psi}$
(13)

where $\mathcal{L}(\cdot)$ denotes the projection from
$\mathbb{X}\times\mathbb{L}$ to $\mathbb{L}$,

$\bigtriangleup({\cdot})=\delta_{|\cdot|}(\mathcal{L}(|\cdot|))$ (14)

is a distinct label indicator to ensure distinct labels, and

$\displaystyle\Phi(X;\cdot)=\left\\{\begin{array}[]{ll}1-r^{(l)},&\mbox{if
$\alpha(l)\notin\mathcal{L}(\mathbf{X})$},\\\ r^{(l)}p^{(l)}(x),&\mbox{if
$(x,\alpha(l))\in\mathbf{X}$}.\end{array}\right.$ (17)

To simplify notations, defining $\alpha$ as an identity mapping, then (13) can
be written as the following compact form (Reuter et al, 2014)

$\pi(\mathbf{X})=\bigtriangleup(\mathbf{X})\omega(\mathcal{L}(\mathbf{X}))p^{\mathbf{X}}$
(18)

where

$\displaystyle\omega(L)$ $\displaystyle=$
$\displaystyle\prod_{j\in\mathbb{L}}(1-r^{(j)})\prod_{l\in
L}\frac{1_{\mathbb{L}}(l)r^{(l)}}{1-r^{(l)}},$ (19) $\displaystyle p(x,l)$
$\displaystyle=$ $\displaystyle p^{(l)}(x).$ (20)

For simplicity, we usually use the parameter set
$\pi=\\{(r^{(l)},p^{(l)})\\}_{l\in\mathbb{L}}$ to abbreviate the above PDF.

####  2.3.2 Generalized Labeled Multi-Bernoulli RFS

A generalized labeled multi-Bernoulli RFS (GLMB) is an LRFS with the following
PDF (Vo and Vo, 2013)

$\pi(\mathbf{X})=\bigtriangleup(\mathbf{X})\sum_{c\in\mathbb{C}}{\omega^{(c)}(\mathcal{L}(\mathbf{X}))[p^{(c)}]^{\mathbf{X}}}$
(21)

where $\mathbb{C}$ is a discrete index set, and

$\displaystyle\sum_{L\subseteq\mathbb{L}}\sum_{c\subseteq\mathbb{C}}{\omega(L)}$
$\displaystyle=$ $\displaystyle 1,$ (22) $\displaystyle\int p^{(c)}(x,l)dx$
$\displaystyle=$ $\displaystyle 1.$ (23)

###  2.4 Augmented LRFSs

To address the issue of group target tracking, in this paper, we propose the
concept of the augmented LRFSs, which can be taken as the extension of the
concept of LRFSs.

Firstly, for the label of each target, defined in an LRFS, $l=(k,i)$, we
augment it by integrating the group information of the group that the target
belongs to. Specifically, an augmented label is defined as

$\rho=(l,g,c)=(k,i,g,c)$ (24)

where $l=(k,i)$, and

  * •

$k$ still denotes the birth time of this target.

  * •

$i\in\mathbb{N}$ still denotes the distinct indice to distinguish the targets
born at the same time.

  * •

$g\in\mathbb{Z}$ denotes the unique index of the group that the target belongs
to. Specially, $g=0$ denotes the target does not belong to any group.

  * •

${c}\in\mathbb{X}$ denotes the center of the group $g$.

Secondly, for $\mathsf{x}=(x,\rho)$, let
$\mathbb{L}^{1},\mathbb{L}^{2},\mathbb{L}^{3},\mathbb{L}^{4}$ denote the
spaces to which $k,i,g,{c}$ belong, respectively. Let
$\mathbb{L}^{m}\times\mathbb{L}^{n}$ denotes the space to which the $m$-th
element and $n$-th element jointly belong. For example,
$\mathbb{L}^{3}\times\mathbb{L}^{4}$ denotes the space to which $g$ and $c$
jointly belong. Let
$\mathbb{L}=\mathbb{L}^{1}\times\mathbb{L}^{2}\times\mathbb{L}^{3}\times\mathbb{L}^{4}$,
then, we have $l\in\mathbb{L}^{1}\times\mathbb{L}^{2}$, $\rho\in\mathbb{L}$,
and $\mathsf{x}\in\mathbb{X}\times\mathbb{L}$. For the sake of convenience,
the following operators are defined.

  * •

$\mathcal{L}(\cdot)$: The projection from $\mathbb{X}\times\mathbb{L}$ to
$\mathbb{L}$, e.g., $\mathcal{L}(\mathsf{x})=\rho$.

  * •

$\mathcal{L}^{i}(\cdot)$: The projection from $\mathbb{X}\times\mathbb{L}$ to
$\mathbb{L}^{i}$, e.g., , $\mathcal{L}^{1}(\mathsf{x})=k$.

  * •

$\mathcal{L}^{i\times j}(\cdot)$: The projection from
$\mathbb{X}\times\mathbb{L}$ to $\mathbb{L}^{i}\times\mathbb{L}^{j}$, e.g.,
$\mathcal{L}^{1\times 2}(\mathsf{x})=l=(k,i)$, $\mathcal{L}^{1\times
3}(\mathsf{x})=l=(k,g)$.

  * •

$\mathcal{X}(\cdot)$: The projection from $\mathbb{X}\times\mathbb{L}$ to
$\mathbb{X}$, e.g., $\mathcal{L}(\mathsf{x})=x$.

  * •

$\mathcal{X}\mathcal{L}^{i\times j}(\cdot)$: The projection from
$\mathbb{X}\times\mathbb{L}$ to
$\mathbb{X}\times\mathbb{L}^{i}\times\mathbb{L}^{j}$, e.g.,
$\mathcal{X}\mathcal{L}^{1\times 2}(\mathsf{x})=(x,l)$.

Thirdly, the state $x$ is augmented by the above label $\rho$, i.e.,
$\mathsf{x}=(x,\rho)$. Lastly, an augmented LRFS is an RFS who has the
following form

$\mathsf{X}=\\{\mathsf{x}_{1},\mathsf{x}_{2},\cdots,\mathsf{x}_{n}\\}.$ (25)

For each element $\mathbf{x}=(x,l)$ in the LMB $\mathbf{X}$, if we subsitute
$\mathbf{x}$ by $\mathsf{x}=(x,\rho)$, where $\rho$ is defined in (24), we
will obtain a new LRFS $\mathsf{X}$, which can be called as an agugmented LMB,
and can be defined as follows.

####  2.4.1 Agugmented LMB RFS

An agugmented LMB RFS is an RFS with the following PDF

$\pi(\mathsf{X})=\bigtriangleup(\mathsf{X})1_{\alpha(\Psi)}(\mathcal{L}^{1\times
2}(\mathsf{X}))[\Phi(\mathsf{X};\cdot)]^{\Psi}$ (26)

where $l=\mathcal{L}^{1\times 2}(\mathsf{x})$, and

$\displaystyle\Phi(\mathsf{X};\cdot)=\left\\{\begin{array}[]{ll}1-r^{(l)},&\mbox{if
$\alpha(l)\notin\mathcal{L}^{1\times 2}(\mathsf{X})$},\\\
r^{(l)}p^{(l)}(x),&\mbox{if $(x,\alpha(l))\in\mathcal{X}\mathcal{L}^{1\times
2}(\mathsf{X})$}.\end{array}\right.$ (29)

If we define $\alpha$ as an identity mapping, similar to (18), (26) can be
simplified to

$\pi(\mathsf{X})=\bigtriangleup(\mathsf{X})\omega(\mathcal{L}^{1\times
2}(\mathsf{X}))p^{\mathsf{X}}$ (30)

where

$\displaystyle p(x,\rho)$ $\displaystyle=$ $\displaystyle
p^{(\mathcal{L}^{1\times 2}(\mathsf{x}))}=p^{(l)}(x),$ (31)
$\displaystyle\omega(L)$ $\displaystyle=$
$\displaystyle\prod_{j\in\mathbb{L}^{1\times 2}}(1-r^{(j)})\prod_{l\in
L}\frac{1_{\mathbb{L}^{1\times 2}}(l)r^{(l)}}{1-r^{(l)}}.$ (32)

For simplicity, we still use the parameter set
$\pi=\\{(r^{(\rho)},p^{(\rho)})\\}_{\rho\in\mathbb{L}}$ to abbreviate the
above PDF.

Similar to $X$ and $\mathbf{X}$, we also call $\mathsf{X}$ as a multi-target
state, since it also describes the multi-state of multiple targets. By
integrating the augmented label $\rho$ into the multi-target state, one can
find that the group attribute information has been incorporated, making
possible infer the dynamic state, the identity and group attribute information
of individual targets. The intensive analysis will be presented in sections
3-4.

###  2.5 Statistical Characteristic of Augmented LRFSs

This subsection will present some preliminary analysis of the statistical
characteristic of augmented LRFSs.

######  Proposition 1.

An augmented LRFS with state space $\mathbb{X}$ and label space $\mathbb{L}$,
i.e., $\mathsf{X}$ denoted in (25), essentially is an RFS on the space
$\mathbb{X}\times\mathbb{L}$.

Since the individual elements in $\mathsf{X}$ are random and unordered
variables on the space $\mathbb{X}\times\mathbb{L}$, and the number of
elements in $\mathsf{X}$ is a random variable, according to the definition of
RFS (see Definition 1), $\mathsf{X}$ is an RFS.

######  Proposition 2.

An augmented LRFS with state space $\mathbb{X}$ and label space $\mathbb{L}$,
i.e., $\mathsf{X}$ denoted in (25), essentially is an LRFS on the space
$\mathbb{X}\times\mathbb{L}$.

The proof is direct. Since for each element
$\mathsf{x}=(x,\rho)\in\mathsf{X}$, it follows that $\mathcal{L}^{1\times
2}(\mathsf{x})=l=(k,i)$ is distinct. Thus, the label $\rho=(l,g,c)$ also is
distinct for each element $\mathsf{x}=(x,\rho)\in\mathsf{X}$, which is in
accordance with the definition of a labeled RFS (see Definition 2).

######  Proposition 3.

For an augmented LRFS $\mathsf{X}$, its corresponding RFS $X$ can be obtained
by imposing the operator $\mathcal{X}(\cdot)$, i.e.,
${X}=\mathcal{X}(\mathsf{X})$, its corresponding LRFS $\mathbf{X}$ can be
obtained by imposing the operator $\mathcal{X}\mathcal{L}^{1\times 2}(\cdot)$,
i.e., $\mathcal{X}\mathcal{L}^{1\times 2}(\mathsf{X})=\mathbf{X}$. An
augmented LRFS $\mathsf{X}$, its corresponding RFS $X$, and its corresponding
LRFS $\mathbf{X}$ have the same cardinility distribution.

Since the projection operator does not change the number of elements in
$\mathsf{X}$, it is straightforward that the above proposition is true.

From propositions 1-2, it can be inferred that the mathematical operators for
both RFSs and LRFSs, are also applicable for the augmented LRFSs. Thus,
inspired by the definition of the integral operators for both RFSs and LRFSs
(Mahler, 2011; Vo and Vo, 2013), the definition of the integral of an
augmented LRFS can be denoted as

$\displaystyle\int f(\mathsf{X})\delta\mathsf{X}=$
$\displaystyle\sum_{i=0}^{\infty}{\frac{1}{i!}\int_{\rho^{i}}\int_{\mathbb{X}^{i}}f(\\{(x_{1},\rho_{1}),\cdots,(x_{i},\rho_{i})\\})d(x_{1},\cdots,x_{i})d\rho_{i}}.$

For each element $\mathbf{x}=(x,l)$ in the GLMB $\mathbf{X}$, if we subsitute
$\mathbf{x}$ by $\mathsf{x}=(x,\rho)$, where $\rho$ is defined in (24), we
will obtain a new LRFS $\mathsf{X}$, which can be called as an agugmented
GLMB. Thus, an agugmented GLMB can be defined as follows.

####  2.5.1 Agugmented GLMB RFS

An agugmented GLMB RFS is an LRFS with the following PDF

$\pi(\mathsf{X})=\bigtriangleup(\mathsf{X})\sum_{c\in\mathbb{C}}{\omega^{(c)}(\mathcal{L}^{1\times
2}(\mathsf{X}))[p^{(c)}]^{\mathsf{X}}}$ (33)

where $\mathbb{C}$ is a discrete index set, and

$\displaystyle\sum_{L\subseteq\mathbb{L}^{1\times
2}}\sum_{c\subseteq\mathbb{C}}{\omega(L)}$ $\displaystyle=$ $\displaystyle 1,$
(34) $\displaystyle\int p^{(c)}(x,\rho)dx$ $\displaystyle=$ $\displaystyle\int
p^{(c)}(x,\mathcal{L}^{1\times 2}(x))dx=1.$ (35)

##  3 System Model

###  3.1 Dynamic Model

Figure 1: The considered group target tracking scenario.

The considered group target tracking scenario is shown in Fig. 1. There are
multiple targets in the field of view (FoV) of a radar, some of which are
closely positioned and move in a coordinated manner, forming several group
targets. To capture the entire dynamic information of all of these targets, it
is necessary to model both the dynamics and group attributes of individual
targets. Here, we use the proposed augmented LRFS to denote the multi-state of
these targets at time $k$, i.e.,

$\mathsf{X}_{k}=\\{\mathsf{x}_{k,1},\mathsf{x}_{k,2},\cdots,\mathsf{x}_{k,n(k)}\\}$
(36)

where $\mathsf{x}_{k,j}=(x_{k,j},\rho_{k,j})$ denotes the individual state of
the $j$-th target at time $k$, consisting of dynamic state $x_{k,j}$ and label
state $\rho_{k,j}$. Note that the label state $\rho_{k,j}$ has been defined in
(24), and $n(k)$ denotes the number of these targets in the FoV at this time.

Thanks to the introduction of the augmented LRFS, it can be found that the
dynamics, labels, and group attributes of all of these targets at time $k$,
are simultaneously contained in the multi-state, i.e., the augmented LRFS
$\mathsf{X}_{k}$. Thus, if $\mathsf{X}_{k}$ can be accurately estimated, the
dynamics, labels, and group attributes of individual targets also can be
accurately inferred.

For the dynamic state $x_{k}\in\mathsf{x}_{k}$, it is worthing noting that its
dynamic state transition function is correlated with its corresponding group
attribute $\mathcal{L}^{3\times 4}(\mathsf{x}_{k})=(g_{k},c_{k})$. On one
hand, if $\mathcal{L}^{3}(\mathsf{x}_{k})=g_{k}=0$, which means that the
target associated with $\mathsf{x}_{k}$ dose not belong to any group. Its
dynamics only depend on itself, and is independent with the dynamics of other
targets. Thus, we can denote its dynamic state transition function as

$f(x_{+}|x,\mathcal{L}^{1\times 2}(\mathsf{x}))=f(x_{+}|x,l)$ (37)

where $x_{+}$ denotes the dynamic state at the next time, and the time
subscript such as $k$, has been omitted. On the other hand, if
$\mathcal{L}^{3}(\mathsf{x}_{k})=g_{k}\neq 0$, which means that the target
associated with $\mathsf{x}_{k}$ belongs to the group indexed by $g_{k}$, it
is reasonable to assume that its dynamic state not only relies on itself, but
also depends on the dynamic state of this group. Thus, it is reasonable to
denote its dynamic state transition function as

$f(x_{+}|x,\mathcal{L}^{4}(\mathsf{x}),\mathcal{L}^{1\times
2}(\mathsf{x}))=f(x_{+}|x,c,l).$ (38)

###  3.2 Measurement Model

Suppose that the radar receives $m(k)$ measurements at time $k$, and these
measurements can be also represented by an RFS ${Z}_{k}$, i.e.,

$Z_{k}=\\{\boldsymbol{z}_{k,1},\boldsymbol{z}_{k,2},\cdots,\boldsymbol{z}_{k,m(k)}\\}$
(39)

where $\boldsymbol{z}_{k,j}$ denotes the $j$-th measurement at time $k$.

###  3.3 The Objective of This Paper

After formulating the dynamics of these multiple targets and the measurements
of the radar, the objectives of this paper are two-fold:

  * •

The first is to simultaneously estimate the number of targets and the dynamic
states of individual targets, which is in accordance with the objective of the
traditional multi-target tracking (MTT) problem.

  * •

The second is to estimate the number and structure of groups present among
these targets, and, at the same time, infer the group attribute for individual
targets.

Since we have used the augmented LRFS $\mathsf{X}_{k}$ to completely capture
all information of these targets, which includes the dynamic state, identity
and group attribute of individual targets. Then, the objectives of this paper
essentially equal to the calculation of the posterior multi-target PDF of
$\mathsf{X}_{k}$ conditioned on the measurements at current time, $Z_{k}$.
After denoting the posterior multi-target PDF as
$\pi_{k}(\mathsf{X}_{k}|Z_{k})$, we can iteratively calculate it via the
following multi-target Bayes filter,

$\displaystyle\pi_{k|k-1}(\mathsf{X}_{k}|Z_{k-1})\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\int
f_{k|k-1}(\mathsf{X}_{k}|\mathsf{X})\pi_{k-1}(\mathsf{X}|Z_{k-1})\delta\mathsf{X}$
(40) $\displaystyle\pi_{k}(\mathsf{X}_{k}|Z_{k})$ $\displaystyle=$
$\displaystyle\frac{g_{k}(Z_{k}|\mathsf{X}_{k})\pi_{k|k-1}(\mathsf{X}_{k}|Z_{k-1})}{\int
g_{k}(Z_{k}|\mathsf{X})\pi_{k|k-1}(\mathsf{X}|Z_{k-1})\delta\mathsf{X}}$ (41)

where $\pi_{k-1}(\mathsf{X}|Z_{k-1})$ is assumed to given at time $k-1$,
$\pi_{k|k-1}(\mathsf{X}_{k}|Z_{k-1})$ represents the predicted multi-target
PDF at time $k-1$, $f_{k|k-1}(\mathsf{X}_{k}|\mathsf{X})$ and
$g_{k}(Z_{k}|\mathsf{X}_{k})$ denote the multi-target transition kernel from
time $k-1$ to the current time $k$, and the likelihood function at the current
time, respectively.

However, before applying the above multi-target Bayes filter, two key problems
needs to be solved. The first is the specific expression of
$f_{k|k-1}(\mathsf{X}_{k}|\mathsf{X})$ and $g_{k}(Z_{k}|\mathsf{X}_{k})$, and
the second is the feasible solution to the above multi-target Bayes filter
that is intractable due to the involvement of intricate integral calculations.

##  4 Application of Augmented LRFS to Group Target Tracking

In this section, after presenting the specific expression of
$f_{k|k-1}(\mathsf{X}_{k}|\mathsf{X})$ and $g_{k}(Z_{k}|\mathsf{X}_{k})$ in
(40)-(41), a feasible solution to the above multi-target Bayes filter will be
provided, which achieves the objectives of this paper.

###  4.1 Multi-target Likelihood Function

For each state $\mathsf{x}$ in a given multi-state $\mathsf{X}$, it can be
either detected with probability $p_{D}(\mathsf{x})$ or not detected with
probability $1-p_{D}(\mathsf{x})$. Moreover, if detected, it will generate a
measurement $z$ with likelihood $g(z|\mathsf{x})$. Thus, the measurement
generated by the state $\mathsf{x}$ can be formulated by a Bernoulli RFS with
parameters $p_{D}(\mathsf{x})$ and $g(\cdot|\mathsf{x})$. Furthermore, under
the assumption that all states in $\mathsf{X}$ are mutually independent, it
can be inferred that all measurements generated by these states form a multi-
Bernoulli RFS $W$ distributed according to

$\pi_{D}(W|\mathsf{X})=\\{p_{D}(\mathsf{x}),g(\cdot|\mathsf{x})\\}_{\mathsf{x}\in\mathsf{X}}$
(42)

In addition to measurements originating from $\mathsf{X}$, $Z$ also includes
measurements from clutters denoted as $Y$, i.e., $Z=W\cup C$. Usually, $C$ is
modeled by a Possion RFS distributed (Reuter et al, 2014) according to

$\pi_{K}(C)=e^{-<c,1>}\kappa^{C},$ (43)

where $\kappa(\cdot)$ is a given the intensity function. Then, according to
FISST (Mahler, 2011; Vo and Vo, 2013), the multi-target likelihood function
can be calculated as

$g_{k}(Z_{k}|\mathsf{X}_{k})=\sum_{W\subseteq
Z_{k}}{\pi_{D}(W|\mathsf{X}_{k})\pi_{K}(Z_{k}-W)}$ (44)

which is exactly the equation (18) in Ref. Vo and Vo (2013), regardless of the
difference between an augmented LRFS and an LRFS.

###  4.2 Multi-target Transition Kernel

For the target $\mathsf{x}\in\mathsf{X}$, it is identified by a unique label
$\mathcal{L}^{1\times 2}(\mathsf{x})=l=(k,i)$. As mentioned in Proposition 2,
for each target $l=(k,j)$ is unique and distinct. It is straightforward that
$\mathcal{L}^{1\times 2}(\mathsf{x})=l=(k,i)$ wil be perserved during the
surviving time of this target. In other words, for each state
$\mathsf{x}=(x,\rho)=(x,l,g,c)$, if it continues to exist at the next time
with probability $p_{S}(x,l)$ and move to a new state
$\mathsf{x}_{+}=(x_{+},\rho_{+})=(x_{+},l_{+},g_{+},c_{+})$, we have
$\delta_{l}(l_{+})=1$.

Furthermore, if it continues to exist at the next time with probability
$p_{S}(x,l)$ and move to a new state, depending on whether the target belongs
to a group or not, its transition function has different forms. Specifically,
according to section 3.1, if it does not belong to any group at current time,
its transition function will be $f(x_{+}|x,l)\delta_{l}(l_{+})$, i.e.,
equation (37). Inversely, if it belongs to a group with group label $g$ at the
current time, it should follow the transition function
$f(x_{+}|\mathcal{L}^{4}(\mathsf{x}),l)\delta_{l}(l_{+})=f(x_{+}|c,l)\delta_{l}(l_{+})$,
i.e., equation (38). In summary, for the target with current state
$\mathsf{x}=(x,\rho)=(x,l,g,c)$, if it continues to exist at the next time
with probability $p_{S}(x,l)$ and moves to a new state
$\mathsf{x}_{+}=(x_{+},\rho_{+})=(x_{+},l_{+},g_{+},c_{+})$, the transition
PDF can be represented as

$\displaystyle f(\mathsf{x}_{+}|\mathsf{x})=$ $\displaystyle
f(x_{+}|x,l)\delta_{l}(l_{+})\delta_{0}(g)+f(x_{+}|c,l)\delta_{l}(l_{+})(1-\delta_{0}(g)).$
(45)

On the other hand, except surviving, the state $\mathsf{x}=(x,\rho)=(x,l,g,c)$
also has the probability $q_{S}(x,l)=1-p_{S}(x,l)$ to die at the next time.
Therefore, the set of the state $\mathsf{x}_{+}$ is a Bernoulli RFS with
argumented label $\rho_{+}$.

On premise of that the state transition of all targets in the current multi-
state $\mathsf{X}$ are multually independent, it follows that the set of
surviving targets at the next time is an augmented LMB RFS with parameter set
$\\{p_{S}(x,l),f(\cdot|\mathsf{x})\\}_{\mathsf{x}\in\mathsf{X}}$, denoted by
$\mathsf{W}$. Hence, according to (26), $\mathsf{W}$ is subjected to the
distribution

$f_{S}(\mathsf{W}|\mathsf{X})=\Delta(\mathsf{W})\Delta(\mathsf{X})1_{\mathcal{L}^{1\times
2}(\mathsf{X})}(\mathcal{L}^{1\times
2}(\mathsf{W})[\Phi(\mathsf{W};\cdot)]^{\mathsf{X}}$ (46)

where

$\displaystyle\Phi(\mathsf{W};\mathsf{X})=\left\\{\begin{array}[]{ll}q_{S}(x,l),&\mbox{if
$l\notin{\mathcal{L}^{1\times 2}(\mathsf{W})}$},\\\
p_{S}(x,l)f(\mathsf{x}_{+}|\mathsf{x}),&\mbox{if
$(x,l)\in\mathsf{W}$},\end{array}\right.$ (49)

If we take the augmented LRFS $\mathsf{W}$ as a special LRFS, one can find
that the above equation is exactly the equation (25) in Ref. Vo and Vo (2013)
.

Apart from the surviving targets, at the next time, there maybe exist the
newborn targets. According to Ref. Reuter et al (2014), the set of newborn
targets at the next time can be formulated as an LMB RFS $\mathbf{Y}$
distributed according to

$f_{B}(\mathbf{Y})=\Delta(\mathbf{Y})w_{B}(\mathcal{L}(\mathbf{Y}))[p_{B}]^{\mathbf{Y}}$
(50)

where $p_{B}$ is the newborn probability. However, the above formulation does
not consider the group information of the newborn targets. Thus, for
$\mathbf{x}=(x,l)\in\mathbf{Y}$, we extend $\mathbf{x}$ to
$\mathsf{x}=(x,\rho)=(x,l,g,c)$, and then we obtain a new augmented LRFS for
newborn targets, i.e., $\mathsf{Y}$, that follows the PDF

$f_{B}(\mathsf{Y})=\bigtriangleup(\mathsf{Y})\omega(\mathcal{L}^{1\times
2}(\mathsf{Y}))p^{\mathsf{Y}}.$ (51)

Assume that each newborn target is independent and does not belong to any
group at the borntime, then for any $\mathsf{x}\in\mathsf{Y}$, we set
$\mathcal{L}^{3}(\mathsf{x})=g=0$.

It follows that the multi-target state at the next time, $\mathsf{X}_{+}$, is
the combination of newborn targets and surviving targets, i.e.,
$\mathsf{X}_{+}=\mathsf{W}\cup\mathsf{Y}$. Then, according to FISST (Mahler,
2011; Vo and Vo, 2013), the multi-target transition kernel is

$f(\mathsf{X}_{+}|\mathsf{X})=\sum_{\mathsf{W}\subseteq\mathsf{X}}{f_{S}(\mathsf{W}|\mathsf{X})f_{B}(\mathsf{X}_{+}-\mathsf{W})}.$
(52)

which is exactly equation (31) in Ref. Vo and Vo (2013), regardless of the
difference between an augmented LRFS and an LRFS.

######  Lemma 1.

If the current multi-target prior $\pi_{k-1}(\mathsf{X}|Z_{k-1})$ in (40) is
an agumented GLMB RFS of the form (33), then both the predicted and posterior
multi-target PDF are aslo agumented GLMB RFSs of the form (33). In other
words, the augmented GLMB RFS is closed under the multi-target Bayes filter
(40)-(41) with respect to the multi-target likelihood function (44) and multi-
target transition kernel (52). Moreover, the multi-target Bayes filter can be
analytically solved by the GLMB filter proposed in Ref. Vo and Vo (2013), and
further can be approximatively solved by the LMB filter proposed in Ref.
Reuter et al (2014).

The proof can be explained from two aspects. First, an agumented GLMB RFS
still is essentially a GLMB RFS, we can consider it as a special GLMB RFS with
agumented label space. Moreover, based on this consideration, the multi-target
likelihood function (44) and multi-target transition kernel (52) are
equivalent to equation (18) and (31) in Ref. Vo and Vo (2013). Second,
according to Ref. Vo and Vo (2013), the GLMB RFS is closed under the multi-
target Bayes filter (40)-(41) with respect to the multi-target likelihood
function (i.e., equation (18) in Ref. Vo and Vo (2013)) and multi-target
transition kernel (i.e., equation (31) in Ref. Vo and Vo (2013)), and the
multi-target Bayes filter can be analytic solved by the GLMB filter. Moreover,
from Ref. Reuter et al (2014), the multi-target Bayes filter can be
approximatively solved by the LMB filter. Thus, we can infer that this lemma
holds true.

###  4.3 LMB filter with Augmented LRFS for Group Target Tracking

According to Lemma 1, after formulating the problem of group target tracking
via the agumented LRFS, this problem can be approximatively solved by the LMB
filter. Thus, in this subsection, we will combine the LMB filter with the
agumented LRFSs, simultaneously integrating group information into the LMB
filter. Differing from the original LMB filter (Reuter et al, 2014) that
includes the prediction and update steps, the new LMB filter consists of three
steps: state prediction, state update and group information update.

####  4.3.1 State Prediction

First, let’s begin with the state prediction step of the new LMB filter, whch
can be presented by the following proposition.

######  Proposition 4.

Suppose that the multi-target posterior PDF is an agumented LMB RFS with state
space $\mathbb{X}$, label space $\mathbb{L}$ and parameter set
$\pi=\\{r^{(\rho)},p^{(\rho)}\\}_{\rho\in\mathbb{L}}$, and the multi-target
birth model is an agumented LMB RFS with state space $\mathbb{X}$, label space
$\mathbb{B}$ and parameter set
$\pi=\\{r^{(\rho)}_{B},p^{(\rho)}_{B}\\}_{\rho\in\mathbb{B}}$, then the multi-
target predicted PDF is also an agumented LMB RFS with state space
$\mathbb{X}$, label space $\mathbb{L}_{+}=\mathbb{B}\cup\mathbb{L}$, which has
the parameter set

$\pi_{+}=\\{(r_{+,S}^{(\rho)},p_{+,S}^{(\rho)}\\}_{\rho\in\mathbb{L}}\cup\\{(r_{B}^{(\rho)},p_{B}^{(\rho)})\\}_{\rho\in\mathbb{B}},$
(53)

where

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle(l,g,c),$ (54)
$\displaystyle r_{+,S}^{(\rho)},$ $\displaystyle=$
$\displaystyle\eta_{S}(\rho)r^{(\rho)},$ (55) $\displaystyle\eta_{S}(\rho)$
$\displaystyle=$ $\displaystyle<p_{S}(\cdot,l),p(\cdot,\rho)>,$ (56)
$\displaystyle p_{+,S}^{(\rho)}$ $\displaystyle=$
$\displaystyle<p_{S}(\cdot,l)f(\mathsf{x}_{+}|\mathsf{x}),p(\cdot,\rho)>/\eta_{S}(\rho),$
(57)

and $f(\mathsf{x}_{+}|\mathsf{x})$ is defined in (4.2).

####  4.3.2 State Update

The state update step of the new LMB filter can be presented by the following
proposition.

######  Proposition 5.

Suppose that the multi-target predicted PDF is an agumented LMB RFS with state
space $\mathbb{X}$, label space $\mathbb{L}_{+}$ and parameter set
$\pi_{+}=\\{r^{(\rho)}_{+},p^{(\rho)}_{+}\\}_{\rho\in\mathbb{L}_{+}}$, then
the agumented LMB RFS that matches exactly the first moment of the multi-
target posterior PDF is
$\pi(\mathsf{X}|Z)=\\{r^{(\rho)},p^{(\rho)}(\cdot)\\}_{\rho\in\mathbb{L}_{+}}$
where

$p^{(\rho)}\\!\\!=\frac{1}{r^{(\rho)}}\\!\\!\\!\\!\\!\\!\sum_{(I_{+},\theta)\in\mathcal{F}(\mathbb{L}_{+})\times\Theta_{I_{+}}}{\omega^{(I_{+},\theta)}(Z)1_{I_{+}}(\rho)p^{(\theta)}(x,\rho)},$
(58) $\displaystyle r^{(\rho)}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{(I_{+},\theta)\in\mathcal{F}(\mathbb{L}_{+})\times\Theta_{I_{+}}}{\omega^{(I_{+},\theta)}(Z)1_{I_{+}}(\rho)},$
(59) $\displaystyle\omega^{(I_{+},\theta)}(Z)$ $\displaystyle\varpropto$
$\displaystyle\omega_{+}(I_{+})[\eta^{(\theta)}_{Z}]^{I_{+}},$ (60)
$\displaystyle p^{(\theta)}(x,\rho|Z)$ $\displaystyle=$
$\displaystyle\frac{p_{+}(x,\rho)\psi_{Z}(x,\rho;\theta)}{\eta_{Z}^{(\theta)}(\rho)},$
(61) $\displaystyle\eta_{Z}^{(\theta)}(\rho)$ $\displaystyle=$
$\displaystyle<p_{+}(\cdot,\rho),\psi_{Z}(\cdot,\rho;\theta)>,$ (62)
$\displaystyle\omega_{+}(I_{+})$ $\displaystyle=$
$\displaystyle\prod_{j_{+}\in\mathbb{L}_{+}}(1-r_{+}^{(j)})\prod_{\rho\in
L_{+}}\frac{1_{\mathbb{L}_{+}}(\rho)r_{+}^{(\rho)}}{1-r_{+}^{(\rho)}}.$ (63)

$\displaystyle\psi_{Z}(x,\rho;\theta)=\left\\{\begin{array}[]{ll}\frac{p_{D}(x)g(z_{\theta(\rho)}|x,l)}{\kappa(z_{\theta(\rho)})},&\mbox{if
$\theta(\rho)>0$},\\\ 1-p_{D}(x),&\mbox{if
$\theta(\rho)=0$},\end{array}\right.$ (66)

and $\Theta_{I_{+}}$ is the space of mappings $\theta$:
$I_{+}\rightarrow\\{0,1,\cdots,|Z|\\}$, such that
$\theta(i)=\theta(i^{{}^{\prime}})>0$ implies $i=i^{{}^{\prime}}$.
$g(z|x,\rho)$ is the single target likelihood for $z$ given $(x,\rho)$, and
$\kappa(\cdot)$ is the intensity of Possion clutter (Reuter et al, 2014).

The proof of propositions 4-5 can be explained as follows. The two propostions
with LRFS version, i.e., propostions 2 and 4 in Ref. Reuter et al (2014), have
been proved in Ref. Reuter et al (2014). Since these propostions can be taken
as the propostions 2-4 in Ref. Reuter et al (2014) with agumented LRFSs, and
an agumented LRFS is essentially an LRFS, thus, it follows that propositions
4-5 hold true.

After the state of state update, the number of targets $\hat{n}_{k}$ can be
extracted by calculating the maximum a posteriori (MAP) estimation of the
cardinality of the estimated agumented LMB RFS $\mathsf{X}$ (Yang et al, 2023;
Vo and Vo, 2013). Then, we pick up $\hat{n}_{k}$ Bernoulli items with the
highest weight, calculate the corresponding mean via $\hat{x}^{({\rho})}=\int
xp^{(\rho)}(x)dx$ for each Bernoulli item $(r^{(\rho)},p^{(\rho)})$, and we
take this mean as the estimated dynamic state of the potential target
represented by this Bernoulli item. Finally, we obtain the estimation of the
dynamic states of individual targets, which achieves the first objective of
this paper.

####  4.3.3 Group Information Update

After obtaining the approximated multi-target posterior PDF
$\pi(\mathsf{X}|Z)=\\{r^{(\rho)},p^{(\rho)}(\cdot)\\}_{\rho\in\mathbb{L}_{+}}$,
we can further extract the group information of each Bernoulli item
$(r^{(\rho)},p^{(\rho)})$. Note that if the number of Bernoulli item in
$\pi(\mathsf{X}|Z)$ equals to one, which means that there exist at most only
one target at this time, then the update of group information can be skipped,
since one target can not form a group.

If the number of the Bernoulli items in $\pi(\mathsf{X}|Z)$ is $n$, where
$n>1$, first, for each Bernoulli item $(r^{(\rho)},p^{(\rho)})$, we calculate
its corresponding mean via $\hat{x}^{({\rho})}=\int xp^{(\rho)}(x)dx$, which
can be taken as the estimated dynamic state of the potential target
represented by this Bernoulli item. Second, for all Bernoulli items in
$\pi(\mathsf{X}|Z)$, we calculate the adjacent matrix given by

$A_{k}=\left[\begin{array}[]{cccc}0&a_{k}(1,2)&\cdots&a_{k}(1,n)\\\
a_{k}(2,2)&0&\cdots&a_{k}(2,n)\\\ \vdots&\vdots&\ddots&\vdots\\\
a_{k}(n,1)&a_{k}(n,2)&\cdots&0\end{array}\right]$ (67)

where

$\displaystyle a_{k}({i,j})=\left\\{\begin{array}[]{ll}1,&\mbox{if $i\neq j$
and $d_{k}({i,j})\leq\varepsilon$},\\\ 0,&\mbox{otherwise},\end{array}\right.$
(70)

and $d_{k}({i,j})=||\hat{x}^{i}-\hat{x}^{j}||_{2}$ and $\varepsilon$ is a
given threshold. According to the adjacent matrix $A_{k}$, we can construct a
graph $G_{k}$.

Then, we calculate all the connected components111The definition of connected
component: A connected component is a set of vertices within a graph such that
there is a path between every pair of vertices in the set, and no path exists
to connect any vertex in the set to a vertex outside the set (Trudeau, 2003).
of the graph $G_{k}$. We take each connected component as a group, where all
Bernoulli items belonging to this connected component are regarded as the
members of this group. Furthermore, we interpret the graph structure of this
connected component as the group’s structure. For this group, we assign a new
distinct integer $g^{\prime}$ as its index, and take the spatial centroid of
this group as its center $c^{\prime}$.

Lastly, for each Bernoulli item $(r^{(\rho)},p^{(\rho)})$, we update its group
information and integrate them into its augmented label $\rho=(l,g,c)$.
Specifically, $g$ is updated by $g^{\prime}$, which is the index of the group
that the Bernoulli item belongs to. $c$ is updated by $c^{\prime}$, which is
the center of the group that the Bernoulli item belongs to.

In summary, the entire schematic of the step of group information update is
shown in Fig. 2. For illustration, we take Fig. 3 for example. Suppose that
there are six Bernoulli items after the state update step of the proposed LMB
filter with augmented LRFSs, which represent the six targets in Fig. 3. First,
we calclulate the adjacent matrix according to (67), yielding

$A_{k}=\left[\begin{array}[]{cccccc}0&1&1&0&0&0\\\ 1&0&1&0&0&0\\\
1&1&0&0&0&0\\\ 0&0&0&0&1&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&0&0\end{array}\right].$
(71)

Then, we calculate all the connected components. As shown in Fig. 3, three
connected components are calculated, which divides these targets into three
groups. Lastly, we assign the group index $g$ and the group center $c$ to each
group, update the group information and integrate them into its augmented
label $\rho=(l,g,c)$.

Figure 2: The entire schematic of the step of group information update.
Figure 3: An example for group information extraction, where T$i$ denotes the
$i$-th target.

After the state of group information update, The number of groups and the
group structure can be extracted from
$\mathcal{L}(\mathsf{X})=\\{\rho_{1},\rho_{2},\cdots,\rho_{n(k)}\\}$. First,
the number of groups equals to the number of non-zero elements in
$\mathcal{L}^{3}(\mathsf{X})$. Then, we can find all the same and non-zero
elements in $\mathcal{L}^{3}(\mathsf{X})$, and take them as a group, and the
corresponding group center can be extracted from
$\mathcal{L}^{4}(\mathsf{X})$.

##  5 Performance Evaluation

###  5.1 Parameters Settings

Consider a two-dimensional scenario with up to 6 targets. For each target, its
state is described by the vector
${x}_{k}=[p_{x,k},\dot{p}_{x,k},p_{y,k},\dot{p}_{y,k}]^{T}$, where
$(p_{x,k},p_{y,k})$ and $(\dot{p}_{x,k},\dot{p}_{y,k})$ are the position and
velocity of this target at time $k$, respectively. On the one hand, for the
target (with label $l$) that does not belong to any group, its transition
density is given by $f(x_{+}|x,l)=N(x_{+};Fx,Q)$, where

$F=\left[\begin{array}[]{cccc}0&\bigtriangleup t&0&0\\\ 0&1&0&0\\\
0&0&1&\bigtriangleup t\\\ 0&0&0&1\end{array}\right]$ (72)

$\bigtriangleup t$ is time interval and $Q=\varrho^{2}GG^{T}$, $\varrho=5$,
and

$G=\left[\begin{array}[]{cc}(\bigtriangleup t)^{2}/2&0\\\ \bigtriangleup
t&0\\\ 0&(\bigtriangleup t)^{2}/2\\\ 0&\bigtriangleup t\end{array}\right]$
(73)

On the other hand, for the target (with label $l$) that belongs to a group
with center $c$, its transition density is denoted by $f(x_{+}|x,c,l)$.
Suppose the transition density of the group center follows
$f(c_{+}|c)=N(c_{+};Fc,Q)$, then we have $f(x_{+}|x,c,l)=N(x_{+};(F-I)c+x,Q)$
222The proof can be found in Appendix.. The survival probability of the
targets is set to be $p_{S}=0.99$. The birth LMB RFS is set to be
$\pi_{B}=\\{(r_{B}^{(i)},p_{B}^{(i)})\\}_{i=1}^{6}$, where $r_{B}^{(i)}=0.03$,
$p_{B}^{(i)}=N(x;m_{B}^{(i)},P_{B})$, $m_{B}^{(1)}=[-800,0,600,0]^{T}$,
$m_{B}^{(2)}=[-800,0,-200,0]^{T}$, $m_{B}^{(3)}=[-850,0,-200,0]^{T}$,
$m_{B}^{(4)}=[-750,0,-200,0]^{T}$, $m_{B}^{(2)}=[-650,0,670,0]^{T}$,
$m_{B}^{(3)}=[-750,0,530,0]^{T}$, $P_{B}=10^{2}I_{4\times 4}.$

The radar receives range measurements over the region
$[-1000,1000]m\times[-1000,1000]m$. The standard deviations of the measurement
noise is set to be $\varrho_{r}=10$. The detection probability of this radar
is set to be $p_{D}=0.98$. The clutter measurements follow the Possion RFS
with $\kappa(z)=\lambda U(z)/V$, where $\lambda=30$, $U(\cdot)$ represents a
uniform density over the observed region and $V=\int U(z)dz$.

The performance of the proposed LMB filter with agumented LRFSs is compared
with the traditional LMB filter (Reuter et al, 2014). In both the two filters,
the parameters such as the maximal number of components and the pruning
threshold are the same. In addition, for the proposed LMB filter with
agumented LRFSs, the group threshold $\varepsilon$ in (70) is set to be
$100m$.

###  5.2 Simulation Results

The trajectories of these targets are presented in Fig. 4. One can find that
these targets can be divided into two groups, where targets with label 1, 2,
and 3 form group 1, and the rest belongs to group 2.

Figure 4: The trajectories of targets.

Fig. 5 and Fig. 6 shows the tracking results of the proposed LMB filter with
the agumented LRFSs and the LMB filter over one trail, respectively. It can be
seen that both the two filters can successfully tracks all of the targets.
Fig. 7 shows the corresponding cardinality estimates of the two filters.
Obviously, both the two filters can accurately estimate the number of targets
in most time.

However, due to the integration of group information, the proposed LMB filter
with agumented LRFSs is expected to perform better performance than the LMB
filter, which can be verified by Fig. 8. Fig. 8 illustrates the corresponding
average OSPA distance of the two filters over 100 Monte Carlo (MC) trials. It
can be seen that the proposed LMB filter with agumented LRFSs performs better
than the LMB filter. In other words, comparing with the LMB filter, the
proposed LMB filter with agumented LRFSs achieves smaller OSPA distance, which
means better tracking performance.

Lastly, we investigate the estimated results of group number. As shown in Fig.
9, the proposed filter with the agumented LRFSs can accurately estimate the
number of groups in most time. As a comparison, the LMB filter does not have
the ability to estimate the number of groups due to the lack of the
intergation of group information.

Figure 5: Measurements, true trajectories, and the tracking results of the
proposed LMB filter with agumented LRFS.
Figure 6: Measurements, true trajectories, and the tracking results of the
proposed LMB filter.
Figure 7: Cardinality estimates.
Figure 8: Average OSPA distance of order $p=1$ and cut-off $c=100$ over 100 MC
trails.
Figure 9: Group number estimates.

##  6 Conclusion

This paper has addressed the issue of group target tracking. By integrating
the group information of individual target into an LRFS, a new kind of LRFS,
named augmented LRFSs, are developed. Furthermore, by the means of the LMB
filter with the proposed augmented LRFSs, the group information, along with
the multi-target state and track labels, have been propagated and updated in
the tracking process. In this way, the estimation of group structure and
multi-target state are integrated, and a holistic group target tracking are
achieved. At last, numerical simulations have been presented to show the
application of the proposed augmented LRFSs in the solving of the issue of
group target tracking.

## Appendix

Suppose that the transition density of the group center follows

$f(c_{+}|c,l)=N(c_{+};Fc,Q),$ (74)

then (74) is equivalent to

$c_{k}=Fc_{k-1}+v_{k},$ (75)

where $v_{k}$ follows the Gaussian distribution with zero mean and covariance
$Q$. Assume that the evolution of group follows the leader follower model,
which means that the deterministic state of any target is a translational
offset of the center (leader) of the group (Zhang et al., 2022; Li and
Godsill, 2020), then, we define the offset as

$\bigtriangleup=x_{k}-c_{k}=x_{k-1}-c_{k-1},$ (76)

and we further have,

$\displaystyle x_{k}$ $\displaystyle=$ $\displaystyle c_{k}+\bigtriangleup$
(77) $\displaystyle=$ $\displaystyle Fc_{k-1}+v_{k}+x_{k-1}-c_{k-1}$ (78)
$\displaystyle=$ $\displaystyle(F-I)c_{k-1}++x_{k-1}+v_{k}.$ (79)

Note that (79) is equivalent to $f(x_{+}|x,c,l)=N(x_{+};(F-I)c+x,Q)$. Thus, if
the transition density of the group center follows $f(c_{+}|c)=N(c_{+};Fc,Q)$,
$f(x_{+}|x,c,l)=N(x_{+};(F-I)c+x,Q)$ holds true.

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[1]IsobelParry [1]PaulRitchie [1]PeterCox

1]College of Engineering, Mathematics and Physical Sciences, University of
Exeter, Exeter, UK, EX4 4QE

Isobel Parry<EMAIL_ADDRESS>
# Evidence of Amazon rainforest dieback in CMIP6 models

###### Abstract

Amazon forest dieback is seen as a potential tipping point under climate
change. These concerns are partly based-on an early coupled climate-carbon
cycle simulation, that produced unusually strong drying and warming in
Amazonia. In contrast, the 5th generation Earth System Models (CMIP5) produced
few examples of Amazon dieback under climate change. Here we examine results
from seven 6th generation models (CMIP6) which include vegetation dynamics,
and in some cases interactive forest fires. Although these models typically
project increases in area-mean forest carbon across Amazonia under CO2-induced
climate change, five of the seven models also produce abrupt reductions in
vegetation carbon which indicate localised dieback events. The Northern South
America region (NSA), which contains most of the rainforest, is especially
vulnerable in the models. These dieback events, some of which are mediated by
fire, are preceded by an increase in the amplitude of the seasonal cycle in
near surface temperature, which is consistent with more extreme dry seasons.
Based-on the ensemble mean of the detected dieback events we estimate that
7+/-5 $\%$ of the NSA region will experience abrupt downward shifts in
vegetation carbon per oC of global warming above 1.5oC.

A ‘tipping point’ commonly refers to small changes to input levels causing a
system to abruptly transition to some alternative (often less desirable)
stable state (Lenton et al., 2008). Future tipping points pose a risk to both
natural ecosystems and, by extension, human activities, as they produce abrupt
system wide changes that are often difficult or even impossible to reverse
(Lenton et al., 2013). The Amazon rainforest is one example in the climate
system that is at risk of experiencing a tipping event, with the possibility
of abrupt forest dieback in response to rising global temperatures (Cox et
al., 2004). Amazon dieback has the potential to accelerate global warming
through reducing the Amazon’s ability to act as a carbon sink, and releasing
carbon dioxide that would lead to additional global warming (Cox et al.,
2000). Tipping points may play an important role in the future of our changing
climate (Jørgensen et al., 2014; Lenton et al., 2013), with previous analysis
of CMIP5 models suggesting that multiple regional abrupt transitions could
occur for global warming levels less than 2 degree Celsius (Drijfhout et al.,
2015).

There are several factors which could contribute to a decline in vegetation in
the Amazon, including a lengthened dry season, increased fire frequency, and
reduced precipitation (Malhi et al., 2009). The number of extreme hot and dry
days in the Amazon is predicted to increase with global warming (Vogel et al.,
2020) and the length and intensity of the dry season expected to intensify
(Malhi et al., 2009). Further drying in the Amazon is anticipated from the
slowdown of the Atlantic Meridional Overturning circulation due to ice melt
causing an influx of fresh water into the North Atlantic (Lenton et al.,
2019). Moisture stress resulting from severe droughts in the Amazon is likely
to result in a degree of tree mortality (Phillips et al., 2009). As the
rainforest dries it becomes more vulnerable to fire which, coupled with the
increased frequency of fires seen in the Amazon over recent years, could lead
the rainforest to pass a tipping point and result in vegetation dieback (Malhi
et al., 2009; Aragão et al., 2018). Anthropogenic deforestation also
contributes to this by reducing dry season rainfall and decreasing the
resilience of the forest to climate change, potentially leading to permanent
forest loss in some regions of the Amazon (Zemp et al., 2017). Mechanisms
which result in the drying of the Amazon rainforest can therefore be
considered to be the main cause of vegetation dieback.

Dieback tipping events are primarily thought of as bifurcation type tipping
points (Ritchie et al., 2021), which occur when external climatic factors
reduce the resilience of a state (e.g. forest) and ultimately cause the system
to tip into a new contrasting state (e.g. savannah) (Scheffer et al., 2009).
For some bifurcation type tipping points there are generic features of a
system that can be detected to indicate the approach of a tipping point
(Scheffer et al., 2009; Dakos et al., 2008). The most common example is
‘critical slowing down’, where a system becomes increasingly slow at
recovering from small perturbations as negative feedbacks become overwhelmed
by positive ones. Critical slowing down can be observed by increases to the
autocorrelation and variance in a state variable (Scheffer et al., 2009; Dakos
et al., 2008). A recent study which looked at such signals in satellite-
retrieved vegetation greenness, reports evidence of reducing resilience of the
Amazon rainforest since 2005 (Boulton et al., 2022).

However, previous research into projections of Amazonian vegetation dieback
has also suggested that generic EWS such as these fail in the Amazon, but that
more system specific indicators may be found (Boulton et al., 2013). For
example, the interannual variability of the atmospheric CO2 concentration, as
a function of tropical temperature variability, has been shown to be connected
to the sensitivity of tropical carbon to climate change (Cox et al., 2013;
Wenzel et al., 2014). This metric shows trends which are also consistent with
reducing resilience of tropical forests (Wang et al., 2014; Luo and Keenan,
2022).

These recent studies (Boulton et al., 2022; Luo and Keenan, 2022) focus on
fairly short observational records. In this paper we look instead at
projections from the latest CMIP6 Earth System Models for evidence of Amazon
dieback, and identify a precursor which is based-on longer-term temperature
records.

##  1 Methods

###  1.1 CMIP6 models, experiment runs and data used.

Climate models that incorporate dynamic vegetation and are from the 6th Phase
of the Coupled Model Intercomparison Project CMIP6 were utilised in this
study. See Table 1 for the corresponding seven CMIP6 climate models. For the
purpose of this study we wanted to focus on the climatic drivers alone
impacting vegetation and therefore make use of the 1pctCO2 runs. Data from the
PIControl runs were also used to determine each model’s internal variability.
Primarily, we use model output data of the vegetation carbon and surface
temperature for the seven climate models in the NSA region. The amplitude of
the temperature seasonal cycle in this study is defined as the difference
between the maximum and minimum monthly mean for each year. All anomalies
presented correspond to the yearly mean relative to the mean of the first ten
years, aside from temperature anomalies which correspond to the ten year
running mean relative to the mean of the first ten years.

Table 1: CMIP6 models used within this study Model | Institution
---|---
EC-Earth3-Veg | EC-Earth-Consortium
GFDL-ESM4 | NOAA-GFDL
MPI-ESM1-2-LR | Max-Planck-Institut für Meteorologie
NorCPM1 | EarthClim
TaiESM1 | AS-RCEC
SAM0-UNICON | Seoul National University
UKESM1-0-LL | Met Office Hadley Centre

###  1.2 Abrupt shift detection algorithm.

The algorithm used to detect abrupt shifts is relatively simple by design.
Three criteria must be be fulfilled for a grid point to be identified as
containing an abrupt shift in the vegetation carbon. Namely, the vegetation
carbon must change by at least 2 kgC/m2 over a 15-year period and that this
must contribute to at least 25% of the overall change in vegetation carbon.
Finally, to remove detected abrupt shifts that might be due to a model’s high
internal variability the mean annual rate of change of the abrupt shift must
be at least three times larger than the variability of the rates of change in
the unforced control run.

Grid points where abrupt shifts were detected are subsequently sorted based on
the direction of the abrupt shift (positive or negative) and the direction of
overall trend (positive or negative). This results in four classifications of
abrupt shifts, where analysis focused on the dieback abrupt shifts
corresponding to the overall trend and direction of the abrupt shift being
negative. This type of abrupt shift can be used as an analogy for a tipping
event where a region changes equilibrium state from rainforest to savannah.

###  1.3 EWS for Amazon dieback.

Our system specific EWS for Amazon dieback is to observe high sensitivities of
the amplitude of the seasonal cycle to global warming. This is defined as the
gradient of a linear regression fit to the amplitude of the temperature
seasonal cycle against global warming. For a comparison between grid points
with abrupt shifts and those that do not, the regression is fitted against the
first 73 years of data corresponding to when CO2 has doubled from pre-
industrial levels (noting that most abrupt shifts occur after a doubling of
CO2). Area weighted histograms of the sensitivities are calculated to compare
the distributions for areas with abrupt vegetation dieback (red) and areas
with no abrupt shifts (purple). The mean of the seven model histograms was
taken to produce the compiled models histogram in Fig. 4h. The percentage risk
of an abrupt dieback shift occurring (Fig. 4i) is calculated as the percentage
of abrupt dieback grid points to all grid points for the specified
sensitivity.

##  2 Results

Figure 1: Abrupt shifts detected in the Amazon by the described algorithm and
example time series for dieback shifts. (a-h) Maps of detected abrupt shifts.
Red grid points indicate detected dieback shifts where the direction of the
overall trend (T) and the abrupt shift (AS) are both negative. Green points
indicate that the direction of trend and abrupt shift are both positive. Blue
points indicate a positive abrupt shift but a negative trend, while orange
points indicate a negative abrupt shift associated with a positive trend. (h)
Example time series for detected dieback shifts in each model, corresponding
to grid points highlighted by black squares.

###  2.1 Detection of abrupt shifts

We focus our analysis on detecting Amazon dieback abrupt shifts in seven
state-of-the-art climate models, which all enable dynamic vegetation, from the
6th Phase of the Climate Model Intercomparison Project (CMIP6). Specifically,
we are interested in climate change induced dieback (rather than direct
deforestation) and therefore consider the idealised scenario of $CO_{2}$
increasing 1% per year starting from pre-industrial levels. We have defined
that for an abrupt shift to be detected a $2kgC/m^{2}$ change must be observed
within a 15-year period and this change must contribute to over a quarter of
the change observed in the entire simulation run. We additionally require that
the mean annual rate of change is more than 3 times as large as the
variability in the rates of change seen in the unforced control runs (see
Methods for further details).

The algorithm categorises abrupt shifts into four types dependent on the
directions of the abrupt shift (AS) itself and the overall trend (T; change in
5-year means between start and end of the run). Figure 1a–g depicts the types
of abrupt shifts detected spatially for the CMIP6 models. Conventional dieback
abrupt shifts (red; $T<0$, $AS<0$), which indicate a move towards a savannah
state, are predominantly detected. Three models; NorCPM1, TaiESM1 and
SAM0-UNICON all show a clustering of dieback abrupt shifts in the north of the
Amazon. GFDL-EMS4 also presents a coherent structure with abrupt shifts
clustered in central-west Amazonia. Contrast to this, EC-Earth3-Veg shows many
abrupt shifts scattered across the Amazon basin. This may be due to the high
natural variability that is inherent in this model and which is difficult to
distinguish from climate forced abrupt shifts. Interestingly, UKESM1-0-LL
displays no dieback events, despite showing large scale dieback in previous
CMIP generations (Cox et al., 2004). Similarly, very few abrupt shifts are
detected in the MPI-ESM1-2-LR model.

Some sample time series of detected dieback abrupt shifts across the models
are shown in Fig. 1h. Between the models there is some variation in the
general shape of abrupt shift time series, however, most exhibit a change of
state from one equilibrium to another that would be expected of a tipping
event.

Figure 2: Plots showing the evolution of abrupt dieback shifts and vegetation
carbon with global warming. (a) IPCC AR6 defined regions of the Amazon. (b)
Plot showing the evolution of the percentage of the NSA region that has
experienced a dieback shift with increasing global warming. The plume
indicates the error in the averaged compiled models line (black). (c) Plot
showing how the vegetation carbon anomaly relative to the mean of the first
ten years evolves with global warming. The plume indicates the error in the
averaged compiled model line (black).

###  2.2 Evolution and impact of abrupt shifts in the NSA region

Henceforth, we restrict our analysis to the IPCC AR6 defined North South
America (NSA) region, which contains the majority of the Amazon basin (see
Fig. 2a), and features many of the detected abrupt dieback shifts (red points
in Fig. 1). Figure 2b shows how the fractional area of the NSA region to
experience an abrupt dieback shift evolves for increasing global warming. Some
models show clear jumps in the NSA area to experience an abrupt shift
reflecting multiple gird points featuring an abrupt shift at a similar level
of warming. Strikingly, TaiESM1 shows about 20% of the NSA region suffering an
abrupt dieback event at about $1.7^{o}C$ warming and by $3^{o}C$ global
warming about 40% of the NSA region would experience an abrupt shift. The bold
black line represents the mean behaviour and the plume variability of all
seven models. Although some of the large jumps from individual models can
still be identified, the model mean shows a smoother increase in fractional
NSA area to undergo an abrupt shift under global warming. There is not a
singular temperature threshold, instead the risk of tipping increases
(approximately linearly) between $1.3^{o}C$ and $3^{o}C$ of warming and
reaches approximately 20% of the NSA region to undergo an abrupt dieback
shift.

Interestingly, when evaluating at the regional scale many of the abrupt shifts
do not appear to materialise, despite identifying a significant number of
local abrupt shifts, (see Fig. 2c). Only the cluster of abrupt shifts at
approximately $1.3^{o}C$ warming for GFDL-ESM4 appear in the total vegetation
carbon anomaly for the NSA region. Furthermore, the CMIP6 models do not even
agree on the sign change in vegetation carbon.

Figure 3: Time series showing how the temperature seasonal cycle amplitude
evolves over time for grid points which experience abrupt dieback shifts.
(a-c) Time series of selected dieback shifts for three models at grid points
5∘S 65∘W, 0∘ 60∘W and 0∘ 60∘ respectively. The red dotted line indicates the
midpoint of the 15 year period where the abrupt shift is detected by the
algorithm. (d-f) The change in the amplitude of the temperature seasonal cycle
with time and $CO_{2}$ for these dieback shifts in each model. The solid lines
represents the 10 year running average of the seasonal cycle amplitude, while
the dotted lines are the yearly data.

###  2.3 An EWS for Amazonian dieback

Three identified abrupt shifts, which all occur around a doubling of $CO_{2}$
in different models, are shown in Fig. 3a–c and all show a change of
equilibrium state after the abrupt shift. Initially, vegetation carbon may
increase due to the $CO_{2}$ fertilization effect (c.f. Fig. 3a,b), however
there exists a critical threshold in the $CO_{2}$ concentration at which
increased temperature and drying overwhelm the positive effect of $CO_{2}$ and
results in an abrupt dieback shift. Fig. 3d–f show the trend in the
temperature seasonal cycle associated with these three grid points. An
increasing trend is observed in the amplitude of the seasonal cycle in the
lead up to an abrupt shift for each point, suggesting that this increase in
variability may be an EWS for abrupt dieback events. This behaviour can be
expected in the lead up to a vegetation dieback shift because the length and
intensity of the dry season has been shown to increase due to drying in the
Amazon (Vogel et al., 2020; Malhi et al., 2009).

Figure 4 investigates the robustness of using the temperature seasonal cycle
amplitude as an EWS for an impending dieback event. Specifically, we compare
the distributions of grid points possessing an abrupt shift and those without
for the temperature seasonal cycle amplitude sensitivity to global warming.
The sensitivity is calculated up to a doubling of $CO_{2}$ (most abrupt shifts
are detected after doubling of $CO_{2}$) for all grid points and models (Fig.
4a-g). The grid point are colour coded according to whether an abrupt shift
occurs and its type. Four of the five models that contain abrupt shifts within
the NSA region (GFDL-ESM4, NorCPM1, SAM0-UNICON, TaiESM1; Fig. 4b,d,e,f)
display clear thresholds in the sensitivity, such that above the thresholds
only grid points with an abrupt shift feature. EC-Earth3-Veg (Fig. 4a)
provides an exception, however due to the high stochasticity of the model some
of the detected abrupt shifts are likely to be simply natural variation.
Promisingly, MPI-ESM1-2-LR and UKESM1-0-LL (Fig. 4c,g) do not possess grid
points with high sensitivities and could therefore offer an explanation for
not exhibiting any abrupt shifts in the NSA region.

Figure 4: Plots demonstrating the efficacy of the temperature seasonal cycle
amplitude as an EWS and how it may be used to indicate the risk of an oncoming
abrupt dieback shift. (a-g) Histograms with bins of width 0.2 showing the
percentage area of the NSA region that have different sensitivities of the
temperature seasonal cycle amplitude to global warming. The bar colors
correspond to the type of abrupt shift detected at the grid point with a
specific sensitivity, as in Figure 1, where purple bars indicate grid points
where no abrupt shifts are detected. (h) The mean of histograms a-g. (i) A bar
chart showing how the percentage risk of a grid point, in any of the 7
analysed models, experiencing a dieback shift changes with increasing
sensitivity of the seasonal cycle amplitude to global warming

Taking an ensemble mean of all the models, shows that grid points without an
abrupt shift tend to have sensitivities centred around zero, whereas abrupt
shift grid points are positively skewed to higher sensitivities (see Fig. 4h).
This means that the risk of a grid point having an abrupt shift (defined as
the ratio of gird points with an abrupt dieback shift to all grid points for
each sensitivity) generally increases for grid points with higher
sensitivities to global warming as shown in Fig. 4i. The minimum risk of a
grid point experiencing an abrupt shift is for a sensitivity close to 0, where
the seasonal cycle amplitude is unaffected by global warming. As the
sensitivity increases from 0.5 to 1.0$K/K$ the risk of a grid point containing
an abrupt shift increases approximately linearly from 10% to 60%. For
sensitivities greater than 1.0$K/K$ the risk remains between 60% and 80%
providing a potential EWS.The risk also increases to 35% for negative
sensitivities, however this is largely from the EC-Earth3-Veg model in which
it is not clear how many of the detected shifts are indeed abrupt.

##  3 Discussion

The effects of abrupt shifts observed in the NSA region may be limited if
anthropogenic climate change is restricted to below 1.5 degrees, as set out in
the aims of the Paris Agreement (UNFCCC, 2015). Exceeding 1.5°C warming is
likely to result in sharp increases in the areas experiencing abrupt shifts.
Despite large areas of the Amazon experiencing tipping events with warming,
the abrupt shifts observed in Fig. 2 may be considered localised events. These
are largely balanced out by the increase in vegetation carbon seen elsewhere
in the NSA region, likely resulting from CO2 fertilization, where an increase
in photosynthesis rate results in an increase in biomass (Cox et al., 2004;
Huntingford et al., 2013). This appears to indicate that large scale regional
dieback, as observed in previous generations of models, are not present in
CMIP6 and the impacts will be more localised. Thus, despite the CMIP6 models
failing to agree on the overall impact of vegetation carbon in the Amazon,
abrupt shifts remain a threat to local communities and ecosystems.

Our analysis shows that typically the sensitivity of the amplitude of the
temperature seasonal cycle to global warming is higher for grid points
subsequently featuring an abrupt dieback shift, compared with grid points with
no abrupt dieback (Fig. 4). This therefore offers the possibility of using
this sensitivity as a system specific early-warning signal for future dieback
in the Amazon. The increase in risk observed for negative sensitivities also
could mean that any changes in moisture and temperature cycles in the Amazon
suggest an increased risk of an abrupt shift occurring.

We find evidence of clustered localised abrupt dieback shifts in over half the
CMIP6 models analysed, however, this analysis is limited by the number of
CMIP6 models containing dynamic vegetation. Additionally, we use the idealised
1% per year increase of CO2 run to focus on abrupt dieback shifts caused
solely by anthropogenic climate change, though abrupt dieback can also be
caused through land use changes such as deforestation.

##  4 Conclusions

Anthropogenic climate change could result in localised tipping events
occurring in the Amazon rainforest, as observed in several CMIP6 models. The
dieback events detected would have severe consequences for local communities
and ecosystems. This study suggests that 7+/-5 $\%$ of the Northern South
America region would experience abrupt downward shifts in vegetation carbon
per oC of global warming above 1.5oC.

Further research is needed to assess the risk of tipping events under climate
change and to identify forewarning methods that can be applied to
observational data. However, our results indicate that the sensitivity of the
amplitude of the temperature seasonal cycle to global warming is a promising
potential early warming signal for local Amazon forest dieback.

The CMIP6 model output datasets analysed during this study are available
online at [https://esgf-node.llnl.gov/search/cmip6/]

IP carried out the data analysis and drafted the paper. PR and PC advised on
the stufy. All authors contributed to the submitted paper.

No competing interests

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