# UAV Trajectory Tracking via RNN-enhanced IMM-KF with ADS-B Data

Yian Zhu∗, Ziye Jia∗, Qihui Wu∗†, Chao Dong∗, Zirui Zhuang‡, Huiling Hu§ and
Qi Cai§
∗The Key Laboratory of Dynamic Cognitive System of Electromagnetic Spectrum
Space, Ministry of Industry and Information Technology, Nanjing University of
Aeronautics and Astronautics, 210016
‡State Key Laboratory of Networking and Switching Technology, Beijing
University of Posts and Telecommunications, Beijing, China, 100876
§Middle-south Regional Air Traffic Management Bureau of CAAC, 510403
†Corresponding author, email<EMAIL_ADDRESS>
###### Abstract

With the increasing use of autonomous unmanned aerial vehicles (UAVs), it is
critical to ensure that they are continuously tracked and controlled,
especially when UAVs operate beyond the communication range of ground stations
(GSs). Conventional surveillance methods for UAVs, such as satellite
communications, ground mobile networks and radars are subject to high costs
and latency. The automatic dependent surveillance-broadcast (ADS-B) emerges as
a promising method to monitor UAVs, due to the advantages of real-time
capabilities, easy deployment and affordable cost. Therefore, we employ the
ADS-B for UAV trajectory tracking in this work. However, the inherent noise in
the transmitted data poses an obstacle for precisely tracking UAVs. Hence, we
propose the algorithm of recurrent neural network-enhanced interacting
multiple model-Kalman filter (RNN-enhanced IMM-KF) for UAV trajectory
filtering. Specifically, the algorithm utilizes the RNN to capture the
maneuvering behavior of UAVs and the noise level in the ADS-B data. Moreover,
accurate UAV tracking is achieved by adaptively adjusting the process noise
matrix and observation noise matrix of IMM-KF with the assistance of the RNN.
The proposed algorithm can facilitate GSs to make timely decisions during
trajectory deviations of UAVs and improve the airspace safety. Finally, via
comprehensive simulations, the total root mean square error of the proposed
algorithm decreases by 28.56%, compared to the traditional IMM-KF.

###### Index Terms:

UAV trajectory tracking, ADS-B, RNN-enhanced IMM-KF.

##  I Introduction

Unmanned aerial vehicles (UAVs) draw a large amount of attentions due to
leading economic development in the low-altitude field [1], [2]. In civilian
scenarios, such as industrial inspection and goods delivery, the autonomous
flight technology is widely applied to UAVs [3]. In detail, UAVs follow the
pre-planed flight path by the ground station (GS) to carry out the mission.
However, due to the autonomy of UAVs, real-time flight monitoring information
can not be transmitted to GSs when UAVs exceed the communication range of GSs.
Therefore, it is challenging to monitor UAVs in real time to ensure flight
safety. In response to this challenge, the satellite communication, ground
mobile network communication and radar surveillance emerge as viable options
for monitoring flight trajectories. However, the aforementioned techniques are
limited by their properties. For example, satellite communications have higher
latency and are susceptible to adverse weather conditions [4]. Ground mobile
network communications may suffer from signal interference in densely
populated areas and their coverage is affected by the distribution of GSs [5].
In addition, radar deployment and construction are constrained by cost and
terrain. Compared to the above methods, the automatic dependent surveillance-
broadcast (ADS-B) presents an efficient aircraft monitoring solution with low
cost, low latency and wide coverage [6]. Specifically, ADS-B consists of two
components: ADS-B OUT and ADS-B IN. ADS-B OUT can be installed on UAVs as a
mobile unit to broadcast real-time flight status information, such as the
position and speed. ADS-B IN is used on the ground to receive and decode the
received data from ADS-B OUT [7]. Such real-time information can assist GSs to
make timely decisions when anomalous behaviors occur, such as deviations from
the flight path. However, UAVs possess small size, high maneuverability and
strong autonomous flight capabilities, imposing higher accuracy requirements
on their flight trajectory monitoring [8]. Moreover, there exists noise caused
by signal transmission interference, atmospheric disturbance and other factors
during data transmission process via ADS-B. It is intractable to make the
analysis and assessment of the true trajectory status of UAVs due to the
noise.

Considering the challenges posed by noise during data transmission via ADS-B,
it becomes crucial to employ effective filtering techniques to reduce the
impact of noise. In this context, the Kalman filter (KF) is employed as a key
technology in various fields such as control, estimation and navigation, and
is widely used in target tracking [9]. Leveraging the dynamic model and the
observation equation of the target system, KF obtains accurate state
estimation. However, when tracking maneuvering targets, it is challenging due
to the model mismatch, leading to performance degradation or result divergence
[10]. In this context, [11] introduces the interacting multiple model (IMM)
algorithm, which is commonly used to track maneuvering targets. By defining
multiple dynamic models of target and switching among them according to
probabilities, IMM enables continuous tracking. Furthermore, deep learning
technologies show remarkable effectiveness in improving the performance of
filters by learning information from large amounts of data [12]. For example,
[13] proposes an adaptive tracking method based on long short-term memory
(LSTM), but the designed algorithm lacks interpretability of the model due to
the abandonment of the traditional filtering framework. [14] proposes a
KalmanNet neural network (NN) based on gated recurrent unit cells, and
combines with KF to achieve real-time state estimation. However, these works
do not consider the scenario of applying ADS-B to UAV trajectory tracking.
Besides, the system model for UAVs in three-dimensional space is ignored.

Therefore, in this paper, we emphasis on utilizing ADS-B as a method of
transmitting UAV trajectory information to achieve real-time situation
awareness of UAV flight. To filter the noise in ADS-B data, we adopt an IMM-KF
algorithm and define dynamic models corresponding to different flight modes of
maneuvering UAVs. To tackle issues from the ever-changing noise and
uncertainties of accurate models, we design a recurrent neural network-
enhanced (RNN-enhanced) IMM-KF algorithm. By conducting simulation results
compared to traditional IMM-KF, the superiority of the proposed algorithm is
verified in overcoming performance degradation caused by model mismatch and
inaccuracy. Also, the approach realizes precise monitoring of UAV
trajectories.

The rest of the paper is organized as follows. The system model and problem
formulation are presented in Section II. Then, Section III proposes the RNN-
enhanced IMM-KF algorithm. Numerical results are provided in Section IV and
conclusions are drawn in Section V.

##  II System Model And Problem Formulation

###  II-A Trajectory and Observation Model Based on ADS-B

As depicted in Fig. 1, a UAV equipped with ADS-B executes autonomous flight
tasks in a specific operational area. The UAV transmits key state information
data obtained from the on-board sensing devices to the GSs via the ADS-B. The
state information of the UAV received by the GSs at discrete-time step
$k\in\mathbb{T}$ is denoted as
$S_{k}=\left\\{\operatorname{lon}_{k},\operatorname{lat}_{k},\operatorname{alt}_{k},v_{k},\psi_{k},\theta_{k}\right\\}$.
$\operatorname{lon}_{k}$, $\operatorname{lat}_{k}$, $\operatorname{alt}_{k}$,
$v_{k}$, $\psi_{k}$ and $\theta_{k}$ represent the longitude, latitude,
altitude, speed, heading angle, and pitch angle, respectively. It is observed
that the data transmitted by ADS-B provides comprehensive, real-time flight
state information of the UAV. For computational convenience and uniformity in
data processing, we standardize the units and coordinate system of the
observed data. In other words, the latitude and longitude information of the
UAV, which is obtained from ADS-B and operated in the World Geodetic System
1984 coordinate system, is firstly mapped into the Cartesian coordinate system
using the Gauss-Kruger projection [13]. Furthermore, the velocity $v_{k}$ is
decomposed into the velocity components $(v_{x_{k}},v_{y_{k}},v_{z_{k}})$ in
the Cartesian coordinate system by heading angle $\psi_{k}$ and pitch angle
$\theta_{k}$. According to the coordinate transformation and velocity
decomposition, the final observation data from ADS-B at time instance $k$ is
given as
$\mathbf{o}_{k}=\left[x_{k},y_{k},z_{k},v_{x_{k}},v_{y_{k}},v_{z_{k}}\right]^{\text{\tiny$T$}}$.
Additionally, the observation equation for the system is expressed as:

$\mathbf{o}_{k}=\mathbf{H}\cdot\mathbf{x}_{k}+\mathbf{v}_{k},\quad\mathbf{v}_{k}\sim\mathcal{N}(0,\mathbf{R}),\quad\mathbf{o}_{k}\in\mathbb{R}^{n},$
(1)

where $\mathbf{H}$ is the observation matrix linking the state
$\mathbf{x}_{k}$ to the observed data, and $\mathbf{v}_{k}$ is the Gaussian
white noise with covariance matrix $\mathbf{R}$. In this scenario, the
covariance matrix $\mathbf{R}$ is expressed as
$\mathbf{R}=diag\left(\sigma_{x}^{2},\sigma_{y}^{2},\sigma_{z}^{2},\sigma_{vx}^{2},\sigma_{vy}^{2},\sigma_{vz}^{2}\right)$.

Figure 1: UAV trajectory tracking system with ADS-B information.

###  II-B Dynamic System Model

We consider the discrete-time state transition equation to describe the
dynamic system of UAVs, denoted as

$\mathbf{x}_{k}=\mathbf{F}\cdot\mathbf{x}_{k-1}+\bm{\omega}_{k},\quad\bm{\omega}_{k}\sim\mathcal{N}(0,\mathbf{Q}),\quad\mathbf{x}_{k}\in\mathbb{R}^{m},$
(2)

where $\mathbf{F}$ is the state transition matrix and $\bm{\omega}_{k}$
represents Gaussian white noise with covariance matrix $\mathbf{Q}$.

Considering that the flight patterns of UAVs are unable to be characterized
accurately by a single model, we define two types of flight states: stable
cruising and high-agility maneuvering, respectively. The corresponding two
dynamic models are described as follows:

####  II-B1 Constant Velocity (CV) model

The state of UAVs in stable cruising state scenario is characterized by the CV
model. In detail, the state of UAVs is captured with the position
$\left(x_{k},y_{k},z_{k}\right)$ and velocity
$\left(v_{xk},v_{yk},v_{zk}\right)$. Accordingly, we derive the state vector
$\mathbf{x}_{k}=\left[x_{k},y_{k},z_{k},v_{xk},v_{yk},v_{zk}\right]^{\text{\tiny$T$}}$.
The associated state transition matrix $\mathbf{F}_{CV}$ and process noise
covariance matrix $\mathbf{Q}_{CV}$ are respectively given as:

$\mathbf{F}_{CV}=\left[\begin{array}[]{ll}\mathbf{I}_{3\times
3}&T\mathbf{I}_{3\times 3}\\\ 0_{3\times 3}&T\mathbf{I}_{3\times
3}\end{array}\right],$ (3)

and

$\mathbf{Q}_{CV}=\sigma_{vw}^{2}\left[\begin{array}[]{cc}\frac{T^{3}}{2}\mathbf{I}_{3\times
3}&\frac{T^{2}}{2}\mathbf{I}_{3\times 3}\\\ \frac{T^{2}}{2}\mathbf{I}_{3\times
3}&T\mathbf{I}_{3\times 3}\end{array}\right],$ (4)

where $T$ represents the periodic time interval at which ADS-B OUT system
broadcasts the state, and $\mathbf{I}$ denotes the identity matrix.
$\sigma_{vw}^{2}$ indicates the variance associated with the velocity of UAVs,
which can also be set separately in the dimensions of $x$, $y$, and $z$. In a
physical sense, the matrix $\mathbf{F}_{CV}$ indicates the UAVs move in a
straight line with a uniform speed in all directions over time interval $T$.
In other words, UAVs are in stable cruising state. Additionally, the process
noise matrix $\mathbf{Q}_{CV}$ enables the filter to better cope with
potential nonlinear motions of UAVs. In practice, it improves the robustness
of unpredictable changes in the velocity of UAVs that cannot be directly
captured by a linear model.

####  II-B2 Constant Jerk (CJ) model

For high-agility maneuvers, CJ model can be used to represent the motion of
UAVs. The state vector of UAVs is given by
$\mathbf{x}_{k}=\left[x_{k},y_{k},z_{k},v_{xk},v_{yk},v_{zk},a_{xk},a_{yk},a_{zk},j_{xk},j_{yk},j_{zk}\right]^{\text{\tiny$T$}}$,
where $(a_{xk},a_{yk},a_{zk})$ represents the accelerations in directions of
$x$, $y$ and $z$, respectively, and $(j_{xk},j_{yk},j_{zk})$ denotes the
jerks, i.e., the change rate of accelerations. According to
[mehrotra1997jerk], the corresponding state transition matrix
$\mathbf{F}_{CJ}$ and the process noise covariance matrix $\mathbf{Q}_{CJ}$
are described as:

$\mathbf{F}_{CJ}=\left[\begin{array}[]{cccc}\mathbf{I}_{3\times
3}&T\mathbf{I}_{3\times 3}&\frac{T^{2}}{2}\mathbf{I}_{3\times
3}&\frac{T^{3}}{6}\mathbf{I}_{3\times 3}\\\ 0_{3\times 3}&\mathbf{I}_{3\times
3}&T\mathbf{I}_{3\times 3}&\frac{T^{2}}{2}\mathbf{I}_{3\times 3}\\\ 0_{3\times
3}&0_{3\times 3}&\mathbf{I}_{3\times 3}&T\mathbf{I}_{3\times 3}\\\ 0_{3\times
3}&0_{3\times 3}&0_{3\times 3}&\mathbf{I}_{3\times 3}\end{array}\right],$ (5)

and

$\mathbf{Q}_{CJ}=\sigma_{jw}^{2}\left[\begin{array}[]{llll}\frac{T^{7}}{252}\mathbf{I}_{3\times
3}&\frac{T^{6}}{72}\mathbf{I}_{3\times 3}&\frac{T^{5}}{72}\mathbf{I}_{3\times
3}&\frac{T^{4}}{24}\mathbf{I}_{3\times 3}\\\
\frac{T^{6}}{72}\mathbf{I}_{3\times 3}&\frac{T^{5}}{20}\mathbf{I}_{3\times
3}&\frac{T^{4}}{8}\mathbf{I}_{3\times 3}&\frac{T^{3}}{6}\mathbf{I}_{3\times
3}\\\ \frac{T^{5}}{72}\mathbf{I}_{3\times
3}&\frac{T^{4}}{8}\mathbf{I}_{3\times 3}&\frac{T^{3}}{3}\mathbf{I}_{3\times
3}&\frac{T^{2}}{2}\mathbf{I}_{3\times 3}\\\
\frac{T^{4}}{24}\mathbf{I}_{3\times 3}&\frac{T^{3}}{6}\mathbf{I}_{3\times
3}&\frac{T^{2}}{2}\mathbf{I}_{3\times
3}&\mathrm{\mathit{T}\mathbf{I}}_{3\times 3}\end{array}\right].$ (6)

Recall that $T$ is defined as the frequency at which the ADS-B OUT system
broadcasts its state. $\sigma_{jw}^{2}$ denotes the variance with respect to
the jerk of UAVs, reflecting the uncertainty of fast UAV maneuvers. Similarly,
$\sigma_{jw}^{2}$ can be set on dimensions of $x$, $y$, and $z$. The physical
implication of the matrix $\mathbf{F}_{CJ}$ is that UAVs move in all
directions with the varying accelerations over a time interval $T$. For
example, $\mathbf{F}_{CJ}$ applies to UAVs in situations involving swift
obstacle avoidance or sharp turns. Taking into account the uncertainty of the
jerk estimation during the movement of UAVs, the robustness of the state
estimation is further enhanced by matrix $\mathbf{Q}_{CJ}$.

###  II-C Problem Formulation

####  II-C1 Significance of $\mathbf{Q}$ and $\mathbf{R}$

In the field of dynamic traget tracking, robust algorithms are important. The
IMM-KF emerges as a frontrunner for its trajectory tracking capabilities. In
practice, numerous implementations leverage IMM-KF for effective dynamic
target tracking. Among them, a key factor affecting the filter efficiency is
the precise setting of the parameters, especially the process noise covariance
matrix $\mathbf{Q}$ and the observation noise covariance matrix $\mathbf{R}$.
Specifically, the martices $\mathbf{Q}$ and $\mathbf{R}$ capture the
uncertainties concerning dynamic models and the observation system,
respectively. In detail, the matrices are employed to obtain the Kalman gain
in KF, which determines the weight assigned to the observation in the update
step. Additionally, $\mathbf{Q}$ plays an important role in IMM during the
model transition process. A smaller $\mathbf{Q}$ value improves the accuracy
of the KF, it may cause delay in the model transition phase, resulting in
weaker timely tracking.

####  II-C2 Adaptive Tuning of $\mathbf{Q}$ and $\mathbf{R}$

Traditionally, in standard KF applications, $\mathbf{Q}$ and $\mathbf{R}$ are
often derived from empirical knowledge and are set as fixed parameters. This
approach may work well for systems with consistent and predictable dynamics.
However, facing system environment changes and target motion nonlinearity, the
static matrices lead to filter divergence and performance degradation, thus
reducing the reliability of the tracking system. Therefore, adaptive tuning
the accurate parameters of the matrices is a primary challenge. It becomes
tricky when domain knowledge is limited, such as partially known models and
unknown statistical properties of noise.

####  II-C3 Data-driven Adaptive Tuning

Considering the challenge of setting suitable $\mathbf{Q}$ and $\mathbf{R}$ in
scenarios with limited domain knowledge, a data-driven approach emerges as a
promising alternative. Based on the information learned by the NN from the
data, the dynamic adjustment of $\mathbf{Q}$ and $\mathbf{R}$ can compensate
for the inaccuracies of system modeling and ensure that the filter adapts to
the noise characteristics. In this way, the traditional filtering method can
be enhanced.

####  II-C4 Implementation with RNNs

RNNs with their inherent capabilities to capture sequential patterns, present
a compelling approach to the problem of adaptively tuning $\mathbf{Q}$ and
$\mathbf{R}$. These networks can be trained in a supervised manner with
groundtruth data. In the context of UAVs, high-precision sensor, such as real-
time kinematic, can be deployed to obtain groundtruth datasets. As the
filtering process continues, matrices $\mathbf{Q}$ and $\mathbf{R}$ are
continuously updated with the help of RNN to ensure that IMM-KF operates under
optimal settings.

In summary, while $\mathbf{Q}$ and $\mathbf{R}$ are fundamental parameters for
KF, it is challenging to obtain their optimal values. The utilization of RNNs
presents a promising solution. During the process of UAV trajectory filtering,
the tracking errors are key indicators for evaluating the performance of
different methods. Therefore, we resort to the root mean square error (RMSE),
a measure to illustrate the magnitude of the state estimation error. The
optimization problem is defined as follows:

$\footnotesize\begin{gathered}\min_{\Theta_{\mathrm{R}},\Theta_{\mathrm{Q}_{CV}},\Theta_{\mathrm{Q}_{CJ}}}\sqrt{\frac{1}{3N}\sum_{k=1}^{N}\left(\left(x_{k}-\hat{x}_{k}\right)^{2}+\left(y_{k}-\hat{y}_{k}\right)^{2}+\left(z_{k}-\hat{z}_{k}\right)^{2}\right)},\end{gathered}\normalsize$
(7)

where $N$ is the length of the ADS-B sampled sequence in the entire flight
trajectory. $(\hat{x}_{k},\hat{y}_{k},\hat{z}_{k})$ and $(x_{k},y_{k},z_{k})$
represent the three-dimensional position state estimation from the filter and
the realistic state values at time $k$, respectively.
$\Theta_{\mathrm{Q}_{CV}}=\left\\{\sigma_{vwx},\sigma_{vwy},\sigma_{vwz}\right\\}$,
$\Theta_{\mathrm{Q}_{CJ}}=\left\\{\sigma_{jwx},\sigma_{jwy},\sigma_{jwz}\right\\}$
and
$\Theta_{\mathrm{R}}=\left\\{\sigma_{x},\sigma_{y},\sigma_{z},\sigma_{vx},\sigma_{vy},\sigma_{vz}\right\\}$
represent the real-time matrix parameter sets to be estimated in the filtering
algorithm.

In order to obtain the minimum RMSE across all dimensions, the filtering
algorithm is expected to be equipped with the ability to adjust the optimal
$\Theta_{Q}$ and $\Theta_{R}$ in real time. In Section III, we propose the
RNN-enhanced IMM-KF algorithm to deal with the optimization problem.

##  III RNN-enhanced IMM-KF Algorithm

In this section, we present the RNN-enhanced IMM-KF algorithm. The diagram of
the algorithm is depicted in Fig. 2. The right part is the KF with the
specific RNN, which has the ability to adjust noise matrices dynamically,
illustrated in Section III-A. On the left part of Fig. 2, the IMM structure
incorporates KF1 and KF2 with CV model and CJ model, respectively, illustrated
in Section III-B. The design of the specific RNN and the overall algorithmic
workflow are demonstrated in Section III-C.

###  III-A Kalman Filter with Dynamic Noise Matrices

In this work, we improve the performance of the IMM-KF by employing an
specific RNN to dynamically generate time-varying noise matrices. Such
matrices replace the process and observation noise matrices that are fixed in
the prediction and the state update process of KF.

####  III-A1 Prediction Phase

The prediction phase predicts the current prior state estimation and error
covariance based on previous state estimations. The formulations are given as
follows:

$\mathbf{\hat{x}}_{k\mid k-1}=\mathbf{F}\mathbf{\hat{x}}_{k-1\mid k-1},$ (8)

and

$\mathbf{P}_{k\mid k-1}=\mathbf{F}\mathbf{P}_{k-1\mid
k-1}\mathbf{F}^{T}+\mathbf{Q}_{k},$ (9)

where $\mathbf{F}$ and $\mathbf{Q}_{k}$ denote the state transition matrix and
the process noise matrix generated by the RNN, respectively.

Figure 2: RNN-enhanced IMM-KF block diagram.

####  III-A2 Update Phase

In the update phase, the updated state estimation and error covariance are
computed as:

$\mathbf{\hat{x}}_{k\mid k}=\mathbf{\hat{x}}_{k\mid
k-1}+\mathbf{K}_{k}\left(\mathbf{o}_{k}-\mathbf{H}\mathbf{\hat{x}}_{k\mid
k-1}\right),$ (10)

and

$\mathbf{P}_{k\mid
k}=\left(\mathbf{I}-\mathbf{K}_{k}\mathbf{H}\right)\mathbf{P}_{k\mid k-1},$
(11)

where the Kalman gain $\mathbf{K}_{k}$ is derived from prior error covariance
$\mathbf{P}_{k\mid k-1}$, observation matrix $\mathbf{H}$, and observation
noise matrix $\mathbf{R}_{k}$ generated by the RNN, given by:

$\mathbf{K}_{k}=\mathbf{P}_{k\mid
k-1}\mathbf{H}^{T}\left(\mathbf{H}\mathbf{P}_{k\mid
k-1}\mathbf{H}^{T}+\mathbf{R}_{k}\right)^{-1}.$ (12)

###  III-B IMM for Fusion of CV and CJ Models

The IMM algorithm is used to combine the two dynamic models defined in Section
II-B. As depicted in Fig. 2, the CV and CJ models are used for KF1 and KF2,
respectively. Specifically, the IMM consists of four primary components:

####  III-B1 Interaction

In the interaction phase, the mixing probability $\mu_{ij}^{k}$ is introduced
to prepare the initial states for the filters corresponding to different
models. In detail,

$\mu_{ij}^{k}=\frac{\mu_{i}^{k-1}\lambda_{ij}}{\sum_{l=1}^{2}\mu_{l}^{k-1}\lambda_{lj}},\forall
i,j\in\\{1,2\\},$ (13)

where $\lambda_{ij}$ and $\mu_{j}^{k-1}$ represent the probability of model
transition and the posterior model, respectively. $i$ and $j$ denote the
dynamic system models in IMM-KF. The state estimation and error covariance are
the input to the KF corresponding to model $j$, computed as:

$\mathbf{\hat{x}}_{0j,k\mid
k-1}=\sum_{i=1}^{2}\mu_{ij}^{k}\mathbf{\hat{x}}_{i,k-1\mid k-1},$ (14)

and

$\displaystyle\mathbf{P}_{0j,k\mid k-1}$
$\displaystyle=\sum_{i=1}^{2}\mu_{ij}^{k}\Big{(}\mathbf{P}_{i,k-1\mid
k-1}+\left(\mathbf{\hat{x}}_{i,k-1\mid k-1}-\mathbf{\hat{x}}_{0j,k\mid
k-1}\right)$ $\displaystyle\quad\times\left(\mathbf{\hat{x}}_{i,k-1\mid
k-1}-\mathbf{\hat{x}}_{0j,k\mid k-1}\right)^{T}\Big{)}.$ (15)

####  III-B2 Filtering

Two KFs operate in parallel, executing the state prediction and update
process, as described in Section III-A.

####  III-B3 Model Probability Update

$\mu_{j}^{k}$ represents the probability that the target is in model $j$ at
time $k$. $\mu_{j}^{k}$ serves as the basis for the weights of each model
considered in the final estimation result, i.e.,

$\mu_{j}^{k}=\frac{\mathcal{L}_{j}\left(\mathbf{o}_{k}\right)\sum_{i=1}^{2}\lambda_{ij}\mu_{i}^{k-1}}{\sum_{l=1}^{2}\mathcal{L}_{l}\left(\mathbf{o}_{k}\right)\sum_{i=1}^{2}\lambda_{il}\mu_{i}^{k-1}},$
(16)

where the likelihood fucntion $\mathcal{L}_{j}$ for model $j$ at time $k$ is
for measuring the similarity between the predicted observation of the model
and the actual observed data [15].

####  III-B4 Combination

The combination step in the IMM algorithm combines the outputs from multiple
models to produce a unified state estimation, denoted as

$\mathbf{\hat{x}}_{k}=\sum_{i=1}^{2}\mu_{i}^{k}\mathbf{\hat{x}}_{i,k\mid k}.$
(17)

###  III-C Neural Network Design for IMM-KF Enhancement

It is significant for NN to store and utilize multi-step historical data due
to the handling of time series data involved in the filtering process. Hence,
we consider employing the NN architecture, depicted in Fig. 3, to obtain the
matrix parameters essential for the IMM-KF filtering process.

####  III-C1 Architecture Design

Specifically, we adopt the LSTM cell to capture the historical sequence
features. Subsequently, a fully connected (FC) layer is employed to map the
hidden state of the LSTM cell to a fixed output dimension. Additionally, a
parametric rectified linear unit (PReLU) activation function is introduced to
incorporate non-linearity. Finally, three individual FC layers, each followed
by a $tanh$ activation function, are employed to output the estimated
parameters $\Theta_{\mathrm{Q}_{CV}}$, $\Theta_{\mathrm{Q}_{CJ}}$ and
$\Theta_{\mathrm{R}}$.

Figure 3: RNN-enhanced IMM-KF flowchart.

####  III-C2 Input Feature and Algorithm Workflow

The designed NN is enabled to learn noise matrix parameters from data, which
is used in the process of IMM-KF filtering. The workflow of the RNN-enhanced
IMM-KF algorithm is illustrated in Fig. 3. We employ a sliding window to
realize the parameter estimation for adjusting and updating the noise matrices
at a specific time interval. As shown in Fig. 3, the length of the sliding
window is $\tau$. For the sequence in the first time window, i.e., the
received UAV flight trajectory sequence for the first $\tau$ time lengths,
three noise matrices $\mathbf{Q}_{0}^{CV}$, $\mathbf{Q}_{0}^{CJ}$ and
$\mathbf{R}_{0}$ are initialized with empirical knowledge. Then, we conduct
IMM-KF filtering to obtain the state estimation with length of $\tau$. We
choose the difference between the observation value at each time step
$k\in\mathbb{T}$ and the corresponding state estimation value, and then
provide it as the input feature to the NN, denoted by
$\mathbf{\delta}_{k}=\mathbf{o}_{k}-\hat{\mathbf{x}}_{k}$. In detail,
$\mathbf{\delta}_{k}$ contains the possible model mismatch information in the
IMM-KF filtering process. Since IMM-KF is a multi-model interactive filter, it
considers different models and corresponding weights to give the final state
estimation. Leveraging the mismatch information can help IMM-KF to correct the
model probability in time. Besides, $\mathbf{\delta}_{k}$ contains the noise
level information of observation data. The information can assist IMM-KF to
adapt to different levels of observation noise.

In the subsequent time window, IMM-KF utilizes the noise matrix parameters
$\Theta_{\mathrm{Q}_{CV}}$, $\Theta_{\mathrm{Q}_{CJ}}$ and
$\Theta_{\mathrm{R}}$ to generate the noise matrices $\mathbf{Q}_{CV}$,
$\mathbf{Q}_{CJ}$ and $\mathbf{R}$, to complete subsequent filtering.

####  III-C3 Training Algorithm

Our training algorithm adopts a supervised learning approach, in which we use
high-precision trajectories as label data. In such way, the end-to-end
training NN is derived. The advantage of this approach is that it is not
necessary to provide the real values of the outside parameters related to
noise matrices. Instead, it enables NN to automatically learn these parameters
from the labeled data. Additionally, the proposed method is applicable to
datasets with different trajectory lengths. The length of the time window is
utilized as the time interval for conducting the backpropagation, and the mean
absolute error (MAE) with $L_{1}$ regularization is chosen as the loss
function, since it is more robust to tolerate outliers in the data than RMSE.

##  IV Simulation Result

We use the fixed-wing UAV parser in MATLAB Toolbox to generate a dataset
containing 10 UAV trajectories, in which the simulated ADS-B sampling
frequency is 1Hz, and the observation trajectory data includes the three-
dimensional position and speed of the UAV at each sampling moment. The NN
training parameters are set as: the hidden layer size of LSTM is 128, the
sliding time window length is 3, training epoch is 15. Additionally, there are
7 trajectories used for training, 2 for validation and 1 for testing.

Figure 4: Tracking diagram of RNN-enhanced IMM-KF versus IMM-KF.

Fig. 4 shows UAV tracking trajectories using RNN-enhanced IMM-KF and IMM-KF
under different matrix parameter settings. Among them, IMM-KF (empirical) is
the trajectory result obtained by directly using the noise matrix parameters
of IMM-KF with good tracking performance on other trajectories and filtering.
It is obvious that severe filter divergence occurs. The IMM-KF (perfect) is
the filtering result when the optimal noise matrix setting is obtained via
reverse verification after all time data of the current trajectory sequence is
obtained. It is observed that the reasonable configuration of the noise
matrices $\mathbf{Q}_{CV}$, $\mathbf{Q}_{CJ}$ and $\mathbf{R}$ has a positive
impact on the filtering performance of IMM-KF. RNN-enhanced IMM-KF
consistently maintains excellent tracking performance by continuously
adjusting the noise matrix parameters.

(a) 3D MAE at each time step.

(b) Comparison of RMSE across different dimensions.

Figure 5: Error performance of RNN-enhanced IMM-KF versus IMM-KF.

Fig. 5 depicts the error performance of RNN-enhanced IMM-KF and IMM-KF
algorithms. In detail, Fig. 5a shows the three-dimensional MAE at each time
step of the estimated values obtained using different filtering methods. It is
obvious that within the first 400s, all three filtering methods maintain good
target tracking and filtering performance. However, around 400s, the UAV
performs a large degree of turning maneuvers during flight. Due to the
incorrect setting of noise matrix parameters, the process of matching and
calculation of correct model weights of IMM-KF (empirical) are affected, which
leads to result divergence. By observing IMM-KF (perfect) and RNN-enhanced,
there exists no filtering divergence in both, but the MAE performance of the
proposed algorithm is better. It is explained that we set different levels of
observation noise according to the distance between the UAV and GS in the
simulation dataset generation stage. Then, we can simulate and verify the
filtering performance in changing environments. Obviously, RNN-enhanced IMM-KF
can better adapt to changes in the environment and achieve better filtering
performance. Fig. 5b shows the RMSE among different filtering methods in
different dimensions. The minimum RMSE achieved by the RNN-enhanced IMM-KF
algorithm in all three spatial dimensions demonstrates its performance.
Compared with the original ADS-B data and IMM-KF (perfect) on the test
trajectory, the total RMSE of the proposed algorithm decrease by 58.98% and
28.56%, respectively.

##  V Conclusions

In this paper, we use ADS-B technology as a method of UAV target tracking.
Considering the impact of receiving a large amount of ADS-B signal noise on
tracking performance, we design the RNN-enhanced IMM-KF algorithm to execute
real-time filtering of the target trajectory. The simulation results
effectively verify the significance of the proposed algorithm in improving
filtering performance and overcoming filtering divergence in UAV target
tracking scenarios. This work can help improve the safety of UAV flight as
well as airspace management.

## References

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# Improved Automated Machine Learning from Transfer Learning

###### Abstract

In this paper, we propose a neural architecture search framework based on a
similarity measure between some baseline tasks and a target task. We first
define the notion of the task similarity based on the log-determinant of the
Fisher Information matrix. Next, we compute the task similarity from each of
the baseline tasks to the target task. By utilizing the relation between a
target and a set of learned baseline tasks, the search space of architectures
for the target task can be significantly reduced, making the discovery of the
best candidates in the set of possible architectures tractable and efficient,
in terms of GPU days. This method eliminates the requirement for training the
networks from scratch for a given target task as well as introducing the bias
in the initialization of the search space from the human domain.

Index Terms—  Transfer Learning, Neural Architecture Search, autoML

##  1 Introduction

Most neural architecture search (NAS) methods focus on reducing the complexity
of search by using a combination of explicit architecture search domains and
specific properties of the given task at hand. These requirements make use of
the existing architecture for a specific task less applicable to architecture
search for other tasks. However, only a few of the above techniques [1] have
considered the similarity of tasks in the search space of architectures. This
paper is motivated by a common assumption made in transfer and lifelong
learning. In transfer learning literature [2, 3, 4, 5], it is often believed
that similar tasks can share similar architectures. In this light, using the
architecture(s) of similar tasks as the potential architecture search space
for a new task may reduce the dependence on prior knowledge and significantly
accelerate the search of the final architecture. Building on this intuition,
this paper proposes a NAS framework that learns an appropriate architecture
for a given task based on its similarity to other tasks. A measure of
similarity between pairs of tasks is proposed based on log-determinant of the
Fisher information matrix of the loss function with respect to the parameters
of the models under consideration. By definition, our measure of the task
distance is asymmetric. Having an asymmetric measure is important since it
might be easier to transfer the knowledge of a complex task to a simple task,
but not the other way around.

In our approach, we find the closest task to a target task in a given set of
base tasks, and the architecture corresponding to the closest task is used to
construct an architectural search space for the target task without requiring
any prior knowledge from the human domain. Subsequently, the FUSE [1]
gradient-based search algorithm is applied to discover an appropriate
architecture for the target task. Experimental results for $8$ classification
tasks on MNIST [6] and CIFAR-10 [7] datasets demonstrate the efficacy of the
proposed approach over the other state-of-the-art methods.

##  2 Related Works

Recent NAS techniques have been shown to offer competitive-to-superior
performance to that of hand-crafted architectures. These techniques include
approaches based on reinforcement learning (RL) [8], and random search
techniques [9, 10]. However, many of the existing NAS methods such as
techniques based on the evolutionary algorithm [11] often requires a lot of
computational resources and thousands of GPU days to perform a search for a
single task. Recently, differentiable search methods [12, 13] have been
proposed and, combined with random search methods and sampling sub-networks
from one-shot super-networks [10, 14], accelerated the underlying search time
significantly. Additionally, random search [9, 10], and network
transformations [15] have shown to be very promising. Most recent techniques
define the network architectures by stacking simple cells together [12, 16];
as a result, the search space is also small and limited. Consequently, these
techniques are often applicable to standard datasets, where the search space
corresponding to each dataset is known. There are other approaches on
combining task similarity with NAS [17, 1] in order to automatically generate
a suitable search space for a given target task. Additionally, the task
similarity is often been studied in transfer learning [2, 3, 4, 5], and is
used as regularization during learning to prevent catastrophic forgetting [18]
in continual learning. However, the introduced measure of similarity in the
above papers is assumed to be symmetric which limits their applicability in
the neural architecture search.

##  3 Neural Architecture Search Framework

In this section, we propose a neural architecture search (NAS) framework that
considers the knowledge of the past learned tasks in order to construct a
suitable search space for the incoming target task. This approach minimizes
the requirement for prior knowledge from the human domain. Consider a set $B$
consisting of $K$ learned baseline tasks $T_{i},$ and its corresponding
dataset $X_{i},$ denoted jointly by pairs $(T_{i},X_{i})$ for
$i=1,2,\ldots,K$. Assume that for each baseline task, the best architecture
and search space are known. In practice, the search for the best architecture
for the baseline task can be perform using the random search method[10].
Below, a NAS framework, whose pseudo-code is provided in Algorithm 1 is
presented for finding a well-performing architecture for a target task and
data set pair $(T_{t},X_{t})$ based on the knowledge of architectures of these
$K$ baseline learned tasks.

  1. 1.

Task Similarity. First, the dissimilarity of each learned task to the target
task using the log-determinant task distance is computed. The closest baseline
task based on the computed dissimilarities is returned.

  2. 2.

Neural Architecture Search. Next, a suitable search space for the target task
is determined based on the closest task architecture. Subsequently, a search
within this space is performed to find a well-performing architecture for the
target task.

1

Data: A set of baseline tasks: $B=\\{(T_{1},X_{1}),...,(T_{K},X_{K})\\}$

Input: $\varepsilon$-approx. network N, # of candidates $C$, $\alpha=1/|C|$,
the incoming target $(T_{t},X_{t})$

Output: Best architecture for the target task $t$

2 Function  _Distance(_$X_{b},X_{t},N_{b},N_{t}$_)_:

3 Train $N_{b}$ with $X_{b}$

4 Train $N_{t}$ with $X_{t}$

5 Compute $F_{b,t}$ using $X_{t}$ on $N_{b}$

6 Compute $F_{t,t}$ using $X_{t}$ on $N_{t}$

7 return $\displaystyle
d[b,t]=\frac{1}{n}\bigg{|}\sum_{i}\log(\frac{\lambda_{b,t}^{i}+\sigma^{2}}{\lambda_{t,t}^{i}+\sigma^{2}})\bigg{|}$

8

9

10 Function  _FUSE(_candidates $C$, data $X$_)_:

11 Define the relaxed output of C: $\displaystyle\bar{c}(X)=\underset{c\in
C}{\sum}\frac{\exp{(\alpha_{c})}}{\underset{c^{\prime}\in
C}{\sum}\exp{(\alpha_{c^{\prime}})}}c(X)$

12 while  _$\alpha$ not converge_ do

13 Update $C$ by descending $\nabla_{w}\mathcal{L}_{tr}(w;\alpha,\bar{c})$

14 Update $\alpha$ by descending
$\nabla_{\alpha}\mathcal{L}_{val}(\alpha;w,\bar{c})$

15

16 return $c^{*}=\underset{c\in C}{\mathrm{argmin}}\ \alpha_{c}$

17

18

19 Function  _Main_ :

20 for  _$b\in B$_ do

21 $d[b,t]=\textnormal{{Distance}}(X_{b},X_{t},N_{b},N_{t})$

22 Select closest task: $b^{*}=\underset{b\in B}{\mathrm{argmin}}\ d[b,t]$

23 while  _criteria not met_ do

24 Sample $C$ candidates $\in$ search space $S=S_{b^{*}}$

25 $c^{*}=\textnormal{{FUSE}}\big{(}\\{C,c^{*}\\},X_{t}\big{)}$

26

27 return best architecture $c^{*}$

28

Algorithm 1 NAS with related search space

###  3.1 Log-Determinant Task Distance

As described above, we need to find the closest task to the target task
between the learned baseline tasks. To this end, we define a dissimilarity
measure between tasks based on the log-determinant of the Fisher information
matrix. Let $\mathcal{P}_{N}(T,X)$ be a function that measures the performance
of a given architecture $N$ on task $T$ with data $X$. An architecture $N$ is
an $\varepsilon$-approximation network for $(T,X)$ if
$\mathcal{P}_{N}(T,X)\geq 1-\varepsilon$, for a given $0<\varepsilon<1$. These
for example may be well-known hand-designed networks. Next, the empirical
Fisher information matrices of the loss function of these
$\varepsilon$-approximation networks are computed as follows:

$F=\frac{1}{M}\sum_{i=1}^{M}\nabla_{\theta}\log(p(x_{i}|\theta))\nabla_{\theta}\log(p(x_{i}|\theta))^{T},$
(1)

where $M$ is the number of data points, $\theta$ are the parameters of the
neural network. Let $N_{t}$ be an $\varepsilon$-approximation network for the
target task. Our goal is to evaluate how well the baseline approximation
networks perform on the target task’s data $X_{t}$. That is, the data from the
target task is used to compute the empirical Fisher information matrices for
all of the approximation networks. After computing these Fisher information
matrices, we define the dissimilarity between tasks as follows. Let $b\in B$
and $N_{b}$ be one of the baseline tasks and its corresponding
$\varepsilon$-approximation network, respectively. Let $F_{b,t}$ be the Fisher
information matrix of $N_{b}$ with data $X_{t}$ from task $t$, and $F_{t,t}$
be the Fisher information matrix of $N_{t}$ with data $X_{t}$ from task $t$.
We define the dissimilarity from task b to task t as:

$d[b,t]=\bigg{|}\frac{\log(\det(F_{b,t}+\sigma^{2}*I_{n\times n}))}{n}\\\
-\frac{\log(\det(F_{t,t}+\sigma^{2}*I_{m\times m}))}{m}\bigg{|},$ (2)

where $I$ is the identity matrix, $\sigma$ is a pre-selected small constant,
$n$ and $m$ are the number of parameters in $N_{b}$ and $N_{t}$, respectively.
If $m=n$ or $N_{b},N_{t}$ have a similar structure, then the distance can be
expressed as:

$\displaystyle{}d[b,t]=\frac{1}{n}\bigg{|}\sum_{i}\log(\frac{\lambda_{b,t}^{i}+\sigma^{2}}{\lambda_{t,t}^{i}+\sigma^{2}})\bigg{|},$
(3)

where $\lambda^{i}$ is the ith eigenvalue of the Fisher information matrix.
This proposed dissimilarity is greater than or equal to $0$, with the distance
$d=0$ indicating perfect similarity. Note that this dissimilarity is
inherently asymmetric since it might be easier to transfer the knowledge of a
comprehensive task to a simple task, but not vice versa.

###  3.2 Neural Architecture Search

Similar to recent NAS techniques [12, 1], our search space is defined by cells
and skeletons. A cell is a densely connected graph of nodes, where nodes are
connected by operations. The operations (e.g., identity, convolution) are set
so that the dimension of the output is the same as that of the input. A
skeleton is a structure consisting of multiple cells stacked together, forming
a complete architecture. Next, the Fusion Search (FUSE) algorithm [1] is
applied to the reduced search space. The FUSE is a gradient-based search
algorithm that evaluates all of the network candidates as a whole. It is based
on the continuous relaxation of the outputs from all of the network
candidates. Its goal is to search through all candidates without fully
training them. Let $C$ be the set of network candidates from the given search
space. For some $c\in C$ and data $X$, $c(X)$ denotes the output of the
network candidate $c$. The relaxed output $\bar{c}$ is the convex combination
of the outputs from all candidates in $C$:

${}\bar{c}(X)=\sum_{c\in C}\frac{\exp{(\alpha_{c})}}{\sum_{c^{\prime}\in
C}\exp{(\alpha_{c^{\prime}})}}c(X),$ (4)

where $\alpha_{c}$ is a continuous variable assigned to network $c$’s output.
Next, we conduct the evaluation by jointly training the network candidates and
optimizing their $\alpha$ coefficients. Let $X_{train}$, $X_{val}$ be the
training and validation data. The training procedure is based on alternative
minimization and can be divided into: (i) freeze $\alpha$ coefficients,
jointly train network candidates and their weights, (ii) freeze network
candidates’ weights, update $\alpha$ coefficients. The best candidate in $C$
will be selected by: $c^{*}=\arg\max_{c\in C}\alpha_{c}$. To search through
the entire search space, this process is repeated until certain criteria are
met. Since the search space is restricted (reduced) only to the space of the
most related tasks, the search algorithm is efficient and requires a reduced
number of GPU-days.

(a) Mean

(b) Standard Deviation

Fig. 1: Distances between $8$ classification tasks from MNIST and CIFAR-10
datasets.

##  4 Experimental Study

For our experiment, we initialize $4$ tasks in MNIST (i.e., Tasks $0-3$) and
$4$ tasks in CIFAR-10 (i.e., Tasks $4-7$). Task $0$ and $1$ are defined as the
binary classification tasks of detecting digits $0$ and $6$, respectively.
Task $2$ is a binary classification of odd versus even digits. Task $3$ is the
full $10$ digits classification. Task $4$ is a binary classification of three
objects in CIFAR-10: automobile, cat, ship, i.e., the goal is to decide if the
given input image consists of one of these three objects or not. Task $5$ is
analogous to Task $4$ but with different objects (cat, ship, truck). Task $6$
is a multi-classification of four classes with labels bird, frog, horse, and
anything else. Finally, Task $7$ is the standard $10$ object classification in
CIFAR-10. We use the VGG-16 architecture [19] as the
$\varepsilon$-approximation for both baseline and target tasks. To apply
VGG-16 to both MNIST and CIFAR-10 tasks without architecture modification, we
convert MNIST data from $1$ channel to $3$ channels and reshape images into
$32\times 32$ in order to match the dimension of CIFAR-10 data.

We first represent each task by an $\varepsilon$-approximation network by
training the network with the training data corresponding to that task. Next,
for each task, we compute $8$ empirical Fisher Information matrices using $8$
different datasets. For simplicity, the Fisher information matrices are
approximated by only their diagonal entries. In other words, we only compute
the diagonal entries of the Fisher Information Matrix, and set the non-
diagonal entries to zero. In particular, for the network of size $n$
parameters, the number of Fisher information entries needed to be computed is
$n$ instead of $n^{2}$ entries. This approach of simplification is often used
in transfer learning [20], and continual learning [18]. Next, we use Equation
(3) to compute the distance for each pair of tasks. Note that the distance
from a task to itself will be zero since the difference between two Fisher
Information matrices is zero. We repeat the experiment $10$ times with
different initialization settings for the $\varepsilon$-approximation
networks. The mean and standard deviation tables of the distance between tasks
are illustrated in Figure 1. The ith column of the mean table represents the
average distance from other tasks to the target Task $i$. Our results suggest
that two tasks from the same dataset (e.g., MNIST or CIFAR-10) are often more
similar than tasks involving different datasets. It is perhaps surprising that
the closest task to Task $3$ is Task $7$. Since other MNIST tasks are binary
classification tasks, they do not intuitively appear as similar to the
$10$-class classification, even though Task $3$ and Task $7$ are using
different datasets.

We consider the problem of learning an architecture for Task $6$ from MNIST
dataset, using the other aforementioned tasks as our baseline. From the sixth
column of the mean table in Figure 1, it is observed that Task $7$ is the
closest one to Task $6$. Thus, we apply cell structure and the operations of
Task $7$ to generate a suitable search space for the target task. We use a
cell structure, including $4$ nodes and $6$ operations. The list of related
operations includes identity, sep-conv3x3, conv(7x1)(1x7), and maxpool3x3. The
FUSE algorithm [1] is then used to find the best architecture in this search
space. Initially, five candidate architectures are randomly generated from
this search space. At each iteration, the search algorithm evaluates these
candidates and only saves the most promising architecture for the next
iteration. The search stops when all criteria (e.g., a number of iterations or
the best architecture converges) are met. The results in Table 1 shows the
best test accuracy of the optimal architecture found by our framework after
$200$ iterations in comparison with state-of-art NAS methods (e.g., ENAS [21],
DARTS [12]), a random search algorithm, and well-known handcrafted
architectures, such as VGG [19], ResNet [22], DenseNet [23]. The architecture
produced by our framework is competitive with the hand-designed networks while
having a significantly smaller number of parameters. When compared with the
random search method, our approach achieves a higher-performing model with
less running time. Our resulted architecture also outperforms ENAS [21] by a
large margin in terms of classification performance, while having a much
smaller number of parameters and requires fewer GPU days. When compare with
DARTS [12], our architecture produces a competitive result, with a fewer
number of parameters.

Table 1: Comparison of the NAS performance with hand-designed classifiers and state-of-the-art methods on Task $3$ in MNIST based on the discovered closest task, Task $7$. Architecture | Accuracy | Parameter | GPU
---|---|---|---
| (Task 3) | (Millions) | days
VGG-16 [19] | 99.55 | 14.72 | n/a
ResNet-18 [22] | 99.56 | 11.44 | n/a
DenseNet-121 [23] | 99.61 | 6.95 | n/a
Random Search | 99.59 | 2.23 | 4
ENAS [21] | 97.77 | 4.60 | 4
DARTS [12] | 99.51 | 2.37 | 2
LD-NAS (ours) | 99.67 | 2.28 | 2

Similarly, we consider Task $3$ from CIFAR-10 dataset as the incoming target
task, and the other tasks as the baseline tasks. As observed from the mean
table given in Figure 1, Task $7$ is the closest one to the target Task $3$.
Table 2 presents results indicating that the constructed architecture for Task
$3$ has higher test accuracy with a significantly fewer number of parameters
compared to hand-designed architectures. Our architecture also outperforms
ENAS [21] by a large margin. When compare with DARTS [12] and random search,
our resulted network perform better in term of classification accuracy, and
requires less search time. Our resulted model also has fewer parameters.
Evidently, the proposed NAS framework can utilize the knowledge of the most
similar task to efficiently find the optimal network architecture for the
target task.

Table 2: Comparison of the NAS performance with hand-designed classifiers and state-of-the-art methods on Task $6$ in CIFAR-10 based on the discovered closest task, Task $7$. Architecture | Accuracy | Parameter | GPU
---|---|---|---
| (Task 6) | (Millions) | days
VGG-16 [19] | 86.75 | 14.72 | n/a
ResNet-18 [22] | 86.93 | 11.44 | n/a
DenseNet-121 [23] | 88.12 | 6.95 | n/a
Random Search | 88.55 | 3.65 | 5
ENAS [21] | 75.22 | 4.60 | 4
DARTS [12] | 90.11 | 3.12 | 2
LD-NAS (ours) | 90.87 | 3.02 | 2

##  5 Conclusions

A task similarity measure from a source task to a target task is given in this
paper. By definition, this measure is asymmetric since applying knowledge of a
comprehensive task into a simple task is easy but not vice versa. We apply
this non-commutative distance measure in the neural architecture search
framework. Using this task distance, a focus search space of architectures for
a target task can be constructed using the closest task to the target from a
set of learned baseline tasks. This reduces the complexity of the search space
and increases the search’s efficiency, results in the architecture with higher
accuracy and a smaller number of parameters.

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11institutetext: Department of Physics and Astronomy, University of Rochester,
Rochester, NY 14627-0171 USA

# Pauli Blocking for a Relativistic Fermi Gas in Quasielastic Lepton Nucleus
Scattering

Arie Bodek

(Received: date / Revised ; Received: date /- version 1. November 5, 2021 )

###### Abstract

The expressions for the overall effect of Pauli blocking in quasielastic (QE)
lepton scattering from nuclear targets within the framework of the
Relativistic Fermi Gas (RFG) given in several publications are incorrect. For
example, the expressions published by Bell and Llewelyn SmithBLS  in 1972 are
incorrect (probably a typographical error). The expressions for Pauli blocking
presented in several subsequent publications including the paper of C.H.
Llewelyn SmithLS  in 1972 and the paper of Paschos and YuPaschos  in 2002 have
the same error. Another example is a 1992 paper by Singh and OsetSingh  which
has a different (probably also a typographical) error. Other papers such as
the papers of TsaiTsai  in 1974 and Bosted and Mamyan Bosted  in 2012 use the
correct expressions. In this short preprint we review the geometrical
derivation of the overall reduction of the QE cross sections due to Pauli
blocking for electron and neutrino scattering for targets with equal numbers
of neutrons and protons and for targets with unequal numbers of neutrons and
protons and derive the correct expressions. However, we note that the effects
of Pauli blocking within the framework of the RFG model are much larger than
in other more realistic models of QE scattering such as $\psi^{\prime}$
superscalingMegias  or a spectral function approach.

###### pacs:

13.15.+gNeutrino interactions and 25.30.PtNeutrino scattering and 25.30.Dh,
25.30.Fj electron scattering inelastic

##  1 Introduction

The expressions for the overall effect of Pauli blocking in quasielastic (QE)
lepton scattering from nuclear targets within the framework of the
Relativistic Fermi Gas (RFG) given in several published papers are incorrect
(probably due to typographical errors). In this short preprint we review the
geometrical derivation of the overall reduction of the QE electron and
neutrino cross sections due to Pauli blocking for the RFG model for targets
with equal numbers of neutrons and protons and for targets with unequal
numbers of neutrons and protons.

Figure 1: Illustration of a momentum transfer to a nucleon in a nuclear target
in QE lepton scattering within the framework of the RFG model for the case in
which the Fermi momentum $K_{F}$ for the ensemble of nucleons in the initial
state is the same as $K_{F}$ of the ensemble of spectator nucleons in the
final state. This is the case of QE electron scattering from neutrons or
protons in any nucleus. It is also the case for QE scattering of neutrinos and
antineutrinos from nuclear targets in which the number of protons and neutrons
is the same. The QE scattering process transfers a 3-momentum $\bf q$ to any
of the nucleons in the initial state. Figure 2:  A comparison of the RFG Pauli
blocking factors for for $\rm{}^{12}_{6}C$ with $K_{F}$ = 0.221 GeV (dashed
red line) and with $K_{F}$ = 0.228 GeV (dotted blue line) to the Pauli factor
used in geniegenie-ref  version 2 for the Bodek-Ritchie Fermi GasBodek-Ritchie
with high momentum tail (solid black line) as a function of the square of the
square of the 4-momentum transfer $Q^{2}$.

###  1.1  Simple Geometrical Derivation of RFG Pauli Blocking

The first calculations of Pauli blocking in quasielastic (QE) electron and
neutrino scattering within the framework a Relativistic Fermi Gas (RFG) were
published by GattoGatto  in 1953. A more complete expression for the case of
neutrino scattering from nuclei with an unequal number of neutrons and protons
was reported in a CERN conference proceeding by BermanBerman  in 1961.

Fig.1 Illustrates the momentum transfer to a nucleon in a nuclear target in QE
lepton scattering within the framework of the RFG model for the case in which
the the Fermi momentum $K_{F}$ for the ensemble of nucleons in the initial
state is the same as $K_{F}$ for the ensemble of spectator nucleons in the
final state. This is the case of QE electron scattering from neutrons or
protons in any nucleus. It is also the case for QE scattering of neutrinos and
antineutrinos from nuclear targets in which the number of protons and neutrons
is the same. In the QE scattering process a 3-momentum $|\bf q|$ is transfered
to one of the nucleons in the initial state.

The Pauli suppression factor is equal to 1.0 minus fraction of the overlap
volume between the left sphere of the spectator nucleons and the right sphere
of the interacting nucleons which were given a 3-momentum transfer $|\bf q|$.
The volume of a dome of radius $R$ and hight $h$ is $\frac{1}{3}\pi
h^{2}(3R-h)$. The overlap volume is twice that or $\frac{2}{3}\pi h^{2}(3R-h)$
and the total volume of the final state nucleon sphere is
$V_{FS}=\frac{4}{3}\pi R^{3}$.

In Fig. 1: $R=K_{F}$ and $h=K_{F}-\frac{|\bf q|}{2}$. For electron scattering
an for neutrino scattering on nuclear targets with equal number of neutrons
and protons this leads to an overall Pauli blocking factor
$P_{RFG}=P_{pauli}^{electrons}=P_{pauli}^{neutrinos}=P_{pauli}^{antineutrinos}$
where:

$\displaystyle P_{RFG}$ $\displaystyle=$ $\displaystyle\frac{3}{4}\frac{|\bf
q|}{K_{F}}-\frac{1}{16}(\frac{|\bf q|}{K_{F}})^{3}(for~{}{|\bf q|}<2K_{F})$
$\displaystyle P_{RFG}$ $\displaystyle=$ $\displaystyle
1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(for~{}{|\bf
q|}>2K_{F})$ (1)

If we define $x=\frac{|\bf q|}{2K_{F}}$ we obtain:

$\displaystyle P_{RFG}$ $\displaystyle=$
$\displaystyle\frac{3}{4}2x-\frac{1}{2}x^{3}~{}~{}~{}~{}~{}~{}(for~{}{|\bf
q}|<2K_{F})$ (2)

The above equation is identical to the expression published by GattoGatto  in
1953. It is also indentical to the expression published in a CERN conference
proceeding by BermanBerman  in 1961 for the case in which the number of
protons Z equal to the number of neutrons N. This expression is also used in
the paper by TsaiTsai  in 1974, and in recent analyses Bosted  of Jefferson
Lab electron scattering data.

However, the expression published in by Bell and Llewelyn SmithBLS  in 1972 is
incorrect (probably due to a typographical error). For the case of a nuclear
target with equal numbers of protons it gives a Pauli blocking factor
$P_{incorrect}=\frac{3}{4}2x-\frac{1}{3}x^{3}$. The expressions for Pauli
blocking presented in several subsequent publications including C.H. Llewelyn
SmithLS  in 1972 and Paschos and YuPaschos  in 2002 have the same (probably
typographical) error. A 1992 paper by Singh and OsetSingh  has a different
(probably typographical) error.

Fig. 2 shows a comparison of the RFG overall Pauli blocking factor for
$\rm{}^{12}_{6}C$ (as a function of the square of the four-momentum transfer
$Q^{2}$) for $K_{F}$ = 0.221 GeV (dashed red line) and for $K_{F}$ = 0.228 GeV
(dotted blue line). Also shown is the overall Pauli blocking factor for the
Bodek-Ritchie Fermi GasBodek-Ritchie  with high momentum tail as implemented
in geniegenie-ref  version 2 (solid black line). There is reasonable agreement
with the genie calculation (exact agreement is not expected since slightly
different momentum distributions are used).

Figure 3:  Neutrino QE scattering within the framework of the RFG model for
nuclei with different numbers of neutrons (N) and Protons (Z). The top panel
shows the case for neutrino QE scattering on the neutrons in the nucleus and
the bottom panel shows the case for antineutrino QE scattering on the protons
in the nucleus.

##  2 Scattering of neutrinos and antineutrinos from nuclei with N protons and
Z neutrons

Fig. 3 Illustrates QE scattering within the framework of the RFG model for QE
scattering from nuclei with different numbers of neutrons (N) and Protons (Z)
(with Fermi momenta $K_{F}^{N}$ and $K_{F}^{P}$, respectively). The top panel
shows the case for neutrino QE scattering on the neutrons in the nucleus and
the bottom panel shows the case for antineutrino QE scattering on the protons
in the nucleus.

Fig. 4 shows the expression2spheres  for the overlap volume V of two spheres
of radii $R$ and $r$ and distance $d$.

The effects of the Pauli exclusion principle is then given by the overlap
volume of the displaced neutron sphere with the spectator proton sphere. In
this case the overlap volume V is calculated for $d=|\bf q|$, $R=K_{F}^{N}$
and $r=K_{F}^{P}$. Therefore, for neutrino QE scattering from the neutrons in
the nucleus ($N>Z$) we obtain:

$\displaystyle P_{N}$ $\displaystyle=$
$\displaystyle[1-\frac{(K_{F}^{P})^{3}}{(K_{F}^{N})^{3}}]~{}~{}~{}~{}~{}~{}~{}~{}~{}{|\bf
q|}<K_{F}^{N}-K_{F}^{P}$ $\displaystyle P_{N}$ $\displaystyle=$ $\displaystyle
1-[\frac{3}{4\pi}\frac{V}{(K_{F}^{N})^{3}}]~{}~{}~{}~{}K_{F}^{N}-K_{F}^{P}<{|\bf
q|}<K_{F}^{N}+K_{F}^{P}$ $\displaystyle P_{N}$ $\displaystyle=$ $\displaystyle
1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{|\bf
q|}>(K_{F}^{N}+K_{F}^{P})$ (3)

For the case of antineutrinos, the QE scattering occurs on the protons in the
nucleus. The effects of the Pauli exclusion principle is then given by the
overlap volume of the displaced proton sphere with the spectator neutron
sphere. He also the overlap volume V is calculated for $d=|\bf q|$,
$R=K_{F}^{N}$ and $r=K_{F}^{P}$.

$\displaystyle P_{Z}$ $\displaystyle=$ $\displaystyle
0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{|\bf
q|}<K_{F}^{N}-K_{F}^{P}$ $\displaystyle P_{Z}$ $\displaystyle=$ $\displaystyle
1-[\frac{3}{4\pi}\frac{V}{(K_{F}^{P})^{3}}]~{}~{}~{}~{}K_{F}^{N}-K_{F}^{P}<{|\bf
q|}<K_{F}^{N}+K_{F}^{P}$ $\displaystyle P_{Z}$ $\displaystyle=$ $\displaystyle
1~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{|\bf
q|}>(K_{F}^{N}+K_{F}^{P})$ (4)

Numerically, the above exact expressions give similar results to earlier
expression by Berman Berman , and the (corrected) expression of J.S. Bell, and
C. H. Llewellyn Smith BLS . Those two calculations make additional assumptions
about the relationship between the Fermi momenta of neutrons of protons.

Figure 4:  Overlap volume of two spheres of radii R and r and distance d. |  $k_{F}^{P}$,$k_{F}^{N}$ | $k_{F}^{P}$ | $E_{\mathrm{shift}}$
---|---|---|---
${}_{Z}^{A}Nucl$ | RFG |  $\psi^{\prime}$ fit |  $\psi^{\prime}$ fit
Nucl. |  $\pm$5 | ref.Donnelly  | ref.Donnelly
| updated |  |
| MeV/c | MeV/c | MeV
${}_{3}^{6}Li$ | 169,169 | 165 | 15.1
${{}_{6}^{12}C}$ | 221,221 | 228 | 20.0
${{}_{8}^{16}O}$ | 225,225 |  |
${}_{13}^{27}Al$ | 238,241 | 236 | 18.0
${}_{14}^{28}Si$ | 239,241 |  |
${{}_{18}^{40}Ar}$ | 251,263 |  |
${{}_{20}^{40}Ca}$ | 251,251 | 241 | 28.8
${}_{23}^{50}V$ | 253,266 |  |
${}_{26}^{56}Fe$ | 254,268 | 241 | 23.0
${}_{28}^{58.7}Ni$ | 257,269 | 245 | 30.0
${}_{79}^{197}Au$ | 275,311 | 245 | 25.0
${}_{82}^{208}Pb$ | 275,311 | 248 | 31.0
Table 1:  Fermi momenta $K_{F}$ for protons and neutrons for various nuclei.
The relativistic Fermi gas values are from Ref. Removal electron . The results
from a $\psi^{\prime}$ superscaling fit are from reference Donnelly .

Table 1 shows the Fermi momenta $K_{F}$ for protons and neutrons for various
nuclei. The relativistic Fermi gas values are from Ref. Removal  electron .
The results from a $\psi^{\prime}$ superscaling fit are from reference
Donnelly .

The Pauli blocking factors for several nuclei are given in Table 2. Here, for
$\rm{}^{12}_{6}C$ we compare genie v2 (for a Bodek-Ritchie momentum
distribution) to RFG calculation with two values of $K_{F}$. (In this case the
factors are the same for electrons and neutrinos). Also shown are the RFG
Pauli blocking factors for neutrino QE scattering on neutrons and antineutrino
QE scattering on protons for $\rm{}^{40}_{18}Ar$, $\rm{}^{56}_{26}Fe$ and
$\rm{}^{208}_{82}Pb.$

##  3 Conclusions

In conclusion, we present the correct expressions for the overall Pauli
blocking within the framework of the RFG model for electron scattering, for
neutrino scattering on targets with the same number of neutrons and protons,
and for neutrino and antineutrino scattering on nuclei with unequal numbers of
neutrons and protons. The Pauli blocking factors for $\rm{}^{12}_{6}C$ in
genie v2 are consistent with the geometrical calculations for the Relativistic
Fermi Gas. However, we note that the effects of Pauli blocking within the
framework of the RFG model are much larger than in other more realistic models
of QE scattering, such as $\psi^{\prime}$ superscalingMegias  or a spectral
function approach. In addition to Pauli blocking, there are other sources of
suppression of QE scattering at low values of $\bf q$. Our experimental
investigation of the suppression of low $\bf q$ QE electron scattering cross
sections within the framework of superscaling is presented in another
publication lowq .

##  4 Acknowledgements

Research supported by the U.S. Department of Energy under University of
Rochester grant number DE-SC0008475.

| nucleus | $\rm{}^{12}_{6}C$ | $\rm{}^{12}_{6}C$ | $\rm{}^{12}_{6}C$ | $\rm{}^{40}_{18}Ar$ | $\rm{}^{40}_{18}Ar$ | $\rm{}^{56}_{26}Fe$ | $\rm{}^{56}_{26}Fe$ | $\rm{}^{208}_{82}Pb$ | $\rm{}^{208}_{82}Pb$
---|---|---|---|---|---|---|---|---|---|---
| $K_{F}$ | 0.221 | 0.221 | 0.228 | 0.263 | 0.251 | 0.268 | 0.254 | 0.311 | 0.275
Q2 | $|\bf q|$ |  genie v2 | RFG | RFG | $\nu-N$ | $\bar{\nu}-P$ | $\nu-N$ | $\bar{\nu}-P$ | $\nu-N$ | $\bar{\nu}-P$
0.00 | 0.000 | 0.000 | 0.000 | 0.000 | 0.131 | 0.000 | 0.149 | 0.000 | 0.309 | 0.000
0.01 | 0.100 | 0.375 | 0.334 | 0.324 | 0.339 | 0.239 | 0.342 | 0.227 | 0.395 | 0.125
0.02 | 0.142 | 0.494 | 0.465 | 0.451 | 0.445 | 0.361 | 0.445 | 0.349 | 0.473 | 0.237
0.03 | 0.174 | 0.582 | 0.560 | 0.544 | 0.523 | 0.452 | 0.522 | 0.439 | 0.533 | 0.325
0.04 | 0.201 | 0.652 | 0.635 | 0.619 | 0.587 | 0.525 | 0.585 | 0.512 | 0.583 | 0.397
0.05 | 0.225 | 0.710 | 0.698 | 0.681 | 0.641 | 0.587 | 0.638 | 0.574 | 0.627 | 0.460
0.06 | 0.247 | 0.759 | 0.751 | 0.733 | 0.688 | 0.641 | 0.684 | 0.628 | 0.665 | 0.515
0.07 | 0.267 | 0.802 | 0.796 | 0.778 | 0.729 | 0.689 | 0.724 | 0.676 | 0.699 | 0.565
0.08 | 0.286 | 0.838 | 0.835 | 0.818 | 0.766 | 0.730 | 0.760 | 0.718 | 0.730 | 0.609
0.09 | 0.304 | 0.870 | 0.869 | 0.852 | 0.798 | 0.768 | 0.792 | 0.756 | 0.758 | 0.650
0.10 | 0.321 | 0.897 | 0.897 | 0.881 | 0.827 | 0.801 | 0.821 | 0.789 | 0.783 | 0.687
0.11 | 0.337 | 0.920 | 0.922 | 0.906 | 0.853 | 0.831 | 0.846 | 0.820 | 0.807 | 0.721
0.12 | 0.352 | 0.939 | 0.942 | 0.928 | 0.876 | 0.857 | 0.870 | 0.847 | 0.828 | 0.752
0.13 | 0.367 | 0.956 | 0.959 | 0.947 | 0.897 | 0.881 | 0.890 | 0.871 | 0.848 | 0.780
0.14 | 0.382 | 0.969 | 0.973 | 0.962 | 0.915 | 0.902 | 0.909 | 0.893 | 0.866 | 0.806
0.15 | 0.395 | 0.980 | 0.984 | 0.975 | 0.931 | 0.921 | 0.925 | 0.912 | 0.883 | 0.830
0.16 | 0.409 | 0.988 | 0.992 | 0.985 | 0.946 | 0.937 | 0.940 | 0.929 | 0.898 | 0.852
0.17 | 0.422 | 0.993 | 0.997 | 0.992 | 0.958 | 0.952 | 0.953 | 0.944 | 0.912 | 0.872
0.18 | 0.435 | 0.996 | 1.000 | 0.997 | 0.969 | 0.964 | 0.964 | 0.957 | 0.924 | 0.890
0.19 | 0.448 | 0.997 | 1.000 | 0.999 | 0.978 | 0.974 | 0.973 | 0.968 | 0.936 | 0.907
0.20 | 0.460 | 0.997 | 1.000 | 1.000 | 0.985 | 0.983 | 0.981 | 0.978 | 0.946 | 0.922
0.21 | 0.472 | 0.998 |  |  | 0.991 | 0.989 | 0.988 | 0.985 | 0.956 | 0.936
0.22 | 0.483 | 0.998 |  |  | 0.995 | 0.994 | 0.993 | 0.991 | 0.964 | 0.948
0.23 | 0.495 | 0.998 |  |  | 0.998 | 0.998 | 0.996 | 0.996 | 0.971 | 0.959
0.24 | 0.506 | 0.999 |  |  | 1.000 | 1.000 | 0.999 | 0.999 | 0.978 | 0.968
0.25 | 0.517 | 0.999 |  |  | 1.000 | 1.000 | 1.000 | 1.000 | 0.984 | 0.976
0.26 | 0.528 | 0.999 |  |  |  |  | 1.000 | 1.000 | 0.988 | 0.983
0.27 | 0.539 | 0.999 |  |  |  |  |  |  | 0.992 | 0.989
0.28 | 0.550 | 0.999 |  |  |  |  |  |  | 0.995 | 0.993
0.29 | 0.560 | 0.999 |  |  |  |  |  |  | 0.998 | 0.997
0.30 | 0.571 | 0.999 |  |  |  |  |  |  | 0.999 | 0.999
0.31 | 0.581 | 1.000 |  |  |  |  |  |  | 1.000 | 1.000
0.32 | 0.591 | 1.000 |  |  |  |  |  |  | 1.000 | 1.000
Table 2:  Paulo blocking factors: For $\rm{}^{12}_{6}C$, we compare genie v2
(for a Bodek-RitchieBodek-Ritchie  momentum distribution) to RFG with two
values of $K_{F}$ (for $\rm{}^{12}_{6}C$ the Pauli blocking factors are the
same for electrons and neutrinos). Also shown are the RFG Pauli blocking
factors for neutrino QE scattering on neutrons and antineutrino QE scattering
on protons for $\rm{}^{40}_{18}Ar$, $\rm{}^{56}_{26}Fe$ and
$\rm{}^{208}_{82}Pb$.

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