# Pilot-Based Unsourced Random Access with a Massive MIMO Receiver in the
Quasi-Static Fading Regime

Alexander Fengler, Peter Jung, Giuseppe Caire The authors are with the
Communications and Information Theory Group, Technische Universität Berlin
({fengler, peter.jung, caire}@tu-berlin.de).

###### Abstract

In this work we treat the unsourced random access problem on a Rayleigh block-
fading AWGN channel with multiple receive antennas. Specifically, we consider
the slowly fading scenario where the coherence block-length is large compared
to the number of active users and the message can be transmitted in one
coherence block. Unsourced random access refers to a form of grant-free random
access where users are considered to be a-priori indistinguishable and the
receiver recovers a list of transmitted messages up to permutation. In this
work we show that, when the coherence block length is large enough, a
conventional approach based on the transmission of non-orthogonal pilot
sequences with subsequent channel estimation and Maximum-Ratio-Combining (MRC)
provides a simple energy-efficient solution whose performance can be well
approximated in closed form. For the finite block-length simulations we use a
randomly sub-sampled DFT matrix as pilot matrix, a low-complexity approximate
message passing algorithm for activity detection and a state-of-the-art polar
code with a successive-cancellation-list decoder as single-user error
correction code. These simulations prove the scalability of the presented
approach and the quality of the analysis.

###### Index Terms:

Internet of Things (IoT), Unsourced Random Access, Massive Multi-User MIMO,
Approximate Message Passing (AMP).

##  I Introduction

Conventional random-access protocols in current mobile communication standards
establish an uplink connection between user and base station (BS) by first
running a multi-stage handshake protocol [1]. During this initial access phase
active users are identified and subsequently a scheduler assigns orthogonal
transmission resources to the active users. One of the paradigms of modern
machine-type communications consists of a very large number of devices (here
referred to as “users”) with sporadic data. Typical examples thereof are
Internet-of-Things (IoT) applications, wireless sensors deployed to monitor
smart infrastructure, and wearable biomedical devices [2]. In such scenarios,
a BS should be able to collect data from a large number of devices. However,
due to the sporadic nature of the data generation and communication, an
initial access procedure is overly wasteful.

An alternative way of communication is known as grant-free random-access,
where users transmit their data without awaiting the grant of transmission
resources by the BS. A commonly discussed grant-free strategy in a massive
multi-user MIMO setting is to assign fixed orthogonal or non-orthogonal pilot
sequences to users [3, 4, 5, 6, 7, 8, 9]. Active users then transmit their
pilot sequence directly followed by their data sequence. The BS identifies the
active users in the first step and estimates their channel vectors.
Subsequently the channel estimates are used to detect the data sequences using
either maximum-ratio-combining (MRC) or zero-forcing [4].

However, as the number of users in a system grows large and the access
frequency becomes small, it gets increasingly inefficient to assign fixed
pilot sequences to all the users. In contrast, unsourced random access (U-RA)
is a novel grant-free paradigm proposed in [10] and motivated by an IoT
scenario where millions of cheap devices have their codebook hardwired at the
moment of production, and are then disseminated into the environment. In this
case, all users make use of the very same codebook and the BS decodes the list
of transmitted messages irrespectively of the identity of of the active users.
The U-RA approach can simplify the random-access protocol design because it
does not required an initial access phase and, in contrast to existing grant-
free approaches, it allows for a system that is completely independent of the
inactive users, which makes it well suited to the IoT scenario with a huge
number of devices with very sporadic activity.

Practical coding schemes for U-RA have been mainly studied on the real AWGN
channel, e.g. [10, 11, 12, 13, 14, 15, 16], and on the Rayleigh quasi-static
fading AWGN channel [17, 18]. The U-RA problem on a Rayleigh block-fading AWGN
channel in a massive MU-MIMO setting was formulated in [19] and it was shown
that a covariance-based activity detection (AD) algorithm combined with a tree
code [13] can achieve sum-spectral-efficiencies that grow proportional to the
coherence block-length $n$, even if the number of active users is
significantly larger then $n$, specifically up to $K_{a}=\mathcal{O}(n^{2})$.

In typical wireless systems the coherence block-length $n$ may range from a
couple of hundred to a couple of thousand, depending mainly on the speed of
the transmitters. At a carrier frequency of $2$ GHz the coherence times,
according to the model $T_{c}\approx 1/(4D_{s})$ where $D_{s}$ is the maximal
Doppler spread [20], may range from $45$ ms at $3$ km/h to $1$ ms at $120$
km/h. The coherence bandwidth depends on the maximal delay spread and, in an
outdoor environment, typically ranges from $100$ to $500$ kHz depending on the
propagation conditions. Therefore, the number of complex symbols in an OFDM
coherence block may range from $n=100$ to $n=20000$, depending mainly on the
assumed speed and the geometry of the environment. Unfortunately, the
covariance-based AD algorithm in [19] has a run-time complexity that scales
with $n^{2}$, which makes it unfeasible to use at $n\gtrsim 300$. So while the
covariance based approach of [19] is well suited to a fast-fading scenario
with $n\lesssim 300$, it becomes unfeasible for large coherence block-lengths
in the order of thousands. On the other hand, algorithms based on hybrid-GAMP
[21] and tensor-based modulation [22] have shown an excellent performance at
large coherence block-length at which the covariance-based approach is no
longer feasible.

In this work we present a conceptually simple algorithm that can be used when
$n>K_{a}$. It is based on pilot transmission, AD, channel estimation, MRC and
single-user decoding, very similar to the state-of-the art approach for
massive MIMO grant-free random access [5, 6, 9]. In contrast to the scheme
with fixed pilots allocated to all users, we use a pool of non-orthogonal
pilots from which active users pick one pseudo-randomly based on the first
bits of their message. We show that a collision of users, i.e. two users
picking the same pilot sequence, can be resolved by using a polar single-user
code with a successive-cancellation-list (SCL) decoder. Note, that multiuser
MRC detection and single-user polar decoding are completely separated, i.e.
succesive cancellation refers purely to the single-user decoding technique
[23]. Finite-length simulations shows that the performance of the coding
scheme can be well predicted by analytical calculations. Despite its
simplicity the suggested scheme has an energy efficiency that is comparable to
existing approaches. Note, that the problem treated here is mathematically
almost equivalent to grant-free random-access with fixed pilots allocated to
each user. Differences arise only in the possibility of collisions and the
associated use of a list decodable single-user code. The error probability of
AD and MRC in the asymptotic limit $K_{a},K_{\text{tot}},n\to\infty$ with
fixed ratios $K_{a}/K_{\text{tot}}$ and $K_{\text{tot}}/n$ has been analysed
in [6]. In this work we focus on the finite-blocklength regime and the
combination of MRC with a single-user polar code.

##  II Channel model

We consider a quasi-static Rayleigh fading channel with a block of $n$ signal
dimensions over which the user channel vectors are constant. In contrast to
the block-fading channel treated in [19], were a message is encoded over
multiple independent fading blocks, here we assume that a message can be
transmitted in a single coherence block. Following the problem formulation in
[10], each user is given the same codebook ${\cal C}=\\{{\bf
c}(m):m\in[2^{nR}]\\}$, formed by $2^{nR}$ codewords ${\bf
c}(m)\in\mathbb{C}^{n}$. The codewords are normalized such that $\|{\bf
c}(m)\|^{2}_{2}\leq n$. An unknown number $K_{a}$ out of $K_{\rm tot}$ total
users transmit their message over the coherence block. Let $\mathcal{K}_{a}$
denote the set of active users, $i_{k}$ denote the index of the message chosen
by user $k$, $h_{k,m}\sim\mathcal{CN}(0,1)$ iid be the Rayleigh channel
coefficient between user $k$ and receive antenna $m$ and let
$g_{k}\in\mathbb{R}_{+}$ denote the large-scale fading coefficient (LSFC) of
user $k$, which captures the path-loss and shadowing components of the fading.
Furthermore, let $\gamma_{k}=g_{k}$ for $k\in\mathcal{K}_{a}$ and zero
otherwise. The received signal at the $m$-th receive antenna takes the form

${\bf y}_{m}=\sum_{k=1}^{K_{\text{tot}}}\sqrt{P\gamma_{k}}h_{k,m}{\bf
c}_{i_{k}}+{\bf z}_{m}=\sum_{k\in\mathcal{K}_{a}}\sqrt{Pg_{k}}h_{k,m}{\bf
c}_{i_{k}}+{\bf z}_{m}$ (1)

where $z_{m,i}\sim\mathcal{CN}(0,N_{0})$ iid. The BS must then produce a list
${\cal L}$ of the transmitted messages $\\{m_{k}:k\in{\cal K}_{a}\\}$ (i.e.,
the messages of the active users). Let
$n_{\text{md}}=\sum_{k\in\mathcal{K}_{a}}\mathds{1}_{\\{m_{k}\notin\mathcal{L}\\}}$
denote the number of transmitted messages missing in the output list. The
system performance is expressed in terms of the Per-User Probability of
Misdetection, defined as
$p_{md}=\mathbb{E}\left\\{n_{\text{md}}\right\\}/K_{a}$. In applications a
slight overhead in the list size may be tolerable if it reduces the
misdetections. Since the number of active users is not necessarily known, it
is practical to let the decoder decide on a list size, which therefore becomes
a random variable. Let $n_{\text{fa}}=|\mathcal{L}\setminus\\{m_{k}:k\in{\cal
K}_{a}\\}|$ denote the number of messages in the output list that were not
transmitted by any user, also called _False Alarms_. $n_{\text{fa}}$ is
related to the list size by

$|\mathcal{L}|=n_{\text{fa}}+K_{a}-n_{\text{md}}$ (2)

To get an empirical performance measure we define the Probability of False-
Alarm as the average fraction of false alarms, i.e.,

$p_{fa}=\mathbb{E}\left\\{\frac{n_{\text{fa}}}{|\mathcal{L}|}\right\\}.$ (3)

Operationally, $p_{\text{fa}}$ is the probability that a randomly chosen
message from the output list is a false alarm. In the special case where
$K_{a}$ is known at the receiver and $|\mathcal{L}|=K_{a}$ is fixed it follows
from (2) that $p_{\text{fa}}=p_{\text{md}}$.

Notice that in this problem formulation the number of total users $K_{\rm
tot}$ is completely irrelevant, as long as it is much larger than the range of
possible active user set sizes $K_{a}$ (e.g., we may consider $K_{\rm
tot}=\infty$). Furthermore, as customary in coded systems, we express energy
efficiency in terms of the standard quantity $E_{b}/N_{0}:=\frac{P}{RN_{0}}$.

In line with the classical massive MIMO setting [4], we assume an independent
Rayleigh fading model for the channel coefficients $h_{k,m}$, such that the
channel vectors for different users are independent from each other and are
spatially white (i.e., uncorrelated along the antennas), that is, ${\bf
h}_{k}=(h_{k,1},...,h_{h,m})^{\top}\sim\mathcal{CN}(0,{\bf I}_{M})$.

##  III Pilot-based massive MIMO U-RA

Let the coherence block be divided into two periods of lengths $n_{p}$ and
$n_{d}$. In the first period each users chooses one of $N=2^{J}$ (non-
orthogonal) pilot sequences based on the first $J$ bits of its message. The
received signal in this phase is given by

$\displaystyle{\bf Y}_{p}={\bf A}\boldsymbol{\Gamma}^{\frac{1}{2}}{\bf H}+{\bf
Z}_{p}\in\mathbb{C}^{n_{p}\times M},$ (4)

where ${\bf A}\in\mathbb{C}^{n_{p}\times N}$ is the matrix of pilot sequences
with columns normalized as $\|{\bf a}_{i}\|_{2}^{2}=n_{p}$, ${\bf H}=({\bf
h}_{1},...,{\bf h}_{N})^{\top}\in\mathbb{C}^{N\times M}$ is the matrix with
the channel vectors as rows and $\boldsymbol{\Gamma}$ is the matrix with
$P_{\text{pilot}}\gamma_{k}$ on the diagonal. Note, that channel vectors are
only defined for those indices which have been chosen by the active users.
Formally, we define the remaining channel vectors as zero. The BS uses an AD
algorithm as in [19] to estimate the indices of the used pilots and the
corresponding LSFCs. Let $\hat{\boldsymbol{\Gamma}}$ denote the matrix with
the estimates $\hat{\gamma}_{k}$ of $\gamma_{k}$ on the diagonal and let
$\hat{\mathcal{I}}$ be the estimate of the set of active users.
$\hat{\mathcal{I}}$ can be obtained by either thresholding the estimated
received powers, i.e.

$\hat{\mathcal{I}}=\\{k:P_{\text{pilot}}\hat{\gamma}_{k}>\theta\\}$ (5)

where $\theta$ is some suitable threshold, or, if the number of active users
is available, by picking the indices corresponding to the $K_{a}$ largest
estimated received powers. Also for an index set $\mathcal{I}$ and for any
matrix ${\bf B}$ let ${\bf B}_{\mathcal{I}}$ denote the matrix that contains
only the columns of ${\bf B}$ with indices in $\mathcal{I}$. Then a linear
MMSE estimate of the channel matrix is computed as

$\hat{{\bf H}}=\hat{\boldsymbol{\Gamma}}_{\hat{\mathcal{I}}}^{1/2}{\bf
A}_{\hat{\mathcal{I}}}^{\sf H}\left({\bf
A}_{\hat{\mathcal{I}}}\hat{\boldsymbol{\Gamma}}_{\hat{\mathcal{I}}}{\bf
A}_{\hat{\mathcal{I}}}^{\sf H}+N_{0}{\bf I}_{n_{p}}\right)^{-1}{\bf
Y}_{p}\in\mathbb{C}^{\hat{K}_{a}\times M}$ (6)

where ${\bf A}_{\hat{\mathcal{I}}}$ denotes a sub-matrix of the pilot matrix
${\bf A}$ which contains only the columns which have been estimated as active
and $\hat{\boldsymbol{\Gamma}}_{\hat{\mathcal{I}}}$ contains the estimated
LSFCs $\hat{\gamma}_{k}$ of the active users on the diagonal. In the second
period each users encodes its remaining $B-J$-bit message with a binary
$(B-J,2n_{d})$ block code and modulates the $2n_{d}$ coded bits via QPSK on a
sequence of $n_{d}$ complex symbols ${\bf s}_{k}$. These are transmitted over
the $n_{d}$ channel uses in the second phase. The matrix of received signals
in the second phase is

${\bf Y}_{d}=\sum_{k\in\mathcal{K}_{a}}\sqrt{P_{\text{data}}g_{k}}{\bf
s}_{k}{\bf h}_{k}+{\bf Z}_{d}\in\mathbb{C}^{n_{d}\times M}.$ (7)

The BS uses the channel estimate $\hat{{\bf H}}$ from the first phase to
perform multiuser detection via MRC, i.e. it computes

$\hat{{\bf S}}=\hat{\boldsymbol{\Gamma}}_{\hat{\mathcal{I}}}^{-1/2}\hat{{\bf
H}}{\bf Y}_{d}^{\sf H}\in\mathbb{C}^{\hat{K}_{a}\times n_{d}}$ (8)

The rows of $\hat{{\bf S}}$ correspond to estimates of the transmitted
sequences ${\bf s}_{k}$. Note, that it is also possible to use zero-forcing
[4] instead of MRC but this would require that $M>K_{a}$. The rows of
$\hat{{\bf S}}$ are individually demodulated, the bit-wise log-likelihood
ratios are computed and fed into a soft-input single-user decoder. If the
decoder finds a valid codeword, the index of the corresponding pilot is
converted back to a $J$ bit sequence and prepended to the codeword. Then the
combination of the two is added to the output list. The use of a polar code
with CRC-bits and a SCL decoder has the additional benefit that we can include
all the valid codewords in the output list of the SCL decoder in the U-RA
output list. This allows to recover the messages of colliding users which have
chosen the same pilot in the first phase. The ability of polar codes to
resolve sums of codewords has been observed and used for U-RA on the AWGN in
combination with spreading sequences [15] and a slotted Aloha approach [24,
18].

###  III-A Activity Detection

For the AD in the pilot phase we use the MMV-AMP algorithm, which was
introduced in [25], and used for AD in a Bayesian setting, where the LSFCs are
either known, or its distribution is known in [5, 7]. The algorithm aims to
recover the unknown matrix ${\bf X}=\boldsymbol{\Gamma}^{1/2}{\bf H}$ from the
linear Gaussian measurements ${\bf Y}_{p}$. Details are omitted due to space
constraints and can be found in [5, 7, 19].

###  III-B DFT Pilots and Fast MMV-AMP

To avoid the $\mathcal{O}(Mn_{p}N)$ complexity of the matrix multiplications
we create the pilot matrix by choosing a random subset of $n_{p}$ rows of a
$N\times N$ DFT matrix. This allows to replace matrix multiplications of the
form ${\bf A}{\bf X}$ and ${\bf A}^{\sf H}{\bf Z}$ in the MMV-AMP algorithm by
FFT operations. Let $\mathcal{S}=(s_{1},...,s_{n_{p}})\subset[N]$ be a
randomly chosen subset. Let ${\bf W}\in\mathbb{C}^{N\times N}$ be a DFT matrix
defined by

$W_{i,j}=\omega^{jk}$ (9)

where $\omega=e^{-2\pi i/N}$. Then we define the pilot matrix ${\bf
A}^{\text{DFT}}\in\mathbb{C}^{n_{p}\times N}$ as the submatrix of ${\bf W}$
with rows defined as ${\bf A}^{\text{DFT}}_{i,:}={\bf W}_{s_{i},:}$. The
matrix multiplication ${\bf A}^{\text{DFT}}{\bf x}$ for some arbitrary vector
${\bf x}\in\mathbb{C}^{N}$ is give by

$({\bf A}^{\text{DFT}}{\bf x})_{i}=(\text{FFT}_{N}({\bf x}))_{s_{i}}$ (10)

for $i=1,...,n_{p}$, where $\text{FFT}:\mathbb{C}^{N}\to\mathbb{C}^{N}$
denotes a properly normalized fast-Fourier transform (FFT) operation. For the
hermitian transpose matrix multiplication ${\bf A}^{\text{DFT},{\sf H}}{\bf
z}$ for some vector ${\bf z}\in\mathbb{C}^{n_{p}}$ first define $\tilde{{\bf
z}}\in\mathbb{C}^{N}$ by $\tilde{z}_{s_{i}}=z_{i}$ and zero otherwise. Then

${\bf A}^{\text{DFT},{\sf H}}{\bf z}=\text{FFT}(\tilde{{\bf z}})$ (11)

For the matrix-matrix multiplications ${\bf A}^{\text{DFT}}{\bf X}$ this
process has to be repeated for each column of ${\bf X}$ leading to a
complexity of $\mathcal{O}(MN\log N)$.

###  III-C Analysis

In this section we calculate an approximate finite-blocklength lower bound on
the error probability and on the energy efficiency of the MRC approach. We
assume that the AD and LSFC estimation can be done without errors, i.e.
$\hat{\boldsymbol{\Gamma}}=\boldsymbol{\Gamma}$. This gives a lower bound on
the error probability and in the regime where $n_{p}>K_{a}$ we expect it to be
tight, as in this regime the AD error rates and the error of the LSFC
estimation are very low [5, 19]. For simplicity we consider
$P_{\text{pilot}}=P_{\text{data}}=:P$ here. The covariance of the channel
estimation error of the LMMSE estimation in (6) is given by

${\bf C}_{e}={\bf I}_{K_{a}}-\boldsymbol{\Gamma}_{\mathcal{I}}^{1/2}{\bf
A}_{\mathcal{I}}^{\sf H}\left({\bf
A}_{\mathcal{I}}\boldsymbol{\Gamma}_{\mathcal{I}}{\bf A}_{\mathcal{I}}^{\sf
H}+N_{0}{\bf I}_{n_{p}}\right)^{-1}{\bf
A}_{\mathcal{I}}\boldsymbol{\Gamma}_{\mathcal{I}}^{1/2}$ (12)

and the MSE of the channel estimate of user $k\in\mathcal{K}_{a}$ is given by
$\sigma^{2}_{k}:=\mathbb{E}\\{|h_{k,m}-\hat{h}_{k,m}|^{2}\\}=({\bf
C}_{e})_{k,k}$. A typically tight approximation of the effective ${\sf SINR}$
of each user after MRC is given by [4, 26]

${\sf
SINR}_{k}=\frac{M(1-\sigma_{k}^{2})g_{k}P}{N_{0}+\sigma_{k}^{2}g_{k}P+\sum_{j=1,j\neq
k}^{K_{a}}g_{j}P}$ (13)

In addition to calculating $\sigma^{2}_{k}$ via (12), a closed form lower
bound on $\sigma_{k}^{2}$ is given by

$\sigma^{2}_{k}\geq\frac{N_{0}}{N_{0}+n_{p}Pg_{k}}$ (14)

which is obtained by assuming orthogonal pilots. We investigate this lower
bound because the evaluation of (14) is much simpler then
$\eqref{eq:error_cov}$ since it does not require the inversion of a possibly
large matrix. Furthermore, we can evaluate the impact of the non-orthogonality
on the channel estimation by comparing (14) to the true channel estimation
errors obtained from (12).

An approximation of the achievable rates of a block-code with block-length
$2n_{d}$ and error probability $p_{e}$ on a real AWGN channel with power ${\sf
SINR}$ is given by the normal approximation [27]

$R\approx 0.5\log(1+{\sf SINR})-\sqrt{\frac{V}{2n_{d}}}Q^{-1}(p_{e})$ (15)

where

$V=\frac{{\sf SINR}}{2}\frac{{\sf SINR}+2}{({\sf SINR}+1)^{2}}\log^{2}e$ (16)

and $Q(\cdot)$ is the Q-function. Using the normal approximation we can find
the required ${\sf SINR}$ to achieve a certain error probability at a given
block-length and then we can find the required input power to achieve the
target SINR.

###  III-D Simulations

For the simulation in Figure 1 we choose $n=3200$, $P_{e}=0.05$, $B=100$,
$n_{p}=1152$, $n_{d}=2048$ and $J=16$. A randomly sub-sampled DFT matrix is
used as pilot matrix and in the AD phase we use the MMV-AMP algorithm with an
approximate calculation of the derivative as described in Sections [19].
Furthermore, all LSFCs are considered to be constant $g_{k}=1$ and known at
the receiver. For simplicity we assume that $K_{a}$ is known at the receiver,
and after the MMV-AMP iterations are finished the active columns are estimated
by picking the $K_{a}$ indices with the largest estimated LSFCs. We use a
polar code [28, 23] with a state-of-the-art SCL decoder with $16$ CRC bits and
a list size of $32$. The polar code is constructed by the Bhattacharyya method
[29]. The resulting code approaches the normal approximation only up to a
$\sim$ 1 dB gap which explains the discrepancy between the theoretical and the
simulation results in Figure 1. The curves obtained by using formula (14) for
the channel estimation error provide a rough but usable approximation which
gets worse as $K_{a}$ grows close to $n_{p}$ (dotted lines). For comparison we
add the reported values of the tensor-based-modulation (TBM) approach [22],
with tensor signature (8,5,5,4,4) and an outer BCH code, although the values
have been obtained with the higher target error rate $P_{e}=0.1$.

Figure 1: Required energy-per-bit with the MRC approach to achieve
$P_{e}=p_{\text{md}}+p_{\text{fa}}<0.05$. Solid lines represent the
achievability estimates from Section III-C. Dotted lines represent the
theoretical results with the assumption of orthogonal pilots. The markers
represent simulation results.

###  III-E Collisions

In this paragrapgh we show how pilot collisions affect the decoding
performance. Specificaly, we explain how the combination of a single-user list
decodable code can efficiently reduce the effect of pilot collisions. The
average number of collisions of $k$ users on one pilot is given by

$\mathbb{E}\\{C_{k}\\}=\frac{{K_{a}\choose k}}{N^{k-1}}.$ (17)

We can safely ignore the collisions of more then two users since their number
is much smaller than $1$ for the considered parameters. For $K_{a}=1000$ and
$N=2^{16}$, (17) gives an average number of $7-8$ collisions of order two. If
all of the colliding messages would result in an error, this would lead to a
per-user error probability of $0.016$ on average. This can be incorporated
into the above analysis by subtracting this values from the target error
probability $p_{e}$ in the normal approximation (15). Nonetheless, if a list
decoder is used as a single-user decoder, it is possible to recover both of
the colliding messages. This can be explained as follows. When there is no
fading, half of the coded bits are erased on average when two codewords are
added. Since the rate of the polar code $R=(B-J)/(2n_{d})$ is much smaller
then $1/2$, these erasure can be recovered, see also [15]. This situation
changes when fading is involved. In Figure 2 we quantify the effect of pilot
collisions by taking the uncertainty of the channel estimation and the MRC
into account via the following simplified two user collision model. Let ${\bf
s}_{1}$ and ${\bf s}_{2}$ denote the QPSK modulated sequences of two users and
${\bf h}_{1},{\bf h}_{2}\in\mathbb{C}^{M}$ their iid Rayleigh channel vectors.
We model the channel estimates as $\hat{{\bf h}}={\bf h}_{1}+{\bf h}_{2}+{\bf
e}$ where ${\bf e}\sim\mathcal{CN}(0,\sigma^{2}_{\text{est}}{\bf I}_{M})$ is
the channel estimation error with variance $\sigma^{2}_{\text{est}}$. The
model for the estimated single-user sequence for both users is then

$\hat{{\bf s}}^{\top}=\frac{1}{M}\hat{{\bf h}}^{\sf H}\left(\sqrt{{\sf
SNR}}{\bf h}_{1}{\bf s}^{\top}_{1}+\sqrt{{\sf SNR}}{\bf h}_{2}{\bf
s}^{\top}_{2}+{\bf z}\right)$ (18)

We fix the per-user ${\sf SNR}$ to -10 dB and vary the variance of the channel
estimation error. The simulation in Figure 2 shows that for $M=50$ when the
channel estimation error $\sigma^{2}_{\text{est}}$ falls below -10 dB both
codewords are correctly recovered in around $75\%$ of the cases. Nonetheless,
the probability that at least one codeword is recovered converges to 1 as the
channel estimation error goes to zero. This effect persist, even when the base
per-user ${\sf SNR}$ is increased, because it is a consequence of the cross-
correlation terms $\sqrt{{\sf SNR}}{\bf h}_{1}^{\sf H}{\bf h}_{2}{\bf
s}_{2}/M$ and $\sqrt{{\sf SNR}}{\bf h}_{2}^{\sf H}{\bf h}_{1}{\bf s}_{1}/M$.
However, in the limit $M\to\infty$ these terms vanish and the model (18)
converges to a 2-user non-fading AWGN unsourced MAC.

Figure 2: Fractions of correctly recovered codewords in the case of a two-user
collision with MRC according to the model (18). Polar code with $B-J=84$
message bits, $n_{d}=2048$ complex QPSK coded symbols.

##  IV Conclusion

In this work we presented a coding scheme for the quasi-static Rayleigh fading
AWGN MAC with a massive MIMO receiver that is compliant with the unsourced
paradigm. Simulations show that the presented scheme performs better than
existing approaches for the quasi-static MIMO MAC despite its conceptual
simplicity. We give a closed form analysis that predicts the achievable limits
of this scheme when a single user code is used which can achieve the normal
approximation. The results show that the presented approach can scale to many
hundreds of users and achieve high sum-spectral efficiencies with low power
requirement and without any coordination between users. Specifically, with
$M=100$ receive antennas over a thousand users can be served concurrently,
which leads to sum-spectral efficiencies beyond 30 bits per channel use.

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# Incremental Transfer Learning: An Approach to Detect COVID-19 Coughs from
Healthy People’s Cough Detection Models and Fewer Patient Coughs

Sudip Vhaduri1 Seungyeon Paik1 and Jessica E Huber*2  3]
1Computer and Information Technology Department
2Speech, Language, and Hearing Sciences Department
Purdue University, West Lafayette, IN 47907, USA
3Communicative Disorders and Sciences Department
University at Buffalo  Buffalo  NY  14214  USA
<EMAIL_ADDRESS>
2jhuber<EMAIL_ADDRESS>
# Incremental Transfer Learning to Detect COVID-19 Coughs from Healthy
People’s Cough Detection Models and Fewer Patient Coughs

Sudip Vhaduri1 Seungyeon Paik1 and Jessica E Huber*2  3]
1Computer and Information Technology Department
2Speech, Language, and Hearing Sciences Department
Purdue University, West Lafayette, IN 47907, USA
3Communicative Disorders and Sciences Department
University at Buffalo  Buffalo  NY  14214  USA
<EMAIL_ADDRESS>
2jhuber<EMAIL_ADDRESS>
# Detect COVID-19 Coughs from Healthy People’s Cough Models and Folds of
Patient Coughs Using Transfer Learning

Sudip Vhaduri1 Seungyeon Paik1 and Jessica E Huber*2  3]
1Computer and Information Technology Department
2Speech, Language, and Hearing Sciences Department
Purdue University, West Lafayette, IN 47907, USA
3Communicative Disorders and Sciences Department
University at Buffalo  Buffalo  NY  14214  USA
<EMAIL_ADDRESS>
2jhuber<EMAIL_ADDRESS>
# Detect COVID-19 Coughs from Healthy People’s Cough Models by Incrementally
Adding Patient Cough-Folds with Transfer Learning

Sudip Vhaduri1 Seungyeon Paik1 and Jessica E Huber*2  3]
1Computer and Information Technology Department
2Speech, Language, and Hearing Sciences Department
Purdue University, West Lafayette, IN 47907, USA
3Communicative Disorders and Sciences Department
University at Buffalo  Buffalo  NY  14214  USA
<EMAIL_ADDRESS>
2jhuber<EMAIL_ADDRESS>
# Transfer Learning to Detect COVID-19 Coughs from Healthy People’s Cough
Models and Incremental Addition of Patient Coughs

Sudip Vhaduri1 Seungyeon Paik1 and Jessica E Huber*2  3]
1Computer and Information Technology Department
2Speech, Language, and Hearing Sciences Department
Purdue University, West Lafayette, IN 47907, USA
3Communicative Disorders and Sciences Department
University at Buffalo  Buffalo  NY  14214  USA
<EMAIL_ADDRESS>
2jhuber<EMAIL_ADDRESS>
# Transfer Learning to Detect COVID-19 Coughs with Incremental Addition of
Patient Coughs to Healthy People’s Cough Detection Models

Sudip Vhaduri1 Seungyeon Paik1 and Jessica E Huber*2  3]
1Computer and Information Technology Department
2Speech, Language, and Hearing Sciences Department
Purdue University, West Lafayette, IN 47907, USA
3Communicative Disorders and Sciences Department
University at Buffalo  Buffalo  NY  14214  USA
<EMAIL_ADDRESS>
2jhuber<EMAIL_ADDRESS>
###### Abstract

Millions of people have died worldwide from COVID-19. In addition to its high
death toll, COVID-19 has led to unbearable suffering for individuals and a
huge global burden to the healthcare sector. Therefore, researchers have been
trying to develop tools to detect symptoms of this human-transmissible disease
remotely to control its rapid spread. Coughing is one of the common symptoms
that researchers have been trying to detect objectively from smartphone
microphone-sensing. While most of the approaches to detect and track cough
symptoms rely on machine learning models developed from a large amount of
patient data, this is not possible at the early stage of an outbreak. In this
work, we present an incremental transfer learning approach that leverages the
relationship between healthy peoples’ coughs and COVID-19 patients’ coughs to
detect COVID-19 coughs with reasonable accuracy using a pre-trained healthy
cough detection model and a relatively small set of patient coughs, reducing
the need for large patient dataset to train the model. This type of model can
be a game changer in detecting the onset of a novel respiratory virus.

###### Keywords:

transfer learning COVID-19 Cough detection.

##  1 Introduction

###  1.1 Motivation

Coronavirus and other human-transmissible respiratory viruses have become
prevalent and have led to human suffering and a large number of deaths in
recent times. According to the World Health Organization (WHO), the novel
coronavirus SARS-CoV-2 (COVID-19) has so far caused a total of over 771
million infections and over 6.9 million deaths globally [8]. Even after the
development of a vaccine, over 300 thousand infections and 1.5 thousand deaths
occur a day [8]. Additionally, COVID-19 created a heavy economic burden on the
health sectors, e.g., the United States incurred a total of $163.4 billion in
direct medical expenses during the pandemic [40]. Early onset detection can
help prevent the rapid spread and its adverse consequences. But, traditional
diagnosis approaches are slow and require resources, such as viral tests
(based on samples from the nose and mouth) or antibody tests [3], chest X-ray
or spirometry tests [4], blood tests, pulse oximetry, and sputum tests [7, 1].
These resources are not readily available in peoples’ homes or at healthcare
access points, such as primary care or urgent care. Therefore, there is a need
for an approach that can be easily deployed to quickly detect the onset and
control disease spread.

###  1.2 Related Work

Researchers have been trying to develop tools/systems to objectively detect
and remotely report typical symptoms of respiratory diseases, such as
coughing. Many of these techniques require the use of wearable technology. For
example, researchers have detected coughing with 0.82 accuracy using
smartwatch accelerometers and audio recordings [31], and 0.94 – 0.95
sensitivity using ECG, thermistor, chest belt, accelerometer, contact
microphone, audio microphone [23] and chest sensor [11]. Some researchers
proposed a respiratory monitoring system using a wearable patch sensor [24]
and a wearable radio-frequency (RF) cough monitoring system [26]. On the other
hand, a group of researchers proposed a COVID-19 symptom tracker utilizing a
headset-like sensor [45].

However, we have found that people’s adherence to wearables drops
significantly over time, compared to smartphone adherence [9, 28]. Therefore,
some researchers have been trying to detect objective symptoms, such as
coughing, using smartphone data [56, 20, 13, 48, 12]. One team of researchers
proposed machine learning-based COVID-19 cough, breath, and speech detection
using smartphone recording files [38, 37]. They achieved up to 0.93 area under
the curve using the k-nearest neighbor classifiers. Other researchers
developed frameworks to diagnose COVID-19 using a smartphone app and built-in
sensors in the smartphone [27, 30].

However, a major limitation of all these existing cough detection models is
the underlying assumption of the availability of a good amount of relevant
data, which is not always possible [13]. For example, during the early stage
of a new outbreak, there is not much data to develop a reasonably good
traditional machine learning model due to the need for a large volume of data.
But coughs from healthy people and patients have similarities, which can be
utilized to detect COVID-19 coughs using a healthy cough detection model and a
relatively small set of coughs from COVID-19 patients, reducing the need for
large amounts of COVID-19 patient coughs. This kind of model would be
invaluable in the detection of a new novel respiratory disease.

###  1.3 Contribution

The main contribution of this work is to present a novel approach leveraging
similarities between healthy people’s coughs and COVID-19 patients’ coughs to
incrementally transfer healthy cough detection models to COVID-19 cough
detection models with smaller batches or folds of COVID-19 coughs. Compared to
large data-driven traditional modeling approaches, incremental transfer
learning approaches can help detect the onset of a novel respiratory virus
early utilizing a relatively small set of cough samples obtained from the
first few people infected and a pre-trained healthy people’s cough detection
model to control the spread of the novel respiratory disease to minimize
adverse consequences.

Figure 1: Proposed modeling approach

##  2 Methods

In this section, we first present our modeling approach. Next, we discuss
cough and non-cough audio recordings obtained from different sources, data
processing steps, and neural network architectures and parameters used in this
work.

###  2.1 Modeling Approaches

In Figure 1, we present our modeling scheme. We start with a pre-trained model
[44] developed to detect 1000 objects, i.e., classes from the ImageNet dataset
[6]. Next, we use transfer learning to develop two base models and one target
model to detect coughs obtained from healthy people and COVID-19 patients
using transfer learning.

  1. 1.

ImageNet to Healthy (I2H) model: We transfer our ImageNet 1000 object
detection model to a binary model detecting healthy cough versus non-cough.
Throughout this manuscript, we name this model as “ImageNet to healthy”, i.e.,
I2H model. This is one of the two base models. This model will be later used
to develop the target COVID-19 cough detection model incrementally.

  2. 2.

ImageNet to COVID-19 (I2C) model: We transfer the ImageNet 1000 object
detection model to a binary model that detects COVID-19 cough versus non-
cough, which we refer to as “ImageNet to COVID-19”, i.e., I2C model in this
manuscript. This is the second base model, which will be used as a reference
model when comparing the performance of our target COVID-19 cough detection
models.

  3. 3.

Healthy to COVID-19 (H2C) model: We transfer the healthy cough detection
model, i.e., I2H model, incrementally by adding smaller batches, i.e., folds
of COVID-19 coughs to detect COVID-19 coughs. This target model is named
“Healthy to COVID-19”, i.e., H2C model.

Our ultimate goal is to utilize the I2H model and smaller batches/folds of
COVID-19 coughs to develop a target H2C model that achieves close performance
to the I2C base model to investigate the capability and feasibility of
incremental transfer learning. The incremental fold addition is the core of
our transfer learning approach. Model performance is analyzed to show how the
target models improve over time and to determine the minimum number of samples
required to identify disease coughs accurately. There may not be a base model
for a new disease for the performance comparison, such as the I2C model.
However, our approach is to demonstrate that incremental transfer learning can
be a pathway to get a valid model even when we have few samples of the new
disease-specific patient data.

Figure 2: Distribution of cough duration

###  2.2 Datasets

This section briefly describes six major datasets used in our experiments,
including three COVID-19, two healthy cough datasets, and an image dataset.

  1. 1.

ImageNet Dataset [6]: This is a publicly available image dataset with a
thousand object classes and millions of images. The dataset contains various
types of objects types, including geese, balloons, and fruits. We use the
ImageNet dataset to develop the pre-trained “ImageNet” model.

  2. 2.

Coswara Dataset [41]: This crowdsourced dataset consists of breathing,
coughing, and voice sound recordings from healthy people and COVID-19
patients. The sampling rate is in the range of 47.82 $\pm$ 0.83 kHz. In this
work, we use the voluntary coughs obtained from 274 COVID-19-positive
patients.

  3. 3.

COUGHVID Dataset [36]: This crowdsourced dataset contains more than 25,000
cough recordings from COVID-19-positive patients or asymptomatic people of
varying ages, genders, and regions. The sampling rate of this dataset is
around 44.1 kHz. We use the voluntary coughs obtained from 719
COVID-19-positive patients or asymptomatic people.

  4. 4.

NoCoCoDa Dataset [17]: This dataset contains natural cough recordings of 13
COVID-19-positive patients collected from public media interviews. Audio
recordings are collected at a 44.1 kHz sampling rate.

  5. 5.

ESC-50 Dataset [39]: The environmental sound classification (i.e., ESC-50)
dataset consists of audio recordings from five categories of sounds (i.e.,
animal, natural soundscapes, human sounds, interior sounds, and exterior
noises) with 10 types of sounds per category recorded at a rate of 44.1 kHz.
There are 40 audio recordings per type (2,000 recordings in total). We use the
voluntary cough recordings from five healthy subjects as the cough class,
i.e., healthy coughs. The remaining 49 sound types are used to create the non-
cough class when developing the binary cough versus non-cough models presented
in Section 2.1.

  6. 6.

AudioSet Dataset [25]: This dataset contains a wide range of 632 sound classes
obtained from YouTube videos, where samples are recorded at 16 kHz and 44.1
kHz. In this work, we use voluntary coughs from 88 healthy subjects.

In this work, we combined all COVID-19 data to create one COVID-19 patient
cough dataset (n = 1006 patients). Similarly, we combined all healthy cough
datasets to develop another cough dataset (n = 83 healthy people). While
combining data from different datasets, we keep the subject information so
that they can be utilized later to create mutually exclusive splits or folds
among the training, validation, and test sets. All non-cough data is obtained
from the ESC-50 dataset.

###  2.3 Data Processing

This section presents our cough ground-truth label collection approach from
long audio recordings, followed by additional processing steps, including
finding an optimal window size, padding, and feature extraction. Finally, we
present our cross-validation approach with incremental training to develop H2C
models from I2H model which requires relatively fewer COVID-19 coughs.

####  2.3.1 Data Cleaning and Ground-Truth Label Collection

Most of the audio data used in this work come from crowd-sourced datasets.
Therefore, we first perform a rigorous cleaning process to remove unwanted
parts, including quiet, speech, and noises using different audio signal
processing libraries and tools, including the Audacity toolbox [2].

After the initial cleaning, we performed data segmentation to extract the
ground-truth cough labels from long audio recordings with multiple coughs
utilizing the Audacity toolbox. Adapted from our previous work [74, 46], we
automated the process by developing an energy threshold-based audio
segmentation followed by a phase classification approach.

We obtained 144 and 252 healthy coughs from the ESC-50 and AudioSet datasets.
Similarly, we obtained 1892, 2690, and 73 COVID-19 coughs from the Coswara,
COUGHVID, and NoCoCoDa datasets, respectively. In total, we used 396 healthy
coughs and 4655 COVID-19 coughs.

(a) I2H or I2C Models

(b) H2C Model

Figure 3: Optimal Architecture of different models

####  2.3.2 Feature Extraction

Since we collected audio recordings from different sources, we first changed
all sampling frequencies to 44.1 kHz frequency before additional processing.

Next, we determined a suitable window size before computing features. In
Figure 2, we use boxplots to represent the distribution of cough duration
obtained from various datasets. In the figure, we find that $75^{th}$
percentile of Coswara healthy cough duration, i.e., 0.3917 seconds, is a
suitable choice for window size since most of the other healthy and patient
coughs, except the NoCoCoDa coughs, have a duration shorter than that.
Compared to other datasets, the NoCoCoDa dataset has relatively fewer samples.

We perform padding (i.e., add 0s) at the end of coughs shorter than the window
size (i.e., 0.3917 seconds). In the case of longer cough and non-cough audio
recordings, we truncate the parts longer than the window size. Finally, we
compute the log mel-spectrogram [34] (logmel) and Mel-frequency cepstral
coefficient (mfcc) image features from each cough and non-cough audio
recording using the Python Librosa library [33].

####  2.3.3 Training, Validation, and Test Split

First, we uniformly split the cough data into 10 mutually exclusive folds
based on subjects. For class balancing, we select the same number of non-cough
samples as we have cough samples for the 10 folds. Thereby, we maintain the
same number of cough (either healthy or COVID-19) and non-cough instances when
training, validating, and testing binary cough (either healthy or COVID-19)
versus non-cough detection models presented in Section 2.1. Next, we follow a
“leave-2-fold-out” for validation and “leave-2-fold-out” for the test
approach, where we use the remaining six folds for training while developing
and validating/testing a I2H or I2C model from the pre-trained ImageNet Model.
We follow a rotational approach, where we perform this mutually exclusive
6-2-2 train-validation-test split 10 times to develop 10 different models.
While developing H2C models, we follow an incremental training approach, where
we add the six training folds of COVID-19 data to the H2C models one-by-one.
We perform this incremental training for one of the 10 6-2-2 splits. During
the incremental training, we distribute COVID-19 subjects in 10 folds using a
snaking approach, where we first sort the subjects in descending order of
cough counts. Then, we put the top 10 subjects into 10 folds. Next, 10
subjects are put into 10 folds. This way, we maximize the number of cough
samples in 10 folds. We end up with 100 subjects in each fold with 450 random
COVID-19 cough samples.

###  2.4 Neural Network Architectures and Parameters

In Figure 3(a), we present the optimal architecture of the two models (i.e.,
I2H or I2C models) transferred from the pre-trained ImageNet model developed
with VGG19 [44]. In Figure 3(b), we present the optimal architecture of the
H2C model transferred from the I2H model by adding folds of COVID-19 coughs.
In the figure, the sequence and meaning of different parameters in each layer
are kernel size and the number of feature maps, i.e., nodes in each layer. The
pre-trained ImageNet VGG19 model had fully connected (FC) three layers with
4096, 4096, and 1000 nodes, followed by a softmax layer to classify 1000
objects. We changed the last four layers with FC 512, 50, and 1 node, followed
by a sigmoid layer for binary classification of healthy cough versus non-cough
(I2H model) or COVID-19 cough versus non-cough (I2C model). Compared to the
I2H or I2C model, we add four additional convolutional layers and a pooling
layer in the case of H2C models.

We used TensorFlow and Keras libraries to develop our models. We used the ReLU
activation function in the hidden layers and the sigmoid decision function in
the final layer for every model. For the loss function, we used binary cross-
entropy. We tried Adam and RMSprop optimizer and found Adam achieves 53%
higher accuracy than the RMSprop. We also tried batch sizes 16 and 32 and
found that batch size 16 is more accurate and has a shorter execution time.
The input size used in this work is (320,320). We considered a range of
learning rates, including 0.00001, 0.0001, and 0.0005, and epochs ranging from
30 to 200. We found a learning rate of 0.0001 works better for the I2H and I2C
models. Similarly, we found a learning rate of 0.00001 is a good compromise
for the H2C models. We developed models on Purdue University’s Gilbreth GPU
server with 8 GPU nodes and 16 cores per node [5]. Each node has 192 GB
memory, 100 Gbps Infiniband interconnects, and 2 P100 GPUs.

Figure 4: I2H and I2C models tested on different coughs (H-H refers to the
case when I2H models are tested on healthy coughs; similarly, H-C and C-C
refer to the cases when I2H and I2C models are tested on COVID-19 coughs)
Figure 5: Performance of H2C models developed using transfer learning
incrementally by adding folds of COVID-19 coughs to I2H models

##  3 Results

Since we use the mutually exclusive folds with the same number of cough and
non-cough instances in each fold (Section 2.3.3) when developing binary
models, our classes are always balanced, and we use classification accuracy
when comparing the performance of different models.

In Figure 4, we present the performance of the two base models, i.e., I2H and
I2C models when tested on different cough datasets. First, we analyze the
performance of the I2H base models, i.e., healthy cough detection models
transferred from the pre-trained ImageNet models using healthy people’s
coughs. The I2H base models are tested on the healthy test coughs, and the
findings are presented using the Healthy-Healthy or H-H box in the figure. We
observe average accuracy values of 0.93 $\pm$ 0.02 (logmel) and 0.92 $\pm$
0.05 (mfcc) with median values of 0.93 (logmel) and 0.93 (mfcc).

Next, we analyze the performance of the second base model, i.e., I2C model
(COVID-19 cough detection models transferred from the pre-trained ImageNet
models using COVID-19 patient coughs). The I2C model is tested on the COVID-19
test coughs, and the findings are presented using the COVID-COVID or C-C box
in the figure. We observe average accuracy values of 0.96 $\pm$ 0.02 (logmel)
and 0.95 $\pm$ 0.01 (mfcc) with median values of 0.96 (logmel) and 0.95
(mfcc).

Next, we analyze the performance of the I2H models (i.e., base models to
detect healthy people’s coughs) when tested on COVID-19 test coughs and
present the findings using the Healthy-COVID or H-C box in the figure. We
observe average accuracy values of 0.88 $\pm$ 0.07 (logmel) and 0.86 $\pm$
0.03 (mfcc) with median values of 0.91 (logmel) and 0.86 (mfcc). In the case
of logmel feature, the drop in average accuracy is 0.05 and 0.08 when compared
with the findings in the Healthy-Healthy (H-H) and COVID-COVID (C-C) boxes,
respectively. In the case of mfcc feature, the drop in average accuracy is
0.06 and 0.09 when compared with the findings in the Healthy-Healthy (H-H) and
COVID-COVID (C-C) boxes, respectively. To improve the accuracy values of the
H-C box (i.e., I2H models tested on COVID-19 test coughs), we incrementally
developed H2C models transferring the I2H models by adding small amounts of
COVID-19 coughs in folds. Since logmel features outperform the mfcc features
across all measures, we consider the logmel features in the next analysis.

In Figure 5, we present the accuracy gap of H2C models with respect to the
average accuracy of the base I2C models by varying amounts of COVID-19 coughs
(i.e., number of folds) included in the training set. In the figure, we
observe that with the addition of only one fold of COVID-19 coughs, the
average accuracy gap drops to 0.037.

The average accuracy gap drops to 0.02 by adding two more folds of COVID-19
coughs to the base I2H models. Thereby, with the addition of three folds of
COVID-19 coughs with the healthy cough detection models, i.e., I2H models, we
can achieve a performance close to that of the base COVID-19 detection model
performance.

As we continue adding more folds of COVID-19 coughs to the base I2H models, we
witness a drop in accuracy gap, and after adding all six folds of COVID-19
patient coughs, the accuracy gap drops to 0.01. Additionally, adding more
folds makes the error bar tighter, reflecting more consistent accuracy values.
Thereby, using this incremental transfer learning approach, we can develop
COVID-19 detection models, such as H2C models, from the base healthy people’s
cough detection models, i.e., I2H models and smaller amounts of COVID-19
coughs to achieve very close performance to that of the base COVID-19
detection models, i.e., I2C models.

##  4 Conclusion and Discussion

This work attempts to utilize the power of transfer learning and similarities
between two types of coughs, i.e., healthy and COVID-19 coughs, to
incrementally develop new models requiring a relatively small set of patient
coughs to achieve similar performance to that of the COVID-19 cough detection
models trained from a bigger COVID-19 cough dataset. Our findings show the
promise of utilizing healthy cough detection models to detect COVID-19 coughs
after training with relatively fewer patient coughs.

This model can be useful to detect an early-onset novel respiratory virus with
a smaller amount of relevant data. However, before generalizing the findings
to similar or other problems, extended studies with a diverse population,
diseases, and stages will be needed. While image feature-based transfer
learning has been adopted in this feasibility work, in the future, other types
of data, e.g., acoustic signals, can be utilized to adapt transfer learning
models and can be compared with this feasibility work. This work and findings
will also impact other domains of predictive modeling, including place of
importance discovery [59, 71, 65, 67, 70], health condition monitoring [42,
66, 57, 14, 73, 43, 72] and well-being tracking [47, 60, 58, 61, 21, 52, 29,
55, 50], securing a user’s cyber-physical space [68, 63, 64, 35, 15, 62, 19,
16, 53, 51, 69, 49, 32, 10, 54, 18, 22], as it presents the feasibility of
developing predictive models with relatively small datasets to alternate the
traditional approaches requiring large-scale datasets.

## Acknowledgement

The authors would like to thank the Clifford B. Kinley Trust for funding this
research.

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