# Deep Learning based Positioning with Multi-task Learning and Uncertainty-
based Fusion

Anastasios Foliadis12, Mario H. Castañeda Garcia1, Richard A. Stirling-
Gallacher1, Reiner S. Thomä2  1Munich Research Center, Huawei Technologies
Duesseldorf GmbH, Munich, Germany
2Electronic Measurements and Signal Processing, Technische Universität
Ilmenau, Ilmenau, Germany
{anastasios.foliadis, mario.castaneda<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Deep learning (DL) methods have been shown to improve the performance of
several use cases for the fifth-generation (5G) New radio (NR) air interface.
In this paper we investigate user equipment (UE) positioning using the channel
state information (CSI) fingerprints between a UE and multiple base stations
(BSs). In such a setup, a single DL model can be trained for UE positioning
using the CSI fingerprints of the multiple BSs as input. Alternatively, based
on the CSI at each BS, a separate DL model can be trained at each BS and then
the output of the different models are combined to determine the UE’s
position. In this work we compare these different fusion techniques and show
that fusing the output of separate models achieves higher positioning
accuracy, especially in a dynamic scenario. We also show that the fusion of
multiple outputs further benefits from considering the uncertainty of the
output of the DL model at each BS. For a more efficient training of the DL
model across BSs, we additionally propose a multi-task learning (MTL) scheme
by sharing some parameters across the models while jointly training all
models. This method, not only improves the accuracy of the individual models,
but also of the final combined estimate. Lastly, we evaluate the reliability
of the uncertainty estimation to ascertain which of the fusion methods
provides the highest quality of uncertainty estimates.

###### Index Terms:

Deep Learning, Wireless Positioning, Late Fusion, Early Fusion, Multi-task
Learning, Uncertainty Estimation

##  I Introduction

Accurate user positioning is one of the enablers of several future services
and technologies [1, 2, 3, 4], e.g., for location-aware communication, vehicle
to everything (V2X) applications, industrial internet of things (IIOT),
cooperating robots, commercial applications, etc. For this purpose, radio-
based positioning of a user equipment (UE) in wireless communication networks
can be considered [5]. Multiple base stations (BSs) deployed in such networks
allow the collection of channel state information (CSI) over distributed
links, which can be exploited for positioning of a UE. The CSI consists of the
channel across the spatial and frequency domain, where the large number of
antennas and large available bandwidth of current and future communication
networks [4], i.e., fifth generation (5G) or upcoming sixth generation (6G),
can provide a high angular and temporal resolution to enable high accuracy
positioning.

Conventional radio-based positioning methods are generally characterized by
being model-driven, i.e., geometry based, and usually follow a two-step
approach. With CSI estimated at one BS [6] or at multiple BSs [7], relevant
parameters or measurements are first determined, e.g., path delay, angle of
arrival (AoA), reference signal receive power (RSRP), time difference of
arrival (TDoA), etc, to subsequently compute the UE’s position in a second
step based on the determined parameters or measurements. Recently, data-driven
techniques have also been proposed for radio-based UE positioning, i.e.,
machine learning (ML) / artificial intelligence (AI)-based methods [8, 9, 10,
11, 12]. In particular, deep learning (DL) methods with convolutional neural
networks (CNNs) have shown promising results [13, 14, 15, 16], being able to
achieve sub-meter accuracy. In such data-driven models, the CSI of a UE at a
given position, i.e., CSI over subcarriers and antennas, is considered as a
fingerprint associated with the UE’s position. By leveraging the ability of
wireless networks to collect large amounts of data, a database of CSI
fingerprints associated with different UE’s positions along with the
respective UE’s position label can be constructed. With the DL-based
positioning methods, a neural network (NN) model can be trained on a given
database, such that afterwards the NN model can be employed for estimating a
UE’s position by providing the CSI of the UE as input to the trained NN model.
Different types of fingerprints have been considered in the literature,
including the received signal strength, the magnitude and/or phase of the CSI
over subcarriers in the frequency domain and across antennas in the spatial
domain [15, 16, 17, 18, 19].

With the CSI of a UE available across multiple BSs, early fusion or late
fusion can be considered for the DL-based positioning methods [20, 21]. In
early fusion, the CSI fingerprints from multiple BSs is collected and bundled
together to constitute a single CSI fingerprint associated with the UE’s
position. Thus with early fusion, only one NN model needs to be trained with a
database comprising fingerprints with the CSI across multiple BSs [20]. On the
other hand, with late fusion, one NN model is assumed at each BS, i.e., per
link, where the CSI collected by a BS is considered as a fingerprint of the
UE’s location associated with the given BS [21]. In late fusion, the NN model
associated with a BS is trained with a database of CSI fingerprints from that
BS, enabling the NN model to determine the UE’s position based on the CSI
estimated by that BS. Afterwards, a final UE’s position estimate is obtained
by combining the position estimates obtained by the NNs across the multiple
BSs [21, 22], e.g., with a weighted average.

The choice between early or late fusion generally depends on the application
[23]. However, when considering changes in the UE-BS channel between the
training phase and deployment phase, e.g., due to a blockage of the line of
sight (LOS) between a UE and a BS, late fusion can benefit from uncertainty
estimation [21]. In particular, the NN model at each BS can be trained to
estimate the uncertainty in the UE’s position determined by each NN model.
This enables the late fusion approach, to determine the final position
estimate for the UE considering the uncertainty of the multiple position
estimates obtained across multiple BSs. In practice the most reliable position
estimates have a larger impact in determining the final UE’s position.
Uncertainty estimation can be computed based on simple approaches like Monte
Carlo Dropout (MCD) [24] and Deep Ensembles (DEs) [25], which characterize the
uncertainty based on the variance of the positioning error obtained with
multiple NNs, i.e., similar position estimates indicates lower uncertainty
estimation.

While conventional positioning approaches may fail or be impaired in channels
with non-line of sight (LOS) or with strong multipath, DL-based methods can
still be employed in such scenarios [26, 20, 27], as the multipath can be
captured in the CSI fingerprint and aid in making the fingerprint more unique
and distinguishable. Despite this fact, with the multipath profile being
susceptible to environmental changes, a DL model trained with CSI fingerprints
from one environment may achieve a poor performance for the UE positioning in
another environment [28]. The lack of direct transferability of the knowledge
acquired in one environment to other environments is one of the challenges of
DL-based positioning [29]. The most straightforward way to address this is to
retrain the NN model from scratch with CSI fingerprints from the new
environment, which may however be resource expensive and may not always be
feasible. On the other hand, several approaches can be considered for
improving the generalizability of a trained model to adapt it to environmental
changes or to a new environment including transfer learning, domain
generalization, multi-task learning and meta learning. With transfer learning,
a previously trained model is used as an initial model that is fine-tuned with
reduced training data from a new environment [19, 28], which allows to speed
up the training and to improve the performance compared to training from
scratch.

Furthermore, with multi-task learning (MTL) the aim is to jointly learn
multiple models by training multiple models which share some parameters,
thereby benefiting from regularization [30]. By considering the positioning in
different environments as different tasks, the positioning across multiple
environments can be improved. When training a MTL scheme the choice of the
relative importance of each task has to be considered. The hardest to learn
tasks should be weighted less, so that the model focuses more on tasks that
are easier to learn. Based on the uncertainty of each task, a method was
proposed in[31] that takes into account the importance of each task. This
method, not only provides a way to tune the importance of different tasks but
also simultaneously learns the uncertainty for each task, which as shown in
[21] is beneficial for the DL-based localization using CSI fingerprints.

Another approach aiming at improving the generalizability of NN models is
meta-learning. With meta-learning, a model which is trained on multiple tasks,
i.e., multiple environments, by considering a meta-level objective, i.e., the
average positioning error across the multiple environments [32, 29], such that
the minimization of the loss function in an unseen task is done more
efficiently. Meta-learning aims at having a trained model that generalizes
better not only across the trained tasks but also facilitates learning an
unseen task with a lower number of training samples, in contrast to multi-task
learning which aims at learning better the trained tasks.

Motivated by the two-step approach of conventional positioning methods, i.e.,
with parameter/measurement extraction from the CSI in a first step and a
position determination in a second step, a two-part model trained with multi-
task learning and a meta-level objective has been recently proposed in [33].
For UE positioning in different environments, i.e., different training tasks,
, different models are assumed with the first part of the models being common
across all task and trained with CSI samples from all tasks (multi-task
learning) aiming at minimizing the sum positioning error across all tasks
(meta-level objective). The second part of the model of each task is trained
to be environment specific, i.e., with training data from each environment.
The proposed approach in [33] is able to improve the positioning accuracy of
the trained environments, as well as achieve a better generalizability when
transferring the first part of the model and fine tuning the two-part model
with CSI samples of a new environment.

###  I-A Contributions

As proposed in [31], MTL benefits from uncertainty estimation. The training in
MTL can be improved by determining the relative weighting of the losses of
each task based on the associated uncertainty estimate [31]. For this reason,
in this paper we combine the results from [21] and [33] to benefit from the
simultaneous training of different tasks and from late fusion using
uncertainty estimation. For a setup with multiple BSs and considering the
positioning of a UE using each BS as a separate task, we show that employing a
multi task learning scheme with uncertainty estimation and late fusion
achieves high positioning accuracy. Additionally, even though this is outside
the scope of the current paper, it was shown in [33] that a model trained with
the MTL scheme can be further used for transfer learning in a new environment,
reducing the time and amount of data that needs to be gathered.

Moreover, we extend the work in [21] by employing a method described in [34]
for sensor fusion that takes into account the possibility that one or more
sensors may be spurious. In the case of DL-based positioning, a model estimate
could be spurious if the purported uncertainty is low but the real error is
high. We employ this method in a late fusion scheme and show that it is
beneficial in improving the positioning accuracy especially in dynamic
environments.

Lastly, we aim not only to minimize the positioning error, but also evaluate
the reliability of the uncertainty estimation. It would be beneficial if the
estimated uncertainty truly reflects the model’s uncertainty about the current
measurement, such that a high uncertainty should indicate high positioning
error and vice-versa. To evaluate the quality of uncertainty estimates we
consider the area under sparsification error (AUSE) metric [35] [36]. In
addition to the AUSE metric, we evaluate the integrity of the positioning
results with respect to the integrity risk (IR) which is used in global
navigation satellite system (GNSS) applications and has been recently proposed
in the Third Generation Partnership Project (3GPP) as a positioning key
performance indicator (KPI) for 5G positioning [37].

The paper is structured as follows. In Section II the considered system model
is described along with the different types of fusion and the MTL scheme is
introduced. In Section III the simulation setup is described and the DL-model
structure. The results and conclusion are then presented in sections IV and V
respectively.

##  II System Model

We consider an uplink setup with $N_{B}$ BSs each with $N_{R}$ receive
antennas and a single transmit antenna at the UE. The UE transmits a reference
signal on $N_{C}$ subcarriers within an orthogonal frequency division
multiplexing (OFDM) symbol. The received uplink signal is used to estimate the
CSI matrix between UE and each BS. The estimated channel at the $n$-th BS over
the $N_{C}$ subcarriers is described as:

$\tilde{\boldsymbol{H}_{n}}=[\boldsymbol{h}_{0}^{n},\boldsymbol{h}_{1}^{n},\ldots,\boldsymbol{h}_{N_{C}-1}^{n}]\in\mathbb{C}^{N_{C}\times
N_{R}},$ (1)

where $\boldsymbol{h}_{l}^{n}\in\mathbb{C}^{N_{R}}$ is a column vector that
describes the estimated uplink channel between the UE and the $N_{R}$ antennas
of the $n$-th BS at the $l$-th subcarrier. The estimated channels can be
considered a unique fingerprint of the position of the UE and it depends on
the multipath between the UE and each BS. As the channel is complex, we simply
stack the matrices $\Re{\tilde{\\{\boldsymbol{H}_{n}\\}}}$ and
$\Im{\tilde{\\{\boldsymbol{H}_{n}\\}}}$ in the third dimension to obtain a new
real-valued 3D matrix $\boldsymbol{H}_{n}\in\mathbb{R}^{N_{C}\times
N_{R}\times 2}$. The symbols $\Re{\\{\cdot\\}}$ and $\Im{\\{\cdot\\}}$ denote
the real and imaginary values of each of the matrix elements respectively.

Figure 1: Early Fusion Figure 2: Late Fusion

###  II-A DL based positioning with fingerprints

Deep learning based localization using CSI fingerprints as inputs consists of
two phases, namely the training and the deployment phase, which are often
alternatively termed as offline and online phase, respectively. During the
training phase, CSI fingerprints are collected throughout the area of interest
along with a label corresponding to the UE position associated with each CSI
fingerprint. Without loss of generality, we assume that the UEs lie on a two
dimensional plane, i.e. on the ground. Subsequently the CSI fingerprints and
the position labels $\boldsymbol{p}=[x,y]\in\mathbb{R}^{2}$ are used to train
the parameters $\epsilon$ of a neural network (NN). Training is accomplished
by minimizing the mean squared error between the position labels and the
output of the NN with the labeled CSI fingerprints as input. Eventually, the
trained NN is then used during the deployment phase to estimate the position
of a UE based on the estimated CSI fingerprint $\boldsymbol{H}$, i.e.,
$f_{\epsilon}(\boldsymbol{H})=\tilde{{\boldsymbol{p}}}$, where
$\tilde{{\boldsymbol{p}}}=[\tilde{x},\tilde{y}]\in\mathbb{R}^{2}$ is the
position estimate for the UE with the CSI fingerprint $\boldsymbol{H}$. One
assumption that must hold is that the propagation environment should not
significantly change between the two phases since that would degrade the
performance of the NN, e.g., the trained NN may be not be useful.

Two different approaches for positioning using CSI fingerprints from multiple
$N_{B}$ BSs can be considered [21], namely early and late fusion.

####  II-A1 Early Fusion

In early fusion , a single NN model is trained for the UE positioning, having
as input the concatenation of the CSI fingerprints from all BSs, i.e., the
single NN model $f_{\epsilon}(\boldsymbol{H})=\tilde{{\boldsymbol{p}}}$ where
$\boldsymbol{H}=[\boldsymbol{H}_{1},\boldsymbol{H}_{2},...,\boldsymbol{H}_{N_{B}}]$.
Although this is a straightforward way to combine the information from all BSs
and perform localization using a DL-model, it has some disadvantages. Firstly,
a large signaling overhead is required in order to transmit the relevant CSI
data to a central server which has the single NN model and second, if the
setup changes (e.g. a BS is removed), then a new NN model has to be trained
from scratch. A block diagram of early fusion is shown on Fig. 1.

####  II-A2 Late Fusion

With late fusion, a UE’s position estimate is determined based on the CSI
fingerprint at each BS. For this purpose, a separate NN model is trained at
each BS based on the CSI at each BS as input. The parameters $\epsilon_{n}$ of
the $n$-th model
$f_{\epsilon_{n}}(\boldsymbol{H}_{n})=\tilde{{\boldsymbol{p}}}_{n}$ are
trained for the $n$-th BSs, with the CSI measurements $\boldsymbol{H}_{n}$ of
that BS. During deployment, the position estimates obtained across all the BSs
are appropriately combined to produce a single position estimate. This type of
fusion is shown in Fig. 2

Compared to early fusion, this method necessitates much less traffic for the
network compared to early fusion. The reason for that is that the data that
needs to be collected is the output of each of the models instead of the whole
CSI fingerprints as in the case of early fusion. On the other hand, an
appropriate model for the combination of the multiple estimates has to be
developed. In this paper we built upon the work in [21] and propose and
compare methods to appropriately combine the position estimates considering
the uncertainty.

###  II-B Uncertainty estimation

Normally, for DL based positioning with CSI fingerprints, the parameters
$\epsilon$ of the NN are optimized with respect to the mean squared error
(MSE) loss:

$\mathcal{L}(f_{\epsilon})=\mathcal{L}_{x}+\mathcal{L}_{y},$ (2)

where $\mathcal{L}_{x}=E[|\tilde{x}-x|^{2}]$,
$\mathcal{L}_{y}=E[|\tilde{y}-y|^{2}]$ and $E[.]$ denotes the mean value over
the samples in the training set.

The drawback of using such a loss function is that the model does not acquire
any knowledge about the uncertainty that is present in the measurements. In
the following, we discuss different types of uncertainties.

####  II-B1 Aleatoric Uncertainty

The data dependent uncertainty is called aleatoric uncertainty and it reflects
the uncertainty that a measurement has about the specific task. In a case of
positioning using CSI fingerprints, a particular CSI measurement would have
high uncertainty if it has low receive SNR for example. This type of aleatoric
uncertainty, i.e. instance-dependent uncertainty, is called heteroscedastic
uncertainty. Since the aleatoric uncertainty in localization using
fingerprints is data-dependent, it can also be learned from the data. In [38]
a modification to the MSE loss was proposed in order to train a model to
simultaneously calculate the position and the aleatoric uncertainty of the
current position estimate. The modified loss which shall be minimized with
respect to the model parameters $\epsilon$ is:

$\mathcal{L}^{\prime}(f_{\epsilon})=\frac{1}{2{(\sigma_{x}^{\alpha})}^{2}}\mathcal{L}_{x}+\frac{1}{2{(\sigma_{y}^{\alpha})}^{2}}\mathcal{L}_{y}+\log({\sigma_{x}^{\alpha}}{\sigma_{y}^{\alpha}}).$
(3)

where ${(\sigma_{x}^{\alpha})}^{2}$ and ${(\sigma_{y}^{\alpha})}^{2}$ are the
aleatoric uncertainties for the outputs $x$ and $y$ respectively.

Subsequently, the output of the model has to be modified to include the
learned aleatoric uncertainty
$\boldsymbol{\sigma}^{\alpha}=[\sigma_{x}^{\alpha}$, $\sigma_{y}^{\alpha}]$ of
$\tilde{\boldsymbol{p}}=[\tilde{x},\tilde{y}]$, i.e.,
$f^{\prime}_{\epsilon}(\boldsymbol{H})=[\tilde{x},\tilde{y},\sigma_{x}^{\alpha},\sigma_{y}^{\alpha}]$.
The modified loss function $\mathcal{L}^{\prime}(f_{\epsilon})$ consists
firstly of two regression terms that describe the inverse relationship between
the MSE loss of each estimated coordinate and its corresponding aleatoric
uncertainty. When the MSE loss for a particular coordinate cannot be minimized
further, the model increases the respective aleatoric uncertainty to
compensate. On the other hand the aleatoric uncertainty remains low for
instances where the MSE loss is low. The last term is a regularization term
that is used to limit the infinite increase of the outputs
$\sigma_{x}^{\alpha}$ and $\sigma_{y}^{\alpha}$.

####  II-B2 Epistemic Uncertainty

The aleatoric uncertainty is not the only type of uncertainty present in a DL
model. The other type is called epsitemic uncertainty and it accounts for
uncertainty in the model’s parameters [38]. Estimates with high epistemic
uncertainty indicate that the input comes from a distribution that was not
learned by the model. In a DL localization model with fingerprints, the
epistemic uncertainty would be high for a region in space where no data were
collected or when a CSI measurement was corrupted.

In [24], an approach for capturing a model’s epistemic uncertainty called
Monte Carlo dropout (MCD) was introduced. When employing dropout, random
neurons in every weight layer of the deep learning model are deactivated with
a predefined probability. Typically, dropout is utilized solely for training
purposes as a regularization technique [39], but with MCD, this same dropout
probability is retained even during the deployment phase. Each successive
forward pass through the deep DL model with MCD generates a unique
configuration, and conducting multiple forward passes is akin to sampling from
an approximate posterior distribution of the model’s parameters given the
dataset [24]. The variance of the estimates from the different model
configurations during these forward passes serves as an indicator of the
model’s epistemic uncertainty.

After $T$ forward passes, the combined aleatoric and epistemic uncertainty of
the coordinate $x$ is [38]:

$\sigma_{x}=\frac{1}{T}\sum_{t=1}^{T}\tilde{x}_{t}^{2}-\Big{(}\frac{1}{T}\sum_{t=1}^{T}\tilde{x}_{t}\Big{)}^{2}+\frac{1}{T}\sum_{t=1}^{T}\sigma_{x}^{\alpha},$
(4)

where $t$ indicates the current forward pass and $\tilde{x}_{t}$ indicates the
estimate of the coordinate x at the $t$-th forward pass. Similarly the
combined aleatoric and epistemic uncertainty is calculated for the $y$
position coordinate.

Even though the epistemic uncertainty estimation is an efficient way for the
DL model to report on its own knowledge about the current measurement, it is
not always accurate. There are cases where the epistemic uncertainty is low
but the mapping to the position is highly inaccurate. In those instances the
model would provide a spurious estimate which has to be identified and
eliminated from the fusion scheme. For the late fusion in [21], the method
employed to combine the results from the different estimates is based on the
assumption that each estimate follows a known Gaussian distribution, whose
variance corresponds to the estimated combined aleatoric and epistemic
uncertainty. As this assumptions does not always hold, we take into account
such model inconsistencies by incorporating a method described in [34] to fuse
measurement from multiple sensors. The basic idea of this method is to weigh
less the estimate that is most inconsistent with the other estimates. We
should note that this method leverages multiple position and uncertainty
estimates from different BSs, and therefore can only be employed in a late
fusion scheme as described next.

In our setup we consider $N_{B}$ different models, corresponding to the models
trained at each of the $N_{B}$ BSs. The authors of [34] assume that the
probability that a measurement from the $n$-th sensor $n\in[1,...,N_{B}]$, is
not spurious with probability

$p_{n}=\exp({\frac{-(x-\tilde{x}_{n})^{2}}{\alpha_{n}}})$ (5)

where $x$ is the true state, $\tilde{x}_{n}$ is the estimate of the $n$-th
sensor and $\alpha_{n}$ is a parameter that depends on the variances of each
separate model and the difference between the output of the $n$-th model with
respect to other sensors:

$\alpha_{n}=\frac{b_{n}}{\prod_{l=1,l\neq
n}^{N_{B}}(\tilde{x}_{n}-\tilde{x}_{l})^{2}}$ (6)

where $b_{n}$ is a hyperparameter. Finally, they show that by making those
assumptions, the variance of the $n$-th model is modified from $\sigma_{x,n}$
to:

$\sigma_{x,n}^{\prime
2}=\frac{\sigma_{n}^{2}b_{n}^{2}}{b_{n}^{2}-2\sigma_{x,n}^{2}\prod_{l=1,l\neq
n}^{N_{B}}(\tilde{x}_{n}-\tilde{x}_{l})^{2}}$ (7)

where it must hold that $b_{n}^{2}>2\sigma_{x,n}^{2}\prod_{l=1,l\neq
n}^{N_{B}}(\tilde{x}_{n}-\tilde{x}_{l})^{2}$, so that $\sigma_{n}^{\prime}>0$.
In our model we choose:

$b_{n}^{2}=2\sigma_{x,n}^{2}\prod_{l=1,l\neq
n}^{N_{B}}(\tilde{x}_{n}-\tilde{x}_{l}+\lambda)^{2}$ (8)

where $\lambda$ is a small valued hyperparameter.

The intuition behind equation (7) and the choice of parameter $b_{n}$ is that
when the estimate of the $n$-th model is very different from the estimates of
the other models, i.e. $\prod_{l=1,l\neq
n}^{N_{B}}(\tilde{x}_{n}-\tilde{x}_{l})^{2}\to\infty$, then
$\sigma_{n}^{\prime}\to\infty$. On the other hand, when $\prod_{l=1,l\neq
n}^{N_{B}}(\tilde{x}_{n}-\tilde{x}_{l})^{2}\approx 0$, meaning that the
estimate of the $n$-th sensor largely agrees with the other models, then
$\sigma_{n}^{\prime}\approx\sigma_{n}$.

###  II-C Multi task learning

When considering the late fusion approach, i.e. a separate NN for each of the
BSs, we propose to share some parameters across models of different BS’s as
described in [33]. The $n$-th model which corresponds to the $n$-th BS
$f_{\theta,\epsilon_{n}}(\boldsymbol{H}_{n})$ is parametrized by the common
parameters $\theta$ and the BS specific parameters $\epsilon_{n}$. By defining
$f_{\theta,\epsilon_{n}}(\boldsymbol{H}_{n})=g_{\epsilon_{n}}(\phi_{\theta}(\boldsymbol{H}_{n}))$
and training the parameters $\theta$ on data from multiple BS we are forcing
$\theta$ to be the same, regardless of the input data. This means that we can
deal with the training of the different models as a multi task learning (MTL)
scheme and jointly minimize the loss for the positioning with the model at
each BS. MTL with parameter sharing allows for information flow between tasks
which eventually may help each individual task [40]. This method of training
is not possible when considering early fusion, since there is only one
available model. The block diagram of a the MTL scheme considered in this work
is shown in Fig. 3. The naive approach of optimizing a MTL scheme is to
minimize a linear sum of the loss for each individual task, i.e. for the
positioning at each BS. Thus, the loss for the training of the models in the
MTL scheme would be would be:

$\mathcal{L}_{\text{MTL}}(f_{\theta,\epsilon_{1}},f_{\theta,\epsilon_{2}},...,f_{\theta,\epsilon_{N_{B}}})=\sum_{n=1}^{N_{B}}\mathcal{L}_{n}(f_{\theta,\epsilon_{n}}),$
(9)

where $\mathcal{L}_{n}(f_{\theta,\epsilon_{n}})$ is the MSE loss of the $n$-th
model as described in (2). The authors of [31] observed that by weighting each
task differently the performance of each task can be greatly improved and
proposed to set the weighting based on the aleatoric uncertainty [38] of each
task.

In the context of DL based localization using fingerprints each model is
generates a 2-dimensional position estimate. By assuming that each output of
the $n$-th DL model follows a Gaussian distribution
$\mathcal{N}(\tilde{x}_{n},{\sigma_{x,n}}^{2})$, and similarly for the $y$
variable, the authors of [38] derive the minimization objective for the multi
task learning as follows:

$\begin{split}\mathcal{L}^{\prime}_{\text{MTL}}(f_{\theta,\epsilon_{1}},f_{\theta,\epsilon_{2}},...,f_{\theta,\epsilon_{N_{B}}})=\sum_{n=1}^{N_{B}}\mathcal{L}^{\prime}_{n}(f_{\theta,\epsilon_{n}})\\\
=\sum_{n=1}^{N_{B}}\bigg{[}\frac{1}{2{(\sigma_{x,n}^{\alpha})}^{2}}\mathcal{L}_{x,n}+\frac{1}{2{(\sigma_{y,n}^{\alpha})}^{2}}\mathcal{L}_{y,n}+\log({\sigma_{x,n}^{\alpha}}{\sigma_{y,n}^{\alpha}})\bigg{]}\end{split}$
(10)

where $\mathcal{L}^{\prime}_{n}(f_{\theta,\epsilon_{n}})$ is the loss of the
$n$-th model as defined in (3) and it includes each model’s aleatoric
uncertainty $\sigma_{x,n}^{\alpha}$ and $\sigma_{y,n}^{\alpha}$ which is
implicitly weighing the losses for each task.

Figure 3: Two-part models with multi-task learning. Models’ output comprise
the UE’s position estimate and the aleatoric uncertainy (when considered)

###  II-D Late fusion with uncertainty estimation

The assumption that the outputs follow a Gaussian distribution, for which we
have estimates of the mean value and variance, can be leveraged during the
data fusion process. The $N_{B}$ estimates all refer to a single UE’s position
in a 2-dimensional plane and the fused probability distribution can be
calculated using Bayesian inference[41]. The resulting maximum likelihood (ML)
fused estimate $\tilde{\boldsymbol{p}}=[\tilde{x},\tilde{y}]$ $\sim$
$\mathcal{N}([\tilde{x}_{\text{ML}},\tilde{y}_{\text{ML}}],\begin{bmatrix}\sigma_{x}^{2},0\\\
0,\sigma_{y}^{2}\end{bmatrix})$ has a variance of:

$\sigma_{x}^{2}=\sum_{n=1}^{N_{B}}1/\sigma_{x,n}^{2},\;\;\;\sigma_{y}^{2}=\sum_{n=1}^{N_{B}}1/\sigma_{y,n}^{2}$
(11)

and a mean value:

$\begin{split}\tilde{x}_{\text{ML}}&=\frac{\sum_{n=1}^{N_{B}}{\tilde{x}_{n}}/{\sigma_{x,n}^{2}}}{\sigma_{x}^{2}},\\\
\tilde{y}_{\text{ML}}&=\frac{\sum_{n=1}^{N_{B}}{\tilde{y}_{n}}/{\sigma_{y,n}^{2}}}{\sigma_{y}^{2}}\end{split}$
(12)

and similarly for $\tilde{y}$. This type of estimate weighting is called
inverse variance weighting. The value of $\sigma_{x,n}$ is given by (4) in the
case of MCD uncertainty estimation, or (7) when considering spurious estimates
(SP). Additionally, if we don’t take into account the uncertainty of each
estimate we can consider $\sigma_{x,n}=\sigma_{y,n}=1$ which this results to a
simple averaging of the position estimates.

###  II-E Quality of uncertainty estimation

After presenting different ways to calculate the uncertainty for each
estimate, we now discuss how to assess the quality of these uncertainty
estimates. As shown in the previous section, instead of providing a single
position estimate, each model provides a different probability distribution
for each individual input, for which the variance corresponds to the
uncertainty. Normally to assess whether the output of the model indeed
conforms to a probability distribution we would repeatedly produce samples
from the model for a single input, calculate the empirical mean and variance
and determine how close they are to the estimated model’s distribution.
However, in the context of localization using CSI fingerprints we have at most
a couple of CSI samples for a given location, therefore any empirical
calculation would be unreliable. Instead we use a method to determine the
reliability of the uncertainty estimation process which is called the area
under sparsification error curve (AUSE).

####  II-E1 Area under sparsification error curve

The idea behind this metric is to use the so-called sparsification plots as a
quality metric and the sparsification error [35]. Before defining the
sparsification error we first need to define the oracle error. We define an
array of the errors of each position sample as:

$e=\left[||\tilde{x}_{0}-x_{0}||_{2},||\tilde{x}_{1}-x_{1}||_{2},\ldots,||\tilde{x}_{N_{\text{test}-1}}-x_{N_{\text{test}-1}}||_{2}\right]$
(13)

where $\tilde{x}_{i}$ is the estimated value of the $i$-th sample and $x_{i}$
is the real value and $N_{\text{test}}$ is the number of samples in the test
set. We also define the function $\text{sort}(.)$ which is used to sort the
elements of an array in descending order, such that
$e^{\prime}=\text{sort}(e)$. With that, the oracle error can be defined as:

$\text{O}_{N}=\sqrt{\frac{\sum
e^{\prime}_{N:N_{\text{test}}-1}}{N_{\text{test}}-N}},$ (14)

where
$e^{\prime}_{N:N_{\text{test}}-1}=[e^{\prime}_{N},e^{\prime}_{N+1},\ldots,e^{\prime}_{N_{\text{test}}-1}]$
and $1\leq N\leq N_{\text{test}}$. The value $O_{N}$ is decreasing
monotonically with $N$ since the errors are removed from the array
$e^{\prime}$ in decreasing order.

To define the sparsification error, we first define an uncertainty vector
$s=[\sigma_{0},\sigma_{1},\ldots,\sigma_{N_{\text{test}}-1}]$ and also, the
function $\text{argsort}_{s}(.)$ which sorts the elements of an array with
respect to the descending order of the elements of the array $s$, and
$e^{S}=\text{argsort}_{s}(e)$. With that, the sparsification error is defined
as:

$S_{N}=\sqrt{\frac{\sum e^{S}_{N:N_{\text{test}}-1}}{N_{\text{test}}-N}},$
(15)

The sparsification error shows how much the estimated uncertainty coincides
with the true errors on the test set. By removing gradually from the test set
the estimates with the highest uncertainty and if the estimated uncertainty is
of high quality the mean error $S_{N}$ should decrease monotonically when
increasing $N$. We compare $S_{N}$ to $O_{N}$, by calculating the curve under
the function $S_{N}-O_{N}$ for $N\in[1,N_{\text{test}}-1]$. The value AUSE is
calculated as:

$\text{AUSE}=\frac{\sum_{N=1}^{N_{\text{test}-1}}S_{N}-O_{N}}{N_{\text{test}-1}}.$
(16)

A small value indicates that the sparsification error is close to the oracle
error, meaning that the uncertainty estimation is a good indicator for the
actual error in the test set.

####  II-E2 Integrity risk

We additionally use the integrity risk (IR) metric which was recently proposed
by 3GPP as a key performance indicator for positioning integrity [37].
Normally, if the uncertainty is high, the system should give a warning that
the respective error is also high. The integrity risk is defined as the
probability that the unknown positioning error exceeds an application specific
alert limit (AL) without warning. The available information from each user is
the position and the uncertainty estimate, therefore we define an indicator
function $\mathbbm{1}_{\text{AL}}[||\boldsymbol{\sigma}_{i}||_{2}]$ which
gives the aforementioned warning when the euclidean norm of the uncertainty
$||\boldsymbol{\sigma}_{i}||_{2}$ vector of the $i$-th measurement is larger
than some threshold $\gamma$ which corresponds to an position error equal to
the AL:

$\mathbbm{1}_{\text{AL}}[||\boldsymbol{\sigma}_{i}||_{2}]=\begin{cases}1&\text{for
}||\boldsymbol{\sigma}_{i}||_{2}\leq\gamma\\\ 0&\text{for
}||\boldsymbol{\sigma}_{i}||_{2}>\gamma\end{cases}.$ (17)

The IR is then defined as:

$\text{IR}=\frac{\sum_{\\{i|\mathbbm{1}_{\text{AL}}(||\boldsymbol{\sigma}_{i}||_{2})\\}}\mathbbm{1}_{\text{AL}}[e_{i}]}{N_{\text{test}}},$
(18)

In words, the IR is calculated as the ratio of samples that exceeded the AL
but no warning was given, to the total number of test samples.

##  III Simulation Setup

###  III-A Database description

To evaluate our proposals we use the Dichasus channel measurements described
in [42], that were collected at four antenna arrays distributed on the corners
of an industrial area shown in Fig. 4. Each of the antenna arrays consists of
a $4\times 2$ uniform rectangular array (URA) with vertical and horizontal
antenna spacing of half a wavelength. The measurements in [42] were collected
with a single-antenna UE transmitting an OFDM signal in the uplink with a
bandwidth of 50 MHz and with a pilot sent at every tenth subcarrier out of
1024 subcarriers, i.e., $Nc=103$. The carrier frequency is $f_{C}=1.272$ GHz.

$\displaystyle{-12.5}$$\displaystyle{-10.0}$$\displaystyle{-7.5}$$\displaystyle{-5.0}$$\displaystyle{-2.5}$$\displaystyle{0.0}$$\displaystyle{2.5}$x
coordinate
(m)$\displaystyle{-14}$$\displaystyle{-12}$$\displaystyle{-10}$$\displaystyle{-8}$$\displaystyle{-6}$$\displaystyle{-4}$$\displaystyle{-2}$y
coordinate (m)ABCDBSsPossible UE positionsScatterer

Figure 4: Layout of considered industrial environment in Dichasus [42]

In a real deployment of a positioning system using fingerprints, the CSI
fingerprints are influenced by variations in the environment such as movement
of objects or people throughout the area of interest. To model a change in the
environment for a UE at a given position, i.e., between the training and
deployment phase, we consider the attenuation of the strongest path from the
UE to a BS, i.e., due to blocking by a nearby person or object. Please note
that the strongest path to a BS may correspond to a NLOS path, as some areas
do not have a LOS to a BS.

To model the attenuation of the strongest path, we firstly transform the CSI
measurement (over antennas and frequency) for a UE at a given position to the
angular-delay domain, i.e., we obtain a CSI matrix across angles and delays,
where the contribution from a path with a particular angle of arrival and time
of arrival can be determined. Given the CSI matrix in the angular-delay
domain, the element with the largest amplitude in the CSI matrix is attenuated
by 20 dB, e.g., to model the attenuation effect caused by a human body at
similar carrier frequency [43]. The CSI matrix with the attenuated strongest
element is then transformed back to the spatial and frequency domain resulting
in a modified CSI fingerprint due to the attenuation.

Consequently, we consider scenarios with or without the above mentioned human
body attenuation. When no attenuation is present we assume that there is no
change in the environment between training and deployment phases, i.e. the
environment is static as shown in Fig. 5. Additionally we consider scenarios
where attenuation affects the signal between UE and each of the BSs, but only
in the deployment phase. This implies a change in the environment between the
training and the deployment phases which we define as a dynamic scenario. We
consider 4 different cases for the dynamic case considering a path attenuation
from the UE to one, two, three or to all BSs. In the dynamic scenario example
in Fig. 5 the strongest paths of BSs C and D are attenuated.

Figure 5: Static and dynamic scenarios example (strongest paths of 2 BSs
attenuated). The UE is depicted as red dot inside the area of interest and the
arrows indicate the strongest paths between UE and BSs. Figure 6: Overview of
different fusion methods and training schemes

Lastly, we compare all the different fusion approaches. An overview of all the
considered approaches is shown in Fig. 6 considering different types of
training loss, i.e., MSE or aleatoric loss, and different fusion methods,
i.e., early or late fusion. For the late fusion specifically we have different
types of model training. Firstly, the separate late fusion which is the same
method as described in [21] where each BS corresponds to a single DL-model and
each model is trained only on data from that BS. We also consider the joint
late fusion which assumes that the model’s of each BS share the parameters of
their initial layers and are trained using the MTL scheme described in Section
II-C. Finally we also consider different types of combining of the estimates
from the multiple models, namely averaging, MCD or SP. With early fusion only
one model is trained, with either MSE or aleatoric loss. In the following we
refer to late fusion with separate and joint training as separate late fusion
and joint late fusion, respectively.

###  III-B Neural Network Configuration

The considered neural network is shown in Fig. 7. In the joint late fusion we
consider that the models across BSs share the parameters of the first four
blocks. Both early and late fusion use the same overall structure but with
different input size, depending on the considered fingerprint.

Figure 7: Complete DL model Figure 8: Pooling Block

The basis of the considered neural network is the convolutional layer as it
has shown promising results for positioning using CSI fingerprints [21, 33,
16, 8, 11, 13, 14]. In general, the convolutional layer is followed by a
pooling layer whose purpose is to downsample its input but recently [44] it
has been shown that using strided convolution instead of a pooling layer may
improve the model’s performance. Therefore, for the DL-model in this work we
only use strided convolution and no pooling layers. A strided convolution can
be thought of as a learned pooling layer, where the input is downsampled but
the method of downsampling is learned during training [39, chapter 9.5].

Additionally, to further reduce information loss during downsampling, we
implement the method of pooling blocks introduced in [45] which was also used
for CSI based positioning in [46]. In a pooling block, a convolutional layer
doubles the number of learned convolutional filters before downsampling and
then the spatial size is reduced by a strided convolution. A final
convolutional layer is used to reduce again the number of learned
convolutional filters to the original size. In this way, it is expected that
the pooling block will learn to transfer the important information from the
spatial dimension to the convolutional filters and preserve it.

TABLE I: Fusion Schemes Training scheme |  Input dim. |  DL models |  Total Parameters
---|---|---|---
Early fusion | $32\times 103\times 2$ | 1 | $357\ 954$
Separate Late Fusion | $8\times 103\times 2$ | 4 | $743\ 688$
Joint Late Fusion | $8\times 103\times 2$ | 4 | $285\ 768$

Lastly, in order to avoid any problems with vanishing gradients we employ skip
connections [47]. By combining all the aforementioned methods we create a
pooling block which is shown in Fig. 8. Two pooling blocks are placed one
after the other and are followed by 3 dense layers with 128 neurons each and
finally with a dense layer with 2 neurons that outputs the estimated
2-dimensional position. When considering the aleatoric uncertainty the last
dense layer has 4 neurons which correspond to the 2-dimensional position plus
the aleatoric uncertainty for the $x$ and $y$ position coordinates. Each
convolutional layer has 32 filters with a $3\times 3$ kernel. Depending on the
method used, i.e., early fusion, separate late fusion or joint late fusion,
the total number of model parameters are different. For the early fusion
method, the input’s dimension is $32\times 103\times 2$ which results from the
CSI over all 32 antennas, i.e. 4 BSs with 8 antennas each, and 103
subcarriers. Therefore the total number of parameters in the early fusion case
is $357\,954$ parameters. When using separate late fusion , the CSI over the 8
antennas of a BS is used as input to each model. As the input dimensionality
is then $8\times 103\times 2$, the number of parameters per model is reduced
to $185\,922$. Since we have 4 models the total number of parameters is
$743\,688$. Lastly when jointly training the multiple models in a MTL scheme
some parameters are shared so in this case the total number of parameters is
$285\,768$. The different configurations are summarized in Table I.

##  IV Simulation Results

We test the different proposed schemes using the Dichasus database [42] in the
deployment area shown in Fig. 4 by incrementally increasing the number of
training samples $N_{\text{train}}$ from $N_{\text{train}}=10\,000$ to
$N_{\text{train}}=60\,000$. The validation set is 10% of the training set i.e.
$N_{\text{val}}=[1000,6000]$. We compare the different schemes with respect to
the mean error (ME), which is given by the mean euclidean distance between the
estimated position and the true position in the test set. The number of test
samples is $N_{\text{test}}=59\,137$, regardless of the number of training and
validation samples. Furthermore we compare the quality of the uncertainty
estimation of the different schemes by the AUSE metric and the IR.

Initially we compare the different results for a static environment, i.e.,
when there is no change between training and deployment phases. Before
comparing the different fusion approaches listed in Table I, we first show the
gain of the training performed with MTL for the joint late fusion approach.
For both the separate and joint late fusion approaches, one model is trained
at each BS by using the MSE loss in Eq. (2) as the objective function.
However, for the separate late fusion the model at each BS is trained only
with data from that BS, while for joint late fusion the first part of the
model at each BS is trained with data from all 4 BSs. In the following, when
mentioning joint training of the models, we refer to joint training of the
first common part of the models. Fig. 9 shows the performance of the models at
each BS with the separate and joint training. The gain of the joint training
can clearly be seen across the models at each BS. By training the models
jointly, their common part incorporates information from multiple BSs
effectively increasing its training size. Thus, for a late fusion scheme,
joint training the models at each BS, instead of separately, leads to a
decreased ME at each BS.

$\displaystyle{10000}$$\displaystyle{20000}$$\displaystyle{30000}$$\displaystyle{40000}$$\displaystyle{50000}$$\displaystyle{60000}$Number
of train
samples$\displaystyle{0.3}$$\displaystyle{0.4}$$\displaystyle{0.5}$$\displaystyle{0.6}$$\displaystyle{0.7}$Mean
Error (m)Separate AJoint ASeparate BJoint BSeparate CJoint CSeparate DJoint D
Figure 9: Comparison of ME of each BS in a static scenario when training
separately or jointly using MSE loss
$\displaystyle{10000}$$\displaystyle{20000}$$\displaystyle{30000}$$\displaystyle{40000}$$\displaystyle{50000}$$\displaystyle{60000}$Number
of train
samples$\displaystyle{0.15}$$\displaystyle{0.20}$$\displaystyle{0.25}$$\displaystyle{0.30}$$\displaystyle{0.35}$$\displaystyle{0.40}$Mean
Error (m)Early fusionSeparate AverageJoint AverageSeparate MCDJoint
MCDSeparate SPJoint SP Figure 10: Comparison of ME for different fusion
methods in a static scenario when training separately or jointly using MSE
loss

Next, we compare the different fusion schemes with respect to the ME in the
test set. Specifically for both separate and joint late fusion we consider
three different methods of combining the estimates of the multiple models as
described in section II-D. The first combining method is averaging where a
simple average of all the model outputs is performed to calculate the overall
estimate. The second combining method uses MCD-based combining where the
variance of each estimate is estimated using the MCD method shown in eq. (4).
Lastly, we also consider the SP-based combining where the variance, shown in
Eq (7), of the different estimates is modified by taking into account spurious
measurements. The fused estimate for both MCD and SP is calculated using eq.
(12). From Fig. 10 we can observe that using early fusion outperforms all late
fusion methods in a static environment, i.e., when there are no changes in the
environment between the training and the deployment phase. This is in contrast
to the result from [21], where it was shown that late fusion outperformed
early fusion in the static case. The difference in the conclusion of the
results of [21] and the ones shown in Fig. 10 can be explained due to the
different considered environments. Whereas in [21] there is always a LOS
between the UE and each BS, in this work the link between the UE and each BS
can either be LOS or NLOS.

We consider the performance of the different fusion approaches when the models
are trained using the aleatoric uncertainty loss function shown in Eq. (3).
Fig. 11 shows the comparison of the models at each BS with separate and joint
learning when using the aleatoric uncertainty loss function. Similar to the
MSE loss function case, there is an improvement in the performance for each
model when training them jointly. As explained in [31], using a MTL scheme
which enables joint late fusion, the aleatoric uncertainty is implicitly used
as a learned weighting parameter between the losses corresponding to each
task, i.e. the positioning at each BS, which can increase the performance for
each task.

$\displaystyle{10000}$$\displaystyle{20000}$$\displaystyle{30000}$$\displaystyle{40000}$$\displaystyle{50000}$$\displaystyle{60000}$Number
of train
samples$\displaystyle{0.3}$$\displaystyle{0.4}$$\displaystyle{0.5}$$\displaystyle{0.6}$$\displaystyle{0.7}$Mean
Error (m)Separate AJoint ASeparate BJoint BSeparate CJoint CSeparate DJoint D
Figure 11: Comparison of ME of each BS in a static scenario when training
separately or jointly using aleatoric loss
$\displaystyle{10000}$$\displaystyle{20000}$$\displaystyle{30000}$$\displaystyle{40000}$$\displaystyle{50000}$$\displaystyle{60000}$Number
of train
samples$\displaystyle{0.15}$$\displaystyle{0.20}$$\displaystyle{0.25}$$\displaystyle{0.30}$$\displaystyle{0.35}$$\displaystyle{0.40}$Mean
Error (m)Early fusionAleatoric, Separate, AverageAleatoric, Joint,
AverageAleatoric Separate, MCDAleatoric, Joint, MCDAleatoric, Separate,
SPAleatoric, Joint, SP Figure 12: Comparison of ME for different fusion
methods in a static scenario when training separately or jointly using the
aleatoric loss
$\displaystyle{10000}$$\displaystyle{20000}$$\displaystyle{30000}$$\displaystyle{40000}$$\displaystyle{50000}$$\displaystyle{60000}$Number
of train
samples$\displaystyle{0.15}$$\displaystyle{0.20}$$\displaystyle{0.25}$$\displaystyle{0.30}$$\displaystyle{0.35}$Mean
Error (m)MSE Early fusionAleatoric, Early FusionMSE, Joint, AverageAleatoric,
Joint, AverageMSE, Joint, MCDAleatoric, Joint, MCDMSE, Joint, SPAleatoric,
Joint, SP Figure 13: Comparison of ME for different fusion methods in a static
scenario when training jointly with MSE or aleatoric loss

We further compare the separate and joint late fusion and the early fusion
methods when training the models using the aleatoric uncertainty loss. The
results are shown in Fig. 12. Similar to the results in Fig. 11 when
considering the MSE loss for the training, jointly training the models shows a
significant improvement in performance compared to when training each model
separately. This effect is particularly strong for a small number of training
samples. We also see that even though the early fusion still outperforms the
late fusion approaches for a small number of training samples, this is no
longer the case when the number of training samples increases, i.e. the late
fusion methods perform similar or better than the early fusion method. Similar
to when training with the MSE loss, the late fusion with averaging is the
worst performing option.

Additionally, we compare the different fusion methods when training the models
using the MSE (2) or the aleatoric uncertainty (3) loss functions. We see in
Fig. 13 that every late fusion scheme benefits when training the models based
on the aleatoric loss function regardless of the number of training samples.
Essentially, the inclusion of the aleatoric uncertainty improves the joint
late fusion schemes with MCD or SP-based combining, such that they are able to
come close to the performance of the early fusion scheme, assuming an adequate
number of training samples. For the early fusion scheme we see the opposite
effect, namely that the inclusion of the aleatoric uncertainty during training
impairs the performance, albeit slightly. The reason for this different
behavior between late and early fusion may come from the fact the early fusion
measurements always have some BSs having a LOS to the UE. This fact limits the
usefulness of the aleatoric uncertainty since in a LOS measurement the
uncertainty is anyway low. For the late fusion this is not the case since
every BS may experience NLOS conditions which have high aleatoric uncertainty
and positioning is then more challenging than the LOS case. As explained in
section II-B1, the model prioritizes cases where the aleatoric uncertainty is
low, i.e. LOS cases.

TABLE II: AUSE in static scenario Training Loss |  Early Fusion |  Separate MCD |  Separate SP |  Joint MCD |  Joint SP
---|---|---|---|---|---
MSE | 0.51345 | 0.42626 | 0.38907 | 0.40984 | 0.38676
Aleatoric | 0.38434 | 0.33162 | 0.30825 | 0.32853 | 0.31335
TABLE III: AUSE in dynamic scenario Blocked BSs |  Training Loss |  Early Fusion |  Separate MCD |  Separate SP |  Joint MCD |  Joint SP
---|---|---|---|---|---|---
A | MSE | 0.49605 | 0.42946 | 0.37902 | 0.41478 | 0.37673
Aleatoric | 0.31853 | 0.33723 | 0.29964 | 0.31908 | 0.30472
A, B | MSE | 0.36338 | 0.42794 | 0.36777 | 0.40224 | 0.35451
Aleatoric | 0.21424 | 0.32243 | 0.29882 | 0.28941 | 0.2768
A, B, C | MSE | 0.25143 | 0.53462 | 0.41211 | 0.41538 | 0.32425
Aleatoric | 0.1995 | 0.36986 | 0.32064 | 0.31521 | 0.26985
A, B, C, D | MSE | 0.28687 | 0.52786 | 0.43868 | 0.38177 | 0.34691
Aleatoric | 0.26857 | 0.44355 | 0.39339 | 0.38311 | 0.35556

Furthermore, we compare the quality of the uncertainty estimation in the
different fusion methods in Table II. During our evaluations, we noticed that
the AUSE value remains mostly constant over training samples and therefore, we
show the average over the training samples in Table II. The averaging late
fusion method it is not included in the table as it does not have uncertainty
information. A lower AUSE value means that the sorting of the positioning
errors across each measurement more closely corresponds with the sorting of
the uncertainty, making the uncertainty a good indicator for the actual
positioning error. We see from the table that for every fusion method,
training using the aleatoric uncertainty loss function improves the quality of
the uncertainty estimates according to AUSE. This makes sense since there is
no aleatoric uncertainty information when training MSE loss function, and
instead only the epistemic uncertainty is used.

Next we explore the positioning in a dynamic scenario, where the channel
between the UE and one or more BSs experiences a change between the training
and deployment phase.phase, , i.e., a 20 dB attenuation of the strongest path
as described in Sec. III-A. In the following, we refer to this attenuation of
the strongest path to a given BS as a change. The effect of this change on the
uncertainty of the estimates can be seen in Fig. 15, which depicts the
aleatoric uncertainty at BS A when the strongest path to BS is attenuated
compared to the aleatoric uncertainty when no attenuation is considered shown
in Fig. 14. While the uncertainty in the static case remains more or less low
throughout the area, i.e., around 0.25, when there is a change the uncertainty
can be up to 8 times larger. We see that the most affected regions are the
ones that there is a LOS path to the BS. This path includes most of the energy
of the CSI thereby by reducing it the CSI is hugely affected. In the NLOS
region we see that the uncertainty remains low since the position information
is included in multiple paths, i.e., no single path contains most of the
energy in the CSI fingerprint.

Figure 14: Uncertainty of model of BS A in a static scenario over all
positions Figure 15: Uncertainty of model of BS A when it experiences a change

Fig. 16 depicts the mean error of different fusion schemes when the channel
between the UE and the BS A experiences the described change above in the
deployment phase. Compared to the static case (see Fig. 13), we observe a huge
degradation in performance for the late fusion with averaging, as well as a
large degradation in performance for the early fusion case. As expected the
late fusion schemes with MCD and SP combining perform much better than the
other fusion methods. Specifically the SP-based late fusion method is able to
outperform all others since it is able to most reliably disregard the spurious
measurements due to the change in the channel to BS A.

Furthermore, we now consider a change in the channels between the UE and
multiple BSs. In Fig. 17, we depict the performance of early fusion and SP-
based late fusion when training with the MSE and aleatoric loss function as a
function of the number of BSs with the attenuation of the strongest path,
considering $40\,000$ training samples. The solid lines show the error over
all positions in the test set. As the uncertainty at different positions
varies, we also propose to consider the performance for the positions when the
uncertainty is below a threshold. By using logistic regression in the static
scenario we determine an uncertainty threshold for each method over which the
positioning error is over 1m. Then for each method we exclude the measurements
with uncertainty over this method-specific uncertainty threshold, and we see
that the error decreases, depicted by the dashed lines in Fig. 17. The
difference is more pronounced for the larger number of BSs with a change and
similarly in those cases more measurements are over the uncertainty threshold
and therefore excluded. Interestingly, even though the difference between
using the MSE loss or the aleatoric loss is relatively high for a small number
of blocked BSs, with aleatoric loss training performing better, the difference
becomes smaller when blocking more BSs. The reason for that is that when more
BSs experience a change then the epistemic uncertainty dominates, since the
measurements differ more from the static scenario.

Next, we investigate the reliability of the uncertainty estimates in a dynamic
scenario with respect to the AUSE as shown in Table III. As in the static
case, we provide the average over all training samples since similarly to the
static case we noticed that the AUSE value remains mostly constant over the
number of training samples. First we see that for almost every late fusion
approach, training with aleatoric uncertainty loss in a joint late fusion
approach results in the most reliable uncertainty estimates.

Lastly, we depict in Fig. 18 the integrity risk for $40\,000$ training samples
considering a channel change to one or more BSs. The integrity risk is
described in equation (18) and we consider $\text{AL}=1\text{m}$ and the
threshold $\gamma$ is calculated using logistic regression in the static
scenario for each fusion method. A low integrity risk shows that the
uncertainty estimation can be used to identify estimates that exceed the alert
limit. We see in Fig. 18 that the SP method is the most reliable in this
regard achieving an IR less than $7\%$ over all cases. On the other hand the
early fusion methods exhibit a quite high integrity risk. The high integrity
risk of those methods combined with the relative low AUSE shows that the early
fusion methods exhibit overconfidence in their estimates. In other words, they
are able to sort their errors based on their uncertainty, as indicated by the
low AUSE, but the error corresponding to each uncertainty estimate is
underestimated. This implies that with early fusion, it is assumed that some
estimates are under the alert limit even though this is not the case.

We note that here we provide only a simple method to calculate the uncertainty
threshold based on the uncertainty vector l2-norm and provide an IR value to
show the effectiveness of the uncertainty estimation. Other methods to better
calculate the threshold can be developed using both uncertainty vector
elements and using other classification methods such as support vector
machines (SVMs). Moreover, the considered metric does not indicate how many
estimates are over the uncertainty threshold for each method which may be
something that needs to be considered in some use cases.

##  V Conclusions

$\displaystyle{10000}$$\displaystyle{20000}$$\displaystyle{30000}$$\displaystyle{40000}$$\displaystyle{50000}$$\displaystyle{60000}$Number
of train
samples$\displaystyle{0.2}$$\displaystyle{0.3}$$\displaystyle{0.4}$$\displaystyle{0.5}$$\displaystyle{0.6}$$\displaystyle{0.7}$$\displaystyle{0.8}$Mean
Error (m)MSE Early fusionAleatoric, Early FusionMSE, Joint, AverageAleatoric,
Joint, AverageMSE, Joint, MCDAleatoric, Joint, MCDMSE, Joint, SPAleatoric,
Joint, SP Figure 16: Comparison of ME for different fusion methods in a
dynamic scenario when training jointly with MSE or aleatoric loss
$\displaystyle{0}$$\displaystyle{1}$$\displaystyle{2}$$\displaystyle{3}$$\displaystyle{4}$Number
of attenuated BSs
$\displaystyle{0.5}$$\displaystyle{1.0}$$\displaystyle{1.5}$$\displaystyle{2.0}$$\displaystyle{2.5}$Mean
Error (m)MSE, Early fusionMSE, Early fusion, UC thresholdAleatoric, Early
FusionAleatoric, Early Fusion, UC thresholdMSE, SPMSE, SP, UC
thresholdAleatoric, SPAleatoric, SP, UC threshold Figure 17: Comparison of the
error of different error fusion for different number of blocked BSs,
considering $40\,000$ training samples
$\displaystyle{0}$$\displaystyle{1}$$\displaystyle{2}$$\displaystyle{3}$$\displaystyle{4}$Number
of attenuated
BSs$\displaystyle{0.00}$$\displaystyle{0.05}$$\displaystyle{0.10}$$\displaystyle{0.15}$$\displaystyle{0.20}$$\displaystyle{0.25}$$\displaystyle{0.30}$Integrity
riskMSE, Early fusionAleatoric, Early FusionMSE, SPAleatoric, SP Figure 18:
Integrity Risk

In this paper, we examined different fusion methods for positioning using deep
learning and CSI fingerprints from multiple BSs. For early fusion, only one
model is used for estimating the UE’s position based on the CSI fingerprint
across multiple base stations. For late fusion, one model per BS is employed
and the overall UE’s position is determined by combining the output of the
models across the BSs. The performance of the trained models was evaluated
considering a static scenario, where the channel between the training phase
and the deployment phase remains the same, as well as in a dynamic scenario,
where the channel between the UE and one or more BSs experience a change
(attenuation of strongest path) in the deployment phase. While early fusion
schemes may normally perform better in static scenarios, changes in the
environment lead to a decrease positioning performance with early fusion, as
the model is not able to adapt in a dynamic scenario. On the other hand, our
results indicate that late fusion approaches are more robust to changes in the
environment, which is an important aspect to be addressed for AI-based
localization with CSI fingerprints in real deployments. Among the different
considered late fusion approaches, we have shown the advantage of multi-task
learning, by jointly training shared parameters of the models across the base
stations, where the common part of the models benefits from a larger number of
training samples.

For the late fusion approaches, different methods for combining the
positioning estimates from the BS models have also been investigated. In
particular, we have considered simple averaging as well as combining based on
considering uncertainty estimation, namely MCD and SP, where the output of the
different models are weighted based on the learned aleatoric uncertainty. We
show that fusing the multiple estimates based on their uncertainty not only
improves the positioning accuracy in both a static and dynamic scenario but
also ultimately gives more reliable uncertainty estimates. The reliability of
the uncertainty estimates is determined in terms of AUSE, which considers
whether the uncertainty corresponds to the real positioning error, and in
terms of the IR, which demonstrates a model’s ability to discard unreliable
estimates. Additionally we consider that some of the estimates may be
spurious, i.e., falsely indicate low uncertainty but with an actual large
positioning error, and we employ a technique to identify and disregard such
estimates.

Overall, we show that late fusion scheme with multi task learning and
uncertainty estimation is the most accurate and reliable in the considered
scenarios. This holds also for the dynamic scenario, which is one of the main
challenges limiting the deployment of AI-based localization with CSI
fingerprints.

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