# High Integrity Localization with Bayesian Optimization for Information Filter Tuning with Fault Detection

Mohammed Salhi, Joelle Al Hage

► To cite this version:

Mohammed Salhi, Joelle Al Hage. High Integrity Localization with Bayesian Optimization for Information Filter Tuning with Fault Detection. 28th International Conference on Information Fusion (FUSION 2025), Jul 2025, Rio de Janeiro, Brazil. pp.1-8, 10.23919/FUSION65864.2025.11124118. hal-05164814

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https://hal.science/hal-05164814v1

Submitted on 1 Sep 2025

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Distributed under a Creative Commons Attribution 4.0 International License
High Integrity Localization with Bayesian
Optimization for Information Filter Tuning with
Fault Detection

Mohammed Salhi and Joelle Al Hage

*Abstract*—Safe navigation of autonomous vehicles relies on high integrity localization system based on a state estimation method, which strongly depends on the choice of parameters, such as the measurement noise covariance matrix. To avoid filter divergence, a fault detection and exclusion (FDE) procedure is also required. Selecting appropriate thresholds for the FDE step is challenging and affects the accuracy and the integrity of localization. This paper presents a multi-sensor data fusion based on the Information Filter (IF) with an auto-tuning method that relies on Bayesian Optimization (BO). The proposed method aims to optimize the measurement covariance matrix and the FDE thresholds. BO is well suited for non-convex and stochastic cost functions with local minima. A novel objective function is also designed to improve the accuracy and guarantee the integrity of the estimates, by ensuring consistent uncertainty regions. The objective function consists of two terms: one related to the error and the other to the quantile. The proposed approach is evaluated with experimental data from a vehicle equipped with wheel speed sensors, fused with Global Navigation Satellite System (GNSS) pseudorange measurements.

# I. INTRODUCTION

In the context of autonomous vehicles, accurate localization with high integrity, obtained from the fusion of data from onboard sensors, is essential for safe navigation. The Kalman Filter (KF) is widely chosen for data fusion due to its ability to operate in real-time and to provide an accurate estimation with an associated uncertainty value [1]. However, KF performance strongly depends on the choice of system and measurement covariances, where an inappropriate choice can negatively impact the estimates.

Several studies have been performed to propose methods for tuning KF parameters. Conventional tuning methods are based on gradient descent optimization to minimize a cost function [2]. However, non-convex and stochastic cost functions often cause solutions to be trapped in local minima, leading to degraded filter performance.

In [3], the authors proposed an adaptive KF for measure-
ment noise covariance based on the innovation between the
predicted and actual measurement. The covariance matrices
are then computed based on a sliding time window. Adapt-
tation of this method was proposed in [4], [5]. Although
the proposed adaptive filters show improved performance,

the noise covariance matrices are directly dependent on the
residuals or innovations calculated during the filter's update
steps. Therefore, any divergence in the filter will directly
impact filter's estimates.

More recently, machine learning-based methods have been applied to parameter tuning. In [6], a convolutional neural network (CNN) was used to learn the covariance matrices of an Invariant Kalman filter. This method requires ground truth data, which can be very expensive to obtain. Another approach presented in [7] uses a recurrent neural network (RNN) with a Gated Recurrent Units (GRU) to output a Kalman gain instead of the system and measurements covariance matrices. Although the filter shows improved performance in terms of accuracy, it is not evaluated at the uncertainty level. An RNN with a GRU was also used in [8] to learn the parameters of an Unscented Kalman filter (UKF). Moreover, approaches based on Reinforcement Learning (RL) can be found in the literature. For example, the authors in [9] used RL to directly correct the estimates rather than explicitly learning the filter parameters. The main challenge with CNN, RNN or RL based approaches is the need of a large amount of data for training.

Bayesian Optimization (BO) is a good candidate for pa-
rameter tuning with limited training data and allows efficient
evaluation of the objective function. It creates a surrogate
model associated with a probability distribution to approximate
a stochastic objective function. To select the next sample to
evaluate, an acquisition function is used to find the candidate
that improves the current optimum. BO is then able to find
the optimum with a small number of objective function
evaluations. Several works have used BO for KF parameter
tuning [10]–[12]. In [10], BO was used to tune the process
and measurement noises of a KF with simulation examples. A
two-stage BO was proposed in [11] to tune the process noise
parameters of an UKF.

In this work, we aim to ensure the accuracy and integrity
of localization through a multi-sensor data fusion, coupled
with parameter tuning using Bayesian optimization. The mea-
surement noise covariance associated with GNSS pseudorange
measurements is auto-tuned for use within the Information
Filter (IF), the informational form of the KF. A Fault De-
tection and Exclusion (FDE) is added to handle erroneous
measurements. Selecting an appropriate threshold remains one
of the main challenges in the FDE step. This choice directly
impacts the quality of the estimation and is typically made
by setting a false alarm probability. However, in the presence

The authors are with Université de Technologie de Compiègne, CNRS, Heudiasyc, France. This work is funded by the ANR TolCar project (ANR-21-CE33-0012) and is part of SIVALab (joint laboratory Re- nault/Heudiasyc/UTC/CNRS).

Email : mohammed.salhi@hds.utc.fr, joelle.al-hage@hds.utc.fr
of non Gaussian noise, the assumed distribution for setting the threshold does not accurately reflect the reality, which degrades detection performance. For this reason, we propose to tune the threshold using the BO.

In the context of intelligent vehicles, integrity denotes the confidence in a navigation solution and is associated with an error bound known as the Protection Level (PL) computed for a given target integrity risk [13]. For safe navigation, this PL must be well estimated and considered during parameter tuning. For tuning KF parameters, objective functions like the normalized estimation error squared (NEES) and normalized innovation squared (NIS) [10] are commonly used. Although these functions are well adapted to improve accuracy, they do not provide consistent PL under small probabilities or when the Gaussian assumption is not verified. In this work, we propose a new objective function that incorporates both errors and quantile. The quantile objective function allows the tuning of parameter values which verify the target integrity risk via the computation of PL without any assumption on the distribution.

This paper is organized as follows: Section II presents the data fusion architecture with the FDE step. The PL computation is also presented. Section III presents the proposed auto-tuning method with BO for high integrity localization. The proposed objective function is presented in details. Experimental results are presented and discussed in section IV.

## II. HIGH INTEGRITY MULTI-SENSOR DATA FUSION

### A. Localization Models

We consider a vehicle equipped with wheel speed sensors, a yaw-rate gyro, and a GNSS receiver for pseudorange measurements from the GPS and Galileo constellations. The vehicle's pose is defined with respect to the vehicle base frame ($R_B$) (figure 1). The GNSS observations are obtained with respect to the antenna (frame $R_G$), requiring a transformation to be applied [14].

<figure><img src="image_2.png" /><figcaption>Fig. 1. $R_0$, $R_G$ and $R_B$ are the local ENU, GNSS receiver and vehicle base frames, respectively.</figcaption></figure>

The state vector contains the vehicle coordinates $[x, y, z]$, the heading angle $\theta$ as well as the receiver clock offset $cdt$ (multiplied by the speed of light $c$) and drift $cdt$ with respect to GPS time. The state vector at time $k$ is therefore given by:

$$ \mathbf{x}_k = [x, y, z, \theta, cdt, c\dot{dt}]_k^T \quad (1) $$

It is worth noting that in the state vector, only one clock offset is added despite using two different constellations. This is made possible by a Galileo to GPS Time Offset (GGTO) conversion.

The evolution model takes advantage of wheel speed sensors and gyro to get the elementary displacement $\Delta_k$ and rotation $\omega_k$ of the vehicle. The evolution model is given by:

$$ \mathbf{x}_{k+1} = f(\mathbf{x}_k, \mathbf{u}_k) + \mathbf{v}_k \quad (2) $$

where $\mathbf{v}_k \sim N(0, Q)$ is the system noise considered as white Gaussian, $Q$ the associated covariance matrix and $\mathbf{u}_k = [\Delta_k, \omega_k]$.

The measurement vector $\mathbf{z}$ contains GNSS pseudorange measurements. The corrected GNSS observation model of pseudorange $i$ is given by:

$$ \mathbf{z}_i = \mathbf{d}_i + cdt + \mathbf{w}_i \quad (3) $$

where $\mathbf{d}_i$ is the Euclidean distance between the GNSS receiver and the satellite $i$ and $\mathbf{w}_i \sim N(0, R_i)$ is the measurement noise considered as white Gaussian with covariance $R_i$. The choice of the covariance matrix $R$ directly impacts the accuracy and integrity of localization. Bayesian Optimization is used for this step, as detailed in the next section.

### B. Sensor data fusion

For the data fusion, an Extended Information Filter (EIF) is applied [15]. IF and KF are mathematically equivalent, but the update step equations of IF are more computationally convenient for multi-sensor data fusion and for fault detection and exclusion [16]. The IF computes an information matrix $Y_k$ and an information vector $\mathbf{y}_k$ instead of a covariance matrix $P_k$ and a state vector $\mathbf{x}_k$:

$$ Y_k = P_k^{-1} \quad (4) $$

$$ \mathbf{y}_k = P_k^{-1} \mathbf{x}_k. \quad (5) $$

The updated information matrix and information vector are obtained as [15]:

$$ Y_{k|k} = Y_{k|k-1} + \sum_{j=1}^{N} \mathbf{I}_{jk} \quad (6) $$

$$ \mathbf{y}_{k|k} = \mathbf{y}_{k|k-1} + \sum_{j=1}^{N} \mathbf{i}_{jk} \quad (7) $$

where $\mathbf{I}_{jk}$ and $\mathbf{i}_{jk}$ are the information contributions of the measurement $j$:

$$ \mathbf{I}_{jk} = H_{j,k}^T R_j^{-1} H_{j,k} \quad (8) $$

$$ \mathbf{i}_{jk} = H_{j,k}^T R_j^{-1} (\mathbf{z}_{j,k} - h_j(\mathbf{x}_{k|k-1}) + H_{j,k} \mathbf{x}_{k|k-1}) \quad (9) $$
H is the Jacobian matrix associated with the observation model, and $R_j$ the covariance of measurement j.

The main advantage of equation 6 and 7 is the summation of the information contributions allowing a simplified FDE step.

### C. Fault detection and exclusion

Measurements from GNSS can be erroneous, potentially affecting the accuracy and integrity of localization and, consequently, the system safety. A Fault Detection and Exclusion (FDE) step is integrated into the data fusion. For this purpose, residuals in the state space are computed using the Mahalanobis distance between the predicted and updated state vectors.

For the fault detection, a global residual using all available measurements is computed as:

$$r_k = (\mathbf{x}_{k|k} - \mathbf{x}_{k|k-1})Y_{k|k}(\mathbf{x}_{k|k} - \mathbf{x}_{k|k-1})^T. \quad (10)$$

If the global residual exceeds a global threshold $th_{glob}$, a bank of $EIF_i$ computes local residuals $r_i$, where each $EIF_i$ uses only one measurement $\mathbf{z}_i$:

$$r_{i,k} = (\mathbf{x}_{i,k|k} - \mathbf{x}_{k|k-1})Y_{i,k|k}(\mathbf{x}_{i,k|k} - \mathbf{x}_{k|k-1})^T. \quad (11)$$

$\mathbf{x}_{i,k|k}$ and $Y_{i,k|k}$ refers to the state vector and information matrix computed using only the information contribution of the $i$th measurement. The local residual is compared with a local threshold $th_{loc}$. If $r_{i,k}$ exceeds that threshold, the corresponding measurement is excluded by simply subtracting its information contributions from equations 6 and 7.

The choice of thresholds for fault detection and exclusion directly impacts the accuracy and integrity of the state estimation. Typically, these thresholds are determined from a Chi-squared distribution, with a given false alarm probability, under assumptions such as white Gaussian noise. In real world applications, these assumptions are not always verified, which can result in poor thresholding. In this work, we aim to tune the thresholds by using data, without relying on prior assumptions. Since the choice of the global threshold is less problematic, and to reduce the complexity of the optimization problem, this paper focuses on tuning the local threshold $th_{loc}$ used in the fault exclusion step via BO, while setting the detection threshold manually.

### D. Protection levels computation

Once the erroneous measurements are excluded from the data fusion, we aim to bound the estimation errors. This is related to the integrity concept, which originates from the aviation domain, and refers to the confidence in a navigation system. It is associated with a Protection Level (PL), an uncertainty bound where the probability of the error exceeding it must be lower than a given Target Integrity Risk (TIR), denoted by $\alpha$:

$$p(PE > PL) \le \alpha. \quad (12)$$

$PE$ is the position error.

Given the filter update step, the PL is expressed in function of the covariance matrix:

$$PL = K(\alpha) \times \sqrt{\max(\text{eigenvalue}(P_{k|k})).} \quad (13)$$

$K(\alpha)$ is obtained from the standard normal distribution according to a given probability $\alpha$. The PL depends on $P_{k|k}$ which is also related to the filter parameters (e.g., noise covariance). Therefore, a bad choice of these parameters can lead to an underestimation of the PL, affecting the system safety.

For the navigation of intelligent vehicles, we are interested by bounding the errors in the Along-Track (AT) and Cross-Track (CT) directions. The errors and the PL are then computed according to these two directions, as described in [16].

## III. BAYESIAN OPTIMIZATION FOR FILTER PARAMETER TUNING

The accuracy and integrity of localization are strongly dependent on filter parameters, and manually setting them is a challenging task that often leads to suboptimal solutions.

### A. Bayesian Optimization

Consider the cost function $J(\cdot) : \mathcal{D} \to \mathbb{R}$ that we seek to minimize, which must take into account both error and integrity performance, as will be explained later. $\mathcal{D}$ is the solution space of dimension $d$. The optimization problem can be written as:

$$\mathbf{q}^* = \underset{\mathbf{q} \in \mathcal{D}}{\operatorname{argmin}} J(\mathbf{q}), \mathcal{D} \in \mathbb{R}^d \quad (14)$$

where $\mathbf{q}$ is the parameter vector and $\mathbf{q}^*$ is the optimum. In our case, the parameter vector includes the measurements noise covariance $R$ as well as the threshold used for fault exclusion.

$J$ is the cost function which may be unknown and expensive to evaluate, but its outputs can be observed. The goal of BO is to find the minimum of the noisy cost function while learning its shape through Bayesian inference [10]. To achieve this, BO models $J$ by a stochastic process and learn it to select the next promising sample that could find a new minimum of $J$.

The following summarizes the two components of BO: the surrogate model and the acquisition function.

a) **Surrogate model:** The surrogate model approximates the true cost function using a probabilistic model. Multiple surrogate models can be found in the literature like Gaussian Process (GP) [10] and t-Student process (tSP) [11]. GP is the most commonly used in BO and is used in this work for convenience.

A GP can be viewed as an infinite collection of random variables, where each finite combination of these variables follows a multivariate Gaussian distribution:

$$g(\mathbf{q}) \sim \text{GP}(m(\mathbf{q}), k(\mathbf{q}, \mathbf{q}')) \quad (15)$$

$$m(\mathbf{q}) = E[g(\mathbf{q})] \quad (16)$$

$$k(\mathbf{q}, \mathbf{q}') = E[(g(\mathbf{q}) - m(\mathbf{q}))(g(\mathbf{q}') - m(\mathbf{q}'))] \quad (17)$$
where $m(\mathbf{q})$ is the mean function of the GP and $k(\mathbf{q}, \mathbf{q}')$ its covariance (kernel) function. A valid kernel leads to a positive semi-definite Gram matrix $K$. Without prior knowledge of the mean function, it can be set to zero [10].

The choice of the kernel is essential for model fitting and it directly influences the optimization problem. In this work, the Matern 3/2 kernel is used since it gives a balance between smoothness and roughness [17]:

$$k(\mathbf{q}, \mathbf{q}') = \alpha_0 \left(1 + \sqrt{3} \frac{v}{l}\right) \exp\left(-\sqrt{3} \frac{v}{l}\right), \quad (18)$$

where $v$ is the Euclidean distance $v = \sqrt{(\mathbf{q} - \mathbf{q}')^T(\mathbf{q} - \mathbf{q}')}$, $l$ and $\alpha_0$ are the length-scale and the amplitude, respectively. The kernel is used with automatic relevance determination (ARD) where a different length-scale $l_i$ can be chosen along each dimension of the solution space, allowing the BO to select relevant dimensions that have a strong influence on the objective function.

The surrogate model $g(.)$ is fitted from the trained data ($D = \{\mathbf{q}_1, ..., \mathbf{q}_n\}, J(D)$) using a GP model. To account for uncertainty in training data, a random variable $\tau$ which follows a normal distribution $\tau \sim N(0, \sigma_{\tau}^2)$ is added.

For new data, $D_T = [\mathbf{q}_{T1}, ..., \mathbf{q}_{Tn_T}]$, the surrogate model allows to predict the test outputs by using the conditional GP mean and covariance [10]:

$$g(D_T) | g(D), D, D_T \sim GP(\mu(D_T), \sigma^2(D_T)) \quad (19)$$

$$\mu(D_T) = K(D_T, D)(K(D, D) + \sigma_{\tau}^2 I)^{-1} g(D) \quad (20)$$

$$\sigma^2(D_T) = K(D_T, D_T) - K(D_T, D)(K(D, D) + \sigma_{\tau}^2 I)^{-1} K(D, D_T) \quad (21)$$

with $K$ the Gram matrix defined by the kernel $k(.,.)$. The surrogate model provides information about the objective function at values in $D_T$ that have not yet been sampled.

The surrogate model is updated and retrained whenever a new observation becomes available. The hyperparameters that describes the GP ($l_i, \alpha_0, \sigma_{\tau}$) are learned in the optimization process using maximum likelihood estimation.

b) Acquisition function: To select the next interesting sample $\mathbf{q}$ to evaluate, the BO algorithm uses an acquisition function having as argument the surrogate model. The idea is to explore new locations of the surrogate model with high uncertainty and exploit it to find the optimum as quickly as possible.

The expected improvement (EI) is one of the most commonly used acquisition function in BO and has the advantage to not rely on any extra hyperparameter [18].

Suppose that $\mathbf{q}^*$ is the current optimum given $n$ sampled points $\mathbf{q}_{1:n}$. The algorithm must now select the next point $\mathbf{q}_{n+1}$ such as $g(\mathbf{q}_{n+1}) < g(\mathbf{q}^*)$. To do so, the sample that maximizes the improvement with respect to $\mathbf{q}^*$ can be selected:

$$I(\mathbf{q}_{n+1}) = \max(g(\mathbf{q}^*) - g(\mathbf{q}_{n+1}), 0) \quad (22)$$

Dealing with surrogate models, $I(.)$ is stochastic. Therefore, the expected improvement can be used instead:

$$EI(\mathbf{q}_{n+1}) = \mathbb{E}_n[I(\mathbf{q}_{n+1}) | D, g(D)] \quad (23)$$

where $\mathbb{E}_n[\cdot]$ denotes the expectation using the posterior GP distribution, given the surrogate model. Maximizing this expectation provides the next sample most likely to improve the current optimum value.

For Gaussian process, equation 23 can be written as [10]:

$$EI(\mathbf{q}) = (g(\mathbf{q}^*) - \mu(\mathbf{q}))\Phi\left(\frac{g(\mathbf{q}^*) - \mu(\mathbf{q})}{\sigma(\mathbf{q})}\right) + \sigma(\mathbf{q})\phi\left(\frac{g(\mathbf{q}^*) - \mu(\mathbf{q})}{\sigma(\mathbf{q})}\right) \quad (24)$$

where $\mu(\mathbf{q})$ and $\sigma(\mathbf{q})$ are the predicted mean and standard deviation of the GP, respectively. $\phi$ and $\Phi$ denote the probability density function and cumulative density function of the standard normal distribution. In BO, the maximization of this function is typically performed using the DIRECT algorithm [19], which is well suited for non-convex optimization and follows a derivative free approach. However, a drawback of the DIRECT algorithm is that it evaluates samples only at the centers of the subdivided rectangles. To overcome this limitation, we apply a Limited-memory BFGS (L-BFGS) optimization, a method from the newton optimization family, to refine the solution returned by the DIRECT method.

## B. Objective function

In the context of safe intelligent vehicles, in addition to ensuring high accuracy, a key objective of tuning filter parameters is to bound estimation errors in real time under small integrity risk ($\alpha \le 10^{-3}$). In real world applications, the noise probability density distribution may be different from KF assumptions due to biases or errors introduced by model linearization, potentially leading to an underestimation of the error bounds (PL). Classical objective functions, such as NEES and NIS, are not well suited for addressing these challenges, particularly under small integrity risk.

In this paper, we are looking for an accurate and consistent estimator. The objective function is designed to deal not only with the Mean Position Error (MPE), but also it incorporates a quantile loss function similar to that used in quantile regression [20], [21], enabling the predict of quantiles. This part is crucial for addressing the integrity part, as it will be detailed in the following paragraph.

a) Quantile objective function: In order to fit the desired statistical uncertainty bound, we will take advantage of the quantile regression principle. It is based on experiment data, without the need to make any assumptions about the underlying distribution [21]. The idea is to estimate a vector of parameters $\mathbf{q}$ which allows verifying the target integrity risk $\alpha$ (e.g. $\alpha = 10^{-3}$) through the computation of PL.

Consider $\mathbf{x}_{k|k}$ the estimated state vector at time $k$ and $\mathbf{x}_k$ the associated ground truth, the norm of the error is defined as:

$$e_k = \| \mathbf{x}_{k|k} - \mathbf{x}_k \|_2 \quad (25)$$
The weighted sum of difference between the position errors e and the uncertainty bound PL under a quantile level (1 − α ) is given by:

$$
QL(PL, e) = (1 - \alpha) \sum_{k, PL_k \le e_k} |e_k - PL_k| + \alpha \sum_{k, PL_k > e_k} |e_k - PL_k| \quad (26)
$$

where both the error and the PL depend on the parameters q that we want to estimate. Similar to quantile regression, optimizing this objective function leads to an estimated rela- tionship between the error and the PL , where a proportion (1 − α ) of errors lies below the PL , and a proportion α lies above the PL . The idea is then to estimate a vector q of parameters that minimizes equation 26 in order to verify the integrity requirement (equation 12).

To account for different possible requirements and sensor contributions in the along track and cross track directions, the QL can be modified to consider each direction separately:

$$
QL(PL, e) = \frac{1}{2} (QL(PL_{AT}, e_{AT}) + QL(PL_{CT}, e_{CT})) \quad (27)
$$

where e$_{AT}$ and e$_{CT}$ are the position errors in the AT and CT directions, respectively.

b) Final objective function: The final objective function combines both the QL and MPE as:

$$
J(\mathbf{q}) = MPE + \frac{1}{N} QL(PL, e) \qquad (28)
$$

where

$$
MPE = \frac{1}{N} \sum_{k} e_{k} \qquad (29)
$$

and N the number of samples.

The MPE and QL functions may produce outcomes with different scales, which can lead the BO to favor minimizing one objective over the other. For this purpose, a standardization is applied to each part of J by subtracting the mean and dividing by the standard deviation of the observed function values.

C. Application to Information filter with Fault detection

In this study, we focus on learning the measurement noise covariance matrix R associated to GNSS pseudoranges and the local thresholds th$_{loc}$ used in the fault exclusion step. This threshold is associated with the residuals that use a single GNSS measurement. As done classically with the KF, we suppose that the measurement noises are uncorrelated which leads to a diagonal global covariance matrix with elements R$_{gnss}$ . We tune R$_{gnss}$ using BO, under the assumption that all pseudorange measurements from a given constellation share the same covariance. This means that equal weight is assigned to all measurements, and the FDE step will handle the erroneous ones. For Galileo measurements, the associated noise covariance is obtained from R$_{gnss}$ , scaled by a constant factor.

Since the optimized parameters depend on the measure- ments only, the optimization is done when measurements are

available. The overall BO architecture with EIF is summarized in figure 2.

<figure><img src="image_3.png" /><figcaption>Fig. 2. BO for EIF parameters optimization.</figcaption></figure>

IV. EXPERIMENTAL RESULTS

In order to evaluate the proposed auto-tuning approach, data were collected using a Renault Zoe vehicle equipped with wheel-speed sensors, a gyro and Septentrio AsteRx SB PRO Connect GNSS receiver for GPS and Galileo pseudorange measurements. A SPAN CPT system is used to obtain the ground truth. Dead-reckoning operates at 50Hz while GNSS measurements are obtained at 1Hz.

Acquisitions were carried out in Compiègne city along a 12.5 km track (figure 3), lasting approximately 40 min. Four different runs were conducted along the same path on different days. Three runs are used for training, covering a distance of 37 km, and one run for testing. The trajectory includes urban and suburban environments with challenging parts for GNSS measurements.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Experimental trajectory of 12.5km in Compiègne (France), repeated four times on different days.</figcaption></figure>
In the following, we examine the performance of the BO with the proposed objective function in two cases:

* We optimize $R_{gnss}$ while manually tuning the thresholds for FDE. We also compare with the case without FDE.

* We optimize $R_{gnss}$ and the threshold $th_{loc}$, and provide a discussion regarding the two components, MPE and QL, of the objective function.

### A. Optimizing $R_{gnss}$ only

In this section, we study the performance of the approach by optimizing $R_{gnss}$ alone, resulting in a simple one-dimensional optimization problem. The optimization is performed with and without FDE to examine the variation in GNSS covariance between the two cases.

Figure 4 illustrates the last iteration of BO based on the proposed objective function. The target integrity risk $\alpha$ is set to $10^{-3}$. When using FDE, the GNSS standard deviation is searched within the range $\sigma_{gnss} \in [0.01, 10]$. The BO uses 10 initial seed samples obtained from Latin hypercube sampling method, with a total of 30 iterations. The optimization is repeated 10 times with different initial sample seeds to assess the reproducibility of the method.

<figure><img src="image_5.png" /><figcaption>Fig. 4. Superposed surrogate model and real objective function at iteration 30 (standardized functions), case with FDE.</figcaption></figure>

Over 10 runs with different seed samples, the BO converges to an average $\sigma_{gnss}$ value of 5.09 with a standard deviation of 0.32, demonstrating the low variability of $\sigma_{gnss}$ and the reproducibility of the method.

<table><caption>TABLE I ERRORS AND IR OBTAINED WITH $R_{GNSS}$ FROM BO ON THE TRAINING DATASETS, WITH AND WITHOUT FDE. $TIR = \alpha = 10^{-3}$</caption><thead><tr><th></th><th>error<sub>AT</sub><br>(m)</th><th>error<sub>CT</sub><br>(m)</th><th>error<br>(m)</th><th>IR<sub>AT</sub><br>(×10<sup>-3</sup>)</th><th>IR<sub>CT</sub><br>(×10<sup>-3</sup>)</th></tr></thead><tbody><tr><td rowspan="2">with FDE</td><td>0.46</td><td>0.69</td><td>0.92</td><td>5.01</td><td>27.09</td></tr><tr><td>±0.001</td><td>±0.001</td><td>±0.001</td><td>±0.52</td><td>±1.71</td></tr><tr><td rowspan="2">No FDE</td><td>0.53</td><td>0.81</td><td>1.08</td><td>5.57</td><td>15.3</td></tr><tr><td>±0.0011</td><td>±0.0003</td><td>±0.0007</td><td>±0.05</td><td>±0.22</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>Fig. 5. Surrogate model when optimizing $\sigma_{GNSS}$ at iteration 30, case without FDE.</figcaption></figure>

In the absence of the FDE procedure, an increase in $R_{gnss}$ is expected to account for faulty measurements. Therefore, the GNSS standard deviation is searched within the extended range $\sigma_{gnss} \in [0.01, 20]$. In this case, The BO converges to a higher value of $\sigma_{gnss} = 13.3 \pm 0.31$ compared to the case with FDE (Figure 5). This indicates that the BO effectively models the presence of faults by increasing the associated uncertainty values.

Regarding the errors and the computed integrity risks IR (proportion of errors exceeding the PL), table I shows that the mean Euclidean error on the training datasets decreases from 1.08 without FDE to 0.92 with FDE, highlighting the importance of the FDE step in reducing errors. However, the integrity risks in the AT (IR<sub>AT</sub>) and CT (IR<sub>CT</sub>) directions exceed the target integrity risk (TIR = $10^{-3}$). This is because the optimization of $R_{gnss}$ alone cannot lead to an optimum that minimizes the errors while simultaneously satisfying the TIR requirement.

### B. Optimization of $R_{gnss}$ and $th_{loc}$

In this section, both $R_{gnss}$ and $th_{loc}$ are optimized. BO explores within the range of [0.01, 10] for the GNSS standard deviation and [0., 0.5] for the threshold. The number of initial seed samples is set to 30, with a maximum of 70 iterations.

Figure 6 shows the parameter values obtained with 10 different initial seeds. It can be observed that the BO converges to two distinct groups of parameters, separated by the red line, with mean $\sigma_{gnss}$ of 9.32 and 5.87, respectively (table II). Indeed, our proposed objective function, which assigns equal weight to both MPE and QL, can be minimized by prioritizing one component over the other. Minimizing both components simultaneously is not always possible using Gaussian filters. The variation in the threshold is less noticeable between the two groups. Figure 7 shows that Group1, characterized by a higher $\sigma_{GNSS}$, reduces the integrity risk on the training dataset compared to Group2 by prioritizing the QL. As a result, the integrity risk is, on average, verified in both directions, as shown in table III (IR < $10^{-3}$). In contrast, Group2, with
<figure><img src="image_7.png" /><figcaption>Fig. 6. The parameters values (GNSS standard deviation ($\sigma_{GNSS}$) and threshold) with 10 different initial seeds, case with FDE. Two groups of parameters separated by red line. Group1 on the left of the red line and Group2 on the right.</figcaption></figure>

<table><caption>TABLE II
MEAN AND STANDARD DEVIATION OF PARAMETERS ($\sigma_{gnss}$ AND $th_{loc}$), USING 10 DIFFERENT INITIAL SEEDS.</caption><thead><tr><th></th><th>Group1</th><th>Group2</th><th>Total</th></tr></thead><tbody><tr><td>$\sigma_{GNSS}$</td><td>9.32 ± 0.83</td><td>5.87 ± 0.71</td><td>7.84 ± 1.93</td></tr><tr><td>$th_{loc}$</td><td>0.033 ± 0.021</td><td>0.012 ± 0.0046</td><td>0.026 ± 0.018</td></tr></tbody></table>

smaller covariance, exhibits lower errors, but at the expense of degraded integrity (e.g., $IR_{AT} = 2.25 \times 10^{-3} > 10^{-3}$).

Regarding the computed uncertainty regions ($PL$), for Group1, they present average values of $3.12 \pm 0.6$m in the AT direction and $4.39 \pm 1.09$m in the CT direction under a $TIR$ of $10^{-3}$, showing a significant increase compared to Group2, which has values of $2.47 \pm 0.51$m and $3.33 \pm 0.86$m in the AT and CT directions, respectively. This increase leads to localization with large uncertainty bounds, which can affect the availability of the autonomous system.

The mean errors and integrity risks obtained on the test trajectory are presented in table IV. The results are averaged using the parameters obtained from 10 runs with different initial seed values. These results are comparable to those from the training dataset, but with lower errors. Ideally, it would be better to add more test trajectories, which will be done in the future. Figure 8 shows the errors and the computed $PL$ using parameters from Group1 and Group2 on the test trajectory. Similar to the results on the training dataset, it can be observed that the $PL$ amplitude values are higher when the parameters are derived from Group1. Figure 9 presents an example of GNSS residuals computed on the test trajectory using parameters from Group2. It should be noted that parameters from Group2 with lower $R_{gnss}$ values result in larger residuals, leading to a higher number of excluded faults.

Depending on the desired localization requirements, weights

<figure><img src="image_8.png" /><figcaption>Fig. 7. Mean errors (AT, CT and Euclidean) and computed integrity risk with 10 different initial seeds on the training dataset using the BO parameters.</figcaption></figure>

<table><caption>TABLE III
MEAN AND STANDARD DEVIATION OF THE EUCLIDEAN ERROR AND IR ON THE TRAINING DATASET. $\sigma_{GNSS}$ AND $th_{loc}$ OBTAINED USING 10 DIFFERENT SEEDS.</caption><thead><tr><th></th><th>Group1</th><th>Group2</th></tr></thead><tbody><tr><td>error (m)</td><td>0.84 ± 0.0117</td><td>0.82 ± 0.007</td></tr><tr><td>IR<sub>AT</sub> (×10<sup>-3</sup>)</td><td>0.25 ± 0.32</td><td>2.25 ± 1.47</td></tr><tr><td>IR<sub>CT</sub> (×10<sup>-3</sup>)</td><td>0.99 ± 0.47</td><td>5.4 ± 2.99</td></tr></tbody></table>

can be added to both parts of the objective function to prioritize one of the two components ($QL$ or $MPE$). By using an objective function based solely on $QL$, the BO converges to a single group of optimum with a mean $\sigma_{gnss}$ around 9. The results are then comparable to those of the first group obtained with the global objective function.

V. CONCLUSION

In this paper, we presented a new approach for tuning the measurement covariance and the associated fault exclusion threshold, using Bayesian Optimization. We introduced a new objective function that considers the errors and integrity. The approach was validated with real experimental data collected from a vehicle navigating in urban and suburban environments. Using the Gaussian filter and the proposed objective function, BO may converge to different optima that minimize the ob-

<table><caption>TABLE IV
MEAN AND STANDARD DEVIATION OF THE EUCLIDEAN ERROR AND IR ON THE TEST DATASET. $\sigma_{GNSS}$ AND $th_{loc}$ OBTAINED USING 10 DIFFERENT SEEDS.</caption><thead><tr><th></th><th>Group 1</th><th>Group 2</th></tr></thead><tbody><tr><td>error (m)</td><td>0.73 ± 0.06</td><td>0.68 ± 0.042</td></tr><tr><td>IR<sub>AT</sub> (×10<sup>-3</sup>)</td><td>2.6 ± 0.15</td><td>4.6 ± 3.9</td></tr><tr><td>IR<sub>CT</sub> (×10<sup>-3</sup>)</td><td>0. ± 0.</td><td>0.64 ± 1.3</td></tr></tbody></table>
<figure><img src="image_9.png" /><figcaption>Fig. 8. Errors and *PLs* in the AT and CT directions, case of the test trajectory. *R<sub>gnss</sub>* and *T<sub>h<sub>loc</sub></sub>* obtained from BO.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9. The residuals used for fault exclusion with the associated threshold value (in blue), test trajectory. Exx_Sat<sub>res</sub> for Galileo and Gxx_Sat<sub>res</sub> for GPS.</figcaption></figure>

jective function. Some optima verify the integrity requirement, while others minimize the errors.

In future work, the Gaussian noise distribution assumption, which is not always justified and can lead to inconsistent state estimation, will be combined to set-membership techniques. Specifically, the Zonotopic and Gaussian Kalman filter [22] will be used, which require tuning a larger number of parameters. Additionally, other sensors will be integrated into the localization process. The balance between the two components of the objective function will be also explored.

VI. ACKNOWLEDGMENTS

The authors would like to thank Maxime Noizet and Manon Desjardins for their valuable assistance during the experiments.

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