# A Bayesian Optimization Approach for Multi-Function Estimation for Environmental Monitoring Using an Autonomous Surface Vehicle: Ypacarai Lake Case Study

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## Abstract

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## 1. Introduction

- A generalized multi-parameter measuring system for environmental monitoring based on Bayesian optimization with acquisition function fusions.
- An experimental study of data sampling distance between measurements using kernel hyper-parameters information.
- A validation and comparison with other approaches in a real-case scenario such as the monitoring of Ypacarai Lake in Paraguay.

## 2. Related Works

#### 2.1. Environmental Monitoring with Autonomous Vehicles

#### 2.2. Multi-Objective Optimization

## 3. Statement of the Problem

#### 3.1. Objective Functions

#### 3.2. Assumptions

**Environment:**The environment consists of a defined region modelled as a matrix $\mathcal{M}$. The matrix $\mathcal{M}$ corresponds to the square grid representation of real-life locations (latitude, longitude). Therefore, the real environment is discrete, meaning that an ASV can be located in a position related to an element of the map. Every element ${\mathcal{M}}_{i,j}$ is a square of side d, and this can be navigable or occupied by an impassable object (such as terrain or obstacles). The full set of navigable elements or squares corresponds to $\mathcal{X}$:$$\mathcal{X}\subseteq \mathcal{M}\subseteq {\mathbb{R}}^{2}$$**Navigation, Guidance, and Control (NGC) System**: The ASV has subsystems that are designed for specific purposes of navigation, guidance, and control. Figure 3 depicts the general system design. The ASV contains the NGC system so that it can autonomously position, decide missions, and move for accomplishing the objective. The navigation subsystem is in charge of locating the vehicle; however, the positioning system is not perfect and can provide the position of the ASV within a circle of radius r. This positioning error leads to guidance inaccuracies. The guidance subsystem is in charge of planning paths. Once a goal is defined using a global path planner (GPP) component, which implements the MEBO approach, this subsystem defines a collision-free path from the current position of the ASV to the goal location. If no obstacles are in the segment, a direct route is planned; otherwise, the guidance system plans a path using RRT*, as it can provide quick, good paths as shown in [23]. Finally, the control subsystem is in charge of reaching the calculated goal, but as the navigation subsystem has an error of $\pm 2r$, the measurement position ${p}_{k}\in \mathbf{x}$ may be shifted by $\pm 3r$. This error also accounts for water and air currents, stopping mechanisms, etc.**Water Quality Sensor System:**The vehicle is supposedly equipped with n water quality sensors named ${s}_{1},{s}_{2},\phantom{\rule{0.277778em}{0ex}}\dots \phantom{\rule{0.277778em}{0ex}},{s}_{n}$. It can be observed in Figure 3 that the ASV communicates the water quality sensor system to the mission planning component through the guidance system. The guidance system commands the performing of measurements, and the sensor system provides the values of water quality parameters. These sensors can measure different variables simultaneously, but the ASV needs to stop to take measurements. This constraint prevents continuous monitoring; i.e., the vehicle cannot be constantly obtaining new information. As shown in Equation (4), the sensors do not perform perfect measurements; therefore, noise is present in every read. For real purposes, the values returned by the sensors are not normally distributed ($\mu \left(\mathbf{x}\right)=0$, $\sigma \left(\mathbf{x}\right)=1$); therefore, additional data pre-processing is needed to ensure that the values that are fed to the MEBO system will have the mentioned characteristics. Preprocessing data is a common procedure in machine learning, and it is used in this work to facilitate the fitting of Gaussian process models with means of zero.**ASV constraints:**The ASV has a smaller size than the size d of a square of matrix $\mathcal{M}$ so that a movement in every direction is always possible. For performance evaluation purposes, the battery autonomy is taken into account. In order to include battery usage, one approach is to consider its level as a function constraint, but since the battery level is independent of the position p of the vehicle (i.e., the input in the proposed MEBO approach), this constraint cannot be approximated to a function of $(x,y)$. Therefore, methods to include the battery level directly in the BO method as a constraint, such as the work in [17], cannot be used. Consequently, in this work, as in a related work [5], we consider that missions finalize whenever the ASV travels a total distance of 15,000 m, which corresponds to approximately $2.1$ h of usage.

## 4. Proposed Approach

#### 4.1. Bayesian Optimization

#### 4.2. Gaussian Processes

#### 4.3. Acquisition Function

#### 4.4. Multi-Function Estimation Generalization

#### 4.4.1. AF Fusion

- Decoupled Evaluation: This is designed to optimize one single GP at a time, so that different objectives are optimized in different steps. The expression responds to select the next measurement position $p\ast $ as the argument of the maximum of the different AFs (Equation (15)). For example, if all the AFs weight the different locations of set $\mathcal{X}$ according to the uncertainty $\sigma \left(\mathbf{x}\right)$, the measurement location will correspond to the position where one of the AFs has the highest value of uncertainty. The decoupled method expression is shown below:$$p\ast ={\mathrm{argmax}}_{\mathbf{x}}\left\{{\alpha}_{i}\left(\mathbf{x}\right)\right\}\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}i=[1,\phantom{\rule{0.277778em}{0ex}}2,\phantom{\rule{0.277778em}{0ex}}\dots \phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}n]$$
- Coupled Evaluation: This is a fusion that acknowledges the importance values of all the AFs. Therefore, the AFs contribute equally to a general or grand AF. It consists of adding all the AFs. Thus, the next measurement location $p\ast $ is calculated as the argument of the maximum of the sum of the AFs, as shown in Equation (16). This location $p\ast $ will be selected as the location where the different AFs will have their values maximized as a combined set.$$p\ast ={\mathrm{argmax}}_{\mathbf{x}}\{\sum _{i=1}^{n}{\alpha}_{i}\left(\mathbf{x}\right)\}$$

#### 4.4.2. Multi-Function Truncated Adaptation

## 5. Performance Evaluation

#### 5.1. Performance Metrics

^{2}score (R2S). The R2S provides a score $(-\infty ,1.0]$ of how well the predicted or approximated values correspond to the real behavior.

#### 5.2. Simulation Setup

#### 5.2.1. Ground Truth and Water Quality Parameters

#### 5.2.2. Simulation Parameters

#### 5.3. Results for AFF and Length Ratio of Truncated Adaptation Selection

#### 5.3.1. Acquisition Functions Fusion Evaluation

#### 5.3.2. Ratio of Length Scale for Truncated-AF

#### 5.4. Comparison with Other Methods

- PESMOC for Environmental Monitoring:In this work, we use the method proposed in [17] but with some modifications in order to ensure monitoring. This consists of obtaining the difference between the logarithm of the uncertainty of a predictive distribution (PD) ${\sigma}_{i}^{PD}$ and the average of logarithms of uncertainties of conditioned PD ${\sigma}_{i}^{CPD}$ (conditioned to $\mathbf{x}|\mathcal{X}{\ast}_{\left(m\right)}$). $\mathcal{X}{\ast}_{\left(m\right)}$ is one of the m different locations of a supposed Pareto Set. For the full explanation of the PESMOC, please refer to [17]. The PES expression is directly taken from the work, and it has the form of:$$\alpha \approx \sum _{i=1}^{n}(\mathrm{log}{\sigma}_{i}^{PD}\left(\mathbf{x}\right)-\frac{1}{M}\sum _{m=1}^{M}\mathrm{log}{\sigma}_{i}^{CPD}\left(\mathbf{x}\right|\mathcal{X}{\ast}_{\left(m\right)}))$$As shown in the expression above, the coupled evaluation sums up the differences for each AF${}_{i}$. The decoupled version considers one difference at a time. Next, we define the parameters that were changed in order to fit the purpose of exploration:
- Pareto Set $\mathcal{X}\ast $: Since the objective of this work can be thought of as minimizing uncertainty, the Pareto Set is taken as the positions where the sum of the predicted standard deviations reaches their maximum values.
- Conditioning $p\left(\mathbf{y}\right|D,\mathbf{x},\mathcal{X}\ast )$: The conditioning is made through a cloned GP model in order to include a supposed evaluation according to the items of the Pareto Set.
- Monte Carlo Sampling M: For efficient evaluations, only one point of the Pareto Set is used. Therefore, the number of Monte Carlo samples is reduced to one. Indeed, this sacrifices accuracy but definitely improves computational efficiency, which has been observed as being less efficient than our method due to the fact that it needs to calculate the GP regression twice for each water quality parameter or objective.

With $M=1$, as foretold, we proceed to test the method using the same GP model and the same simulation sensors as the proposed method evaluation, but with the best parameters, $\lambda =0.375$ and coupled AFF. - TSP-Based Environmental Monitoring:In [16], a set of 60 waypoints were defined in the shore of Ypacarai Lake. Afterwards, the best TSP solution (waypoint visiting order) was found by a GA evolved to optimize exploration of the Ypacarai Lake. Due to the fact that the GA can be trapped in local minimum [28], we randomize the starting waypoint so that the ASV does not start always on the same initial position. Contrary to the continuous measuring approach stated by the mentioned work [16], for comparison, the monitoring system is modified so that the vehicle can perform measurements only while the ASV is not moving. For this method, the proposed system in Figure 3 is the same, with the difference of the global path Planning component.The distance between measurements locations is the same as the one proposed in this work, length scale-based. Therefore, the ASV travels from waypoint to waypoint performing measurements every $l=\lambda \times \mathrm{min}\left\{{\ell}_{i}\right\}$ meters. Whenever the total distance traveled reaches 15,000 m, the ASV stops and performs a last measurement, and the mission ends.

#### 5.5. Discussion

- The proposed MEBO system can efficiently select measurement locations online (i.e., with streaming data) so that multiple surrogate models can be obtained simultaneously. It is notable how the system provides efficient sequential measuring locations taking into account the available information.
- We evaluated two different fusion methods of AF, namely, the decoupled evaluation and the coupled evaluation. The latter proved to be better whenever the number of measurements were sufficient for good R2S values.
- Through simulations, empirical foundations of optimal measurement performing for efficient exploring were laid out. We propose that exploration of unknown functions take into account the underlying hyper-parameters of GPs, such as the length scale. The simulations showed that, similar to the Nyquist–Shannon sampling theorem, the distance between measurement locations for surrogate model acquisition needd to be shorter than half of a supposed frequency of similarity between different locations (namely, length scale ℓ).
- With an appropriate value of $\lambda $, comparisons with other methods were carried out. Our proposed MEBO approach outperforms the other approaches in terms of R2S obtained versus distance and number of measurements, being $2.43\%$ better than the PESMOC implementation and $10.82\%$ better than the GA method. The MEBO approach is also more robust than the other methods and provides better variance of R2S values. Moreover, the proposed system also outperforms the others whenever the data is noisy, obtaining a $17.23\%$ improvement versus PESMOC and a $2.63\%$ versus GA.
- Computational efficiency comparisons also showed that the proposed method is better than other BO approaches and similar to adapted water quality environmental approaches.

## 6. Conclusions and Future Works

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Image representation of the Ypacarai Lake model. Note that, since there exist segments represented with two black squares that cross white squares (obstacles or terrain), $\mathcal{X}$ is a non-convex set.

**Figure 3.**General systems of the designed ASV. Consists mainly on 4 subsystems: water quality sensors, navigation, guidance, and control.

**Figure 4.**Proposed multi-function estimation using Bayesian optimization approach for environmental monitoring using an ASV.

**Figure 5.**Example of different random simulated ground truth maps using the Shekel function. The maximum locations can overlap and be outside of the feasible set $\mathcal{X}$.

**Figure 6.**R2 score for different $\lambda $ values. For each value, the average R2S score is shown, divided in two bars, one for the coupled evaluation and the second for the decoupled.

**Figure 8.**Results for R2 score using three methods. The proposed MEBO approach (blue) is always better when compared with the distance traveled or number of measurements. (

**a**) R2S comparison according to the distance traveled and noiseless observations. (

**b**) R2S comparison according to the number of noiseless measurements performed. (

**c**) R2S comparison according to the distance traveled, with noisy observations. (

**d**) R2S comparison according to the number of noisy measurements performed.

**Figure 9.**Best results for each method when the ASV measures two different parameters. (

**a**) Squared error and uncertainty maps for the proposed MEBO approach. (

**b**) Squared error and uncertainty maps for the PESMOC approach. (

**c**) Squared error and uncertainty maps for GA monitoring.

**Figure 10.**Average variance of R2Ss of multiple surrogate models using different methods. The more the method generalizes the acquisition, the lower the variance.

**Figure 11.**Number of seconds for obtaining new performed measurements for different methods. The variance in each column is large due to the fact that there are different number of surrogate models that needs to be fitted on each set of simulations (Table 2).

**Table 1.**Brief summary of related works. Specifying whether the work is Multi-Objective (MO) or not.

Ref. | Specific Objective | Is MO? | Main Algorithm | Vehicle | Year |
---|---|---|---|---|---|

[9] | Search for targets in risky environments | Yes | Pareto-front point selection | Aerial | 2018 |

[13] | Agricultural field coverage | No | Distributed swarm control | Aerial | 2018 |

[10] | Package delivery pptimization | Yes | $\u03f5$-constraint method | Aerial | 2019 |

[3] | Pure exploration | No | TSP and genetic algorithm | ASV | 2019 |

[16] | Exploration and intensification | No | Hamiltonian and Eulerian circuits | ASV | 2019 |

[8] | Routing algorithms for Comms. | Yes | Q-learning fuzzy logic | Aerial | 2020 |

[12] | Time-efficient path planning | No | Genetic algorithm | - | 2020 |

[4] | Monitoring through patrolling | No | Double deep Q-learning | ASV | 2020 |

[11] | Wireless network coverage | Yes | Simulated annealing | Aerial | 2020 |

[5] | Monitoring and model obtaining | No | Bayesian optimization | ASV | 2021 |

[14] | Multi-agent monitoring | No | Double deep Q-learning | ASV | 2021 |

Sim. ID | Sensors Involved |
---|---|

1 | ($S1,S2$) |

2 | ($S5,S6$) |

3 | ($S1,S2,S3$) |

4 | ($S5,S6,S7$) |

5 | ($S1,S2,S3,S4$) |

6 | ($S5,S6,S7,S8$) |

7 | ($S1,S2,S3,S4,S5$) |

8 | ($S1,S5,S6,S7,S8$) |

**Table 3.**Definition of MEBO proposed model. Note that two hyperparameters, $\lambda $ and AFF, are defined as variables.

Gaussian Processes | Acq. Functions | Adaptation | AFF |
---|---|---|---|

RBF (${\ell}_{i}=100$) | EI ($\xi =1.0$) | tr- ($l=\lambda \times \mathrm{min}\left\{{\ell}_{i}\right\}$) | coupled or decoupled |

**Table 4.**Values of average R2S, quantity of measurements and time for different distances using different $\lambda $ ratios. R2S is always better for $\lambda =0.25$, while quantity of measures and computational time is better with $\lambda =0.5$. $\lambda =0.375$ presents a good balance of these properties.

Average R2S | Quantity of Meas. | Average Time (s) | |||||||
---|---|---|---|---|---|---|---|---|---|

$\mathbf{\lambda}\u27f6$ | 0.5 | 0.375 | 0.25 | 0.5 | 0.375 | 0.25 | 0.5 | 0.375 | 0.25 |

$d=5000$ | 0.301 | 0.355 | 0.409 | 7 | 9 | 12 | 1.021 | 1.131 | 1.435 |

$d=$ 10,000 | 0.542 | 0.589 | 0.614 | 12 | 15 | 21 | 1.443 | 1.811 | 2.746 |

$d=$ 15,000 | 0.693 | 0.734 | 0.754 | 17 | 22 | 32 | 2.063 | 2.799 | 4.736 |

Parameter | Value |
---|---|

Gaussian Processes | RBF (${\ell}_{i}=100$) |

Acq. Functions | EI ($\xi =1.0$) |

Adaptation | tr- ($l=0.375\times \mathrm{min}\left\{{\ell}_{i}\right\}$) |

AFF | coupled |

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**MDPI and ACS Style**

Peralta, F.; Reina, D.G.; Toral, S.; Arzamendia, M.; Gregor, D.
A Bayesian Optimization Approach for Multi-Function Estimation for Environmental Monitoring Using an Autonomous Surface Vehicle: Ypacarai Lake Case Study. *Electronics* **2021**, *10*, 963.
https://doi.org/10.3390/electronics10080963

**AMA Style**

Peralta F, Reina DG, Toral S, Arzamendia M, Gregor D.
A Bayesian Optimization Approach for Multi-Function Estimation for Environmental Monitoring Using an Autonomous Surface Vehicle: Ypacarai Lake Case Study. *Electronics*. 2021; 10(8):963.
https://doi.org/10.3390/electronics10080963

**Chicago/Turabian Style**

Peralta, Federico, Daniel Gutierrez Reina, Sergio Toral, Mario Arzamendia, and Derlis Gregor.
2021. "A Bayesian Optimization Approach for Multi-Function Estimation for Environmental Monitoring Using an Autonomous Surface Vehicle: Ypacarai Lake Case Study" *Electronics* 10, no. 8: 963.
https://doi.org/10.3390/electronics10080963