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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

This paper presents a novel class of self-organizing sensing agents that adaptively learn an anisotropic, spatio-temporal Gaussian process using noisy measurements and move in order to improve the quality of the estimated covariance function. This approach is based on a class of anisotropic covariance functions of Gaussian processes introduced to model a broad range of spatio-temporal physical phenomena. The covariance function is assumed to be unknown a priori. Hence, it is estimated by the maximum a posteriori probability (MAP) estimator. The prediction of the field of interest is then obtained based on the MAP estimate of the covariance function. An optimal sampling strategy is proposed to minimize the information-theoretic cost function of the Fisher Information Matrix. Simulation results demonstrate the effectiveness and the adaptability of the proposed scheme.

In recent years, due to global climate changes, more environmental scientists are interested in the change of ecosystems over vast regions in lands, oceans, and lakes. For instance, for certain environmental conditions, rapidly reproducing harmful algal blooms in the Great Lakes can produce cyanotoxins. Besides such natural disasters, there exist growing ubiquitous possibilities of the release of toxic chemicals and contaminants in the air, lakes, and public water systems. This resulted in the rising demands to utilize autonomous robotic systems that can perform a series of tasks such as estimation, prediction, monitoring, tracking and removal of a scalar field of interest undergoing often complex transport phenomena (Common examples are diffusion, convection, and advection).

Significant enhancements have been made in the areas of mobile sensor networks and mobile sensing vehicles such as unmanned ground vehicles, autonomous underwater vehicles, and unmanned aerial vehicles. Emerging technologies have been reported on the coordination of mobile sensing agents [

The mobility of mobile agents can be designed in order to perform the optimal sampling of the field of interest. Recently in [

The motivation of our work is as follows. Even though there have been efforts to utilize Gaussian processes to model and predict the spatio-temporal field of interest, most of recent papers assume that Gaussian processes are isotropic, implying that the covariance function only depends on the distance between locations. Many studies also assume that the corresponding covariance functions are known a priori for simplicity. However, this is not the case in general as pointed out in literature [

The contribution of this paper is to develop covariance function learning algorithms for the sensing agents to perform nonparametric prediction based on a properly adapted Gaussian process for a given spatio-temporal phenomenon. By introducing a generalized covariance function, we expand the class of Gaussian processes to include the anisotropic spatio-temporal phenomena. The maximum a posteriori probability (MAP) estimator is used to find hyperparameters for the associated covariance function. The proposed optimal navigation strategy for autonomous vehicles minimizes the information-theoretic cost function such as D-or A-optimality criterion using the Fisher Information Matrix (or Cramér-Rao lower bound (CRLB)[

This paper is organized as follows. In Section 2, we briefly review the mobile sensing network model and the notation related to a graph. A nonparametric approach to predict a field of interest based on measurements is presented in Section 3. Section 4 introduces a covariance function learning algorithm for an anisotropic, spatio-temporal Gaussian process. An optimal navigation strategy is described in Section 5. In Section 6, simulation results illustrate the usefulness of our approach and its adaptability for unknown and/or time-varying covariance functions.

The standard notation will be used in the paper. Let ℝ, ℝ_{≥0}, ℤ denote, respectively, the set of real, non-negative real, and integer numbers. The positive semi-definiteness of a matrix

First, we explain the mobile sensing network and the measurement model used in this paper. Let _{s}^{2}. Assume that 𝒬 is a compact set. The identity of each agent is indexed by ℐ := {1, 2,⋯, _{s}_{i}_{≥0}. We assume that the measurement _{i}_{i}_{i}_{i}

The communication network of mobile agents can be represented by a graph with edges. Let _{i}_{i}_{i}

With the spatially distributed sampling capability, agents need to estimate and predict the field of interest by fusing the collective samples from different space and time indices. We show a nonparametric approach to predict a field of interest based on measurements. We assume that a field undergoing a physical transport phenomenon can be modeled by a spatio-temporal Gaussian process, which can be used for nonparametric prediction.

Consider a spatio-temporal Gaussian process:
_{≥0} and _{l}_{x}_{y}_{t}

In the case that the global coordinates are different from the local model coordinates, a similarity transformation can be used to address this issue. For instance, a rotational relationship between the model basis {_{x}_{y}_{x}_{y}_{X}_{Y}

Up to time _{k}_{j}_{m}_{m}_{i}_{m}_{i}_{m}_{m}_{j}_{m}_{m}_{j}_{m}_{m}_{j}_{m}_{j}_{m}_{m}_{≥0}. The measurements are corrupted by the sensor and communication noises represented by Gaussian white noise
_{w}_{w}_{k}_{Y≤k} := 𝔼(_{≤k}) is the mean vector of _{≤k}, Σ_{Y≤k}:= 𝔼((_{≤k} − _{Y≤k}) (_{≤k} − _{Y≤k})^{T}_{≤k} obtained by
_{ij}

If the covariance function is known a priori, the prediction of the random field _{k}_{k}_{z}_{≤k}_{zY≤k}]_{j}_{j}_{j}

Without loss of generality, we use a zero mean Gaussian process

If the covariance function of a Gaussian process is not known a priori, mobile agents need to estimate parameters of the covariance function (Ψ) based on the observed samples. Using Bayes’ rule, the posterior _{≤k}) is proportional to the likelihood _{≤k}|Ψ) times the prior _{k}_{k}_{≤k}), _{≤k}. Notice that if no prior information is given, the MAP estimate in

A gradient ascent algorithm is used to find a MAP estimate of Ψ:
_{x}f_{j}

After finding a MAP estimate of Ψ, agents can proceed the prediction of the field of interest using

Agents should find new sampling positions to improve the quality of the estimated covariance function in the next iteration at time _{k+1}. For instance, to precisely estimate the anisotropic phenomenon,

To this end, we consider a centralized scheme. Suppose that a leader agent (or a central station) knows the communication graph at the next iteration time _{k+1} and also has access to all measurements collected by agents. Let _{k+1} and _{≤k}_{k+1} and the collective measurements up to time _{k}

To derive the optimal navigation strategy, we compute the log likelihood function of observations of _{≤k+1}:
_{≤k+1} is the size of _{≤k+1}.

Since the locations of observations in _{≤k} were already fixed, we represent the log likelihood function in terms of a vector of future sampling points _{k+1} only and the hyperparameter vector Ψ:

Now consider the Fisher Information Matrix (FIM) that measures the information produced by measurements _{≤k+1} for estimating the hyperparameter vector at time _{k+1}. The Cramér-Rao lower bound (CRLB) theorem states that the inverse of the FIM is a lower bound of the estimation error covariance matrix [_{k+1} represents the estimation of Ψ at time _{k+1}. The FIM [_{≤k+1} | Ψ). The analytical closed-form of FIM is given by
_{k}

We can expect that minimizing the CRLB results in a decrease of uncertainty in estimating Ψ [^{(i)} is a small positive number which can be obtained by using a backtracking line search. Alternatively, a control law for the mobile sensor network can be formulated as follows:
_{d}_{j}

However, optimization on ln _{≤k+1}|Ψ) in _{k}_{Y≤k}. One way to deal with this problem is to use a truncated date set
_{≤k}. In addition, this approach based on the truncated observations can be viewed as a strategy to deal with a slowly time-varying parameter vector Ψ, which will further investigated in Section 6.2.

The overall protocol for the sensor network is summarized as in

In this section, we evaluate the proposed approach for a spatio-temporal Gaussian process (Section 6.1) and an advection-diffusion process (Section 6.3). For both cases, we compare the simulation results using the proposed optimal sampling strategy with results using a benchmark random sampling strategy. In this random sampling strategy, each agent was initially randomly deployed in the surveillance region. At each time step, the next sampling position for agent _{i}

We apply our approach to a spatio-temporal Gaussian process. The Gaussian process was numerically generated for the simulation [_{s}_{f}_{x}_{y}_{t}_{w}^{2} in which all possible values are positive. The gradient method was used to find the MAP estimate of the hyperparameter vector.

For simplicity, we assumed that the global basis is the same as the model basis. We considered a situation where at each time, measurements of agents are transmitted to a leader (or a central station) that uses our Gaussian learning algorithm and sends optimal control back to individual agents for next iteration to improve the quality of the estimated covariance function. The maximum distance for agents to move in one time step was chosen to be 1 for both

For both proposed and random strategies, Monte Carlo simulations were run for 100 times and the statistical results are shown in

To illustrate the adaptability of the proposed strategy to time-varying covariance functions, we introduce a Gaussian process defined by the following covariance function. The time-varying covariance function is modeled by a time-varying weighted sum of two known covariance functions 𝒦_{1}(·) and 𝒦_{2}(·) such as
_{1}(·) is constructed with _{f}_{x}_{y}_{t}_{w}_{2}(·) is with _{f}_{x}_{y}_{t}_{w}_{1} and 𝒦_{2} effectively models hyperparameter changes in

To improve the adaptability, the mobile sensor network uses only observations sampled during the last 20 iterations for estimating

We apply our approach to a spatio-temporal process generated by physical phenomena (advection and diffusion). This work can be viewed as a statistical modeling of a physical process, _{0}, _{0}, _{0}). This is then spread by the wind with mean velocity
_{0} = 0), the concentration _{0}, Δ_{0}, and Δ_{0}. The parameters used in the simulation study are shown in _{x}^{2}_{y}^{2}_{s}

The initial values for the algorithm was chosen to be
_{f}_{w}

We again assumed that the global basis is the same as the model basis and assumed all agents have the same level of measurement noises for simplicity. In our simulation study, agents start sampling at _{0} = 100_{k}_{s}

Monte Carlo simulations were run for 100 times, and _{x}_{y}_{t}

In this paper, we presented a novel class of self-organizing sensing agents that learn an anisotropic, spatio-temporal Gaussian process using noisy measurements and move in order to improve the quality of the estimated covariance function. The MAP estimator was used to estimate the hyperparameters in the unknown covariance function and the prediction of the field of interest was obtained based on the MAP estimates. An optimal navigation strategy was proposed to minimize the information-theoretic cost function of the Fisher Information Matrix for the estimated hyperparameters. The proposed scheme was applied to both a spatio-temporal Gaussian process and a true advection-diffusion field. Simulation study indicated the effectiveness of the proposed scheme and the adaptability to time-varying covariance functions. The trade-off between a precise estimation and computational efficiency using truncated observations will be studied in the future work.

This work has been supported by the National Science Foundation through CAREER Award CMMI-0846547. This support is gratefully acknowledged.

Snap shots of the realized Gaussian process at

Monte Carlo simulation results (100 runs) for a spatio-temporal Gaussian process using

The predicted fields along with agents’ trajectories at

Snap shots of the advection-diffusion process at

Simulation results (100 runs) for a advection-diffusion process. The estimated hyperparameters with

An adaptive sampling strategy for mobile sensor networks.

Learning: At time Prediction: For given Sampling: Based on {Ψ̂ Repeat the steps 1–3 until Ψ converges. |

Parameters used in simulation.

Parameter | Notation | Unit | Value |
---|---|---|---|

Number of agents | _{s} |
- | 5 |

Sampling time | _{s} |
min | 5 |

Initial time | _{0} |
min | 100 |

Gas release mass | kg | 10^{6} | |

Wind velocity in |
_{x} |
m/min | 0.5 |

Eddy diffusivity in |
_{x} |
m^{2}/ |
20 |

Eddy diffusivity in |
_{y} |
m^{2}/ |
10 |

Eddy diffusivity in |
_{z} |
m^{2}/ |
0.2 |

Location of explosion | _{0} |
m | 2 |

Location of explosion | _{0} |
m | 5 |

Location of explosion | _{0} |
m | 0 |

Sensor noise level | _{w} |
kg/m^{3} |
0.1 |