MultiType Sensor Placements in Gaussian Spatial Fields for Environmental Monitoring^{ †}
Abstract
:1. Introduction
2. Problem Formulation
2.1. Gaussian Process
2.2. Informative Locations for Single Spatial Field
2.3. Optimal MultiType Sensor Placement
3. Solution Approach
3.1. OnewithAll Case
Algorithm 1 Multitype sensor deployment algorithm for onewithall case 

3.2. General Case
Algorithm 2 Multitype sensor deployment algorithm for the general cost case 

3.3. Assessing the Trade Off
3.4. Speeding up the Algorithms
Algorithm 3 Lazy greedy algorithm for onewithall case 

Algorithm 4 Lazy greedy algorithm for the general cost case 

4. Simulations
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Notation  Definition 

$m(\xb7)$  The mean function of the Gaussian Process 
$k(\xb7,\xb7)$  The kernel function of the Gaussian Process 
${\mathcal{X}}_{A}$  The random variables over the location index set A 
$\sigma $  Try to span the whole column of the table 
T  The total number of types of interest 
$\left[T\right]$  The abbreviation for the set $\{1,2,\cdots ,T\}$ 
V  The set of all indexes, each corresponding to a location/grid 
$\leftV\right$  The number of indexes in the set V 
s  an index in the set V 
${A}_{i}$  The set of the indexes of the selected locations for the ith type 
$\mathbf{A}$  The placement scheme $\{{A}_{1},{A}_{2},\cdots ,{A}_{T}\}$ 
${f}_{i}$  The ithe objective function 
${w}_{i}$  The weight parameter of the ith objective function 
${c}_{i}$  The unit cost for the ith type 
${c}_{site}$  The site construction cost 
B  The total budget constraint 
K  The subset size constraint 
${k}_{i}$  The total number of sensors for the ith type 
$\lfloor x\rfloor $  The floor function mapping x to the greatest integer 
less than or equal to x  
${\delta}_{i,s}$  The information gain of adding location index s of type i 
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Sun, C.; Yu, Y.; Li, V.O.K.; Lam, J.C.K. MultiType Sensor Placements in Gaussian Spatial Fields for Environmental Monitoring. Sensors 2019, 19, 189. https://doi.org/10.3390/s19010189
Sun C, Yu Y, Li VOK, Lam JCK. MultiType Sensor Placements in Gaussian Spatial Fields for Environmental Monitoring. Sensors. 2019; 19(1):189. https://doi.org/10.3390/s19010189
Chicago/Turabian StyleSun, Chenxi, Yangwen Yu, Victor O. K. Li, and Jacqueline C. K. Lam. 2019. "MultiType Sensor Placements in Gaussian Spatial Fields for Environmental Monitoring" Sensors 19, no. 1: 189. https://doi.org/10.3390/s19010189