Master’s Thesis Presentation Adaptive Sampling of Clouds with a Fleet of UAVs : Improving Gaussian Process Regression by Including Prior Knowledge Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 1 / 19
Motivation: SkyScanner Project Adaptive Sampling of Cumulus Clouds with a Fleet of UAVs: Clouds remain an uncertainty in current atmospherical models: Characterize the evolution of parameters (3D wind, liquid water content, etc.) Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 2 / 19
Motivation: SkyScanner Project Adaptive Sampling of Cumulus Clouds with a Fleet of UAVs: Clouds remain an uncertainty in current atmospherical models: Characterize the evolution of parameters (3D wind, liquid water content, etc.) − → dense spatial sampling Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 2 / 19
Motivation: SkyScanner Project Adaptive Sampling of Cumulus Clouds with a Fleet of UAVs: Clouds remain an uncertainty in current atmospherical models: Characterize the evolution of parameters (3D wind, liquid water content, etc.) − → dense spatial sampling Adaptive Sampling vs. Systematic Sampling : Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 2 / 19
Motivation: SkyScanner Project Adaptive Sampling of Cumulus Clouds with a Fleet of UAVs: Clouds remain an uncertainty in current atmospherical models: Characterize the evolution of parameters (3D wind, liquid water content, etc.) − → dense spatial sampling Adaptive Sampling vs. Systematic Sampling : 4D map of parameters, with only 1D manifolds available Information efficiency − → quantification of uncertainty Energy efficiency − → mapping and exploiting vertical wind. Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 2 / 19
Motivation: SkyScanner Project Adaptive Sampling of Cumulus Clouds with a Fleet of UAVs: Clouds remain an uncertainty in current atmospherical models: Characterize the evolution of parameters (3D wind, liquid water content, etc.) − → dense spatial sampling Adaptive Sampling vs. Systematic Sampling : 4D map of parameters, with only 1D manifolds available Information efficiency − → quantification of uncertainty Energy efficiency − → mapping and exploiting vertical wind. → Gaussian Process Regression Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 2 / 19
Table of Contents Motivation: SkyScanner Project 1 Introduction: Simulation and Architecture 2 Gaussian Process Regression 3 Spatial Statistics 4 Implementation 5 Summary and Outlook 6 Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 3 / 19
MesoNH Simulation and Sampling Architecture Large Eddy Simulation(LES) of non-precipitating shallow cumulus clouds. Domain : 3540 s × 4 km × 4 km × 4 km (3TB of data), Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 4 / 19
MesoNH Simulation and Sampling Architecture Large Eddy Simulation(LES) of non-precipitating shallow cumulus clouds. Domain : 3540 s × 4 km × 4 km × 4 km (3TB of data), Grid : 3540 x 161 x 400 x 400 ( t , z , x , y ) and dt = 1 s , dx = dy = 10 m , dz = 10 m ... 100 m ; dz = 10 m for boundary and convective cloud layer. Variables : 3D wind, temperature, pressure, liquid water content(LWC), etc. Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 4 / 19
MesoNH Simulation and Sampling Architecture Atmospheric Simulation 3D Wind 3D Wind Ground Truth Ground Truth @1 Hz UAV Trajectory Wind Sensors UAV Model Model Sequence 3D Wind of @ 0.1 Hz Samples Commands Wind GP Trajectory Regression Planner Wind prediction Models Hyperparameter optimization Large Eddy Simulation(LES) of non-precipitating shallow cumulus clouds. Domain : 3540 s × 4 km × 4 km × 4 km (3TB of data), Grid : 3540 x 161 x 400 x 400 ( t , z , x , y ) and dt = 1 s , dx = dy = 10 m , dz = 10 m ... 100 m ; dz = 10 m for boundary and convective cloud layer. Variables : 3D wind, temperature, pressure, liquid water content(LWC), etc. Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 4 / 19
MesoNH Simulation and Sampling Architecture Atmospheric Simulation 3D Wind 3D Wind Ground Truth Ground Truth @1 Hz UAV Trajectory Wind Sensors UAV Model Model Sequence 3D Wind of @ 0.1 Hz Samples Commands Wind GP Trajectory Regression Planner Wind prediction Models Hyperparameter optimization Large Eddy Simulation(LES) of non-precipitating shallow cumulus clouds. Domain : 3540 s × 4 km × 4 km × 4 km (3TB of data), Grid : 3540 x 161 x 400 x 400 ( t , z , x , y ) and dt = 1 s , dx = dy = 10 m , dz = 10 m ... 100 m ; dz = 10 m for boundary and convective cloud layer. Variables : 3D wind, temperature, pressure, liquid water content(LWC), etc. → Wind predictions needed under real-time constraints Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 4 / 19
Introduction to Gaussian Process Regression Bayesian Machine Learning framework Generalization of the M-dim. Gaussian distribution to stochastic processes(functions), i.e. a Gaussian distribution over functions: Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 5 / 19
Introduction to Gaussian Process Regression Bayesian Machine Learning framework Generalization of the M-dim. Gaussian distribution to stochastic processes(functions), i.e. a Gaussian distribution over functions: Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 5 / 19
Introduction to Gaussian Process Regression Bayesian Machine Learning framework Generalization of the M-dim. Gaussian distribution to stochastic processes(functions), i.e. a Gaussian distribution over functions: Two key ingredients Mean function m ( x ) : Center for the distribution of functions Covariance function, matrix k ( x , x ′ ) , Σ : Defines smoothness and variability. Quantifies similarity. If x , x ′ similar − → outputs similar Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 5 / 19
Introduction to Gaussian Process Regression Making predictions With training data: X , Y | new input vector x ⋆ | mean function m ( x ) | covariance matrices Σ X , X = [ k ( x i , x i )] , i , j = 1 , ..., n | Σ x ⋆ , X = [ k ( x ⋆ , x i )] , i = 1 , ..., n | Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 6 / 19
Introduction to Gaussian Process Regression Making predictions With training data: X , Y | new input vector x ⋆ | mean function m ( x ) | covariance matrices Σ X , X = [ k ( x i , x i )] , i , j = 1 , ..., n | Σ x ⋆ , X = [ k ( x ⋆ , x i )] , i = 1 , ..., n | p ( y ⋆ | x ⋆ , X , Y ) = N ( y ⋆ , V [ y ⋆ ]) , (1) y ⋆ = m ( x ⋆ ) + Σ x ⋆ , X Σ − 1 X , X ( Y − m ( X )) , (2) V [ y ⋆ ] = k ( x ⋆ , x ⋆ ) − Σ x ⋆ , X Σ − 1 X , X Σ T (3) x ⋆ , X Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 6 / 19
Introduction to Gaussian Process Regression Making predictions With training data: X , Y | new input vector x ⋆ | mean function m ( x ) | covariance matrices Σ X , X = [ k ( x i , x i )] , i , j = 1 , ..., n | Σ x ⋆ , X = [ k ( x ⋆ , x i )] , i = 1 , ..., n | p ( y ⋆ | x ⋆ , X , Y ) = N ( y ⋆ , V [ y ⋆ ]) , (1) y ⋆ = m ( x ⋆ ) + Σ x ⋆ , X Σ − 1 X , X ( Y − m ( X )) , (2) V [ y ⋆ ] = k ( x ⋆ , x ⋆ ) − Σ x ⋆ , X Σ − 1 X , X Σ T (3) x ⋆ , X Advantages of GPR Inbuilt estimation of uncertainty adapted to test inputs Limitations Mean function and covariance function are parameterized − → Expensive optimization, usually Bayesian Marginal Log-Likelihood (several iterations of O ( n 3 ) ) With no prior knowledge about process, “off-the-shelf”: � − 0 . 5 | x − x ′ | 2 � → m ( x ) = 0, k ( x , x ′ ) = σ 2 exp − l 2 Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 6 / 19
Introduction to Gaussian Process Regression Types of prior knowledge to improve GPR: Determining the mean function m ( x ) 1 Determining type and parameter distribution of covariance function 2 k ( x , x ′ ) If output multidimensional, then determine and exploit correlations 3 between outputs Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 7 / 19
Introduction to Gaussian Process Regression Types of prior knowledge to improve GPR: Determining the mean function m ( x ) 1 Determining type and parameter distribution of covariance function 2 k ( x , x ′ ) If output multidimensional, then determine and exploit correlations 3 between outputs Approaches to determine prior knowledge Brute Force : Cross-validate implementations that combine several mean-functions, covariance functions and output-correlation structures − → No real understanding about the process → Computational complexity O ( n 3 ) − Diego Selle (RIS @ LAAS-CNRS, RT-TUM) Master’s Thesis Presentation October 12, 2016 7 / 19
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