Weak Constraint Gaussian Processes for optimal sensor placement

Abstract We present a Weak Constraint Gaussian Process (WCGP) model to integrate noisy inputs into the classical Gaussian Process (GP) predictive distribution. This model follows a Data Assimilation approach (i.e. by considering information provided by observed values of a noisy input in a time window). Due to the increased number of states processed from real applications and the time complexity of GP algorithms, the problem mandates a solution in a high performance computing environment. In this paper, parallelism is explored by defining the parallel WCGP model based on domain decomposition. Both a mathematical formulation of the model and a parallel algorithm are provided. We use the algorithm for an optimal sensor placement problem. Experimental results are provided for pollutant dispersion within a real urban environment.

[1]  Neil D. Lawrence,et al.  Probabilistic Non-linear Principal Component Analysis with Gaussian Process Latent Variable Models , 2005, J. Mach. Learn. Res..

[2]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[3]  A. O'Hagan,et al.  Curve Fitting and Optimal Design for Prediction , 1978 .

[4]  Chris Bailey-Kellogg,et al.  Gaussian Processes for Active Data Mining of Spatial Aggregates , 2005, SDM.

[5]  Michalis K. Titsias,et al.  Variational Learning of Inducing Variables in Sparse Gaussian Processes , 2009, AISTATS.

[6]  Andreas Krause,et al.  Near-optimal sensor placements in Gaussian processes , 2005, ICML.

[7]  Henry P. Wynn,et al.  Maximum entropy sampling , 1987 .

[8]  A. Robins,et al.  Enhancing CFD-LES air pollution prediction accuracy using data assimilation , 2019, Building and Environment.

[9]  Jiyun Song,et al.  Natural ventilation in cities: the implications of fluid mechanics , 2018, Building Research & Information.

[10]  Carl E. Rasmussen,et al.  Distributed Variational Inference in Sparse Gaussian Process Regression and Latent Variable Models , 2014, NIPS.

[11]  Héctor H. González-Baños,et al.  A randomized art-gallery algorithm for sensor placement , 2001, SCG '01.

[12]  Almerico Murli,et al.  On the variational data assimilation problem solving and sensitivity analysis , 2017, J. Comput. Phys..

[13]  Wolfgang Maass,et al.  Approximation schemes for covering and packing problems in image processing and VLSI , 1985, JACM.

[14]  Marc Peter Deisenroth,et al.  Distributed Gaussian Processes , 2015, ICML.

[15]  Carl E. Rasmussen,et al.  Gaussian Process Training with Input Noise , 2011, NIPS.

[16]  W. F. Caselton,et al.  Optimal monitoring network designs , 1984 .

[17]  Ionel M. Navon,et al.  Machine learning-based rapid response tools for regional air pollution modelling , 2019, Atmospheric Environment.

[18]  Andreas Krause,et al.  Near-Optimal Sensor Placements in Gaussian Processes: Theory, Efficient Algorithms and Empirical Studies , 2008, J. Mach. Learn. Res..

[19]  Christopher C. Pain,et al.  A domain decomposition non-intrusive reduced order model for turbulent flows , 2019, Computers & Fluids.

[20]  Yike Guo,et al.  Scalable Weak Constraint Gaussian Processes , 2019, ICCS.

[21]  Christopher C. Pain,et al.  Optimal reduced space for Variational Data Assimilation , 2019, J. Comput. Phys..

[22]  Yu Ding,et al.  Domain Decomposition Approach for Fast Gaussian Process Regression of Large Spatial Data Sets , 2011, J. Mach. Learn. Res..

[23]  H. ApSimon,et al.  Synthetic-Eddy Method for Urban Atmospheric Flow Modelling , 2010 .

[24]  Cheng Soon Ong,et al.  Mathematics for Machine Learning , 2020, Journal of Mathematical Sciences & Computational Mathematics.

[25]  Mike Rees,et al.  5. Statistics for Spatial Data , 1993 .

[26]  Matthias W. Seeger,et al.  Using the Nyström Method to Speed Up Kernel Machines , 2000, NIPS.

[27]  Maurice Queyranne,et al.  An Exact Algorithm for Maximum Entropy Sampling , 1995, Oper. Res..

[28]  Salvatore Cuomo,et al.  An error estimate of Gaussian recursive filter in 3Dvar problem , 2014, 2014 Federated Conference on Computer Science and Information Systems.

[29]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[30]  Yike Guo,et al.  A reduced order model for turbulent flows in the urban environment using machine learning , 2019, Building and Environment.

[31]  Ionel Michael Navon,et al.  Computational Methods for Data Evaluation and Assimilation , 2019 .

[32]  Christopher C. Pain,et al.  A comparison of mesh-adaptive LES with wind tunnel data for flow past buildings: Mean flows and velocity fluctuations , 2009 .

[33]  Sarvapali D. Ramchurn,et al.  2008 International Conference on Information Processing in Sensor Networks Towards Real-Time Information Processing of Sensor Network Data using Computationally Efficient Multi-output Gaussian Processes , 2022 .

[34]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[35]  Agathe Girard,et al.  Learning a Gaussian Process Model with Uncertain Inputs , 2003 .

[36]  Neil D. Lawrence,et al.  Bayesian Gaussian Process Latent Variable Model , 2010, AISTATS.

[37]  C.R.E. de Oliveira,et al.  Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations , 2001 .