Sensor placement is an optimisation problem that has recently gained great relevance. In order to achieve accurate online updates of a predictive model, sensors are used to provide observations. When sensor location is optimally selected, the predictive model can greatly reduce its internal errors. A greedy-selection algorithm is used for locating these optimal spatial locations from a numerical embedded space. A novel architecture for solving this big data problem is proposed, relying on a variational Gaussian process. The generalisation of the model is further improved via the preconditioning of its inputs: Masked Autoregressive Flows are implemented to learn nonlinear, invertible transformations of the conditionally modelled spatial features. Finally, a global optimisation strategy extending the Mutual Information-based optimisation and fine-tuning of the selected optimal location is proposed. The methodology is parallelised to speed up the computational time, making these tools very fast despite the high complexity associated with both spatial modelling and placement tasks. The model is applied to a real three-dimensional test case considering a room within the Clarence Centre building located in Elephant and Castle, London, UK.
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Computational Fluid Dynamics
Evidence Lower BOund
Indoor Air Quality
Large Eddy Simulation
London South Bank University
Masked Autoencoder for Distribution Estimation
Masked Autoregressive Flows
Markov-Chain Monte Carlo
Sparse Variational Process
Stochastic Variational Inference
Variational Gaussian Process
Variational Gaussian Process optimal sensor placement
K. Abhishek, M. P. Singh, S. Ghosh, A. Anand: Weather forecasting model using artificial neural network. Procedia Technology 4 (2012), 311–318.
Applied Modelling and Computation Group: Fluidity manual (Version 4.1). Available at https://figshare.com/articles/Fluidity_Manual/1387713 (2015), 329 pages.
R. Arcucci, L. D’Amore, J. Pistoia, R. Toumi, A. Murli: On the variational data assimilation problem solving and sensitivity analysis. J. Comput. Phys. 335 (2017), 311–326.
R. Arcucci, D. McIlwraith, Y.-K. Guo: Scalable weak constraint Gaussian processes. Computational Science — ICCS 2019. Lecture Notes in Computer Science 11539. Springer, Cham, 2019, pp. 111–125.
R. Arcucci, L. Mottet, C. Pain, Y.-K. Guo: Optimal reduced space for variational data assimilation. J. Comput. Phys. 379 (2019), 51–69.
E. Aristodemou, R. Arcucci, L. Mottet, A. Robins, C. Pain, Y.-K. Guo: Enhancing CFD-LES air pollution prediction accuracy using data assimilation. Building and Environment 165 (2019), Article ID 106383, 15 pages.
M. J. Beal: Variational Algorithms for Approximate Bayesian Inference: A Thesis Submitted for the Degree of Doctor of Philosophy of the University of London. University of London, London, 2003.
J. H. T. Bentham: Microscale Modelling of Air Flow and Pollutant Dispersion in the Urban Environment: Doctoral Thesis. University of London, London, 2004.
D. M. Blei, A. Kucukelbir, J. D. McAuliffe: Variational inference: A review for statisticians. J. Am. Stat. Assoc. 112 (2017), 859–877.
B. Bócsi, P. Hennig, L. Csató, J. Peters: Learning tracking control with forward models. IEEE International Conference on Robotics and Automation (ICRA). IEEE, New York, 2012, pp. 259–264.
D. Cornford, I. T. Nabney, C. K. I. Williams: Adding constrained discontinuities to Gaussian process models of wind fields. Advances in Neural Information Processing Systems 11 (NIPS 1998). MIT Press, Cambridge, 1999, pp. 861–867.
N. Cressie: Statistics for spatial data. Terra Nova 4 (1992), 613–617.
L. D’Amore, R. Arcucci, L. Marcellino, A. Murli: A parallel three-dimensional variational data assimilation scheme. Numerical Analysis and Applied Mathematics, IC-NAAM 2011. AIP Conference Proceedings 1389. AIP, Melville, 2011, pp. 1829–1831.
C. Doersch: Tutorial on variational autoencoders. Available at https://arxiv.org/abs/1606.05908 (2016), 23 pages.
T. H. Dur, R. Arcucci, L. Mottet, M. Molina Solana, C. Pain, Y.-K. Guo: Weak constraint Gaussian processes for optimal sensor placement. J. Comput. Sci. 42 (2020), Article ID 101110, 12 pages.
M. Germain, K. Gregor, I. Murray, H. Larochelle: MADE: Masked Autoencoder for Distribution Estimation. Proc. Mach. Learn. Res. 37 (2015), 881–889.
H. González-Banos: A randomized art-gallery algorithm for sensor placement. SCG’01: Proceedings of the 17th Annual Symposium on Computational Geometry. ACM, New York, 2001, pp. 232–240.
I. Goodfellow, Y. Bengio, A. Courville: Deep Learning. Adaptive Computation and Machine Learning. MIT Press, Cambridge, 2016.
Google Brain Team: TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems. Available at https://www.tensorflow.org/ (2015).
C. Guestrin, A. Krause, A. P. Singh: Near-optimal sensor placements in Gaussian processes. ICML’05: Proceedings of the 22nd International Conference on Machine Learning. ACM, New York, 2005, pp. 265–272.
J. Hagan, A. R. Gillis, J. Chan: Explaining official delinquency: A spatial study of class, conflict and control. Sociological Quarterly 19 (1978), 386–398.
J. Hensman, N. Fusi, N. D. Lawrence: Gaussian processes for big data. Available at https://arxiv.org/abs/1309.6835 (2013), 9 pages.
N. Jarrin, S. Benhamadouche, D. Laurence, R. Prosser: A synthetic-eddy-method for generating inflow conditions for large-eddy simulations. Int. J. Heat Fluid Flow 27 (2006), 585–593.
F. J. Kelly, J. C. Fussell: Improving indoor air quality, health and performance within environments where people live, travel, learn and work. Atmospheric Environment 200 (2019), 90–109.
D. P. Kingma, M. Welling: Auto-encoding variational Bayes. Available at https://arxiv.org/abs/1312.6114 (2013), 14 pages.
A. Krause, A. Singh, C. Guestrin: Near-optimal sensor placements in Gaussian processes: Theory, efficient algorithms and empirical studies. J. Mach. Learn. Res. 9 (2008), 235–284.
S. Kullback, R. A. Leibler: On information and sufficiency. Ann. Math. Stat. 22 (1951), 79–86.
C.-C. Lin, L. L. Wang: Forecasting simulations of indoor environment using data assimilation via an ensemble Kaiman filter. Building and Environment 64 (2013), 169–176.
H. Liu, Y.-S. Ong, X. Shen, J. Cai: When Gaussian process meets big data: A review of scalable GPs. Available at https://arxiv.org/abs/1807.01065 (2018), 20 pages.
D. J. C. MacKay: Introduction to Gaussian processes. Neural Networks and Machine Learning. NATO ASI Series F Computer and Systems Sciences 168. Springer, Berlin, 1998, pp. 133–166.
M. I. Mead, O. A. M. Popoola, G. B. Stewart, P. Landshoff, M. Calleja, M. Hayes, J. J. Baldovi, M. W. McLeod, T. F. Hodgson, J. Dicks, A. Lewis, J. Cohen, R. Baron, J. R. Saffell, R. L. Jones: The use of electrochemical sensors for monitoring urban air quality in low-cost, high-density networks. Atmospheric Environment 70 (2013), 186–203.
C. C. Pain, A. P. Umpleby, C. R. E. de Oliveira, A. J. H. Goddard: Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations. Comput. Methods Appl. Mech. Eng. 190 (2001), 3771–3796.
G. Papamakarios, T. Pavlakou, I. Murray: Masked autoregressive flow for density estimation. Advances in Neural Information Processing Systems 30 (NIPS 2017). MIT Press, Cambridge, 2017, pp. 2338–2347.
D. Pavlidis, G. J. Gorman, J. L. M. A. Gomes, C. C. Pain, H. ApSimon: Synthetic-eddy method for urban atmospheric flow modelling. Boundary-Layer Meteorology 136 (2010), 285–299. I
J. Quiñonero-Candela, C. E. Rasmussen: A unifying view of sparse approximate Gaussian process regression. J. Mach. Learn. Res. 6 (2005), 1939–1959.
N. Ramakrishnan, C. Bailey-Kellogg, S. Tadepalliy, V. N. Pandey: Gaussian processes for active data mining of spatial aggregates. Proceedings of the 2005 SIAM International Conference on Data Mining. SIAM, Philadelphia, 2005, pp. 427–438.
C. E. Rasmussen: Gaussian processes in machine learning. Advanced Lectures on Machine Learning. Lecture Notes in Computer Science 3176. Springer, Berlin, 2003, pp. 63–71.
D. J. Rezende, S. Mohamed: Variational inference with normalizing flows. Available at https://arxiv.org/abs/1505.05770 (2015), 10 pages.
J. Smagorinsky: General circulation experiments with the primitive equations I. The basic experiment. Mon. Wea. Rev. 91 (1963), 99–164.
J. Song, S. Fan, W. Lin, L. Mottet, H. Woodward, M. Davies Wykes, R. Arcucci, D. Xiao, J.-E. Debay, H. ApSimon, E. Aristodenou, D. Birch, M. Carpentieri, F. Fang, M. Herzog, G. R. Hunt, R. L. Jones, C. Pain, D. Pavlidis, A. G. Robins, C. A. Short, P. F. Linden: Natural ventilation in cities: The implications of fluid mechanics. Building Research & Information 46 (2018), 809–828.
M. K. Titsias: Variational learning of inducing variables in sparse Gaussian processes. Proc. Mach. Learn. Res. 5 (2009), 567–574.
M. K. Titsias: Variational Model Selection for Sparse Gaussian Process Regression. Technical report, University of Manchester, Manchester, 2009.
V. H. Tran: Copula variational Bayes inference via information geometry. Available at https://arxiv.org/abs/1803.10998 (2018), 23 pages.
D. Tran, R. Ranganath, D. M. Blei: The variational Gaussian process. Available at https://arxiv.org/abs/1511.06499 (2015), 14 pages.
H. Wickham: ggplot2: Elegant Graphics for Data Analysis. Use R! Springer, Cham, 2016.
This work is supported by the EPSRC Grand Challenge grant Managing Air for Green Inner Cities (MAGIC) EP/N010221/1, the EP/T003189/1 Health assessment across biological length scales for personal pollution exposure and its mitigation (INHALE), the EP/T000414/1 PREdictive Modelling with QuantIfication of UncERtainty for MultiphasE Systems (PREMIERE) and the Leonardo Centre for Sustainable Business at Imperial College London.
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Tajnafoi, G., Arcucci, R., Mottet, L. et al. Variational Gaussian Process for Optimal Sensor Placement. Appl Math 66, 287–317 (2021). https://doi.org/10.21136/AM.2021.0307-19
- sensor placement
- variational Gaussian process
- mutual information