# Linear Latent Force Models Using Gaussian Processes

Mauricio A. Álvarez, Universidad Tecnológica de Pereira, Colombia
Neil D. Lawrence, University of Sheffield

IEEE Transactions on Pattern Analysis and Machine Intelligence 35, pp 2693-2705

#### Abstract

Purely data driven approaches for machine learning present difficulties when data is scarce relative to the complexity of the model or when the model is forced to extrapolate. On the other hand, purely mechanistic approaches need to identify and specify all the interactions in the problem at hand (which may not be feasible) and still leave the issue of how to parameterize the system. In this paper, we present a hybrid approach using Gaussian processes and differential equations to combine data driven modelling with a physical model of the system. We show how different, physically-inspired, kernel functions can be developed through sensible, simple, mechanistic assumptions about the underlying system. The versatility of our approach is illustrated with three case studies from motion capture, computational biology and geostatistics.

  @Article{alvarez-llfm13, title = {Linear Latent Force Models Using Gaussian Processes}, journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence}, author = {Mauricio A. Álvarez and David Luengo and Neil D. Lawrence}, pages = {2693}, year = {2013}, volume = {35}, number = {11}, month = {00}, edit = {https://github.com/lawrennd//publications/edit/gh-pages/_posts/2013-05-13-alvarez-llfm13.md}, url = {http://inverseprobability.com/publications/alvarez-llfm13.html}, abstract = {Purely data driven approaches for machine learning present difficulties when data is scarce relative to the complexity of the model or when the model is forced to extrapolate. On the other hand, purely mechanistic approaches need to identify and specify all the interactions in the problem at hand (which may not be feasible) and still leave the issue of how to parameterize the system. In this paper, we present a hybrid approach using Gaussian processes and differential equations to combine data driven modelling with a physical model of the system. We show how different, physically-inspired, kernel functions can be developed through sensible, simple, mechanistic assumptions about the underlying system. The versatility of our approach is illustrated with three case studies from motion capture, computational biology and geostatistics.}, key = {Alvarez-llfm13}, note = {}, pubmedid = {24051729}, doi = {10.1109/TPAMI.2013.86}, linkpdf = {http://arxiv.org/pdf/1107.2699}, linksoftware = {https://github.com/SheffieldML/multigp}, OPTgroup = {} }
 %T Linear Latent Force Models Using Gaussian Processes %A Mauricio A. Álvarez and David Luengo and Neil D. Lawrence %B %C IEEE Transactions on Pattern Analysis and Machine Intelligence %D %F alvarez-llfm13 %J IEEE Transactions on Pattern Analysis and Machine Intelligence %P 2693--2705 %R 10.1109/TPAMI.2013.86 %U http://inverseprobability.com/publications/alvarez-llfm13.html %V 35 %N 11 %X Purely data driven approaches for machine learning present difficulties when data is scarce relative to the complexity of the model or when the model is forced to extrapolate. On the other hand, purely mechanistic approaches need to identify and specify all the interactions in the problem at hand (which may not be feasible) and still leave the issue of how to parameterize the system. In this paper, we present a hybrid approach using Gaussian processes and differential equations to combine data driven modelling with a physical model of the system. We show how different, physically-inspired, kernel functions can be developed through sensible, simple, mechanistic assumptions about the underlying system. The versatility of our approach is illustrated with three case studies from motion capture, computational biology and geostatistics. 
 TY - CPAPER TI - Linear Latent Force Models Using Gaussian Processes AU - Mauricio A. Álvarez AU - David Luengo AU - Neil D. Lawrence PY - 2013/05/13 DA - 2013/05/13 ID - alvarez-llfm13 SP - 2693 EP - 2705 DO - 10.1109/TPAMI.2013.86 L1 - http://arxiv.org/pdf/1107.2699 UR - http://inverseprobability.com/publications/alvarez-llfm13.html AB - Purely data driven approaches for machine learning present difficulties when data is scarce relative to the complexity of the model or when the model is forced to extrapolate. On the other hand, purely mechanistic approaches need to identify and specify all the interactions in the problem at hand (which may not be feasible) and still leave the issue of how to parameterize the system. In this paper, we present a hybrid approach using Gaussian processes and differential equations to combine data driven modelling with a physical model of the system. We show how different, physically-inspired, kernel functions can be developed through sensible, simple, mechanistic assumptions about the underlying system. The versatility of our approach is illustrated with three case studies from motion capture, computational biology and geostatistics. ER - 
 Álvarez, M.A., Luengo, D. & Lawrence, N.D.. (2013). Linear Latent Force Models Using Gaussian Processes. IEEE Transactions on Pattern Analysis and Machine Intelligence 35(11):2693-2705