Multi-step prediction of nonlinear Gaussian Process dynamics models with adaptive Gaussian mixtures

Abstract

This paper presents an adaptive Gaussian Mixture Model (aGMM) formulation for performing multiple-step probabilistic state predictions using a nonparametric Gaussian Process (GP) regression model. The presented prediction algorithm is applicable to any dynamic system that is challenging to model parametrically, but where data is available. Gaussian mixture elements are propagated through the GP by analytically evaluating expectation integrals for the moments of the output distribution. Two metrics are presented and compared for adaptively splitting the initial state distribution into a sum of Gaussians to reduce the effect of nonlinearities on prediction accuracy: (1) an analytical evaluation of the excess kurtosis which measures the non-Gaussianity of the output distribution, and (2) a weighted least-squares regression model which evaluates the local nonlinearity of the GP mapping with respect to the input distribution. In addition, an on-the-fly data selection method is presented to reduce the computational complexity associated with analytically evaluating the higher-order moments of the GP output distribution. The proposed adaptive GP-aGMM formulation is applied to the case of anticipating driver behavior at road intersections using a GP driver behavior model in combination with a parametric vehicle model. Prediction performance for this scenario is evaluated using driving data collected from three human subjects navigating a standard four-way intersection. Results demonstrate that the presented prediction algorithm is capable of accurately capturing multimodal behavior in the GP training data.

Keywords

Gaussian Process machine learning driver modeling obstacle anticipation Gaussian Mixture Models

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References

Althoff

Stursberg

Buss

(2009) Model-based probabilistic collision detection in autonomous driving. IEEE Transactions on Intelligent Transportation Systems 10(2): 299–310.

Candela

Girard

Larsen

. (2003) Propagation of uncertainty in bayesian kernel models-application to multiple-step ahead forecasting. In: Acoustics, speech, and signal processing, 2003. proceedings. (ICASSP’03). 2003 IEEE international conference on, volume 2, Hong Kong, pp. II–701. IEEE.

Deisenroth

(2009) Efficient Reinforcement Learning Using Gaussian Processes. KIT Scientific Publishing.

Deisenroth

Rasmussen

Fox

(2011) Learning to control a low-cost manipulator using data-efficient reinforcement learning. In: Robotics science and systems (RSS), Los Angeles, USA.

Doucet

Godsill

Andrieu

(2000) On sequential monte carlo sampling methods for bayesian filtering. Statistics and Computing 10(3): 197–208.

Espié

Guionneau

Wymann

. (2005) Torcs-the open racing car simulator. Available at: http://torcs.sourceforge.net.

Ferguson

Darms

Urmson

. (2008) Detection, prediction, and avoidance of dynamic obstacles in urban environments. In: Proceedings of the IEEE intelligent vehicles symposium (IV), Eindhoven, The Netherlands.

Girard

Rasmussen

Quinonero-Candela

. (2003) Gaussian process priors with uncertain inputs? application to multiple-step ahead time series forecasting. In: Neural information processing systems conference (NIPs), Vancouver, Canada.

Groot

Lucas

van den Bosch

(2011) Multiple-step time series forecasting with sparse gaussian processes. In: Proceedings of the 23rd Benelux conference on artificial intelligence (BNAIC), Ghent, Belgium, pp. 105–112.

10.

Havlak

Campbell

(2010) Discrete and continuous, probabilistic anticipation for autonomous robots in urban environments. In: Society of Photo-Optical Instrumentation Engineers (SPIE) conference series, vol. 7833, Toulouse, France, p. 12.

11.

Havlak

Campbell

(2013) Discrete and continuous, probabilistic anticipation for autonomous robots in urban environments. IEEE Transactions on Robotics 30(2): 461–474.

12.

Hollinger

Englot

Hover

Mitra

Sukhatme

(2012) Active planning for underwater inspection and the benefit of adaptivity. The International Journal of Robotics Research 32(1): 3–18.

13.

Huber

(2011) Adaptive gaussian mixture filter based on statistical linearization. In: Information fusion (FUSION), 2011 proceedings of the 14th international conference on, pp. 1–8. IEEE.

14.

Hwang

Seah

(2008) Intent-based probabilistic conflict detection for the next generation air transportation system. Proceedings of the IEEE 96(12): 2040–2059.

15.

Hyvärinen

Oja

(2000). Independent component analysis: algorithms and applications. Neural Networks 13(4): 411–430.

16.

Joseph

Doshi-Velez

Huang

Roy

(2011) A bayesian nonparametric approach to modeling motion patterns. Autonomous Robots 31(4): 383–400.

17.

Julier

Uhlmann

(1996) A general method for approximating nonlinear transformations of probability distributions. Technical Report, Robotics Research Group, Department of Engineering Science, University of Oxford, UK.

18.

Kim

Lee

Essa

(2011) Gaussian process regression flow for analysis of motion trajectories. In: Computer vision (ICCV), 2011 IEEE international conference on, pp. 1164–1171. IEEE.

19.

Fox

(2009) Gp-bayesfilters: Bayesian filtering using gaussian process prediction and observation models. Autonomous Robots 27(1): 75–90.

20.

Kotecha

Djuric

(2003) Gaussian sum particle filtering. IEEE Transactions on Signal Processing 51(10): 2602–2612.

21.

Laugier

Paromtchik

Perrollaz

. (2011) Probabilistic analysis of dynamic scenes and collision risks assessment to improve driving safety. IEEE Intelligent Transportation Systems Magazine 3(4): 4–19.

22.

Lawrence

Seeger

Herbrich

. (2003) Fast sparse gaussian process methods: The informative vector machine. Advances in Neural Information Processing Systems 625–632.

23.

Miura

Shirai

(2002) Probabilistic uncertainty modeling of obstacle motion for robot motion planning. Journal of Robotics and Mechatronics 14(4): 349–356.

24.

Psiaki

Schoenberg

Miller

(2010) Gaussian mixture approximation by another gaussian mixture for “blob” filter re-sampling. In: 2010 AIAA guidance, navigation, and control conference, Toronto, Canada.

25.

Quinonero-Candela

Girard

Rasmussen

(2003) Prediction at an Uncertain Input for Gaussian Processes and Relevance Vector Machines Application to Multiple-Step Ahead Time-Series Forecasting. IMM, Informatik og Matematisk Modelling, DTU.

26.

Quiñonero-Candela

Rasmussen

(2005) A unifying view of sparse approximate gaussian process regression. The Journal of Machine Learning Research 6: 1939–1959.

27.

Rasmussen

Williams

(2006) Gaussian Processes for Machine Learning, vol. 1. Cambridge, MA: MIT Press.

28.

Rasmussen

Ghahramani

(2002) Infinite mixtures of gaussian process experts. Advances in Neural Information Processing Systems 2: 881–888.

29.

Runnalls

(2007) Kullback–Leibler approach to gaussian mixture reduction. IEEE Transactions on Aerospace and Electronic Systems 43(3): 989–999.

30.

Salzmann

Urtasun

(2010) Implicitly constrained gaussian process regression for monocular non-rigid pose estimation. Advances in Neural Information Processing Systems 23: 2065–2073.

31.

Schubert

Adam

Obst

. (2011) Empirical evaluation of vehicular models for ego motion estimation. In: 2011 IEEE intelligent vehicles symposium (IV), pp. 534–539. IEEE.

32.

Snelson

Ghahramani

(2006) Sparse gaussian processes using pseudo-inputs. Advances in Neural Information Processing Systems 18: 1257.

33.

Tam

Plataniotis

Hatzinakos

(1999) An adaptive gaussian sum algorithm for radar tracking. Signal Processing 77(1): 85–104.

34.

Urtasun

Fleet

Fua

(2006) 3d people tracking with gaussian process dynamical models. In: 2006 IEEE computer society conference on Computer vision and pattern recognition, vol. 1, pp. 238–245. IEEE.

35.

Vasudevan

Ramos

Nettleton

. (2009) Gaussian process modeling of large-scale terrain. Journal of Field Robotics 26(10): 812–840.

36.

Wang

Fleet

Hertzmann

(2006) Gaussian process dynamical models. Advances in Neural Information Processing Systems 18: 1441.

37.

Williams

Maybeck

(2003) Cost-function-based gaussian mixture reduction for target tracking. In: Proceedings of the sixth international conference of Information fusion, 2003, vol. 2, Cairns, Australia, pp. 1047–1054.