AASS, Örebro University

Transcription

1 Mobile Robotics Achim J. and Lilienthal Olfaction Lab, AASS, Örebro University 1

2 Contents 1. Gas-Sensitive Mobile Robots in the Real World 2. Gas Dispersal in Natural Environments 3. Gas Quantification with MOX Sensors in an OSS o Gaussian Processes (GP)» Weight-Space View Bayesian Linear Regression» Weight-Space View Bayesian Non-Linear Regression» Function-Space View» Examples, Drawing Samples from a GP» Predictions with Noise-Free Observations» Predictions with Noisy Observations 3. Gas Quantification with MOX Sensors in an OSS Part 2 4. Gas Dispersal Simulation 5. Literature 2

3 1 Gas-Sensitive Mobile Robots in the Real World 3

4 1. Gas-Sensitive Mobile Robots Basic Idea o to combine autonomous robots with... o gas sensing technology ("electronic nose") and o... eventually other relevant sensors + + = 4

5 1. Gas-Sensitive Mobile Robots Covered in this Lecture Chemical Sensing o under water o airborne (land, air) Airborne Chemical Sensing with Mobile Robots o first publications at the beginning of the 1990s 5

6 1. Gas-Sensitive Mobile Robots Basic Idea o autonomous robots + gas sensors + other relevant sensors 6

7 1. Gas-Sensitive Mobile Robots Basic Idea o autonomous robots + gas sensors + other relevant sensors Applications: "Smelling" Robots for... o pollution monitoring o security applications (detecting leaks, for example) o non-military surveillance, search and rescue o automatic humanitarian demining,... Basic Research: Learn More... o about the physics of gas distribution o smell related biological solutions (homing, communication,...) 7

8 1. Gas-Sensitive Mobile Robots Subtasks in Mobile Robot Olfaction o gas detection (gas finding)» detecting an increased concentration of a target gas 8

9 1. Gas-Sensitive Mobile Robots Subtasks in Mobile Robot Olfaction o gas detection (gas finding) o "odour" discrimination and concentration estimation» sensor calibration» pattern recognition 9

10 1. Gas-Sensitive Mobile Robots Subtasks in Mobile Robot Olfaction o gas detection (gas finding) o "odour" discrimination and concentration estimation o gas source tracking» following the cues from the sensed gas distribution (eventually using also other sensor modalities) towards the source [Lilienthal et al. 2006] 10

11 1. Gas-Sensitive Mobile Robots Subtasks in Mobile Robot Olfaction o gas detection (gas finding) o "odour" discrimination and concentration estimation o gas source tracking o gas source declaration» determining the certainty that the source has been found 11

12 1. Gas-Sensitive Mobile Robots Subtasks in Mobile Robot Olfaction o gas detection (gas finding) o "odour" discrimination and concentration estimation o gas source tracking o gas source declaration gas source localisation 12

13 1. Gas-Sensitive Mobile Robots Subtasks in Mobile Robot Olfaction o gas detection (gas finding) o "odour" discrimination and concentration estimation o gas source tracking o gas source declaration o trail guidance» trail following [Lilienthal et al. 2006] 13

14 1. Gas-Sensitive Mobile Robots Subtasks in Mobile Robot Olfaction o gas detection (gas finding) o "odour" discrimination and concentration estimation o gas source tracking o gas source declaration o trail guidance» trail following» trail avoiding strategies [Lilienthal et al. 2006] 14

15 1. Gas-Sensitive Mobile Robots Subtasks in Mobile Robot Olfaction o gas detection (gas finding) o "odour" discrimination and concentration estimation o gas source tracking o gas source declaration o trail guidance» trail following» trail avoiding strategies o gas distribution modelling (gas distribution mapping) 15

16 1. Gas-Sensitive Mobile Robots From "Electronic Nose" to "Mobile Nose" o space, power, weight restrictions o varying environmental conditions (temperature, humidity, ) o open sampling system» direct exposition of gas sensors to the environment» less controlled gas sampling» typically continuous sampling? 16

17 1. Gas-Sensitive Mobile Robots Differences to Other Sensors in Mobile Robotics o point measurement» sensitive sensor surface is typically small (often < 1cm 2 ) 17

18 1. Gas-Sensitive Mobile Robots Differences to Other Sensors in Mobile Robotics o point measurement» sensitive sensor surface is typically small (often < 1cm 2 )» effective sampling region also small and approx. spherical (fan) 18

19 1. Gas-Sensitive Mobile Robots Differences to Other Sensors in Mobile Robotics o point measurement o sensor dynamics» long response time, very long recovery time 19

20 1. Gas-Sensitive Mobile Robots Differences to Other Sensors in Mobile Robotics o point measurement o sensor dynamics o calibration» complicated "sensor response concentration" relation» dependent on other variables (temperature, humidity,...)» has to consider sensor dynamics» variation between individual sensors» long-term drift 20

21 1. Gas-Sensitive Mobile Robots Differences to Other Sensors in Mobile Robotics o point measurement o sensor dynamics o calibration o very dynamic reality» constantly changing chaotic concentration field» real-time gas distribution mapping changes at different time-scales rapid fluctuations slow changes of the overall structure of the average distribution 21

22 2 Gas Dispersal in Natural Environments 22

23 2. Gas Dispersal in Natural Environments Chaotic Gas Dispersal o diffusion [Smyth and Moum 2001] 23

24 2. Gas Dispersal in Natural Environments Chaotic Gas Dispersal o diffusion [Smyth and Moum, 2001] [Smyth and Moum 2001] 24

25 2. Gas Dispersal in Natural Environments Chaotic Gas Dispersal o diffusion o advective transport o turbulent transport [Smyth and Moum, 2001] [Smyth and Moum 2001] 25

26 2. Gas Dispersal in Natural Environments Chaotic Gas Dispersal o diffusion o advective transport o turbulent transport [Smyth and Moum, 2001] [Smyth and Moum 2001] 26

27 2. Gas Dispersal in Natural Environments Chaotic Gas Dispersal o diffusion o advective transport o turbulent transport video courtesy of Hiroshi Ishida 27

28 2. Gas Dispersal in Natural Environments Chaotic Gas Dispersal data collected by Yuichiro Fukazawa, Marco Trincavelli, Yuta Wada and Hiroshi Ishida 29

29 2. Gas Dispersal in Natural Environments Chaotic Gas Dispersal data collected by Yuichiro Fukazawa, Marco Trincavelli, Yuta Wada and Hiroshi Ishida 31

30 2. Gas Dispersal in Natural Environments Chaotic Gas Dispersal data collected by Yuichiro Fukazawa, Marco Trincavelli, Yuta Wada and Hiroshi Ishida 32

31 2. Gas Dispersal in Natural Environments Chaotic Gas Dispersal data collected by Yuichiro Fukazawa, Marco Trincavelli, Yuta Wada and Hiroshi Ishida 33

32 2. Gas Dispersal in Natural Environments Turbulent Flow Turbulent Flow o caused by shear forces» shear forces e.g. through obstacles o shear forces generate torque vortices» strong viscous forces can prevent vortices to form (rotational energy heat) o Reynold's number describes the balance between viscous and shear forces 35

33 2. Gas Dispersal in Natural Environments Turbulent Flow o caused by shear forces» shear forces e.g. through obstacles o shear forces generate torque vortices» strong viscous forces can prevent vortices to form (rotational energy heat) o Reynold's number describes the balance between viscous and shear forces Re = V L ν 36

34 2. Gas Dispersal in Natural Environments Turbulent Flow o caused by shear forces» shear forces e.g. through obstacles o shear forces generate torque vortices» strong viscous forces can prevent vortices to form (rotational energy heat) o Reynold's number describes the balance between viscous and shear forces rms fluctuation of velocity at one point Re = V L ν 37

35 2. Gas Dispersal in Natural Environments Turbulent Flow o caused by shear forces» shear forces e.g. through obstacles o shear forces generate torque vortices» strong viscous forces can prevent vortices to form (rotational energy heat) o Reynold's number describes the balance between viscous and shear forces Re = V L ν integral scale = correlation length of the velocity field (relates to dimensions of available space) 38

36 2. Gas Dispersal in Natural Environments Turbulent Flow o caused by shear forces» shear forces e.g. through obstacles o shear forces generate torque vortices» strong viscous forces can prevent vortices to form (rotational energy heat) o Reynold's number describes the balance between viscous and shear forces Re = V L ν kinematic viscosity ( m 2 /s for air at 20 C) 39

37 2. Gas Dispersal in Natural Environments Turbulent Flow o caused by shear forces» shear forces e.g. through obstacles o shear forces generate torque vortices» strong viscous forces can prevent vortices to form (rotational energy heat) o Reynold's number describes the balance between viscous and shear forces Re» laminar flow for Re < 2000» turbulent flow for Re > 10000» laminar or turbulent in between = V L ν 40

38 2. Gas Dispersal in Natural Environments Turbulent Flow o caused by shear forces» shear forces e.g. through obstacles o shear forces generate torque vortices» strong viscous forces can prevent vortices to form (rotational energy heat) o energy dissipates from larger to smaller eddies o largest eddy size given by Kolmogorov scale 41

39 2. Gas Dispersal in Natural Environments Turbulent Flow o caused by shear forces» shear forces e.g. through obstacles o shear forces generate torque vortices» strong viscous forces can prevent vortices to form (rotational energy heat) o energy dissipates from larger to smaller eddies o largest eddy size given by Kolmogorov scale L k 3 ν = ε 1 4 rate at which energy is passed from larger to smaller vortices (in simple cases V 3 / L) 42

40 2. Gas Dispersal in Natural Environments Turbulent Flow o caused by shear forces» shear forces e.g. through obstacles o shear forces generate torque vortices» strong viscous forces can prevent vortices to form (rotational energy heat) o energy dissipates from larger to smaller eddies o largest eddy size given by Kolmogorov scale o lower limit of vortex size» layer of laminar flow next to surfaces of solid objects» chemical compounds can only pass through this laminar layer by diffusion 43

41 2. Gas Dispersal in Natural Environments Turbulent Flow Characteristics [Roberts, Webster 2000] o turbulent flow is chaotic/unpredictable» instantaneous velocity/concentration at some instant of time is generally insufficient to predict the velocity some time later o high degree of vortical motion» large-scale eddies cause a meandering dispersal» small scale eddies stretch and twist the gas distribution resulting in a complicated patchy structure o turbulent transport is much faster than molecular diffusion» gaseous ethanol at 25 C and 1 atm: diffusion constant: cm 2 /s diffusion velocity: 20.7 cm/h 44

42 3 Gas Quantification with an Array of MOX Sensors in an Open Sampling System 46

43 3. Gas Quantification with MOX Sensors in Open Sampling Configuration Gas Quantification in an Open Sampling System o challenge 1: highly dynamic characteristic of turbulent airflow Gas Quantification with MOX Sensors o challenge 2: slow dynamics of the MOX sensors traditional three-phase sampling continuous (open) sampling with a mobile robot 47

44 3. Gas Quantification with MOX Sensors in Open Sampling Configuration Gas Quantification in an Open Sampling System o challenge 1: highly dynamic characteristic of turbulent airflow Gas Quantification with MOX Sensors o challenge 2: slow dynamics of the MOX sensors» steady state virtually never reached (!)» we must use transients 48

45 3. Gas Quantification with MOX Sensors in Open Sampling Configuration Gas Quantification in an Open Sampling System o challenge 1: highly dynamic characteristic of turbulent airflow Gas Quantification with MOX Sensors o challenge 2: slow dynamics of the MOX sensors» steady state virtually never reached (!)» we must use transients» we cannot use a deterministic calibration function that requires steady state information g i : steady state conductance of sensor c(t): gas concentration A i, α i : parameters of an exponential model 49

46 3. Gas Quantification with MOX Sensors in Open Sampling Configuration A Probabilistic Approach to MOX-OSS Gas Quantification o MOX-OSS: MOX sensors in an Open Sampling System o key idea: relate measurements with a posterior over concentrations find p(c r 1,..., r N ) 50

47 3. Gas Quantification with MOX Sensors in Open Sampling Configuration A Probabilistic Approach to MOX-OSS Gas Quantification o MOX-OSS: MOX sensors in an Open Sampling System o key idea: relate measurements with a posterior over concentrations» more general (consider non-chemosensor measurements) find p(c r 1,..., r N, T, r H ) 51

48 3. Gas Quantification with MOX Sensors in Open Sampling Configuration A Probabilistic Approach to MOX-OSS Gas Quantification o MOX-OSS: MOX sensors in an Open Sampling System o key idea: relate measurements with a posterior over concentrations» more general (consider last k+1 readings) find p(c(t n ) r 1 n-k:n,..., r N n-k:n, T n-k:n, r H n-k:n ) 52

49 3. Gas Quantification with MOX Sensors in Open Sampling Configuration A Probabilistic Approach to MOX-OSS Gas Quantification o MOX-OSS: MOX sensors in an Open Sampling System o key idea: relate measurements with a posterior over concentrations» here find p(c(t n ) r 1 n-k:n,..., r N n-k:n, T n-k:n, r H n-k:n ) 53

50 3. Gas Quantification with MOX Sensors in Open Sampling Configuration A Probabilistic Approach to MOX-OSS Gas Quantification o MOX-OSS: MOX sensors in an Open Sampling System o key idea: relate measurements with a posterior over concentrations» has hardly been done so far!» previous works with MOX-OSS typically neglect the problem!! find p(c(t n ) r 1 n-k:n,..., r N n-k:n, T n-k:n, r H n-k:n ) 54

51 3. Gas Quantification with MOX Sensors in Open Sampling Configuration A Probabilistic Approach to MOX-OSS Gas Quantification o MOX-OSS: MOX sensors in an Open Sampling System o key idea: relate measurements with a posterior over concentrations o approach using Gaussian Processes [Monroy et al. 2012]» uncertainty in MOX conductance concentration due to gas transport mechanisms inherent sensor dynamics environmental factors such as temperature or humidity (not explicitly addressed here) 55

52 3. Gas Quantification with MOX Sensors in Open Sampling Configuration A Probabilistic Approach to MOX-OSS Gas Quantification o MOX-OSS: MOX sensors in an Open Sampling System o key idea: relate measurements with a posterior over concentrations o approach using Gaussian Processes [Monroy et al. 2012] o use PID readings as ground truth» we know there is only one compound present (+ sensor responds quickly)» assumption: MOX sensors and PID are close enough so that they are exposed to the same concentrations data collected by Yuichiro Fukazawa, Marco Trincavelli, Yuta Wada and Hiroshi Ishida 56

53 3. Gas Quantification with MOX Sensors in Open Sampling Configuration A Probabilistic Approach to MOX-OSS Gas Quantification o MOX-OSS: MOX sensors in an Open Sampling System o key idea: relate measurements with a posterior over concentrations o approach using Gaussian Processes [Monroy et al. 2012] o use PID readings as ground truth o specific research questions» which sensors contribute most?» how much do we gain from using multiple sensors?» how much do we gain from using previous sensor readings? 57

54 3. Gas Quantification with MOX Sensors in Open Sampling Configuration Related Work o calibration on steady-state values» multivariate linear regression methods Principal Component Regression (PCR) Partial Least Squares Regression (PLSR)» non-linear methods Artificial Neural Networks (ANN)» kernel methods Support Vector Regression (SVR) o estimation of steady-state values based on transients o continuous recalibration on integral values C. D. Natale, S. Marco, F. Davide, A. D Amico. Sensor-array calibration time reduction by dynamic modelling. Sensors and Actuators B: Chemical 25 (1-3) (1995) » hourly regression using an ANN» use hourly average of conventional monitoring station with spectrometer as ground truth (compare to hourly average of sensor response) S. D. Vito, E. Massera, M. Piga, L. Martinotto, G. D. Francia. On field calibration of an electronic nose for benzene estimation in an urban pollution monitoring scenario. Sensors and Actuators B: Chemical 129 (2) (2008)

55 3. Gas Quantification with MOX Sensors in Open Sampling Configuration Related Work in MRO? o authors typically work on conductance directly o only [Ishida 1998] propose using sensor calibration based on steady state values as a rough approximation for an OSS o no method considers the transient nature of the signal o no probabilistic method! [Ishida 1998] H. Ishida, T. Nakamoto, T. Moriizumi. Remote sensing of gas/odor source location and concentration distribution using mobile system. Sensors and Actuators B: Chemical 49 (1-2) (1998)

56 3. Gas Quantification with MOX Sensors in Open Sampling Configuration GP Approach to "Probabilistic Calibration" for MOX-OSS o signal conditioning» compensate for long-term drift: r i (t) = R i (t) / R i (0)» alt 1: learn GP on r i (t)» alt 2: learn GP on log(r i (t)) 61

57 3. Gas Quantification with MOX Sensors in Open Sampling Configuration GP Approach to "Probabilistic Calibration" for MOX-OSS o signal conditioning o Gaussian Process (GP) learning 62

58 + Gaussian Processes (GP) following [Rasmussen, Williams 2006] using own slides refined by Marco Trincavelli 63

59 + Gaussian Processes What is a Gaussian Process? o stochastic process generalization of probability distribution to functions» probability distribution describes a finite-dimensional random variable» stochastic process describes a distribution over functions o a particular stochastic process» GPs are a particularly tractable stochastic process o a Bayesian version of an SVM o an infinitely large neural network o particularly effective method for placing prior distributions over the space of functions or over the space of model parameters» space of model parameters weight-space view» space of functions function-space view 64

60 + Gaussian Processes What is a Gaussian Process? 65

61 + Gaussian Processes (GP) Weight-Space View Bayesian Linear Regression 66

62 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o standard linear model ( ) T ( ) ( 2 x = x w y = f x ε, ε ~ N 0, σ ) f +, ε» additive, i.i.d. Gaussian noise 67

63 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o standard linear model ( ) T ( ) ( 2 x = x w y = f x ε, ε ~ N 0, σ ) f +, ε» non-vector notation ( x) = w0 + w1 x w D xd f + 68

64 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o standard linear model ( ) T ( ) ( 2 x = x w y = f x ε, ε ~ N 0, σ ) f +, ε» non-vector notation (D = 1) ( x) = w0 w1 x1 f + 69

65 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o standard linear model ( ) T ( ) ( 2 x = x w y = f x ε, ε ~ N 0, σ ) f +, ε» non-vector notation (D = 1) ( x) = w0 w1 x1 f + 70

66 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o standard linear model ( ) T ( ) ( 2 x = x w y = f x ε, ε ~ N 0, σ ) f +, ε» non-vector notation (D = 1) ( x) = w0 w1 x1 f + 71

67 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o likelihood of observations (normally distributed error) ε ε ε» given weights w and training data y» X: design matrix collection of D-dimensional input vectors (as column vectors) in the training set ε ε 72

68 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o introduce Gaussian prior over weights w ~ N ( 0, Σ ) p 73

69 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o introduce Gaussian prior over weights w ~ N ( 0, Σ ) p» example (D = 1) ( x) = w0 w1 x1 f + 74

70 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o introduce Gaussian prior over weights w ~ N ( 0, Σ ) p» example (D = 1) ( x) = w0 w1 x1 f + w 1 ( w) N( 0, I ) p = w 0 75

71 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o compute posterior over weights using Bayes rule 76

72 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o compute posterior over weights using Bayes rule» likelihood of training data ε ε 79

73 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o compute posterior over weights using Bayes rule 1 ( ) ( ) T T ( T ) 1 T 1 p w Xy, exp y Xw y Xw exp Σ 2 w p w 2σ n ε 2 81

74 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o compute posterior over weights using Bayes rule ( w Xy, ) ( y Xw, ) ( w) p p p 1 ( ) ( ) T T ( T ) 1 T 1 p w Xy, exp y Xw y Xw exp Σ 2 w p w 2σ n ε 2... p ( ) ( 2 1 1) 2 T 1 w X, y N σ A Xy, A, A = σ X X + Σ» posterior over weights is again Gaussian n ε n ε p 82

75 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o Gaussian prior over weights w ~ N ( 0, Σ ) p» example (D = 1) ( x) = w0 w1 x1 f + w 1 ( w) N( 0, I ) p = w 0 83

76 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o compute posterior over weights using Bayes rule» example: we get 3 observations 84

77 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o compute posterior over weights using Bayes rule» example: we get 3 observations» calculate observation likelihood assuming ε ~ p ( 2 0, ) N ( 0,1) σ ε N = ( ) ( T 2 y X, w = N X w, σ I ) ε w 1 w 0 85

78 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o compute posterior over weights using Bayes rule» example: we get 3 observations» calculate observation likelihood assuming» compute posterior over weights p A ( ) ( w X, y ~ N σ A Xy, A ) ε 2 T 1 = σ ε X X + Σ p w 1 w 0 86

79 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o compute posterior over weights using Bayes rule» example: we get 3 observations» calculate observation likelihood assuming» compute posterior over weights w 1 Prior Likelihood Posterior w 1 w 1 w 0 w 0 w 0 87

80 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o compute posterior over weights using Bayes rule» example: we get 3 observations» calculate observation likelihood assuming» compute posterior over weights w 1 Prior Likelihood Posterior w 1 w 1 intercept reduced ( 0) w 0 w 0 w 0 88

81 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o compute posterior over weights using Bayes rule» example: we get 3 observations» calculate observation likelihood assuming» compute posterior over weights w 1 Prior Likelihood Posterior w 1 w 1 slope left unchanged w 0 w 0 w 0 89

82 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o predictions for a query point» = average predictions f * at x * over all parameter values (weights) weighted by their posterior predictive distribution 90

83 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o predictions for a query point» = average predictions f * at x * over all parameter values (weights) weighted by their posterior» predictive distribution is again Gaussian A 2 T 1 = σ ε X X + Σ p 91

84 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o predictions for a query point» = average predictions f * at x * over all parameter values (weights) weighted by their posterior predictive distribution» posterior distribution over weights predictive distribution over functions w 1 w 0 92

85 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o problem: linear model has limited expressiveness f ( ) T ( ) ( 2 x = x w y = f x + ε, ε N 0, σ ), ε» non-vector notation (D = 1) ( x) = w0 w1 x1 f + 93

86 + Gaussian Processes (GP) Weight-Space View Bayesian Non-Linear Regression 94

87 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o problem: linear model has limited expressiveness o apply Kernel trick 95

88 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o apply Kernel trick» project inputs into some high dimensional space using a set of basis functions and apply the linear model in the high dimensional space x φ, ( ) D N x R R 96

89 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o apply Kernel trick» project inputs into some high dimensional space using a set of basis functions and apply the linear model in the high dimensional space x φ, ( ) D N x R R! the model is still linear in w f ( ) T x x w ( x) ( x) T = f = φ w 97

90 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o apply Kernel trick» project inputs into some high dimensional space using a set of basis functions and apply the linear model in the high dimensional space x φ, ( ) D N x R R! the model is still linear in w f ( ) T x x w ( x) ( x) T = f = φ w! N can be infinite φ ( ) 2 x = 1, x, x,... 98

91 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o apply Kernel trick» project inputs into some high dimensional space using a set of basis functions and apply the linear model in the high dimensional space f ( ) T x x w ( x) ( x) T = f = φ w o apply linear model in feature space (to make predictions) 99

92 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o apply Kernel trick» project inputs into some high dimensional space using a set of basis functions and apply the linear model in the high dimensional space f ( ) T x x w ( x) ( x) T = f = φ w o apply linear model in feature space (to make predictions) o... predictive mean in feature space o... predictive covariance in feature space 105

93 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o predictive mean and covariance in feature space 106

94 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o predictive mean and covariance in feature space o identify dot products of the form φ ( ) T x Σ ( ) pφ 1 x2 107

95 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o predictive mean and covariance in feature space o identify dot products of the form φ ( ) T x Σ ( ) 1 x2» since Σ p is positive definite, we can find a Cholesky decomposition Σ p = pφ L T L 108

96 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o predictive mean and covariance in feature space o identify dot products of the form ( ) T x Σ ( ) 1 x2» since Σ p is positive definite, we can find a Cholesky decomposition» and define a kernel (or covariance function) as φ Σ p = pφ L T L 109

97 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o predictive mean and covariance in feature space o the feature space φ appears only within dot products as φ ( ) T x Σ ( ) 1 x2 o and therefore the feature space φ can be removed using a kernel pφ 110

98 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior o predictive mean and covariance in feature space ( ) 1 T 2 µ k x, X k XX, σ I y = ( ) ( ) + n T 1 2,, X XX, X, ( ) ( ) Σ= ( ) ( ) ( ) + σ n k x x k x k I k x p f x, Xy, = N µ, Σ ( ) ( ) 111

99 + Gaussian Processes, Weight-Space View Bayesian Linear Regression with a Gaussian Prior, Summary o define standard linear model (step 1) o likelihood of observations in standard linear model (step 2) o introduce Gaussian prior over weights (step 3) o compute weight posterior using Bayes rule (step 4) o integrate predictions over weights to get predictive distribution (step 5) o apply Kernel trick (step 6) 113