learning target dynamics while tracking using gaussian...

Learning Target Dynamics WhileTracking Using Gaussian ProcessesClas Veibäck

1 Introduction2 Unknown Influence3 Approximation of Influence4 Time-Varying Influence5 Shared Influence6 State-Dependent Influence7 Estimation8 Applications9 Conclusions

1 Introduction2 Unknown Influence3 Approximation of Influence4 Time-Varying Influence5 Shared Influence6 State-Dependent Influence7 Estimation8 Applications9 Conclusions

Gaussian Process Motion Model Clas Veibäck November 7, 2018 3

Introduction

• General-purpose motion models in tracking

• What if other behaviours or influences are visible?

Like

• Preferred paths in open terrain or indoors• Obstacles that are avoided or common

destinations• Velocity or acceleration profiles in racing events• Wind affecting flying vehicles• Currents in the water affecting animals and ships

• Approximate initial model

Gaussian Process Motion Model Clas Veibäck November 7, 2018 3

Introduction

• General-purpose motion models in tracking

• What if other behaviours or influences are visible?

Like

• Preferred paths in open terrain or indoors• Obstacles that are avoided or common

destinations• Velocity or acceleration profiles in racing events• Wind affecting flying vehicles• Currents in the water affecting animals and ships

• Approximate initial model

Gaussian Process Motion Model Clas Veibäck November 7, 2018 3

Introduction

• General-purpose motion models in tracking

• What if other behaviours or influences are visible?Like

• Preferred paths in open terrain or indoors

• Obstacles that are avoided or commondestinations

• Velocity or acceleration profiles in racing events• Wind affecting flying vehicles• Currents in the water affecting animals and ships

• Approximate initial model

Gaussian Process Motion Model Clas Veibäck November 7, 2018 3

Introduction

• General-purpose motion models in tracking

• What if other behaviours or influences are visible?Like

• Preferred paths in open terrain or indoors• Obstacles that are avoided or common

destinations

• Velocity or acceleration profiles in racing events• Wind affecting flying vehicles• Currents in the water affecting animals and ships

• Approximate initial model

Gaussian Process Motion Model Clas Veibäck November 7, 2018 3

Introduction

• General-purpose motion models in tracking

• What if other behaviours or influences are visible?Like

• Preferred paths in open terrain or indoors• Obstacles that are avoided or common

destinations• Velocity or acceleration profiles in racing events

• Wind affecting flying vehicles• Currents in the water affecting animals and ships

• Approximate initial model

Gaussian Process Motion Model Clas Veibäck November 7, 2018 3

Introduction

• General-purpose motion models in tracking

• What if other behaviours or influences are visible?Like

• Preferred paths in open terrain or indoors• Obstacles that are avoided or common

destinations• Velocity or acceleration profiles in racing events• Wind affecting flying vehicles

• Currents in the water affecting animals and ships

• Approximate initial model

Gaussian Process Motion Model Clas Veibäck November 7, 2018 3

Introduction

• General-purpose motion models in tracking

• What if other behaviours or influences are visible?Like

• Preferred paths in open terrain or indoors• Obstacles that are avoided or common

destinations• Velocity or acceleration profiles in racing events• Wind affecting flying vehicles• Currents in the water affecting animals and ships

• Approximate initial model

Gaussian Process Motion Model Clas Veibäck November 7, 2018 3

Introduction

• General-purpose motion models in tracking

• What if other behaviours or influences are visible?Like

• Preferred paths in open terrain or indoors• Obstacles that are avoided or common

destinations• Velocity or acceleration profiles in racing events• Wind affecting flying vehicles• Currents in the water affecting animals and ships

• Approximate initial model

Gaussian Process Motion Model Clas Veibäck November 7, 2018 4

Introduction

• Joint state estimation and learning of influences

• Model influences as sparse Gaussian processes

• Online learning using sequential algorithm

• Approximations to achieve tractable solution

• Shared influences between multiple targets

• Time-varying influences

Gaussian Process Motion Model Clas Veibäck November 7, 2018 4

Introduction

• Joint state estimation and learning of influences

• Model influences as sparse Gaussian processes

• Online learning using sequential algorithm

• Approximations to achieve tractable solution

• Shared influences between multiple targets

• Time-varying influences

Gaussian Process Motion Model Clas Veibäck November 7, 2018 4

Introduction

• Joint state estimation and learning of influences

• Model influences as sparse Gaussian processes

• Online learning using sequential algorithm

• Approximations to achieve tractable solution

• Shared influences between multiple targets

• Time-varying influences

Gaussian Process Motion Model Clas Veibäck November 7, 2018 4

Introduction

• Joint state estimation and learning of influences

• Model influences as sparse Gaussian processes

• Online learning using sequential algorithm

• Approximations to achieve tractable solution

• Shared influences between multiple targets

• Time-varying influences

Gaussian Process Motion Model Clas Veibäck November 7, 2018 4

Introduction

• Joint state estimation and learning of influences

• Model influences as sparse Gaussian processes

• Online learning using sequential algorithm

• Approximations to achieve tractable solution

• Shared influences between multiple targets

• Time-varying influences

Gaussian Process Motion Model Clas Veibäck November 7, 2018 4

Introduction

• Joint state estimation and learning of influences

• Model influences as sparse Gaussian processes

• Online learning using sequential algorithm

• Approximations to achieve tractable solution

• Shared influences between multiple targets

• Time-varying influences

1 Introduction2 Unknown Influence3 Approximation of Influence4 Time-Varying Influence5 Shared Influence6 State-Dependent Influence7 Estimation8 Applications9 Conclusions

Gaussian Process Motion Model Clas Veibäck November 7, 2018 6

Dynamic System with an Unknown Influence

x0 ∼ N (x0, P0),

yk = Ckxk + ek, k = 1, . . . ,K

xk = Akxk−1 +Bkfk−1 + vk, k = 1, . . . ,K

fk =(f1(zfk), . . . , f

J(zfk))T

f j(z) ∼ GP(0, K(z, z′)), j = 1, . . . , J

The noise is given by

vk ∼ N (0, Qk) and ek ∼ N (0, Rk).

Gaussian Process Motion Model Clas Veibäck November 7, 2018 6

Dynamic System with an Unknown Influence

x0 ∼ N (x0, P0),

yk = Ckxk + ek, k = 1, . . . ,K

xk = Akxk−1 +Bkfk−1 + vk, k = 1, . . . ,K

fk =(f1(zfk), . . . , f

J(zfk))T

f j(z) ∼ GP(0, K(z, z′)), j = 1, . . . , J

The noise is given by

vk ∼ N (0, Qk) and ek ∼ N (0, Rk).

Gaussian Process Motion Model Clas Veibäck November 7, 2018 6

Dynamic System with an Unknown Influence

x0 ∼ N (x0, P0),

yk = Ckxk + ek, k = 1, . . . ,K

xk = Akxk−1 +Bkfk−1 + vk, k = 1, . . . ,K

fk =(f1(zfk), . . . , f

J(zfk))T

f j(z) ∼ GP(0, K(z, z′)), j = 1, . . . , J

The noise is given by

vk ∼ N (0, Qk) and ek ∼ N (0, Rk).

Gaussian Process Motion Model Clas Veibäck November 7, 2018 6

Dynamic System with an Unknown Influence

x0 ∼ N (x0, P0),

yk = Ckxk + ek, k = 1, . . . ,K

xk = Akxk−1 +Bkfk−1 + vk, k = 1, . . . ,K

fk =(f1(zfk), . . . , f

J(zfk))T

f j(z) ∼ GP(0, K(z, z′)), j = 1, . . . , J

The noise is given by

vk ∼ N (0, Qk) and ek ∼ N (0, Rk).

Gaussian Process Motion Model Clas Veibäck November 7, 2018 6

Dynamic System with an Unknown Influence

x0 ∼ N (x0, P0),

yk = Ckxk + ek, k = 1, . . . ,K

xk = Akxk−1 +Bkfk−1 + vk, k = 1, . . . ,K

fk =(f1(zfk), . . . , f

J(zfk))T

f j(z) ∼ GP(0, K(z, z′)), j = 1, . . . , J

The noise is given by

vk ∼ N (0, Qk) and ek ∼ N (0, Rk).

Gaussian Process Motion Model Clas Veibäck November 7, 2018 6

Dynamic System with an Unknown Influence

x0 ∼ N (x0, P0),

yk = Ckxk + ek, k = 1, . . . ,K

xk = Akxk−1 +Bkfk−1 + vk, k = 1, . . . ,K

fk =(f1(zfk), . . . , f

J(zfk))T

f j(z) ∼ GP(0, K(z, z′)), j = 1, . . . , J

The noise is given by

vk ∼ N (0, Qk) and ek ∼ N (0, Rk).

Gaussian Process Motion Model Clas Veibäck November 7, 2018 6

Dynamic System with an Unknown Influence

x0 ∼ N (x0, P0),

yk = Ckxk + ek, k = 1, . . . ,K

xk = Akxk−1 +Bkfk−1 + vk, k = 1, . . . ,K

fk =(f1(zfk), . . . , f

J(zfk))T

F∼ N (0, Kff), F = (fT1 , . . . , fTK)T

The noise is given by

vk ∼ N (0, Qk) and ek ∼ N (0, Rk).

1 Introduction2 Unknown Influence3 Approximation of Influence4 Time-Varying Influence5 Shared Influence6 State-Dependent Influence7 Estimation8 Applications9 Conclusions

Gaussian Process Motion Model Clas Veibäck November 7, 2018 8

Fully Independent Conditional

Inducing points zul , for l = 1, . . . , L with values

ul = (f1(zul ), . . . , fJ(zul ))

T represented by

U = (uT1 , . . . ,u

TL)

T .

whereQab = KauK−1uuKub andΛ = diag(Kff −Qff ).

UsingW = (wT1 , . . . ,w

TL)

T = K−1uuU .

Gaussian Process Motion Model Clas Veibäck November 7, 2018 8

Fully Independent Conditional

Inducing points zul , for l = 1, . . . , L with values

ul = (f1(zul ), . . . , fJ(zul ))

T represented by

U = (uT1 , . . . ,u

TL)

T .

Original GP is(FU

)∼ N

(0,

(Kff Kfu

Kuf Kuu

)⊗ IJ

),

whereQab = KauK−1uuKub andΛ = diag(Kff −Qff ).

UsingW = (wT1 , . . . ,w

TL)

T = K−1uuU .

Gaussian Process Motion Model Clas Veibäck November 7, 2018 8

Fully Independent Conditional

Inducing points zul , for l = 1, . . . , L with values

ul = (f1(zul ), . . . , fJ(zul ))

T represented by

U = (uT1 , . . . ,u

TL)

T .

Fully independent conditional ( ) approximation is(FU

)∼ N

(0,

(Qff +Λ Kfu

Kuf Kuu

)⊗ IJ

),

whereQab = KauK−1uuKub andΛ = diag(Kff −Qff ).

UsingW = (wT1 , . . . ,w

TL)

T = K−1uuU .

Gaussian Process Motion Model Clas Veibäck November 7, 2018 8

Fully Independent Conditional

Inducing points zul , for l = 1, . . . , L with values

ul = (f1(zul ), . . . , fJ(zul ))

T represented by

U = (uT1 , . . . ,u

TL)

T .

Fully independent conditional ( ) approximation is(FW

)∼ N

(0,

(Qff +Λ KfuK

−1uu

K−1uuKuf K−1

uu

)⊗ IJ

),

whereQab = KauK−1uuKub andΛ = diag(Kff −Qff ).

UsingW = (wT1 , . . . ,w

TL)

T = K−1uuU .

Gaussian Process Motion Model Clas Veibäck November 7, 2018 9

Approximation of the Influence Function

Original GP model is

xk = Akxk−1 +Bkfk−1 + vk,

F ∼ N (0, Kff),

where vfk ∼ N (0, Λk) and Λk = [Λ]kk.

Using W0 = K−1uu U0 as the prior.

Gaussian Process Motion Model Clas Veibäck November 7, 2018 9

Approximation of the Influence Function

Sparse GP approximation gives

xk = Akxk−1 +Bkfk−1 + vk,(FW

)∼ N

(0,

(Qff +Λ KfuK

−1uu

K−1uuKuf K−1

uu

)⊗ IJ

)

where vfk ∼ N (0, Λk) and Λk = [Λ]kk.

Using W0 = K−1uu U0 as the prior.

Gaussian Process Motion Model Clas Veibäck November 7, 2018 9

Approximation of the Influence Function

Marginalizing out F results in

xk = Akxk−1 +Bk(Kk−1fu W + vf

k) + vk,

W∼ N (0, K−1uu ),

where vfk ∼ N (0, Λk) and Λk = [Λ]kk.

Using W0 = K−1uu U0 as the prior.

Gaussian Process Motion Model Clas Veibäck November 7, 2018 9

Approximation of the Influence Function

Adding a prior to the GP gives

xk = Akxk−1 +Bk(Kk−1fu W + vf

k) + vk,

W ∼ N (W0, K−1uu ),

where vfk ∼ N (0, Λk) and Λk = [Λ]kk.

Using W0 = K−1uu U0 as the prior.

1 Introduction2 Unknown Influence3 Approximation of Influence4 Time-Varying Influence5 Shared Influence6 State-Dependent Influence7 Estimation8 Applications9 Conclusions

Gaussian Process Motion Model Clas Veibäck November 7, 2018 11

Time-Varying Influence

Original GP model is

xk = Akxk−1 +Bk(Kk−1fu W + vf

k) + vk,

W∼ N (W0, K−1uu ),

Wk= GkWk−1 + vwk

where vwk ∼ N (µk,Σk) and

G = e−αT I,

µk = (1− e−αT )W0,

Σk = (1− e−2αT )K−1uu .

Gaussian Process Motion Model Clas Veibäck November 7, 2018 11

Time-Varying Influence

Time-varying influence is given by adding dynamics to

the function

xk = Akxk−1 +Bk(Kk−1fu Wk−1 + vf

k) + vk,

W0∼ N (W0, K−1uu ),

Wk= GkWk−1 + vwk

where vwk ∼ N (µk,Σk) and

G = e−αT I,

µk = (1− e−αT )W0,

Σk = (1− e−2αT )K−1uu .

Gaussian Process Motion Model Clas Veibäck November 7, 2018 11

Time-Varying Influence

xk = Akxk−1 +Bk(Kk−1fu Wk−1 + vf

k) + vk,

W0∼ N (W0, K−1uu ),

Wk= GkWk−1 + vwk

where vwk ∼ N (µk,Σk)

and

G = e−αT I,

µk = (1− e−αT )W0,

Σk = (1− e−2αT )K−1uu .

Gaussian Process Motion Model Clas Veibäck November 7, 2018 11

Time-Varying Influence

xk = Akxk−1 +Bk(Kk−1fu Wk−1 + vf

k) + vk,

W0∼ N (W0, K−1uu ),

Wk= GkWk−1 + vwk

where vwk ∼ N (µk,Σk) and

G = e−αT I,

µk = (1− e−αT )W0,

Σk = (1− e−2αT )K−1uu .

1 Introduction2 Unknown Influence3 Approximation of Influence4 Time-Varying Influence5 Shared Influence6 State-Dependent Influence7 Estimation8 Applications9 Conclusions

Gaussian Process Motion Model Clas Veibäck November 7, 2018 13

Multiple Systems with Shared Influence

Original GP model is

x0 ∼ N (x0, P0),

yk = Ckxk + ek,

xk = Akxk−1 +Bk(Kk−1fu Wk−1 + vf

k) + vk,

W0 ∼ N (W0, K−1uu ),

Wk = GkWk−1 + vwk

Gaussian Process Motion Model Clas Veibäck November 7, 2018 13

Multiple Systems with Shared Influence

Multiple systems are modelled by the extension

xi0 ∼ N (x0, P0),

yik = Ckx

ik + eik,

xik = Akx

ik−1 +Bk(K

k−1iu Wk−1 + vif

k ) + vik,

W0 ∼ N (W0, K−1uu ),

Wk = GkWk−1 + vwk

Gaussian Process Motion Model Clas Veibäck November 7, 2018 13

Multiple Systems with Shared Influence

Multiple systems are modelled by the extension

xi0 ∼ N (x0, P0),

yik = Ckx

ik + eik,

xik = Akx

ik−1 +Bk(K

k−1iu Wk−1 + vif

k ) + vik,

W0 ∼ N (W0, K−1uu ),

Wk = GkWk−1 + vwk

where zik is the input for process i = 1, . . . , I.

1 Introduction2 Unknown Influence3 Approximation of Influence4 Time-Varying Influence5 Shared Influence6 State-Dependent Influence7 Estimation8 Applications9 Conclusions

Gaussian Process Motion Model Clas Veibäck November 7, 2018 15

State-Dependent Influence

Original GP model is

xi0 ∼ N (x0, P0),

yik = Ckx

ik + eik,

xik = Akx

ik−1 +Bk

(Kk−1

iu Wk−1+

vifk

)+ vi

k,

W0 ∼ N (W0, K−1uu ),

Wk = GkWk−1 + vwk

where zik = Dkxik−1.

Gaussian Process Motion Model Clas Veibäck November 7, 2018 15

State-Dependent Influence

State dependence is modelled by the extension

xi0 ∼ N (x0, P0),

yik = Ckx

ik + eik,

xik = Akx

ik−1 +Bk

(K·u(Dkx

ik−1)Wk−1+

vifk (Dkx

ik−1)

)+ vi

k,

W0 ∼ N (W0, K−1uu ),

Wk = GkWk−1 + vwk

where zik = Dkxik−1.

Gaussian Process Motion Model Clas Veibäck November 7, 2018 15

State-Dependent Influence

Resulting in the Gaussian process motion

model(GPMM)

xi0 ∼ N (x0, P0),

yik = Ckx

ik + eik,

xik = Akx

ik−1 +Bk

(K·u(Dkx

ik−1)Wk−1+

vifk (Dkx

ik−1)

)+ vi

k,

W0 ∼ N (W0, K−1uu ),

Wk = GkWk−1 + vwk

where zik = Dkxik−1.

1 Introduction2 Unknown Influence3 Approximation of Influence4 Time-Varying Influence5 Shared Influence6 State-Dependent Influence7 Estimation8 Applications9 Conclusions

Gaussian Process Motion Model Clas Veibäck November 7, 2018 17

Extended Kalman Filter

• Non-linear dependence on state in kernel

• Almost linear-Gaussian

• Extended Kalman filter(EKF)

• Noisy input compensated for implicitly by filter

• Enormous state space so approximations are

needed

Gaussian Process Motion Model Clas Veibäck November 7, 2018 17

Extended Kalman Filter

• Non-linear dependence on state in kernel

• Almost linear-Gaussian

• Extended Kalman filter(EKF)

• Noisy input compensated for implicitly by filter

• Enormous state space so approximations are

needed

Gaussian Process Motion Model Clas Veibäck November 7, 2018 17

Extended Kalman Filter

• Non-linear dependence on state in kernel

• Almost linear-Gaussian

• Extended Kalman filter(EKF)

• Noisy input compensated for implicitly by filter

• Enormous state space so approximations are

needed

Gaussian Process Motion Model Clas Veibäck November 7, 2018 17

Extended Kalman Filter

• Non-linear dependence on state in kernel

• Almost linear-Gaussian

• Extended Kalman filter(EKF)

• Noisy input compensated for implicitly by filter

• Enormous state space so approximations are

needed

Gaussian Process Motion Model Clas Veibäck November 7, 2018 17

Extended Kalman Filter

• Non-linear dependence on state in kernel

• Almost linear-Gaussian

• Extended Kalman filter(EKF)

• Noisy input compensated for implicitly by filter

• Enormous state space so approximations are

needed

1 Introduction2 Unknown Influence3 Approximation of Influence4 Time-Varying Influence5 Shared Influence6 State-Dependent Influence7 Estimation8 Applications9 Conclusions

Gaussian Process Motion Model Clas Veibäck November 7, 2018 19

Velocity Field Simulation

0 200 400 600 800

Horizontal Position (m)

200

400

600

800

1000

1200

Vert

ical P

ositio

n (

m) • 200 targets

• 150 time steps

• Velocity given by function

• Targets are identified

• Velocity field estimated well

Gaussian Process Motion Model Clas Veibäck November 7, 2018 19

Velocity Field Simulation

0 200 400 600 800

Horizontal Position (m)

200

400

600

800

1000

1200

Vert

ical P

ositio

n (

m) • 200 targets

• 150 time steps

• Velocity given by function

• Targets are identified

• Velocity field estimated well

Gaussian Process Motion Model Clas Veibäck November 7, 2018 19

Velocity Field Simulation

0 200 400 600 800

Horizontal Position (m)

200

400

600

800

1000

1200

Vert

ical P

ositio

n (

m) • 200 targets

• 150 time steps

• Velocity given by function

• Targets are identified

• Velocity field estimated well

Gaussian Process Motion Model Clas Veibäck November 7, 2018 19

Velocity Field Simulation

0 200 400 600 800

Horizontal Position (m)

200

400

600

800

1000

1200

Vert

ical P

ositio

n (

m) • 200 targets

• 150 time steps

• Velocity given by function

• Targets are identified

• Velocity field estimated well

Gaussian Process Motion Model Clas Veibäck November 7, 2018 19

Velocity Field Simulation

0 200 400 600 800

Horizontal Position [m]

200

400

600

800

1000

1200

Vert

ical P

ositio

n [m

]

1

2

3

4

5

6

7

8

9

• 200 targets

• 150 time steps

• Velocity given by function

• Targets are identified

• Velocity field estimated well

Gaussian Process Motion Model Clas Veibäck November 7, 2018 20

Sea Ice Tracking

• Radar station detecting ice

• Measurements for long

tracks are extracted from

simple tracker

• Acceleration caused by

currents are modelled by GP

• Prediction errors are

reduced

Gaussian Process Motion Model Clas Veibäck November 7, 2018 20

Sea Ice Tracking

• Radar station detecting ice

• Measurements for long

tracks are extracted from

simple tracker

• Acceleration caused by

currents are modelled by GP

• Prediction errors are

reduced

Gaussian Process Motion Model Clas Veibäck November 7, 2018 20

Sea Ice Tracking

• Radar station detecting ice

• Measurements for long

tracks are extracted from

simple tracker

• Acceleration caused by

currents are modelled by GP

• Prediction errors are

reduced

Gaussian Process Motion Model Clas Veibäck November 7, 2018 20

Sea Ice Tracking

• Radar station detecting ice

• Measurements for long

tracks are extracted from

simple tracker

• Acceleration caused by

currents are modelled by GP

• Prediction errors are

reduced

1 Introduction2 Unknown Influence3 Approximation of Influence4 Time-Varying Influence5 Shared Influence6 State-Dependent Influence7 Estimation8 Applications9 Conclusions

Gaussian Process Motion Model Clas Veibäck November 7, 2018 22

Conclusions

Theory is presented on

• a Gaussian process motion model

• a number of variations of the model

• tractable estimation for the model

Demonstration through two applications

Gaussian Process Motion Model Clas Veibäck November 7, 2018 22

Conclusions

Theory is presented on

• a Gaussian process motion model

• a number of variations of the model

• tractable estimation for the model

Demonstration through two applications

Gaussian Process Motion Model Clas Veibäck November 7, 2018 22

Conclusions

Theory is presented on

• a Gaussian process motion model

• a number of variations of the model

• tractable estimation for the model

Demonstration through two applications

Gaussian Process Motion Model Clas Veibäck November 7, 2018 22

Conclusions

Theory is presented on

• a Gaussian process motion model

• a number of variations of the model

• tractable estimation for the model

Demonstration through two applications

Gaussian Process Motion Model Clas Veibäck November 7, 2018 23

Future Work

Possible future work is

• splitting up the input space

• analysis of the impact of approximations

• marginalized particle filter implementation

Gaussian Process Motion Model Clas Veibäck November 7, 2018 23

Future Work

Possible future work is

• splitting up the input space

• analysis of the impact of approximations

• marginalized particle filter implementation

Gaussian Process Motion Model Clas Veibäck November 7, 2018 23

Future Work

Possible future work is

• splitting up the input space

• analysis of the impact of approximations

• marginalized particle filter implementation

Thank you for listening!www.liu.se