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

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?

• 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

Introduction

• What if other behaviours or influences are visible?

Introduction

• 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

Introduction

destinations

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

Introduction

destinations• Velocity or acceleration profiles in racing events

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

Introduction

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

• Currents in the water affecting animals and ships

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

Introduction

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).

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

vk ∼ N (0, Qk) and ek ∼ N (0, Rk).

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

vk ∼ N (0, Qk) and ek ∼ N (0, Rk).

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

vk ∼ N (0, Qk) and ek ∼ N (0, Rk).

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

vk ∼ N (0, Qk) and ek ∼ N (0, Rk).

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

vk ∼ N (0, Qk) and ek ∼ N (0, Rk).

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

vk ∼ N (0, Qk) and ek ∼ N (0, Rk).

Fully Independent Conditional

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

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

T represented by

U = (uT1 , . . . ,u

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

UsingW = (wT1 , . . . ,w

T = K−1uuU .

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

T represented by

U = (uT1 , . . . ,u

Original GP is(FU

)∼ N

(Kff Kfu

Kuf Kuu

)⊗ IJ

UsingW = (wT1 , . . . ,w

T = K−1uuU .

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

T represented by

U = (uT1 , . . . ,u

Fully independent conditional ( ) approximation is(FU

)∼ N

(Qff +Λ Kfu

Kuf Kuu

)⊗ IJ

UsingW = (wT1 , . . . ,w

T = K−1uuU .

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

T represented by

U = (uT1 , . . . ,u

Fully independent conditional ( ) approximation is(FW

)∼ N

(Qff +Λ KfuK

−1uu

K−1uuKuf K−1

)⊗ IJ

UsingW = (wT1 , . . . ,w

T = K−1uuU .

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.

Sparse GP approximation gives

xk = Akxk−1 +Bkfk−1 + vk,(FW

)∼ N

(Qff +Λ KfuK

−1uu

K−1uuKuf K−1

)⊗ IJ

Marginalizing out F results in

xk = Akxk−1 +Bk(Kk−1fu W + vf

k) + vk,

W∼ N (0, K−1uu ),

Adding a prior to the GP gives

k) + vk,

W ∼ N (W0, K−1uu ),

Time-Varying Influence

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 .

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

G = e−αT I,

µk = (1− e−αT )W0,

Σk = (1− e−2αT )K−1uu .

k) + vk,

W0∼ N (W0, K−1uu ),

Wk= GkWk−1 + vwk

where vwk ∼ N (µk,Σk)

G = e−αT I,

µk = (1− e−αT )W0,

Σk = (1− e−2αT )K−1uu .

k) + vk,

W0∼ N (W0, K−1uu ),

Wk= GkWk−1 + vwk

G = e−αT I,

µk = (1− e−αT )W0,

Σk = (1− e−2αT )K−1uu .

Multiple Systems with Shared Influence

x0 ∼ N (x0, P0),

yk = Ckxk + ek,

k) + vk,

W0 ∼ N (W0, K−1uu ),

Wk = GkWk−1 + vwk

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

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.

State-Dependent Influence

xi0 ∼ N (x0, P0),

yik = Ckx

ik + eik,

xik = Akx

ik−1 +Bk

(Kk−1

iu Wk−1+

W0 ∼ N (W0, K−1uu ),

Wk = GkWk−1 + vwk

where zik = Dkxik−1.

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)

W0 ∼ N (W0, K−1uu ),

Wk = GkWk−1 + vwk

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)

W0 ∼ N (W0, K−1uu ),

Wk = GkWk−1 + vwk

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

Velocity Field Simulation

0 200 400 600 800

Horizontal Position (m)

ical P

ositio

m) • 200 targets

• 150 time steps

• Velocity given by function

• Targets are identified

• Velocity field estimated well

0 200 400 600 800

ical P

ositio

m) • 200 targets

• 150 time steps

0 200 400 600 800

ical P

ositio

m) • 200 targets

• 150 time steps

0 200 400 600 800

ical P

ositio

m) • 200 targets

• 150 time steps

0 200 400 600 800

Horizontal Position [m]

ical P

ositio

• 200 targets

• 150 time steps

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

Sea Ice Tracking

simple tracker

reduced

Sea Ice Tracking

simple tracker

reduced

Sea Ice Tracking

simple tracker

reduced

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

Conclusions

Future Work

Possible future work is

• splitting up the input space

• analysis of the impact of approximations

• marginalized particle filter implementation

Future Work

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