Course Calendar Class DATE Contents

1 Sep. 26 Course information & Course overview

2 Oct. 4 Bayes Estimation

3 〃 11 Classical Bayes Estimation - Kalman Filter -

4 〃 18 Simulation-based Bayesian Methods

5 〃 25 Modern Bayesian Estimation Particle Filter

6 Nov. 1 HMM(Hidden Markov Model)

Nov. 8 No Class

7 〃 15 Supervised Learning

8 〃 29 Bayesian Decision

9 Dec. 6 PCA(Principal Component Analysis)

10 〃 13 ICA(Independent Component Analysis)

11 〃 20 Applications of PCA and ICA

12 〃 27 Clustering, k-means et al.

13 Jan. 17 Other Topics 1 Kernel machine.

14 〃 22(Tue) Other Topics 2

Lecture Plan

Classical Bayesian Estimation Kalman Filter

1. Introduction

What is Kalman filter

Dynamical system representation

2. Discrete-time Gauss-Markov Model

Models, State and covariance transition

3. Bayesian Approach to Kalman Filter

State estimation problem, Deviation

1. Introduction

Kalman filter is an optimal recursive data processing algorithm

It processes available measurement data to estimate the current value

of the variables of interest, by using

1) dynamics of system and measurement models,

2) statistical knowledge of system and measurement noises,

3) available information about initial conditions of state variables

Recursive: Not require all previous data by keeping in storage,

repeatedly process every time a new measurement is acquired

Real-time processing

3

system

measurement device

System error noise : w(t)

Measurements

y(t) Optimal estimate

of state variables

Measurement

error noise : v(t)

unknown state

variables : x(t)

Deterministic

Input : u(t)

(Control)

Figure 1 Role of Kalman filter

Kalman filter

4

ˆ tx

5

1 2 3 4

T

x t x t x t x t x t

Linearized continuous-time state-space model for motion in the z-direction

dispacement angle

( ) z t z t t t

M : Mass of carriage, m : mass of pendulum l : length of pendulum

g : gravitational constant

1 1

23 43232 2 2

3 3

2 4

434 4 4

1

2

3

4

0 1 0 0 0,

0 0 0

0 0 0 1 0 1 1,

0 0 0

1 0 0 0 ( )

x x M m gmga a

ax x bd M Mlu tx xdt

b bax x b M Ml

x

xy t z t

x

x

2）Dynamical state-space representation

3）Measurement equation

１）State space variables （Vector）

Continuous-time state space model Deterministic linear time-invariant state space model is given by

In time-varing systems, all elements of matrices , , are function

(State Dynamic Model)

(Measurement Model)

where : n-vector, : m-vector, : -vector,

A B C

x tAx t Bu t

dt

y t Cx t

x u y l

of .t

6

Discrete-time state space model

1 1 (State Dynamic Model)

(Measurement Model)

where is integers (discrete-time index)

x t Ax t Bu t

y t Cx t

t

2.1 Random signals(noises) are applied to a discrete-time state-space

system with random initial conditions. The input u(t) is deterministic

and the system noise w(t) to be zero-mean, white*1), random Gaussian

First-order Markov Process: State at t depends only on the previous

state at t－1. (State-space representation)

:mean value of 0

1 1 1

where 1 0, 1 and

0 0 0 , 0 0

,

where . :

ww

T

xx

x x

x t Ax t Bu t Ww t

w t R t

x Pr x x P

x t Pr x t x t P t

P t R t Cov x t E x t x t

2. Discrete-time Gauss-Markov Models

*1 white signal :=signal value is not correlated in time, thus this implies signal has equal

power at all frequencies. 7

:

where, 0, vv

y t Cx t v t

v N R t

Measurement Model

The model is shown as the following block diagram.

u t

w t

1x t x t

v t

y tB

A

C

W

1z I

1 : delay operatorz

Figure 3 System diagram 8

9

1

/2 1/2

1

1 1 1, exp

22

is D-dimensional random vector

:

: :

: Determinant of

Linear Transformation

, ,

T

xx xxD

xx

T

D

xx

xx xx

T

yy xx

Pr x x R x x R x xR

x x x

x E x

R Cov x E x x x x

R R

y Ax

Pr y y R A x AR A

Appendix: Multivariable Gaussian Density Distribution

Since the Gauss-Markov model is characterized by a Gaussian

distribution, it is completely specified statistically by its mean and

variance.

1 1x t Ax t Bu t

y t Cx t

Mean

(State Variance)

1 1

(Measurement)

T T

ww

T

yy vv

P t AP t A WR t W

R t CP t C R t

Variance

2.2 Mean and Variance Transitions

10

3. Bayesian Approach to Kalman filter [Problem]

/State-space system model

/Gauss-Markov model of the state transition mechanism.

/Bayesian optimal estimation (MAP solution)

Extracting the unobserved or hidden dynamic state variables x(t)

from noisy measurement data {y(τ): τ=0,1, …, t}.

We see that the dynamic state variable x(t) at time t is obtained

through the transition probability based on the previous state

(Markovian property), and the knowledge of the underlying

conditional probability.

11

In a random process the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is called the Markov property. A stochastic process has the Markov property if the conditional probability distribution of future states of the process depends only upon the present state, not on the sequence of events that preceded it. A process with this property is called a Markov process. Quote from Wikipedia

Bayesian Approach – Prediction + Correction Scheme-

( )

1

1

: : 0 , 1 , , 1 ,

Bayesian Estimate:

ˆ arg

+ approach for computing

0 , 1 , , 1 ,

t

tx t

t

t

t

Data Y y y y t y t

x t Max Pr x t Y

Pr x t Y

y y y t y t

Y

Pr y t x tPr x t Y

Pr y t

Prediction Correction

1

t

t

Pr x t YY

12

Prediction Phase

Correction Phase

13

1

1

(1)

, ( ) ( 1), 1 ? , ?

ˆ 1 ,

t

t

t

vv

ee

Pr y t x t Pr x t YPr x t Y

Pr y t Y

N Cx t R t N x t t P t t

N y t t R t

1

1

From the Gauss-Markov model, we have

,

ˆ 1 , 1

ˆ 1 ,

vv

t

t ee

Pr y t x t C t x t R t

Pr x t Y x t t P t t

Pr y t Y y t t R t

1ˆ ˆPrediction: 1 1 1

ˆPrediction error: 1 1

1 1 1

tx t t E x t Y Ax t t

x t t x t x t t

Ax t t Ww t

1 1 1

1 1 1

T

T

ww

P t t E x t t x t t

AP t t A R t

Using the process model, we have (ignore the input term Bu(t-1))

State prediction error covariance

14

1

1

1

Substituting these into Eq (1)gives

1exp

2

1ˆ ˆexp 1 1 1

2

1ˆ ˆexp 1 1

2

T

vv

T

T

ee

p x t Y t y t Cx t R y t Cx t

x t x t t P t t x t x t t

y t y t t R y t y t t

Rewrite the terms above as follows:

:

ˆ 1 1

ˆ : 1

ˆ 1

measurement noise v t y t Cx t

state estimation error x t t x t x t t

innovation e t y t y t t

y t Cx t t

15

Optimal Bayesian estimator is the processor that maximizes the

posterior above. Taking natural logarithms of both sides of Eq.(2),

1

1 1

1

ln Pr ( ) ln Pr ln Pr

1 11 1 1

2 2

1R

2

t t t

T T

vv

T

ee

ln x t Y y t x t x t Y x t Y

LnK v t R t v t x t t p t t x t t

e t e t

The MAP estimate is obtained by differentiating eq.(3), setting it to

zero, and solving the equation. That is,

1

1

1

1exp

2

1exp 1 1 1

2

1exp (2)

2

T

vv

T

T

ee

p x t Y t v t R v t

x t t P t t x t t

e t R e t

16

(3)

ˆ

1 1

ln Pr 0

ln Pr

1 1 0

MAP

x tx X

x t

T

vv

x t Y

x t Y

C R t y t Cx t P t t x t t

11 1

1 1

11 1

1 1

1

ˆ1 1

then,

ˆˆ 1

ˆ 1 1 (4)

T

vv

T

vv

T

MAP vv

T

vv

C R t C P t t x t

P t t x t t C R t y t

x t t X t C R t C P t t

P t t x t t C R t y t

This representation can be simplified by using a matrix inversion

lemma

1 1

1 1 1 1T T T

Matrix Inversion Lemma

A BD A A B I D A B D A

17

11 1

1 1

1 1

1 1

It is possible to show 1

thus, 1

1

Substituting above into eq. (4),

ˆˆ ˆ1 1

ˆ =

T

vv

T

vv

T

vv

T

map vv

P t t C R t C P t t

P t t P t t C R t C I

P t t P t t I P t t C R t C

x t t X t P t t P t t x t t C R t y t

x t

1 1

1

ˆ1 1

ˆ ˆ = 1 1

ˆ = 1

where

is called .

T T

vv vv

T

vv

t P t t C R t Cx t t P t t C R t y t

x t t K t y t Cx t t

x t t K t e t

K t P t t C R t

　

　

Kalman gain

18

We can derive the following covariance-update relation.

1P t t I K t C P t t

19

ˆ 0 0 , 0 0

ˆ ˆ1 1 1 1

1 1 1 1

ˆ ˆ1 1

1

ww

T

ee vv

x P

x t t Ax t t Bu t

P t t AP t t A R t

e t y t y t t y t Cx t t

R t CP t t C R

Initial Conditions :

Prediction :

Innovation :

Kalman Filter Algorithm

11

ˆ ˆ 1

1

T

ee

t

K t P t t C R t

x t t x t t K t e t

P t t I K t C P t t

Gain :

Update :

1t

t y t

1y t

ˆ 1 1 , 1 1x t t p t t

ˆ 1 ,

1

x t t

p t t

ˆ ,x t t p t t

Measurement

Measurement

Correction process

Correction process

Prediction by Model

Prediction by Model

Figure 4

Understanding Kalman filter operation and its Gain

( :Kalman gain)new old newˆ ˆX X KE K

Prediction: Use state space model

Correction: Use measurement

is small

is large

smallReliable model is small

large

largeReliable measurement is large

small

T

vv

new old

new new

vv

vv

K t P t t C R t

ˆ ˆK X X

ˆK X KE

P t tK

R t

P t tK

R t

1

22

References：

[1] J. Candy, “Model-based Signal Processing”, John Wiley/IEEE Press, 2006

[2] J. Candy, “ Bayesian Signal Processing Classical, Modern, and Particle Filtering Methods”,

John Wiley/IEEE Press, 2009

[3] S. Maybeck, “Stochastic Models, Estimation, and Control”, Vol. 1, Academic Press, 1979,

1999

[4] R. Brown, “Introduction to random signal analysis and Kalman filtering”, John Weily, 1983