[PPT] - Time Series Bhiksha Raj Class 22. 14 Nov 2013 14 Nov 2013 PowerPoint Presentation

SLIDE 1

Machine Learning for Signal Processing

Predicting and Estimation from Time Series

Bhiksha Raj Class 22. 14 Nov 2013

14 Nov 2013 11-755/18797 1

SLIDE 2

Administrivia

No class on Tuesday..
Project Demos: 5th December (Thursday).

– Before exams week

14 Nov 2013 11-755/18797 2

SLIDE 3

An automotive example

Determine automatically, by only listening to a running

automobile, if it is:

– Idling; or – Travelling at constant velocity; or – Accelerating; or – Decelerating

Assume (for illustration) that we only record energy level

(SPL) in the sound

– The SPL is measured once per second

14 Nov 2013 11-755/18797 3

SLIDE 4

What we know

An automobile that is at rest can accelerate, or

continue to stay at rest

An accelerating automobile can hit a steady-

state velocity, continue to accelerate, or decelerate

A decelerating automobile can continue to

decelerate, come to rest, cruise, or accelerate

A automobile at a steady-state velocity can

stay in steady state, accelerate or decelerate

14 Nov 2013 11-755/18797 4

SLIDE 5

What else we know

The probability distribution of the SPL of the

sound is different in the various conditions

– As shown in figure

In reality, depends on the car
The distributions for the different conditions
verlap

– Simply knowing the current sound level is not enough to know the state of the car

14 Nov 2013 11-755/18797 5

45 70 65 60 P(x|idle) P(x|decel) P(x|cruise) P(x|accel)

SLIDE 6

The Model!

The state-space model

– Assuming all transitions from a state are equally probable

14 Nov 2013 11-755/18797 6

45 P(x|idle) Idling state 70 P(x|accel) Accelerating state 65 Cruising state 60 Decelerating state 0.5 0.5 0.33 0.33 0.33 0.33 0.33 0.25 0.25 0.25 0.33 0.25 I A C D I 0.5 0.5 A 1/3 1/3 1/3 C 1/3 1/3 1/3 D 0.25 0.25 0.25 0.25

SLIDE 7

Estimating the state at T = 0-

A T=0, before the first observation, we know

nothing of the state

– Assume all states are equally likely

14 Nov 2013 11-755/18797 7

Idling Accelerating Cruising Decelerating

0.25 0.25 0.25 0.25

SLIDE 8

The first observation

At T=0 we observe the sound level x0 = 67dB SPL

– The observation modifies our belief in the state of the system

P(x0|idle) = 0
P(x0|deceleration) = 0.0001
P(x0|acceleration) = 0.7
P(x0|cruising) = 0.5

– Note, these don’t have to sum to 1 – In fact, since these are densities, any of them can be > 1

14 Nov 2013 11-755/18797 8

45 70 65 60 P(x|idle) P(x|decel) P(x|cruise) P(x|accel)

SLIDE 9

Estimating state after at observing x0

P(state | x0) = C P(state)P(x0|state)

– P(idle | x0) = 0 – P(deceleration | x0) = C 0.000025 – P(cruising | x0) = C 0.125 – P(acceleration | x0) = C 0.175

Normalizing

– P(idle | x0) = 0 – P(deceleration | x0) = 0.000083 – P(cruising | x0) = 0.42 – P(acceleration | x0) = 0.57

14 Nov 2013 11-755/18797 9

SLIDE 10

Estimating the state at T = 0+

At T=0, after the first observation, we must

update our belief about the states

– The first observation provided some evidence about the state of the system – It modifies our belief in the state of the system

14 Nov 2013 11-755/18797 10

Idling Accelerating Cruising Decelerating

0.0 0.57 0.42 8.3 x 10-5

SLIDE 11

Predicting the state at T=1

Predicting the probability of idling at T=1

In general, for any state S

– P(ST=1 | x0) = SST=0 P(ST=0 | x0) P(ST=1|ST=0)

14 Nov 2013 11-755/18797 11

I A C D I 0.5 0.5 A 1/3 1/3 1/3 C 1/3 1/3 1/3 D 0.25 0.25 0.25 0.25 I A C D

Idling Accelerating Cruising Decelerating 0.0 0.57 0.42 8.3 x 10-5

SLIDE 12

Predicting the state at T = 1

14 Nov 2013 11-755/18797 12

Idling Accelerating Cruising Decelerating

0.0 0.57 0.42 8.3 x 10-5 2.1x10-5 0.33 0.33 0.33 P(ST=1 | x0) = SST=0 P(ST=0 | x0) P(ST=1|ST=0)

SLIDE 13

Updating after the observation at T=1

At T=1 we observe x1 = 63dB SPL
P(x1|idle) = 0
P(x1|deceleration) = 0.2
P(x1|acceleration) = 0.001
P(x1|cruising) = 0.5

14 Nov 2013 11-755/18797 13

45 70 65 60 P(x|idle) P(x|decel) P(x|cruise) P(x|accel)

SLIDE 14

Update after observing x1

P(state | x0:1) = C P(state| x0)P(x1|state)

– P(idle | x0:1) = 0 – P(deceleration | x0,1) = C 0.066 – P(cruising | x0:1) = C 0.165 – P(acceleration | x0:1) = C 0.00033

Normalizing

– P(idle | x0:1) = 0 – P(deceleration | x0:1) = 0.285 – P(cruising | x0:1) = 0.713 – P(acceleration | x0:1) = 0. 0014

14 Nov 2013 11-755/18797 14

SLIDE 15

Estimating the state at T = 1+

The updated probability at T=1 incorporates

information from both x0 and x1

– It is NOT a local decision based on x1 alone – Because of the Markov nature of the process, the state at T=0 affects the state at T=1

x0 provides evidence for the state at T=1

14 Nov 2013 11-755/18797 15

Idling Accelerating Cruising Decelerating

0.0 0.713 0.0014 0.285

SLIDE 16

Estimating a Unique state

What we have estimated is a distribution over

the states

If we had to guess a state, we would pick the

most likely state from the distributions

State(T=0) = Accelerating
State(T=1) = Cruising

14 Nov 2013 11-755/18797 16

Idling Accelerating Cruising Decelerating 0.0 0.713 0.0014 0.285 Idling Accelerating Cruising Decelerating 0.0 0.57 0.42 8.3 x 10-5

SLIDE 17

Overall procedure

At T=0 the predicted state distribution is the initial state

probability

At each time T, the current estimate of the distribution over

states considers all observations x0 ... xT

– A natural outcome of the Markov nature of the model

The prediction+update is identical to the forward computation

for HMMs to within a normalizing constant

14 Nov 2013 11-755/18797 17

Predict the distribution of the state at T Update the distribution of the state at T after observing xT T=T+1

PREDICT UPDATE

SLIDE 18

Comparison to Forward Algorithm

Forward Algorithm:

– P(x0:T,ST) = P(xT|ST) SST-1 P(x0:T-1, ST-1) P(ST|ST-1)

Normalized:

– P(ST|x0:T) = (SS’T P(x0:T,S’T))-1 P(x0:T,ST) = C P(x0:T,ST)

14 Nov 2013 11-755/18797 18

Predict the distribution of the state at T Update the distribution of the state at T after observing xT T=T+1 PREDICT UPDATE PREDICT UPDATE

SLIDE 19

Decomposing the forward algorithm

 P(x0:T,ST) = P(xT|ST) SST-1 P(x0:T-1, ST-1) P(ST|ST-1)

Predict:

 P(x0:T-1,ST) = SST-1 P(x0:T-1, ST-1) P(ST|ST-1)

Update:

 P(x0:T,ST) = P(xT|ST) P(x0:T-1,ST)

14 Nov 2013 11-755/18797 19

SLIDE 20

Estimating the state

The state is estimated from the updated

distribution

– The updated distribution is propagated into time, not the state

14 Nov 2013 11-755/18797 20

Estimate(ST) Predict the distribution of the state at T Update the distribution of the state at T after observing xT T=T+1

SLIDE 21

Predicting the next observation

The probability distribution for the observations at the

next time is a mixture:

– P(xT|x0:T-1) = SST P(xT|ST) P(ST|x0:T-1)

The actual observation can be predicted from P(xT|x0:T-1)

14 Nov 2013 11-755/18797 21

Predict P(xT|x0:T-1) Predict the distribution of the state at T Update the distribution of the state at T after observing xT T=T+1 Predict xT

SLIDE 22

Predicting the next observation

MAP estimate:

– argmaxxT P(xT|x0:T-1)

MMSE estimate:

– Expectation(xT|x0:T-1)

14 Nov 2013 11-755/18797 22

SLIDE 23

Difference from Viterbi decoding

Estimating only the current state at any time

– Not the state sequence – Although we are considering all past observations

The most likely state at T and T+1 may be such

that there is no valid transition between ST and ST+1

14 Nov 2013 11-755/18797 23

SLIDE 24

A known state model

HMM assumes a very coarsely quantized state

space

– Idling / accelerating / cruising / decelerating

Actual state can be finer

– Idling, accelerating at various rates, decelerating at various rates, cruising at various speeds

Solution: Many more states (one for each

acceleration /deceleration rate, crusing speed)?

Solution: A continuous valued state

14 Nov 2013 11-755/18797 24

SLIDE 25

The real-valued state model

A state equation describing the dynamics of the system

– st is the state of the system at time t – et is a driving function, which is assumed to be random

The state of the system at any time depends only on the state at

the previous time instant and the driving term at the current time

An observation equation relating state to observation

– ot is the observation at time t – gt is the noise affecting the observation (also random)

The observation at any time depends only on the current state of

the system and the noise

14 Nov 2013 11-755/18797 25

) , (

1 t t t

s f s e





) , (

t t t

s g

g



SLIDE 26

Continuous state system

The state is a continuous valued parameter that is not directly

seen

– The state is the position of navlab or the star

The observations are dependent on the state and are the only

way of knowing about the state

– Sensor readings (for navlab) or recorded image (for the telescope)

) , (

t t t

s g

g

 ) , (

1 t t t

s f s e





SLIDE 27

Statistical Prediction and Estimation

Given an a priori probability distribution for

the state

– P0(s): Our belief in the state of the system before we observe any data

Probability of state of navlab
Probability of state of stars
Given a sequence of observations o0..ot
Estimate state at time t

14 Nov 2013 11-755/18797 27

SLIDE 28

Prediction and update at t = 0

Prediction

– Initial probability distribution for state – P(s0) = P0(s0)

Update:

– Then we observe o0 – We must update our belief in the state

P(s0|o0) = C.P0(s0)P(o0|s0)

14 Nov 2013 11-755/18797 28

) ( ) | ( ) ( ) ( ) | ( ) ( ) | (

P

s

P

s P

P

s

P

s P

s

P  

SLIDE 29

The observation probability: P(o|s)

– This is a (possibly many-to-one) stochastic function
f state st and noise gt

– Noise gt is random. Assume it is the same dimensionality as ot

Let Pg(gt) be the probability distribution of gt
Let {g:g(st, g)=ot} be all g that result in ot

14 Nov 2013 11-755/18797 29

) , (

t t t

s g

g







t t t

s

g t s g t t

J

P s

P

) , ( : ) , (

| ) ( | ) ( ) | (

g g g g g

SLIDE 30

The observation probability

P(o|s) = ?
The J is a jacobian
For scalar functions of scalar variables, it is simply a

derivative:

14 Nov 2013 11-755/18797 30

) , (

t t t

s g

g







t t t

s

g t s g t t

J

P s

P

) , ( : ) , (

| ) ( | ) ( ) | (

g g g g g ) ( ) ( ) 1 ( ) ( ) ( ) 1 ( ... ) 1 ( ) 1 ( | ) ( |

) , (

n n

n
n
J

t t t t t s g

t

g g g g

g

             g

g

  

t t s g

J

t

| ) ( |

) , (

SLIDE 31

Predicting the next state

Given P(s0|o0), what is the probability of the state

at t=1

State progression function:

– et is a driving term with probability distribution Pe(et)

P(st|st-1) can be computed similarly to P(o|s)

– P(s1|s0) is an instance of this

14 Nov 2013 11-755/18797 31

 

 

} { 1 } { 1 1

) | ( ) | ( ) | , ( ) | (

s s

ds

s

P s s P ds

s

s P

s

P ) , (

1 t t t

s f s e





SLIDE 32

And moving on

P(s1|o0) is the predicted state distribution for

t=1

Then we observe o1

– We must update the probability distribution for s1 – P(s1|o0:1) = CP(s1|o0)P(o1|s1)

We can continue on

14 Nov 2013 11-755/18797 32

SLIDE 33

P(s1 | O0,O1)  C P(s1 | O0) P(O1|s1) P(s1| O0 ,O1)  C P(s1| O0) P(O1| s1) Update after O1:

Discrete vs. Continuous state systems

P(s0)  P(s) P(s0 | O0)  C p (s0)P(O0| s0) Update after O0: Prediction at time 1:



 ) | ( ) O | ( ) O | (

1 1 s

s s P s P s P

1 1

) | ( ) O | ( ) O | ( ds s s P s P s P



  

 P(s0)  p (s0) P(s0| O0)  C P(s0) P(O0| s0) Prediction at time 0: P(s) s

p 

0.2 0.3 0.4 0.1 1 2 3 1 2 3

) , (

t t t

s g

g

 ) , (

1 t t t

s f s e





SLIDE 34

Update after Ot:

Discrete vs. Continuous State Systems

Prediction at time t:





 



1

) | ( ) O | ( ) O | (

1 1

t

: 1 1

t

:

t

s t t t t

s s P s P s P

1 1 1

t

: 1 1

t

:

) | ( ) O | ( ) O | (

     



t t t t t

ds s s P s P s P

) | O ( ) O | ( ) O | (

1

t

: t : t t t t

s P s CP s P 

) , (

t t t

s g

g

 ) , (

1 t t t

s f s e





) | O ( ) O | ( ) O | (

1

t

: t : t t t t

s P s CP s P 

1 2 3

SLIDE 35

Initial state prob.

Discrete vs. Continuous State Systems

Parameters

p

) (s P

) | O ( s P

) | (

1  t t s

s P

) , (

t t t

s g

g

 ) , (

1 t t t

s f s e





) | ( } {

1

i s j s P T

t t ij

  



) | ( s

P

Transition prob Observation prob

1 2 3

SLIDE 36

Special case: Linear Gaussian model

A linear state dynamics equation

– Probability of state driving term e is Gaussian – Sometimes viewed as a driving term me and additive zero-mean noise

A linear observation equation

– Probability of observation noise g is Gaussian

At, Bt and Gaussian parameters assumed known

– May vary with time

14 Nov 2013 11-755/18797 36

t t t t

s B

g

 

t t t t

s A s e  

1

   

 

e e e e

m e m e p e      

1

5 . exp | | ) 2 ( 1 ) (

T d

P

   

 

g g g g

m g m g p g      

1

5 . exp | | ) 2 ( 1 ) (

T d

P

SLIDE 37

The initial state probability

We also assume the initial state distribution to

be Gaussian

– Often assumed zero mean

14 Nov 2013 11-755/18797 37

   

 

T d

s s R s s R s P    

1

5 . exp | | ) 2 ( 1 ) ( p ) , ; ( ) ( R s s Gaussian s P 

t t t t

s A s e  

1 t t t t

s B

g

 

SLIDE 38

The observation probability

The probability of the observation, given the state, is

simply the probability of the noise, with the mean shifted

– Since the only uncertainty is from the noise

The new mean is the mean of the distribution of the

noise + the value of the observation in the absence of noise

14 Nov 2013 11-755/18797 38

t t t t

s B

g

 

) , ; ( ) (

g g

m g g   Gaussian P

) , ; ( ) | (

g g

m   

t t t t t

s B

Gaussian

s

P

SLIDE 39

The updated state probability at T=0

P(s0| o0) = C P(s0) P(o0| s0)

14 Nov 2013 11-755/18797 39

) , ; ( ) ( R s s Gaussian s P 

) , ; ( ) | (

g g

m    s B

Gaussian

s

P

) , ; ( ) , ; ( ) | (

g g

m    s B

Gaussian

R s s CGaussian

s

P

t t t t

s B

g

 

) , ; ( ) (

g g

m g g   N P

SLIDE 40

Note 1: product of two Gaussians

The product of two Gaussians is a Gaussian

14 Nov 2013 11-755/18797 40

) , ; ( ) , ; (   Bs

Gaussian

R s s Gaussian m

   

) ( ) ( 5 . exp ) ( ) ( 5 . exp

1 2 1 1

Bs

Bs
C

s s R s s C

T T

        

 

m m

     

 

1 1 1 1 1 1 1 1

, ) ( ; .

       

       B B R

B

s R B B R s Gaussian C

T T T

m

Not a good estimate --

SLIDE 41

The updated state probability at T=0

P(s0| o0) = C P(s0) P(o0| s0)

14 Nov 2013 11-755/18797 41

) , ; ( ) ( R s s Gaussian s P 

) , ; ( ) | (

g g

m    s B

Gaussian

s

P

 ) | (

0 o

s P

     

 

1 1 1 1 1 1 1 1

, ) ( ;

       

       B B R

B

s R B B R s Gaussian

T T T g g g g

m

 

ˆ , ˆ ; ) | ( R s s Gaussian

s

P 

SLIDE 42

The state transition probability

The probability of the state at time t, given the

state at time t-1 is simply the probability of the driving term, with the mean shifted

14 Nov 2013 11-755/18797 42

) , ; ( ) (

e e

m e e   Gaussian P

) , ; ( ) | (

1 1 e e

m   

  t t t t t

s A s Gaussian s s P

t t t t

s A s e  

1

SLIDE 43

Note 2: integral of product of two Gaussians

The integral of the product of two Gaussians is a Gaussian

14 Nov 2013 11-755/18797 43

   

 

       

              dx b Ax y b Ax y C x x C dx b Ax y Gaussian x Gaussian

y T x x T x y x x

) ( ) ( 5 . exp ) ( ) ( 5 . exp ) , ; ( ) , ; (

1 2 1 1

m m m

 

T x y x

A A b A y Gaussian      , ; m

SLIDE 44

Note 2: integral of product of two Gaussians

P(y) is the integral of the product of two Gaussians is a

Gaussian

14 Nov 2013 11-755/18797 44

 

     

     dx b Ax y Gaussian x Gaussian dx x y P y P

y x x

) , ; ( ) , ; ( ) , ( ) ( m

 

T x y x

A A b A y Gaussian      , ; m

e Ax y   ) , ( ~

x x

N x  m ) , ( ~

g

 b N e

 

T x y x

A A b A N y P      , ) ( m

SLIDE 45

The predicted state probability at t=1

Remains Gaussian

14 Nov 2013 11-755/18797 45

 

     

 

1 1 1

) | ( ) | s ( ) | s , ( )

|

( ds s s P

P

ds

s

P s P

) , ; ( ) | (

1 1 1 e e

m    s A s Gaussian s s P

 

ˆ , ˆ ; ) | ( R s s Gaussian

s

P 

 



  

  

1 1 1

) , ; ( ˆ , ˆ ; )

|

( ds s A s Gaussian R s s Gaussian s P

e e

m

 

T

A R A s A s Gaussian

s

P

1 1 1 1 1

ˆ , ˆ ; ) | (    

e e

m

t t t t

s A s e  

1

SLIDE 46

The updated state probability at T=1

P(s1| o0:1) = C P(s1 |o0) P(o1| s1)

14 Nov 2013 11-755/18797 46

) , ; ( ) | (

1 1 1 1 1 g g

m    s B

Gaussian

s

P

 

1 1 1 1 : 1

ˆ , ˆ ; ) | ( R s s Gaussian

s

P 

 

T

A R A s A s Gaussian

s

P

1 1 1 1 1

ˆ , ˆ ; ) | (    

e e

m

SLIDE 47

The Kalman Filter!

Prediction at T
Update at T

14 Nov 2013 11-755/18797 47

     

 

1 1 1 1 1 1 1 1

, ) ( ;

       

      

t T t t t T t t t t T t t t

B B R

B

s R B B R s Gaussian

g g g g

m

 

t t t t t

R s s Gaussian

s

P ˆ , ˆ ; ) | (

:



 ) | (

: 0 t t o

s P

 

T t t t t t t t t

A R A s A s Gaussian

s

P

1 1 1 :

ˆ , ˆ ; ) | (

  

   

e e

m

 

t t t t t

R s s Gaussian

s

P , ; ) | (

1 :





t t t t

s A s e  

1 t t t t

s B

g

 

SLIDE 48

Linear Gaussian Model

P(s0| O0)  C P(s0) P(O0| s0)

1 1

) | ( ) O | ( ) O | ( ds s s P s P s P



  

 P(s1| O0:1)  C P(s1| O0) P(O1| s0)

1 1 2 1 : 1 1 : 2

) | ( ) O | ( ) O | ( ds s s P s P s P



  



a priori Transition prob. State output prob

t t t t

s B

g

 

t t t t

s A s e  

1

SLIDE 49

The Kalman filter

The actual state estimate is the mean of the

updated distribution

Predicted state at time t
Updated estimate of state at time t

14 Nov 2013 11-755/18797 49

e

m   

  1 1 :

ˆ )] | ( [

t t t t t

s A

s

P mean s

   

) ( )] | ( [ ˆ

1 1 1 1 1 : g g g

m       

     t T t t t t T t t t t t

B

s R B B R

s

P mean s

SLIDE 50

Stable Estimation

The above equation fails if there is no
bservation noise

– g = 0 – Paradoxical? – Happens because we do not use the relationship between o and s effectively

Alternate derivation required

– Conventional Kalman filter formulation

14 Nov 2013 11-755/18797 50

   

) ( )] | ( [ ˆ

1 1 1 1 1 : g g g

m       

     t T t t t t T t t t t t

B

s R B B R

s

P mean s

SLIDE 51

Conditional Probability of y|x

The conditional probability of y given x is also Gaussian

– The slice in the figure is Gaussian

The mean of this Gaussian is a function of x
The variance of y reduces if x is known

– Uncertainty is reduced

12 Nov 2013 11755/18797 51

) ), ( ( ) | (

1 1 xy xx T yx yy x xx yx y

C C C C x C C N x y P

 

    m m

If P(x,y) is Gaussian:

) , ( ) , , (

, , , , , ,

            

yy yx xy xx y x

x y

k k k k k k

C C C C N k P m m

SLIDE 52

A matrix inverse identity

– Work it out..

14 Nov 2013 11-755/18797 52

       

                     

              1 1 1 1 1 1 1 1 1 1 1 1 1 1

B A B C A B B A B C B A B C B A A B B A B C B A A C B B A

T T T T T T T

SLIDE 53

For any jointly Gaussian RV

Using the Matrix Inversion Identity

14 Nov 2013 11-755/18797 53

       Y X Z       

Y X Z

m m m       

YY T XY XY XX Z

C C C C C        

               

              1 1 1 1 1 1 1 1 1 1 1 1 1 1 XY XX T XY YY XX T XY XY XX T XY YY XY XX T XY XY XX XX T XY XY XX T XY YY XY XX XX Z

C C C C C C C C C C C C C C C C C C C C C C C C C C

SLIDE 54

For any jointly Gaussian RV

Using the Matrix Inversion Identity

14 Nov 2013 11-755/18797 54

       Y X Z       

Y X Z

m m m       

YY T XY XY XX Z

C C C C C

   

 

 Z Z T Z

Z C Z m m

1

 ) (X Quadratic

     

) ( ) (

1 1 1 1 X XX YX Y XY XX T XY YY T X XX YX Y

X C C Y C C C C X C C Y m m m m       

   

       

               

              1 1 1 1 1 1 1 1 1 1 1 1 1 1 XY XX T XY YY XX T XY XY XX T XY YY XY XX T XY XY XX XX T XY XY XX T XY YY XY XX XX Z

C C C C C C C C C C C C C C C C C C C C C C C C C C

SLIDE 55

For any jointly Gaussian RV

The conditional of Y is a Gaussian

14 Nov 2013 11-755/18797 55

   ) ( 5 . exp( X Quadratic const

     )

) ( ) ( 5 .

1 1 1 1 X XX YX Y XY XX T XY YY T X XX YX Y

X C C Y C C C C X C C Y m m m m        

   

   

 

   

 Z Z T Z

Z C Z Const Y X P m m

1

5 . exp ) , (

     

 

) ( ) ( 5 . exp

1 1 1 1 X XX YX Y XY XX T XY YY T X XX YX Y

X C C Y C C C C X C C Y K m m m m        

   

 ) | ( X Y P

   

XY XX T XY YY X XX YX Y

C C C C X C C Y Gaussian

1 1

), ( ;

 

    m m

SLIDE 56

Conditional Probability of y|x

The conditional probability of y given x is also Gaussian

– The slice in the figure is Gaussian

The mean of this Gaussian is a function of x
The variance of y reduces if x is known

– Uncertainty is reduced

12 Nov 2013 11755/18797 56

) ), ( ( ) | (

1 1 xy xx T yx yy x xx yx y

C C C C x C C N x y P

 

    m m

If P(x,y) is Gaussian:

) , ( ) , , (

, , , , , ,

            

yy yx xy xx y x

x y

k k k k k k

C C C C N k P m m

SLIDE 57

Estimating P(s|o)

Consider the joint distribution of o and s

14 Nov 2013 11-755/18797 57

) , ; ( ) | (

1 :

R s s Gaussian

s

P

t





g   Bs





e e p g

g g 1

5 . exp | | ) 2 ( 1 ) (



   

T d

P Assuming g is 0 mean Dropping subscript t and o0:t-1 for brevity

       s

O

 O is a linear function of s

 Hence O is also Gaussian

) , ; ( ) (

O O

O Gaussian O P   m

SLIDE 58

The joint PDF of o and s

o is Gaussian. Its cross covariance with s:

14 Nov 2013 11-755/18797 58

g   Bs

)

, ; ( ) | (

1 :

R s s Gaussian

s

P

t





) , ( ) (

g

g   Gaussian P

s B



m

g

  

T

BRB

C ,

) , ( ) | (

1 : g

  

 T t

BRB s B Gaussian

P

BR C

s



,

SLIDE 59

The probability distribution of O

14 Nov 2013 11-755/18797 59

       s

O

g   Bs



                     s s B s E

E

s

E

O E

O

] [ ] [ ] [ ] [ m ) , ; ( ) ( R s s Gaussian s P 

) , ; ( ) (

g

g g   Gaussian P

) , ; ( ) (

O O

O Gaussian O P   m        s s B

O

m

SLIDE 60

The probability distribution of O

14 Nov 2013 11-755/18797 60

) , ; ( ) (

O O

O Gaussian O P   m

       s s B

O

m

g   Bs

)

, ; ( ) ( R s s Gaussian s P 

) , ; ( ) (

g

g g   Gaussian P

       

s s

s

s

O

C C C C

, , , ,

          R RB BR BRB

T T O g

g

  

T

BRB

C , BR C

s



,

       s s B

O

m

SLIDE 61

The probability distribution of O

14 Nov 2013 11-755/18797 61

) , ; ( ) (

O O

O Gaussian O P   m        s s B

O

m

g   Bs

)

, ; ( ) ( R s s Gaussian s P 

) , ; ( ) (

g

g g   Gaussian P

          R RB BR BRB

T T O g

       s

O

SLIDE 62

The probability distribution of O

Writing it out in extended form

14 Nov 2013 11-755/18797 62

) , ; ( ) | , ( ) | (

1 : 1 : O O t t

O Gaussian

s
P
O

P   

 

m

   

 

                          



s s s B

R

RB BR BRB s s s B

C

T T T 1

5 . exp

g

SLIDE 63

Recall: For any jointly Gaussian RV

Applying it to our problem (replace Y by s, X by o):

14 Nov 2013 11-755/18797 63

   

XY XX T XY YY X XX YX Y

C C C C X C C Y Gaussian X Y P

1 1

), ( ; ) | (

 

    m m

g

  

T

BRB

C , BR C

s



,

BRB

RB s B BRB RB I

T T T T 1 1

) ( ) ) ( (

 

      

g g

m

s B



m

 

  , ; ) | (

:

m s Gaussian

s

P

t

BR BRB RB R

T T 1

) (



    

g

SLIDE 64

Stable Estimation

14 Nov 2013 11-755/18797 64

 Note that we are not computing g

1 in this

formulation

 

t t

s
s

t

s Gaussian

s

P

: 1 : 1

| | :

, ; ) | (   m

t T T T T

s
BRB

RB s B BRB RB I

t

1 1 |

) ( ) ) ( (

: 1

 

      

g g

m

BR BRB RB R

T T

s

t

1 |

) (

: 1



    

g

SLIDE 65

The Kalman filter

The actual state estimate is the mean of the

updated distribution

Predicted state at time t
Updated estimate of state at time t

14 Nov 2013 11-755/18797 65

e

m    

  1 1 :

ˆ )] | ( [

t t t t pred t t

s A

s

P mean s s

t T t t t T t t t t T t t t T t t t t

B

R B B R s B B R B B R I

s

s

1 1 1 : 1

) ( ) ) ( ( | ˆ

  

      



g g

m

t t t t

s A s e  

1 t t t t

s B

g

 

SLIDE 66

The Kalman filter

Prediction
Update

14 Nov 2013 11-755/18797 66

e

m    

  1 1 :

ˆ )] | ( [

t t t t pred t t

s A

s

P mean s s

t t T t t t T t t t t

R B B R B B R R R

1

) ( ˆ



   

g

T t t t t

A R A R

1

ˆ



  

e

 

t T t t t T t t t t T t t t T t t t

B

R B B R s B B R B B R I s

1 1

ˆ

 

      

g g

SLIDE 67

The Kalman filter

Prediction
Update

14 Nov 2013 11-755/18797 67

e

m  

1

ˆt

t t

s A s

 

t t t t

R B K I R   ˆ

T t t t t

A R A R

1

ˆ



  

e

 

t t t t t t

s B

K

s s    ˆ

 

1 

  

g T t t t T t t t

B R B B R K

t t t t

s A s e  

1 t t t t

s B

g

 

SLIDE 68

The Kalman Filter

Very popular for tracking the state of

processes

– Control systems – Robotic tracking

Simultaneous localization and mapping

– Radars – Even the stock market..

What are the parameters of the process?

14 Nov 2013 11-755/18797 68

SLIDE 69

Kalman filter contd.

Model parameters A and B must be known

– Often the state equation includes an additional driving term: st = Atst-1 + Gtut + et – The parameters of the driving term must be known

The initial state distribution must be known

14 Nov 2013 11-755/18797 69

t t t t

s B

g

 

t t t t

s A s e  

1

SLIDE 70

Defining the parameters

State state must be carefully defined

– E.g. for a robotic vehicle, the state is an extended vector that includes the current velocity and acceleration

S = [X, dX, d2X]
State equation: Must incorporate appropriate

constraints

– If state includes acceleration and velocity, velocity at next time = current velocity + acc. * time step – St = ASt-1 + e

A = [1 t 0.5t2; 0 1 t; 0 0 1]

14 Nov 2013 11-755/18797 70

SLIDE 71

Parameters

Observation equation:

– Critical to have accurate observation equation – Must provide a valid relationship between state and observations

Observations typically high-dimensional

– May have higher or lower dimensionality than state

14 Nov 2013 11-755/18797 71

SLIDE 72

Problems

f() and/or g() may not be nice linear functions

– Conventional Kalman update rules are no longer valid

e and/or g may not be Gaussian

– Gaussian based update rules no longer valid

14 Nov 2013 11-755/18797 72

) , (

t t t

s g

g

 ) , (

1 t t t

s f s e





SLIDE 73

Solutions

f() and/or g() may not be nice linear functions

– Conventional Kalman update rules are no longer valid – Extended Kalman Filter

e and/or g may not be Gaussian

– Gaussian based update rules no longer valid – Particle Filters

14 Nov 2013 11-755/18797 73

) , (

t t t

s g

g

 ) , (

1 t t t

s f s e



