Estimation: chapter 6 - UIC Engineeringdevroye/courses/ECE531/lectures/b1c6.pdf · Estimation: chapter 6 Best Linear Unbiased Estimators Natasha Devroye [email protected] devroye

Estimation: chapter 6

Best Linear Unbiased Estimators

Natasha Devroye

[email protected]

http://www.ece.uic.edu/~devroye

Spring 2010

Finding estimators so far

1. CRLB - may give you the MVUE

2. Linear models - MVUE and its statistics explicitly!

3. Rao-Blackwell-Lehmann-Scheffe (RBLS) theorem -

may give you the MVUE if you can find sufficient and

complete statistics

All assume we know p(x;!)

MVUE still may be tough to find

Finding estimators so far

1. CRLB - may give you the MVUE

2. Linear models - MVUE and its statistics explicitly!

3. Rao-Blackwell-Lehmann-Scheffe (RBLS) theorem -

may give you the MVUE if you can find sufficient and

complete statistics

All assume we know p(x;!)

MVUE still may be tough to find

• only need first and second order moments of = fairly practical!

Best Linear Unbiased Estimator

• simplify fining an estimator by constraining the class of estimators under

consideration to the class of linear estimators, i.e.

• The vector a is a vector of constants, whose values we will design to meet

certain criteria.

• Note that there is no reason to believe that a linear estimator will produce

either an efficient estimator (meeting the CRLB), an MVUE, or an estimator

that is optimal in any sense.

We are trading optimality for practicality!!

BLUE vs. MVUE

• when does the BLUE become the MVUE?

BLUE assumptions

How to pick a?

LINEAR MODEL

without Gaussian noise!

BLUE - scalar

Examples

• Gauss-Markov theorem for BLUEs:

BLUE - vector

Examples

Example 6.3: source localization

9

!"#$%#&'$()*+,-./01$!234405$678.4379

Tx @ (xs,ys)

Rx3

(x3,y3)Rx2

(x2,y2)

Rx1

(x1,y1)

: ;

: <;: =; : &;

Hyperbola:

!12 = t2 – t1 = constant

Hyperbola:

!23 = t3 – t2 = constant

Assume that the ith Rx can measure its TOA: ti

Then… from the set of TOAs… compute TDOAs

Then… from the set of TDOAs… estimate location (xs,ys)

We won’t worry about

“how” they do that.

Also… there are TDOA

systems that never

actually estimate TOAs!

Taken from http://www.ws.binghamton.edu/fowler/fowler%20personal%20page/EE522.htm

http://en.wikipedia.org/wiki/Multilateration

10

!"#$%&'()*&+&,-$%./&0

Assume measurements of TOAs at N receivers (only 3 shown above):

t0, t1, … ,tN-1There are measurement errors

TOA measurement model:

To = Time the signal emitted

Ri = Range from Tx to Rxi

c = Speed of Propagation (for EM: c = 3x108 m/s)

ti = To + Ri/c + !i i = 0, 1, . . . , N-1

%&'()*&+&,-$1.2(&" zero-mean, variance #2, independent (but 345$),6,.7,)

(variance determined from estimator used to estimate ti’s)

Now use: Ri = [ (xs – xi)2 + (ys - yi)

2 ]1/2

iisisossi yyxxc

Tyxft !$%$%$&&22 )()(

1),(

Nonlinear

Model

http://en.wikipedia.org/wiki/Multilateration

11

!"#$%&"'%(")#*)+*,-.*/)0$1

! " = [#x #y]T

So… we linearize the model so we can apply BLUE:

Assume some rough estimate is available (xn, y

n)

xs

= xn

+ #xs ys

= yn

+ #ys

know estimate know estimate

Now use truncated Taylor series to linearize Ri (xn, y

n):

sn

ins

n

inni y

R

yyx

R

xxRR

iBiA

ii

i##

!"!#$!"!#$

$$%%

&'

&'(

Known

isi

si

o

n

ii yc

Bx

c

AT

c

Rtt i

)## '''$&$~

Apply to TOA:

known known known

,2&$$*3#4#)5#*6%&%7$($&8*()*$8("7%($9*To:*#ys: #ys

12

!"#$%&'()$*+,$!-"#$%&'()

Two options now:

1. Use TOA to estimate 3 parameters: To, !ys, !ys

2. Use TDOA to estimate 2 parameters: !ys, !ys

Generally the fewer parameters the better…

Everything else being the same.

But… here “everything else” is not the same:

Options 1 & 2 have different noise models

(Option 1 has independent noise)

(Option 2 has correlated noise)

In practice… we’d explore both options and see which is best.

http://en.wikipedia.org/wiki/Multilateration

13

!"#$%&'("#)*")+,-.)/"0%1 N–1 TDOAs rather

than N TOAsTDOAs: 1,,2,1,

~~1 !"!" ! Nitt iii !#

"#"$%"#"$%"#"$%

noise correlated

1

known

1

known

1!

!! !$!

$!

" iisii

sii y

c

BBx

c

AA%%&&

In matrix form: 2 = 3' + 4

43 A

)()(

)()(

)()(

1

21

12

01

2121

1212

0101

"

((((((

)

*

++++++

,

-

!

!

!

"

((((((

)

*

++++++

,

-

!!

!!

!!

"

!!!!!! NNNNNN BBAA

BBAA

BBAA

c

%%

%%

%%

&

&

&&&

&

&

. /TN 121 !" ### '2 . /Tss yx &&"

See book for structure

of matrix .T

..4!42

}cov{ 0""

http://en.wikipedia.org/wiki/Multilateration

14

!""#$%&'()%*+%,-.!%'/0123/415%6+51#

! "

! "1

12

11ˆ

##

##

$%&

'()*

*

7!!7

7878 9!

TT

T

+

! "

! " ! " :!!77!!7

:87787! 99

11

1

111ˆ

###

###

$%&

'()*

*

TTTT

TT

BLUE

Describes how large

the location error is

Dependence on +2

cancels out!!!

,;/0<=%91%>20%0+9%5+:

1. Explore estimation error cov for different Tx/Rx geometries

• Plot error ellipses

2. Analytically explore simple geometries to find trends

• See next chart (more details in book)

http://en.wikipedia.org/wiki/Multilateration

15

!""#$%&'(!%)*+,#-%-.%/01"#*%2*.1*-3$

Rx1 Rx2 Rx3

d d

! !R

Tx

""""

#

$

%%%%

&

'

(

)

2

222

ˆ

)sin1(

2/30

0cos2

1

!

!* c4Then can show:

'056.75#%833.3%4.9% + !#067*:%833.3%8##0"+*

!7:;%%$<*33.3%5#=5$+%>066*3%-?57%@<*33.3 ex

ey

http://en.wikipedia.org/wiki/Multilateration

16

0 10 20 30 40 50 60 70 80 9010

-1

100

101

102

103

! (degrees )

"x/c"

or

"

y/c"

"x

"y

Rx1 Rx2 Rx3

d d

! !

R

Tx

• Used Std. Dev. to show units of X & Y

• Normalized by c"… get actual values by

multiplying by your specific c" value

• !"#$!%&'($)*+,'$R-$.+/#'*0%+,$)&$12*/%+,$d .32#"4'0$5//6#*/7

• !"#$!%&'($12*/%+,$d-$8'/#'*0%+,$)*+,'$R .32#"4'0$5//6#*/7

http://en.wikipedia.org/wiki/Multilateration