Estimation: chapter 6 - UIC Engineeringdevroye/courses/ECE531/lectures/b1c6.pdf · Estimation:...
Transcript of Estimation: chapter 6 - UIC Engineeringdevroye/courses/ECE531/lectures/b1c6.pdf · Estimation:...
Estimation: chapter 6
Best Linear Unbiased Estimators
Natasha Devroye
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 !#
"#"$%"#"$%"#"$%
noise correlated
1
known
1
known
1!
!! !$!
$!
" iisii
sii y
c
BBx
c
AA%%&&
In matrix form: 2 = 3' + 4
43 A
)()(
)()(
)()(
1
21
12
01
2121
1212
0101
"
((((((
)
*
++++++
,
-
!
!
!
"
((((((
)
*
++++++
,
-
!!
!!
!!
"
!!!!!! NNNNNN BBAA
BBAA
BBAA
c
%%
%%
%%
&
&
&&&
&
&
. /TN 121 !" ### '2 . /Tss yx &&"
See book for structure
of matrix .T
..4!42
}cov{ 0""
http://en.wikipedia.org/wiki/Multilateration
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!""#$%&'()%*+%,-.!%'/0123/415%6+51#
! "
! "1
12
11ˆ
##
##
$%&
'()*
*
7!!7
7878 9!
TT
T
+
! "
! " ! " :!!77!!7
:87787! 99
11
1
111ˆ
###
###
$%&
'()*
*
TTTT
TT
BLUE
Describes how large
the location error is
Dependence on +2
cancels out!!!
,;/0<=%91%>20%0+9%5+:
1. Explore estimation error cov for different Tx/Rx geometries
• Plot error ellipses
2. Analytically explore simple geometries to find trends
• See next chart (more details in book)
http://en.wikipedia.org/wiki/Multilateration
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!""#$%&'(!%)*+,#-%-.%/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
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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