Transcript ppt

Covariance & GLAST
Agenda
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•
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Bill Atwood, August, 2003
Review of Covariance
Application to GLAST
Kalman Covariance
Present Status
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GLAST
Review of Covariance
Ellipse
b
Take a circle – scale the x & y axis:
x2 y2
 2 1
2
a
b
a
Rotate by
q:
x  x cos(q )  y sin( q )
y  y cos(q )  x sin( q )
Results:
2
cos 2 (q ) sin 2 (q )
1 1
cos 2 (q )
2 sin (q )
x (

)  2 xy cos(q ) sin( q )( 2  2 )  y (

) 1
2
2
2
2
a
b
a b
a
b
Rotations mix x & y. Major & minor axis plus rotation angle q complete description.
2
Error Ellipse described by Covariance Matrix:
Distance between a point with an error and another point measured in
ns
(ns ) 2  r T C 1r
Bill Atwood, August, 2003
where

r  ( x  x ) and
C xx1 C xy1 
1
1
1
C  Inverse(C )   1
and
C

C

1
xy
yx
C yx C yy 
2
s’s:
Simply weighting the
distance by 1/s2
GLAST
Review 2
Multiplying it out gives:
1
1


C
C
xx
xy  x 
2
T
1
2 1
1
2 1


(ns )  r C r  ( x, y )  1

x
C

2
xyC

y
C yy

xx
xy
1  
C xy C yy  y 
Where I take
x 0
without loss of generality.
This is the equation of an ellipse! Specifically for 1
s error ellipse (ns = 1) we identify:
1 1
cos 2 (q ) sin 2 (q )
sin 2 (q ) cos 2 (q )
1
1
C

sin(
q
)
cos(
q
)(
 2)
C 

C


xy
yy
2
2
2
2
2
a b
a
b
a
b
1  C yy  Cxy 
2
and C 1 
where det(C )  (Cxx C yy  Cxy )
 C

det(C )  xy Cxx 
1
xx
And the correlation coefficient is defined as:
r2 
C xy2
C xx C yy
Summary: The inverse of the Covariance Matrix describes an ellipse where
the major and minor axis and the rotation angle map directly onto
its components!
Bill Atwood, August, 2003
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GLAST
Review 3
Let the fun begin! To disentangle the two descriptions consider
A
Cxy1
Cxx1  C yy1
 Cxy
cos(q ) sin( q )(b 2  a 2 )


Cxx  C yy
a 2  b2
sin( 2q )  1  r 2  where r = a/b


thus A 
2 
2 1 r 
q=0
q = p/4
q = p/2
q = 3p/4
Also det(C) yields (with a little algebra & trig.):
a  b  det(C )
Now we’re ready to look at results from GLAST!
Bill Atwood, August, 2003
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GLAST
Covariance Matrix from Kalman Filter
Results shown for
AxisAsym  
Tkr1SXY
Tkr1SXX  Tkr1SYY
Binned in
cos(q) and log10(EMC)
Recall however that KF
gives us C in terms of the
track slopes Sx and Sy.
AxisAsym grows like
1/cos2(q)
Peak amplitude ~ .4
1 (1  r 2 )
A(max) 
2 (1  r 2 )
Bill Atwood, August, 2003
r2 
1  2 A(max)
1  2 A(max)
5
r 
b
1  2  .4
1 1



a
1  2  .4
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GLAST
Relationship between Slopes and Angles
For functions of the estimated variables the usual prescriptions is:
f 2
) when the errors are uncorrelated. For correlated

x
variables
i
f f
f
f
2
)(
)s j  ( )Ci , j (
)
errors this becomes s f  s i (

x

x

x

x
i, j
i
j
i
j
s 2f 
 s i2 (
and reduces to the uncorrelated case when
Ci , j  s i2 i , j
The functions of interest here are:
cos(q ) 
1
1 S  S
2
x
2
y
and
tan(  )  
Sy
Sx
A bit of math then shows that:
s q2  cos 4 (q )cos 2 ( )Cxx  2 sin(  ) cos( )Cxy  sin 2 ( )C yy 
and
s 2 
Bill Atwood, August, 2003

1
2
2
sin
(

)
C

2
sin(

)
cos(

)
C

cos
( )C yy
xx
xy
2
tan (q )
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
GLAST
Angle Errors from GLAST
s
sq
However ssin(q) cures this.
1
decreases as cos2(q) - while ssin(q) increases as
cos(q )
We also expect the components of the covariance matrix to increase as
has a divergence at q
 0.
cos(q)
1
due to the dominance of multiple scattering.
cos(q )
q q
Plot measured residuals in terms of Fit s's (e.g. meas MC )
s FIT
log10(EMC)
Bill Atwood, August, 2003
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Angle Errors 2
What's RIGHT:
1) cos(q) dependence
2) Energy dependence in Multiple Scattering dominated range
What's WRONG:
1) Overall normalization of estimated errors (sFIT)
- off by a factor of ~ 2.3!!!
2) Energy dependence as measurement errors begin to dominate
- discrepancy goes away(?)
Both of these correlate with with the fact that the fitted c2's are
much larger then 1 at low energy (expected?).
How well does the Kalman Fit PSF model the event to event PSF?
Bill Atwood, August, 2003
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GLAST
Angle Errors 3
Comparison of
Event-by-Event PSF
vs
FIT Parameter PSF
(Both Energy Compensated)
Difficult to assess
level of correlation
- probably not zero
- approximately same
factor of 2.3
Bill Atwood, August, 2003
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Angle Errors: Conclusions
1) Analysis of covariance matrix gives format for modeling instrument response
2) Predictive power of Kalman Fit?
- Factor of 2.3
- May prove a good handle for CT tree determination of "Best PSF"
Bill Atwood, August, 2003
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GLAST
Present Analysis Status
Bill Atwood, August, 2003
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GLAST
Present Status 2: BGE Rejection
Events left after
Good-Energy Selection:
Events left in VTX Classes:
1904
26
Events left in 1Tkr Classes: 1878
CT BGE Rejection factors obtained:
20:1 (1Tkr)
2:1 (VTX)
NEED FACTOR OF 10X EVENTS BEFORE PROGRESS CAN BE MADE!
Bill Atwood, August, 2003
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GLAST