Bill Atwood, August, 2003 GLASTGLAST1

Covariance & GLAST

Agenda

• Review of Covariance• Application to GLAST• Kalman Covariance• Present Status

Bill Atwood, August, 2003 GLASTGLAST2

Review of Covariance

ab

Ellipse

12

2

2

2

b

y

a

xTake a circle – scale the x & y axis:

Rotate by : )sin()cos(

)sin()cos(

xyy

yxx

Results:

1))(cos)(sin

()11

)(sin()cos(2))(sin)(cos

(2

2

2

22

222

2

2

22

bay

baxy

bax

Rotations mix x & y. Major & minor axis plus rotation angle complete description.

Error Ellipse described by Covariance Matrix:

n

Distance between a point with an error and another point measured in ’s:

rCrn T 12)( where )( xxr

and

11

111 )(

yyyx

xyxx

CC

CCCInverseC and 11 yxxy CC

Simply weighting the distance by 1/2

Bill Atwood, August, 2003 GLASTGLAST3

Review 2

1211211

1112 2),()(

yyxyxx

yyxy

xyxxT CyxyCCxy

x

CC

CCyxrCrn

Multiplying it out gives:

Where I take 0x without loss of generality.

This is the equation of an ellipse! Specifically for 1 error ellipse (n = 1) we identify:

2

2

2

21 )(sin)(cos

baCxx

2

2

2

21 )(cos)(sin

baCyy

)

11)(cos()sin(

221

baCxy

and

xxxy

xyyy

CC

CC

CC

)det(

11 where )()det( 2xyyyxx CCCC

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!

And the correlation coefficient is defined as: yyxx

xy

CC

Cr

22

Bill Atwood, August, 2003 GLASTGLAST4

Review 3Let the fun begin! To disentangle the two descriptions consider

22

22

11

1))(sin()cos(

ba

ab

CC

C

CC

CA

yyxx

xy

yyxx

xy

where r = a/b

= 0 = /4 = /2 = 3/4

2

2

1

1

2

)2sin(

r

rA

Also det(C) yields (with a little algebra & trig.):

)det(Cba

Now we’re ready to look at results from GLAST!

thus

Bill Atwood, August, 2003 GLASTGLAST5

Covariance Matrix from Kalman Filter

Results shown for

SYYTkrSXXTkr

SXYTkrAxisAsym

11

1

Binned in cos() and log10(EMC)

Recall however that KF gives us C in terms of the track slopes Sx and Sy.

AxisAsym grows like 1/cos2()

Peak amplitude ~ .4

)1(

)1(

2

1(max)

2

2

r

rA

(max)21

(max)212

A

Ar

3

1

9

1

4.21

4.21

a

br

Bill Atwood, August, 2003 GLASTGLAST6

Relationship between Slopes and Angles

For functions of the estimated variables the usual prescriptions is:

222 )(ivariables

if x

f

when the errors are uncorrelated. For correlated

errors this becomes )()())(( ,,

2

jji

ijij

jiif x

fC

x

f

x

f

x

f

and reduces to the uncorrelated case when jiijiC ,2

,

The functions of interest here are:

221

1)cos(

yx SS and

x

y

S

S)tan(

A bit of math then shows that:

yyxyxx CCC )(sin)cos()sin(2)(cos)(cos 2242

yyxyxx CCC )(cos)cos()sin(2)(sin)(tan

1 222

2

and

Bill Atwood, August, 2003 GLASTGLAST7

Angle Errors from GLASThas a divergence at . However sin() cures this.

decreases as cos2() - while sin() increases as )cos(

1

We also expect the components of the covariance matrix to increase as

due to the dominance of multiple scattering.)cos(

1

log10(EMC)

cos()

Plot measured residuals in terms of Fit 's (e.g. )FIT

MCmeas

Bill Atwood, August, 2003 GLASTGLAST8

Angle Errors 2What's RIGHT: 1) cos() dependence 2) Energy dependence in Multiple Scattering dominated range

What's WRONG: 1) Overall normalization of estimated errors (FIT) - 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 2'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 GLASTGLAST9

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 GLASTGLAST10

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 GLASTGLAST11

Present Analysis Status

Bill Atwood, August, 2003 GLASTGLAST12

Present Status 2: BGE Rejection

Events left after Good-Energy Selection: 1904

Events left in VTX Classes: 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!