Covariance & GLAST

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# Covariance & GLAST - PowerPoint PPT Presentation

Agenda Review of Covariance Application to GLAST Kalman Covariance Present Status . Covariance & GLAST. b. a. n s. Review of Covariance. Ellipse. Take a circle – scale the x & y axis:. Rotate by q : . Results: .

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## Covariance & GLAST

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Presentation Transcript
Agenda

Review of Covariance

Application to GLAST

Kalman Covariance

Present Status

Covariance & GLAST

b

a

ns

Review of Covariance

Ellipse

Take a circle – scale the x & y axis:

Rotate by q :

Results:

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

Error Ellipse described by Covariance Matrix:

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

where

and

Simply weighting the

distance by 1/s2

and

Where I take

without loss of generality.

Review 2

Multiplying it out gives:

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

and

where

And the correlation coefficient is defined as:

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!

Review 3

Let the fun begin! To disentangle the two descriptions consider

where r = a/b

thus

q = 0

q = p/2

q = p/4

q = 3p/4

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

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

Covariance Matrix from Kalman Filter

Results shown for

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

Relationship between Slopes and Angles

For functions of the estimated variables the usual prescriptions is:

when the errors are uncorrelated. For correlated

errors this becomes

and reduces to the uncorrelated case when

The functions of interest here are:

and

A bit of math then shows that:

and

Angle Errors from GLAST

sfhas a divergence at q = 0. However sfsin(q) cures this.

sqdecreases as cos2(q) - while sfsin(q) decreases as

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

due to the dominance of multiple scattering.

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

cos(q)

log10(EMC)

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?

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

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"

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!