Statistical Methods in High Energy Physicsmenzemer/Stat... · Statistics in High Energy Physics 1 hits in detectors hit in calorimeter: # counts in photomulitplier hit in wire chamber

Statistical Methods in High Energy PhysicsOr: How to align 220 Outer Tracker Modules

J. Blouw

Physikalisches Institut, Universitaet Heidelberg

Heidelberg, February, 4, 2008

Blouw Statistical Methods in High Energy Physics

Statistics in High Energy Physics



1 hits in detectorshit in calorimeter: # counts in photomulitplierhit in wire chamber # of collected electrons on wire

2 reconstructed tracks in detector

particle trajectory is fitted to selected hitsuse test-statistic to determine whether hit belongs to track

3 particle identification

associate tracks to a certain particle typehypothesis testing: is it a muon, kaon, electron etc

4 alignment...

: or where is my detector located?- can be determined by eye...- can be measured optically- ... but also by using tracks!



1 hits in detectors

hit in calorimeter: # counts in photomulitplierhit in wire chamber # of collected electrons on wire

2 reconstructed tracks in detectorparticle trajectory is fitted to selected hitsuse test-statistic to determine whether hit belongs to track



4 alignment...




1 hits in detectors




3 particle identificationassociate tracks to a certain particle typehypothesis testing: is it a muon, kaon, electron etc

4 alignment...




1 hits in detectors






4 alignment...




1 hits in detectors






4 alignment...: or where is my detector located?

- can be determined by eye...- can be measured optically- ... but also by using tracks!



1 hits in detectors






4 alignment...: or where is my detector located?- can be determined by eye...

- can be measured optically- ... but also by using tracks!



1 hits in detectors






4 alignment...: or where is my detector located?- can be determined by eye...- can be measured optically

- ... but also by using tracks!



1 hits in detectors






4 alignment...: or where is my detector located?- can be determined by eye...- can be measured optically- ... but also by using tracks!



1 hits in detectors






4 alignment...: or where is my detector located?- can be determined by eye...- can be measured optically- ... but also by using tracks!

Problem

What about a reference point?


Test Statistic

Usually used to test H0 (Null)hypothesis:

easily calculated quantitiesapproximate χ2 distributionwhenvariables in a sample arerandomly correlated

If null hypothesis is true→test-statistic follow χ2

distribution

e.g. for k independent variables

Q =kX

i=1

X 2i ∼ χ2

k

k : number of degrees of freedom Cumulative χ2

distribution yield the probability P(X < x).

FX (x) = P(X < x)

Probability that x lies in interval (a, b]:

F (b)− F (a)


Test Statistic

χ2 distribution for various degrees offreedom cumulative chi2 distribution


χ2 minimization

well known method to find optimum

optimum defined by calculation of χ2

and

calculation of derivatives wrt variablesof interest

e.g. Pearson’s χ2 test for a countingexperiment:

χ =NX

i=1

(Oi − Ei )2

Ei

with Oi observation i

Ei expectation i

If Ei is a theoretical expectation:“goodness of fit”-test.


Goodness of fit testsLine fitting:

R2(a, b) =nX

i=1

[yi − (a + bxi )]2

1 exact solution2 including errors!



R2(a, b) =nX

i=1

[yi − (a + bxi )]2

Minimize χ2 wrt a,b:

∂R2

∂a= −2

nXi=1

[yi − (a + bxi )] = 0

∂R2

∂b= −2

nXi=1

[yi − (a + bxi )] xi = 0




R2(a, b) =nX

i=1

[yi − (a + bxi )]2


∂R2

∂a= −2

nXi=1

[yi − (a + bxi )] = 0

∂R2

∂b= −2

nXi=1

[yi − (a + bxi )] xi = 0

»n

Pni=1 xiPn

i=1 xiPn

i=1 x2i

– »ab

–=

» Pni=1 yiPn

i=1 xiyi

–




R2(a, b) =nX

i=1

[yi − (a + bxi )]2


∂R2

∂a= −2

nXi=1

[yi − (a + bxi )] = 0

∂R2

∂b= −2

nXi=1

[yi − (a + bxi )] xi = 0

»n

Pni=1 xiPn

i=1 xiPn

i=1 x2i

– »ab

–=

» Pni=1 yiPn

i=1 xiyi

–

1 exact solution

2 including errors!



R2(a, b) =nX

i=1

[yi − (a + bxi )]2


∂R2

∂a= −2

nXi=1

[yi − (a + bxi )] = 0

∂R2

∂b= −2

nXi=1

[yi − (a + bxi )] xi = 0

»n

Pni=1 xiPn

i=1 xiPn

i=1 x2i

– »ab

–=

» Pni=1 yiPn

i=1 xiyi

–




R2(a, b) =nX

i=1

[yi − (a + bxi )]2


∂R2

∂a= −2

nXi=1

[yi − (a + bxi )] = 0

∂R2

∂b= −2

nXi=1

[yi − (a + bxi )] xi = 0

»n

Pni=1 xiPn

i=1 xiPn

i=1 x2i

– »ab

–=

» Pni=1 yiPn

i=1 xiyi

–



Goodness of fit testsPolynomial fitting:


R2 =nX

i=1

hyi − (a0 + a1xi + · · ·+ ak xk

i )i2

∂R2

∂ak= −2

nXi=1

hyi − (a0 + a1xi + · · · ak xk

i )i

xk = 0


Goodness of fit testsPolynomial fitting:


R2 =nX

i=1

hyi − (a0 + a1xi + · · ·+ ak xk

i )i2

∂R2

∂ak= −2

nXi=1

hyi − (a0 + a1xi + · · · ak xk

i )i

xk = 0

26664n

Pni=1 xi · · ·

Pni=1 xk

iPni=1 xi

Pni=1 x2

i · · ·Pn

i=1 xk+1i

......

. . ....Pn

i=1 xki

Pni=1 xk+1

i . . .Pn

i=1 x2ki

3777526664

a0

a1...

ak

37775 =

26664Pn

i=1 yiPn1=1 xiyi

...Pni=1 xk

i yi

37775hence, calculate inverted matrices to solve this equation!


An application of χ2 minimization

Question

What is alignment?

Try to measure where your detector is locatedUse “residuals”; difference between track-position and hit position in plane:

worse track resolution

worse separation between signal and background



Question

What is alignment?

Try to measure where your detector is locatedUse “residuals”; difference between track-position and hit position in plane:





Consequence:

When alignment is not very well known:

Alignment accuracy: 40µm





Consequence:






Consequence:





Alignment by optical measurements

Not so easy

possibly not accurate enough



Not so easy




Not so easy




Not so easy


But: may provide a reference point...


Tracking stations: Outer Tracker

Modules are supported by c-framein turn supported by “Amstel”bridge


Tracking stations: Outer Tracker

wire position known with 20 µm accuracy

200 µm hit resolution in Outer Tracker

Goal: measure module positions with accuracy much better than 100µm.


Alignment: the Iterative Method

A misaligned detector

planes seem to be aligned

residuals did not improve

Expect to find solid track, however found dashed track

re-iterate...



Shift the planes according to their residuals




re-iterate...



Re-fit track




re-iterate...



re-calculate residuals




re-iterate...






re-iterate...






re-iterate...






re-iterate...






re-iterate...






re-iterate...


The Exact Solution. . .Need to consider the following requirements:

a model which describes a track

a model which describes a track: Xtrack

a model which describes the geometry of the plane

a model which describes the geometry of the plane: Xhit



















then we can calculate derivatives of χ2!!!


The Exact Solution. . .





Express residuals using linear track and hit model:

ε = Xhit − Xtrack

= (∆x)− (t2 · zhit + t1)

∆x = f (a1, a2, · · · , an)

∆x : alignment parameters

t1, t2: track parameters in track model


The large MATRIX

Minimize χ2 function e.g. for one track and n planes:

χ2 =nX

i=1

„X meas

i − X tracki

σi

«2

∂χ2

∂ai= 0

∂χ2

∂ti= 0

C matrix contains all derivativesa vector contains track and alignment parametersb vector contains hit positions: solve matrix equation of size

1 sub-matrixP

k Cglobalk correlates

alignment parameters2 sub-matrix Hk correlates alignment

parameters with track parameters3 sub-matrix C local

k error onmeasurement


The large MATRIX

Rewrite resulting equations as matrix equation:

C · ~a = ~b

C matrix contains all derivatives

a vector contains track and alignment parameters

b vector contains hit positions

: solve matrix equation of size

1 sub-matrixP






The large MATRIX


C · ~a = ~b





1 sub-matrixP






The large MATRIX


C · ~a = ~b





1 sub-matrixP






The large MATRIX


C · ~a = ~b





1 sub-matrixP






The large MATRIX


C · ~a = ~b





ntotal = ntracks · ntr.pars. + nalignment.

1 sub-matrixP






The large MATRIX





For instance:

LHCb: 10000× 5 + 6×m(OT : 448) = 50000 + 2688

1 sub-matrixP






The large MATRIX


Large MATRIX, can be inverted using (e.g. Millepede)Matrix is sparse, consist of many smaller matrices

1 sub-matrixP






The large MATRIX



1 sub-matrixP


alignment parameters

2 sub-matrix Hk correlates alignmentparameters with track parameters

3 sub-matrix C localk error on

measurement


The large MATRIX



1 sub-matrixP



parameters with track parameters

3 sub-matrix C localk error on

measurement


The large MATRIX



1 sub-matrixP






Complicated math: a little bit of intuitive perception...

e.g. z-shifts.

depend on size of shift

depend on slope of track


Complicated math: a little bit of intuitive perception...

e.g. z-shifts.

depend on size of shift

depend on slope of track


Results from Simulation Studies

By M. Deissenroth

After inverting the C-matrix


Results from Simulation Studies

By M. Deissenroth

Accuracy for shifts along x: = 40 µm Accuracy for shifts in z: = 0.2 mm


Summary/Conclusion(s)...

1 intricate relationship between physics measurment and statistics!

2 e.g. tracks, particle identifacation... and alignment3 misaligned detector deteriorates quality of physics measurements4 χ2 minimization is powerful tool also for alignment5 Pulling oneself from one’s shoe-strings out of the mud6 not entirely possible: only relative alignment can be achieved7 can even be used for non-linear problems (z shifts...)



1 intricate relationship between physics measurment and statistics!2 e.g. tracks, particle identifacation... and alignment

3 misaligned detector deteriorates quality of physics measurements4 χ2 minimization is powerful tool also for alignment5 Pulling oneself from one’s shoe-strings out of the mud6 not entirely possible: only relative alignment can be achieved7 can even be used for non-linear problems (z shifts...)



1 intricate relationship between physics measurment and statistics!2 e.g. tracks, particle identifacation... and alignment3 misaligned detector deteriorates quality of physics measurements

4 χ2 minimization is powerful tool also for alignment5 Pulling oneself from one’s shoe-strings out of the mud6 not entirely possible: only relative alignment can be achieved7 can even be used for non-linear problems (z shifts...)



1 intricate relationship between physics measurment and statistics!2 e.g. tracks, particle identifacation... and alignment3 misaligned detector deteriorates quality of physics measurements4 χ2 minimization is powerful tool also for alignment

5 Pulling oneself from one’s shoe-strings out of the mud6 not entirely possible: only relative alignment can be achieved7 can even be used for non-linear problems (z shifts...)



1 intricate relationship between physics measurment and statistics!2 e.g. tracks, particle identifacation... and alignment3 misaligned detector deteriorates quality of physics measurements4 χ2 minimization is powerful tool also for alignment5 Pulling oneself from one’s shoe-strings out of the mud

6 not entirely possible: only relative alignment can be achieved7 can even be used for non-linear problems (z shifts...)



1 intricate relationship between physics measurment and statistics!2 e.g. tracks, particle identifacation... and alignment3 misaligned detector deteriorates quality of physics measurements4 χ2 minimization is powerful tool also for alignment5 Pulling oneself from one’s shoe-strings out of the mud6 not entirely possible: only relative alignment can be achieved

7 can even be used for non-linear problems (z shifts...)



1 intricate relationship between physics measurment and statistics!2 e.g. tracks, particle identifacation... and alignment3 misaligned detector deteriorates quality of physics measurements4 χ2 minimization is powerful tool also for alignment5 Pulling oneself from one’s shoe-strings out of the mud6 not entirely possible: only relative alignment can be achieved7 can even be used for non-linear problems (z shifts...)


Statistical Methods in High Energy Physicsmenzemer/Stat... · Statistics in High Energy Physics 1 hits in detectors hit in calorimeter: # counts in photomulitplier hit in wire chamber

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