Statistical Methods in High Energy Physicsmenzemer/Stat... · Statistics in High Energy Physics 1...
Transcript of Statistical Methods in High Energy Physicsmenzemer/Stat... · Statistics in High Energy Physics 1...
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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
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Statistics in High Energy Physics
Blouw Statistical Methods in High Energy Physics
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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!
Blouw Statistical Methods in High Energy Physics
![Page 4: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/4.jpg)
Statistics in High Energy Physics
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
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!
Blouw Statistical Methods in High Energy Physics
![Page 5: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/5.jpg)
Statistics in High Energy Physics
1 hits in detectors
hit 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 identificationassociate 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!
Blouw Statistical Methods in High Energy Physics
![Page 6: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/6.jpg)
Statistics in High Energy Physics
1 hits in detectors
hit 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!
Blouw Statistical Methods in High Energy Physics
![Page 7: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/7.jpg)
Statistics in High Energy Physics
1 hits in detectors
hit 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!
Blouw Statistical Methods in High Energy Physics
![Page 8: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/8.jpg)
Statistics in High Energy Physics
1 hits in detectors
hit 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!
Blouw Statistical Methods in High Energy Physics
![Page 9: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/9.jpg)
Statistics in High Energy Physics
1 hits in detectors
hit 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!
Blouw Statistical Methods in High Energy Physics
![Page 10: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/10.jpg)
Statistics in High Energy Physics
1 hits in detectors
hit 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!
Blouw Statistical Methods in High Energy Physics
![Page 11: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/11.jpg)
Statistics in High Energy Physics
1 hits in detectors
hit 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!
Problem
What about a reference point?
Blouw Statistical Methods in High Energy Physics
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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)
Blouw Statistical Methods in High Energy Physics
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Test Statistic
χ2 distribution for various degrees offreedom cumulative chi2 distribution
Blouw Statistical Methods in High Energy Physics
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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.
Blouw Statistical Methods in High Energy Physics
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Goodness of fit testsLine fitting:
R2(a, b) =nX
i=1
[yi − (a + bxi )]2
1 exact solution2 including errors!
Blouw Statistical Methods in High Energy Physics
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Goodness of fit testsLine fitting:
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
1 exact solution2 including errors!
Blouw Statistical Methods in High Energy Physics
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Goodness of fit testsLine fitting:
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
»n
Pni=1 xiPn
i=1 xiPn
i=1 x2i
– »ab
–=
» Pni=1 yiPn
i=1 xiyi
–
1 exact solution2 including errors!
Blouw Statistical Methods in High Energy Physics
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Goodness of fit testsLine fitting:
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
»n
Pni=1 xiPn
i=1 xiPn
i=1 x2i
– »ab
–=
» Pni=1 yiPn
i=1 xiyi
–
1 exact solution
2 including errors!
Blouw Statistical Methods in High Energy Physics
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Goodness of fit testsLine fitting:
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
»n
Pni=1 xiPn
i=1 xiPn
i=1 x2i
– »ab
–=
» Pni=1 yiPn
i=1 xiyi
–
1 exact solution2 including errors!
Blouw Statistical Methods in High Energy Physics
![Page 20: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/20.jpg)
Goodness of fit testsLine fitting:
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
»n
Pni=1 xiPn
i=1 xiPn
i=1 x2i
– »ab
–=
» Pni=1 yiPn
i=1 xiyi
–
1 exact solution2 including errors!
Blouw Statistical Methods in High Energy Physics
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Goodness of fit testsPolynomial fitting:
1 exact solution2 including errors!
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
Blouw Statistical Methods in High Energy Physics
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Goodness of fit testsPolynomial fitting:
1 exact solution2 including errors!
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!
Blouw Statistical Methods in High Energy Physics
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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
Blouw Statistical Methods in High Energy Physics
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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
Blouw Statistical Methods in High Energy Physics
![Page 25: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/25.jpg)
An application of χ2 minimization
Consequence:
When alignment is not very well known:
Alignment accuracy: 40µm
worse track resolution
worse separation between signal and background
Blouw Statistical Methods in High Energy Physics
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An application of χ2 minimization
Consequence:
When alignment is not very well known:
worse track resolution
worse separation between signal and background
Blouw Statistical Methods in High Energy Physics
![Page 27: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/27.jpg)
An application of χ2 minimization
Consequence:
When alignment is not very well known:
worse track resolution
worse separation between signal and background
Blouw Statistical Methods in High Energy Physics
![Page 28: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/28.jpg)
Alignment by optical measurements
Not so easy
possibly not accurate enough
Blouw Statistical Methods in High Energy Physics
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Alignment by optical measurements
Not so easy
possibly not accurate enough
Blouw Statistical Methods in High Energy Physics
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Alignment by optical measurements
Not so easy
possibly not accurate enough
Blouw Statistical Methods in High Energy Physics
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Alignment by optical measurements
Not so easy
possibly not accurate enough
But: may provide a reference point...
Blouw Statistical Methods in High Energy Physics
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Tracking stations: Outer Tracker
Modules are supported by c-framein turn supported by “Amstel”bridge
Blouw Statistical Methods in High Energy Physics
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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.
Blouw Statistical Methods in High Energy Physics
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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...
Blouw Statistical Methods in High Energy Physics
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Alignment: the Iterative Method
Shift the planes according to their residuals
planes seem to be aligned
residuals did not improve
Expect to find solid track, however found dashed track
re-iterate...
Blouw Statistical Methods in High Energy Physics
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Alignment: the Iterative Method
Re-fit track
planes seem to be aligned
residuals did not improve
Expect to find solid track, however found dashed track
re-iterate...
Blouw Statistical Methods in High Energy Physics
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Alignment: the Iterative Method
re-calculate residuals
planes seem to be aligned
residuals did not improve
Expect to find solid track, however found dashed track
re-iterate...
Blouw Statistical Methods in High Energy Physics
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Alignment: the Iterative Method
planes seem to be aligned
residuals did not improve
Expect to find solid track, however found dashed track
re-iterate...
Blouw Statistical Methods in High Energy Physics
![Page 39: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/39.jpg)
Alignment: the Iterative Method
planes seem to be aligned
residuals did not improve
Expect to find solid track, however found dashed track
re-iterate...
Blouw Statistical Methods in High Energy Physics
![Page 40: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/40.jpg)
Alignment: the Iterative Method
planes seem to be aligned
residuals did not improve
Expect to find solid track, however found dashed track
re-iterate...
Blouw Statistical Methods in High Energy Physics
![Page 41: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/41.jpg)
Alignment: the Iterative Method
planes seem to be aligned
residuals did not improve
Expect to find solid track, however found dashed track
re-iterate...
Blouw Statistical Methods in High Energy Physics
![Page 42: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/42.jpg)
Alignment: the Iterative Method
planes seem to be aligned
residuals did not improve
Expect to find solid track, however found dashed track
re-iterate...
Blouw Statistical Methods in High Energy Physics
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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
Blouw Statistical Methods in High Energy Physics
![Page 44: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/44.jpg)
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
Blouw Statistical Methods in High Energy Physics
![Page 45: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/45.jpg)
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
Blouw Statistical Methods in High Energy Physics
![Page 46: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/46.jpg)
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!!!
Blouw Statistical Methods in High Energy Physics
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The Exact Solution. . .
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
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
Blouw Statistical Methods in High Energy Physics
![Page 48: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/48.jpg)
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
Blouw Statistical Methods in High Energy Physics
![Page 49: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/49.jpg)
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
k Cglobalk correlates
alignment parameters2 sub-matrix Hk correlates alignment
parameters with track parameters3 sub-matrix C local
k error onmeasurement
Blouw Statistical Methods in High Energy Physics
![Page 50: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/50.jpg)
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
k Cglobalk correlates
alignment parameters2 sub-matrix Hk correlates alignment
parameters with track parameters3 sub-matrix C local
k error onmeasurement
Blouw Statistical Methods in High Energy Physics
![Page 51: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/51.jpg)
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
k Cglobalk correlates
alignment parameters2 sub-matrix Hk correlates alignment
parameters with track parameters3 sub-matrix C local
k error onmeasurement
Blouw Statistical Methods in High Energy Physics
![Page 52: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/52.jpg)
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
k Cglobalk correlates
alignment parameters2 sub-matrix Hk correlates alignment
parameters with track parameters3 sub-matrix C local
k error onmeasurement
Blouw Statistical Methods in High Energy Physics
![Page 53: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/53.jpg)
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
ntotal = ntracks · ntr.pars. + nalignment.
1 sub-matrixP
k Cglobalk correlates
alignment parameters2 sub-matrix Hk correlates alignment
parameters with track parameters3 sub-matrix C local
k error onmeasurement
Blouw Statistical Methods in High Energy Physics
![Page 54: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/54.jpg)
The large MATRIX
C matrix contains all derivatives
a vector contains track and alignment parameters
b vector contains hit positions
: solve matrix equation of size
For instance:
LHCb: 10000× 5 + 6×m(OT : 448) = 50000 + 2688
1 sub-matrixP
k Cglobalk correlates
alignment parameters2 sub-matrix Hk correlates alignment
parameters with track parameters3 sub-matrix C local
k error onmeasurement
Blouw Statistical Methods in High Energy Physics
![Page 55: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/55.jpg)
The large MATRIX
C matrix contains all derivativesa vector contains track and alignment parametersb vector contains hit positions: solve matrix equation of size
Large MATRIX, can be inverted using (e.g. Millepede)Matrix is sparse, consist of many smaller matrices
1 sub-matrixP
k Cglobalk correlates
alignment parameters2 sub-matrix Hk correlates alignment
parameters with track parameters3 sub-matrix C local
k error onmeasurement
Blouw Statistical Methods in High Energy Physics
![Page 56: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/56.jpg)
The large MATRIX
C matrix contains all derivativesa vector contains track and alignment parametersb vector contains hit positions: solve matrix equation of size
Large MATRIX, can be inverted using (e.g. Millepede)Matrix is sparse, consist of many smaller matrices
1 sub-matrixP
k Cglobalk correlates
alignment parameters
2 sub-matrix Hk correlates alignmentparameters with track parameters
3 sub-matrix C localk error on
measurement
Blouw Statistical Methods in High Energy Physics
![Page 57: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/57.jpg)
The large MATRIX
C matrix contains all derivativesa vector contains track and alignment parametersb vector contains hit positions: solve matrix equation of size
Large MATRIX, can be inverted using (e.g. Millepede)Matrix is sparse, consist of many smaller matrices
1 sub-matrixP
k Cglobalk correlates
alignment parameters2 sub-matrix Hk correlates alignment
parameters with track parameters
3 sub-matrix C localk error on
measurement
Blouw Statistical Methods in High Energy Physics
![Page 58: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/58.jpg)
The large MATRIX
C matrix contains all derivativesa vector contains track and alignment parametersb vector contains hit positions: solve matrix equation of size
Large MATRIX, can be inverted using (e.g. Millepede)Matrix is sparse, consist of many smaller matrices
1 sub-matrixP
k Cglobalk correlates
alignment parameters2 sub-matrix Hk correlates alignment
parameters with track parameters3 sub-matrix C local
k error onmeasurement
Blouw Statistical Methods in High Energy Physics
![Page 59: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/59.jpg)
Complicated math: a little bit of intuitive perception...
e.g. z-shifts.
depend on size of shift
depend on slope of track
Blouw Statistical Methods in High Energy Physics
![Page 60: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/60.jpg)
Complicated math: a little bit of intuitive perception...
e.g. z-shifts.
depend on size of shift
depend on slope of track
Blouw Statistical Methods in High Energy Physics
![Page 61: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/61.jpg)
Results from Simulation Studies
By M. Deissenroth
After inverting the C-matrix
Blouw Statistical Methods in High Energy Physics
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Results from Simulation Studies
By M. Deissenroth
Accuracy for shifts along x: = 40 µm Accuracy for shifts in z: = 0.2 mm
Blouw Statistical Methods in High Energy Physics
![Page 63: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/63.jpg)
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...)
Blouw Statistical Methods in High Energy Physics
![Page 64: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/64.jpg)
Summary/Conclusion(s)...
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...)
Blouw Statistical Methods in High Energy Physics
![Page 65: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/65.jpg)
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 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...)
Blouw Statistical Methods in High Energy Physics
![Page 66: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/66.jpg)
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 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...)
Blouw Statistical Methods in High Energy Physics
![Page 67: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/67.jpg)
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 mud
6 not entirely possible: only relative alignment can be achieved7 can even be used for non-linear problems (z shifts...)
Blouw Statistical Methods in High Energy Physics
![Page 68: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/68.jpg)
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 achieved
7 can even be used for non-linear problems (z shifts...)
Blouw Statistical Methods in High Energy Physics
![Page 69: 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](https://reader034.fdocuments.in/reader034/viewer/2022050218/5f64392aa43019721811c5f5/html5/thumbnails/69.jpg)
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...)
Blouw Statistical Methods in High Energy Physics