UNDERSTANDING THE KALMAN FILTER
Transcript of UNDERSTANDING THE KALMAN FILTER
FEDERICOBATTISTI
OUTLINE
2
β’ This presentation will cover a fairly complete description of our current Kalman Filter based
reconstruction algorithm together with some early findings from a Toy Monte Carlo study developed to
study the performance of said algorithm
β’ The presentation will be divided into 4 main sections:
β’ Description of what Kinematic Fitting is
β’ Description of what a Kalman Filter is
β’ Description of our own algorithm (T. Junk, DUNE-Doc-13933 https://docs.dunescience.org/cgi-bin/private/ShowDocument?docid=13933), which is a convolution of a Kalman Filter and a
Kinematic Fitting algorithm
β’ Early findings from my Toy Monte Carlo study
FEDERICOBATTISTI
KINEMATIC FITTING
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π₯βπ₯π
(π₯β, π¦β)
π
ππ¦
ππ₯
(π₯π, π¦β)
β’ A new value for the x coordinate π₯π
is calculated as an alternative to the
measured value π₯β
β’ π₯π is taken such that the weighted
distance (in terms of ππ₯ and ππ¦)
between the model π and the
measurement (π₯β, π¦β) is minimal.
β’ In the limit case where ππ¦ is very
large, this is equivalent to using the
measured π₯β
β’ In the limit case where ππ₯ is very
large, this is equivalent to using the π₯from the model (this is the case
shown in the diagram)
FEDERICOBATTISTI
KINEMATIC FITTING
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β’ Kinematic fitting is an algorithm that updates a measurement within error in the presence of a model
constraint.
β’ An example of this sort of fit can be found in https://inspirehep.net/literature/1780811 where
kinematic fitting has been used in the context of neutrino-induced charged-current (CC) π0production on carbon, to improve the neutral pion momentum reconstruction
β’ For more insights you can consult:
β’ https://www.phys.ufl.edu/~avery/fitting/kinfit_talk1.pdf
β’ http://www-hermes.desy.de/notes/pub/TALK/yaschenk.ColloqGlasgow.pdf
FEDERICOBATTISTI
KALMAN FILTER IN GENERAL
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β’ A Kalman filter is an iterative algorithm which uses a system's physical laws of motion, known control inputs and
multiple sequential measurements to form an estimate of the system's varying quantities
β’ At each step of the iteration an estimate of the state of the system is produced as a weighted average of the system's
predicted state and of the new measurement. The weights are calculated from the covariance.
β’ The extended Kalman filter expands the Kalman filter technique to non-linear systems
β’ The models for state transition and measurement can be written as:
π π = π π πβ1, ππβ1
β’ Where π is the function of the previous state, π πβ1 , and the free parameter, ππβ1, that provides the current state π π.
β’ Note that the measurement vector ππβ and the state vector π π do not necessarily have the same number of
components
ππβ
STATE VECTOR
MEASUREMENT
VECTOR
FEDERICOBATTISTI
KALMAN FILTER IN GENERAL
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1. Make a priori predictions for the current stepβs state and covariance matrix using the a posteriori best estimate of
the previous step (i.e. updated using measurement)
π πβ = π π πβ1
+ , ππβ1
ππβ = πΉπβ1ππβ1
+ πΉπβ1π + π
πΉπβ1 = α€ππ
ππ π πβ1+ ,ππβ1
π
JACOBIAN PROCESS NOISE
COVARIANCE
STATE VECTOR
COVARIANCE MATRIX
Note: In the first iteration step we use step 0 estimates for the state vector and the covariance matrix (π 0, π0), which
can be made very roughly
FEDERICOBATTISTI
KALMAN FILTER IN GENERAL
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2. Calculate the measurement residual and the Kalman Gain
π¦π = ππβ β π»(π π
β)
πΎπ = ππβπ»π π + π»ππ
βπ»π β1
3. Update the estimate
π π+ = π π
β + πΎπ π¦
ππ+ = 1 β πΎππ» ππ
β
RESIDUAL
KALMAN GAIN
STATE VECTOR
COVARIANCE MATRIX
π
MEASUREMENT
NOISE COVARIANCE
π»
CONVERSION
MATRIX
Note: the conversion matrix is
needed to make the dimensions
of vectors and matrixes turn out
right. For exemple if π πβ is a 2-
D vector and π πβ is 5-D, then H
would be a 2 Γ 5 matrix:
π» = (1 0 00 1 0
0 00 0
)
Note: in the case
where R is a null
matrix π π+ = π π
β
and ππ+ = 0
FEDERICOBATTISTI
KALMAN FILTER IN GENERAL
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(π₯0β, π¦0
β)
(π₯1β, π¦1
β)
(π₯1β, π¦1
β)
(π₯1β, π¦1
+)
(π₯2β, π¦2
+)
(π₯2β, π¦2
β) (π₯3β, π¦3
β) (π₯4β, π¦4
β)
(π₯3β, π¦3
+) (π₯4β, π¦4
+)
(π₯2β, π¦2
β) (π₯3β, π¦3
β)(π₯4
β, π¦4β)
π₯
π¦
π₯1β π₯2
β π₯4βπ₯3
β
FEDERICOBATTISTI
OUR RECONSTRUCTION ALGORITHMβ’
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T. Junk, DUNE-Doc-13933: https://docs.dunescience.org/cgi-bin/private/ShowDocument?docid=13933
FEDERICOBATTISTI
KINEMATIC FITTING
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ππ₯1
π₯1βπ₯1
π
(π₯1β, π¦1
β)
π
(π₯0π, π¦0
π)
β’ In our case we use kinematic fitting
to determine the free parameter step
ππ₯πβ’ ππ₯π is taken such that a weighted
distance Ξ (in terms of ππ₯ and ππ¦ =
ππ§ = ππ¦π§) between the model π and
the measurement (π₯πβ, π¦π
β) is
minimal.
β’ In the limit case where ππ¦π§ is very
large, this is equivalent to using the
measured π₯β
β’ In the limit case where ππ₯ is very
large, this is equivalent to using the
π₯ from the model (this is the case
shown in the diagram)
ππ¦
ππ₯
(π₯1π, π¦1
β)
FEDERICOBATTISTI
KINEMATIC FITTING
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β’ The quantity minimize in our kinematic fitting is the weighted distance Ξ with ππ₯, ππ¦ = (0.5ππ, 1ππ):
Ξ(ππ₯π; π₯πβ, π¦π
β, π§πβ, π₯πβ1
π, π¦πβ1
+ , π§πβ1+ , ππβ1
+ , ππβ1+ ) =
π₯πβ β π₯π
π
ππ₯
2
+π¦πβ β π¦π
β
ππ¦π§
2
+π§πβ β π§π
β
ππ¦π§
2
=π₯πβ β π₯πβ1
πβ ππ₯π
ππ₯
2
+π¦πβ β π¦πβ1
+ β ππ₯π Γ cot ππβ1 Γ sinππβ1+
ππ¦π§
2
+π§πβ β π§πβ1
+ β ππ₯π Γ cot ππβ1 Γ cosππβ1+
ππ¦π§
2
β’ We determine ππ₯π by imposing that the derivative of Ξ is equal to zero
ππ₯π =
cot ππβ1+
ππ¦π§2 π¦π
β β π¦πβ1+ sinππβ1
+ + π§πβ β π§πβ1
+ cosππβ1+ +
π₯πβ β π₯πβ1
π
ππ₯2
ΰ΅cot2 ππβ1
+
ππ¦π§2 + ΰ΅1 ππ₯
2
dΞ
d(ππ₯π)= 0
FEDERICOBATTISTI
OUR KALMAN FILTER: VISUALIZATION
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π¦0
π¦1β
π¦1β
π¦1+
π¦
π₯1β π₯2
β π₯3βπ₯1
ππ₯2π π₯3
π
Ξπππ
ππ₯π
π₯πβπ₯π
π
(π₯πβ , π¦π
β)
π(π₯πβ1π
, π¦πβ1+ )
(π₯πβ1π
, π¦πβ1+ )
KINEMATIC FITTING
STANDARD KALMAN FILTER
OUR KALMAN FILTERβ’
+
=
π₯
FEDERICOBATTISTI
TOY MONTE CARLO OUTLINE
β’ We constructed a Toy Monte Carlo study as a sanity check for our current Kalman filterβ’
β’ The study will eventually develop in multiple steps:
β’ Use a perfect helix model with fixed step length to check the a priori model
β’ Use a perfect helix model with randomized step length (but no smearing)
β’ Add Gaussian error on π₯π¦π§ (1D at a time, then 2D at a time, then fully 3D) to validate covariance and
calibrate π2
β’ Use Cauchy smearing instead to test sensitivity to noise model
β’ Simulate Energy Loss etc..
β’ The current presentation will cover the first two steps in the study
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FEDERICOBATTISTI
TOY MONTE CARLO
β’ We applied our Kalman Filterβ’ to ideal
measurements following perfectly an helix with
initial coordinates: π 0π =
(π¦0 π§0 1/π0 π0 π0) =π¦0β π§0
β 0.014 ππβ1 6 rad β0.05 radand the free parameter π₯0 = π₯0
β and the step
ππ₯ = 0.04 cm
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Note: even though we have unsmeared
idealized measurement for this first study
we kept the βπ (noise) matrixβ unmodified
with uncertainties Ξ£π₯π¦ = 4ππ
FEDERICOBATTISTI
TOY MONTE CARLO
β’ Plots describing the evolution of the estimate of the measured quantities (π¦, π§) as a function of the free parameter π₯ for the
perfect helix. Note that the TPC points have on the x coordinate the measured value, while for the estimates we use the estimated
π₯ from the kinematic fitting (i.e. π₯π = π₯0 + π Γ ππ₯π)
18FILTER PROPAGATION DIRECTION
KFβ’: ππ₯, ππ¦π§ = (0.5ππ, 1ππ)Note: the βTPC Cluster measured valuesβ are
from a Toy MC, so the red error bars are the
dummy values from the R matrix
FEDERICOBATTISTI
TOY MONTE CARLO
β’ We now replot the results from the previous slide, all identical except replacing the π₯ coordinate by π₯β instead of π₯π. The
plots show that the predicted π¦ and π§ components coincide with the measured ones
19FILTER PROPAGATION DIRECTION
Note: this is not the same as using
the standard Kalman filter since y
and z have been estimated using
the x from kinematic fitting, as in
the previous slideKFβ’: ππ₯, ππ¦π§ = (0.5ππ, 1ππ)
FEDERICOBATTISTI
TOY MONTE CARLO: UNDERSTANDING STEP DETERMINATION
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β’ With the current sigma values (ππ₯ = 0.5ππ and ππ¦π§ = 1ππ ) the ratio ππ₯/ππ¦ is large enough that the evolution of
π¦ and π§ is essentially decoupled from the evolution of the free parameter π₯.
β’ This gives us π¦πβ β π¦π
β and π§πβ β π§π
β and an π₯π that is updated completely disregarding the measured π₯β values
β’ A way to see this is to divide the ππ₯π update formula into two parts: one that depends on the evolution of the π₯ free
parameter, and one on π¦ and π§:
ππ₯π =
cot ππβ1+
ππ¦π§2 π¦π
β β π¦πβ1+ sinππβ1
+ + π§πβ β π§πβ1
+ cos ππβ1+
ΰ΅cot2 ππβ1
+
ππ¦π§2 + ΰ΅1 ππ₯
2
+
π₯πβ β π₯πβ1
π
ππ₯2
ΰ΅cot2 ππβ1
+
ππ¦π§2 + ΰ΅1 ππ₯
2
= ππ₯π¦π§ + ππ₯π₯
β’ With the current sigma values (ππ₯ = 0.5ππ and ππ¦π§ = 1ππ ) ππ₯π¦π§ completely dominates, being often 2 or 3
orders of magnitude larger than ππ₯π₯: this gives us fairly predictions for π¦ and π§ very close to the measured values,
but completely wrong ones for π₯, becuase the fit can never recover from a bad initial estimate for the free
parameter
FEDERICOBATTISTI
TOY MONTE CARLO: UNDERSTANDING STEP DETERMINATION
21FILTER PROPAGATION DIRECTION
β’ In order for the x parameter to have more weight in the update of ππ₯π, changed the value of ππ¦π§ from 1cm to 4cm, leaving ππ₯ = 0.5ππ
β’ The fit initially concentrates on fixing the π₯ prediction until that becomes accurate, and then it focuses on yz, reaching similar levels of
precision by the end
KFβ’: ππ₯, ππ¦π§ = (0.5ππ, 4ππ)
FEDERICOBATTISTI
TOY MONTE CARLO: UNDERSTANDING STEP DETERMINATION
22FILTER PROPAGATION DIRECTION
β’ We now test an extreme case were ππ¦π§/ππ₯ is very large: ππ¦π§ is changed from 1cm to 1000cm, leaving ππ₯ = 0.5ππ
β’ This becomes very close to be using π₯β instead of π₯π
KFβ’: ππ₯, ππ¦π§ = (0.5ππ, 1000ππ)
FEDERICOBATTISTI
TOY MONTE CARLO: UNDERSTANDING STEP DETERMINATION
23FILTER PROPAGATION DIRECTION
β’ We try applying a standard Kalman Filter to our Toy Monte Carlo, such that it is identical to our Kalman Filterβ’ but it avoids the
kinematic fitting algorithm completely
β’ We get essentially the same result as in the last slide: having ππ¦π§/ππ₯ very large is equivalent to a standard Kalman Filter
FEDERICOBATTISTI
RANDOMIZED X STEP
β’ Now we try applying our Kalman Filter to
ideal measurements following perfectly an
helix with initial coordinates: π₯ππ =
(π¦π π§π 1/ππ ππ ππ) =π¦1β π§1
β 0.014 ππβ1 6 rad β0.05 radand the free parameter π₯π = π₯1
β but with a
randomized step ππ₯ uniformely distributed
between 0.02cm and 0.06cm.
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Note: the step progression is rondomized,
but we are still following a perfect helix
i.e. No smearing on the xyz coordinates
is involved
FEDERICOBATTISTI
RANDOMIZED X STEP
β’ Plots describing the evolution of the estimate of the measured quantities (π¦, π§) as a function of the free parameter π₯ for the
perfect helix. Note that the the TPC points have on the x coordinate the measured value, while for the estimates we use the
estimated x (i.e. π₯π = π₯0 + π Γ ππ₯π)
25FILTER PROPAGATION DIRECTION
KFβ’: ππ₯, ππ¦π§ = (0.5ππ, 1ππ)
FEDERICOBATTISTI
RANDOMIZED X STEP
β’ Reapplying the fit with the new ππ¦π§ = 4cm we see that the 3D predictions are more in line with the Monte Carlo truth, past the
first few steps
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FILTER
PROPAGATION
DIRECTION
KFβ’: ππ₯, ππ¦π§ = (0.5ππ, 1ππ) KFβ’: ππ₯, ππ¦π§ = (0.5ππ, 4ππ)
FEDERICOBATTISTI
RANDOMIZED X STEP
β’ Reapplying the fit with the new ππ¦π§ = 4cm we see that the 3D predictions are more in line with the Monte Carlo truth, past the
first few steps
27
FILTER
PROPAGATION
DIRECTION
KFβ’: ππ₯, ππ¦π§ = (0.5ππ, 1ππ) KFβ’: ππ₯, ππ¦π§ = (0.5ππ, 4ππ)
FEDERICOBATTISTI
RANDOMIZED X STEP: TESTING THE RATIO
β’ Since ππ₯ and ππ¦π§ are essentially weights, only their ratio ππ₯/ππ¦π§ should matter. To test this we try the pair ππ₯ , ππ¦π§ =
(1ππ, 8ππ) which has the same ratio as ππ₯ , ππ¦π§ = (0.5ππ, 4ππ)
28
FILTER
PROPAGATION
DIRECTION
KFβ’: ππ₯, ππ¦π§ = (1ππ, 8ππ)KFβ’: ππ₯, ππ¦π§ = (0.5ππ, 4ππ)
FEDERICOBATTISTI
RANDOMIZED X STEP: TESTING THE RATIO
β’ Since ππ₯ and ππ¦π§ are essentially weights, only their ratio ππ₯/ππ¦π§ should matter. To test this we try the pair ππ₯ , ππ¦π§ =
(1ππ, 8ππ) which has the same ratio as ππ₯ , ππ¦π§ = (0.5ππ, 4ππ)
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FILTER
PROPAGATION
DIRECTION
KFβ’: ππ₯, ππ¦π§ = (0.5ππ, 4ππ) KFβ’: ππ₯, ππ¦π§ = (1ππ, 8ππ)
FEDERICOBATTISTI
RANDOMIZED X STEP
β’ Plots comparing the reconstructed tansverse momentum ππ and the reconstructed total momentum ππ from our Kalman filter
algorithmβ’ to the MC truth as a function of the free parameter π₯
30FORWARD FITKFβ’: ππ₯, ππ¦π§ = (0.5ππ, 1ππ)
FEDERICOBATTISTI
RESULTS FROM THE STANDARD KALMAN FILTER
31FITTER PROPAGATION DIRECTION
β’ We try applying a standard Kalman Filter to our Toy Monte Carlo, such that it is identical to our Kalman Filterβ’ but it avoids the
kinematic fitting algorithm completely
β’ This means that our free parameter coincides with the measured value of x for the TPC cluster
FEDERICOBATTISTI
APPLYING A VERY SMALL R A STANDARD KALMAN FILTER
32FITTER PROPAGATION DIRECTION
β’ We try applying a standard Kalman Filter to our Toy Monte Carlo, substituting the values in the π matrix with Ξ£π₯π¦ = 10β6cm ,
where before we had Ξ£π₯π¦ = 4cm: with such a small R the Kalman filter follows the measurements almost exactly
FEDERICOBATTISTI
APPLYING A VERY SMALL R TO OUR KALMAN FILTER
33FITTER PROPAGATION DIRECTION
β’ We try applying our Kalman Filterβ’ to our Toy Monte Carlo, but substituting the values in the π matrix with Ξ£π₯π¦ = 10β6cm ,
where before we had Ξ£π₯π¦ = 4cm: with such a small R the Kalman filter should converge with the measurement very quickly
(this happens immediately with a Standard Kalman filter, as you can see in the previous slide)
KFβ’: ππ₯, ππ¦π§ = (0.5ππ, 4ππ)
FEDERICOBATTISTI
SUMMARY AND FUTURE STEPS
β’ The Kalman Filter now in use is convoluted with Kinematic Fitting which determines the evolution of the free
parameter, minimizing the distance between the measured value and the a priori prediction
β’ The Kinematic Fitting is regulated by two weights ππ¦π§ and ππ₯ whose ratio determines weather the fit is
dominated by corrections related to the π¦π§ or the π₯ measurements and predictions
β’ With the previous values of ππ₯, ππ¦π§ = (0.5ππ, 1ππ) the Kalman Fit disregarded the π₯ measurements in the
update
β’ Increasing the value of ππ¦π§ ,bringing the ratio from Ξ€ππ¦π§ ππ₯ = 2 to Ξ€ππ¦π§ ππ₯ = 8 the fit better takes into
account the x measurements and follows the Monte Carlo truth more closely
β’ Future steps:
β’ Weights with a larger Ξ€ππ¦π§ ππ₯ ratio (i.e. following the x measurement more closely) should maybe be
used
β’ A smeared helix Toy Monte Carlo needs to be investigated
β’ Other options for the free parameter need to be studied
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FEDERICOBATTISTI
KALMAN FILTER APPLICATION
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β’ We apply a Kalman filter to the motion of a charged particle in the magnetic field
α
π₯ = π₯0 + π tanπ (π β π0)π¦ = π¦πΆ β π cosππ§ = π§πΆ + π sinπ
TRACK PARAMETERS
We apply the Kalman filter to track candidates,
consisting of groups of TPC clusters, which are
identified and put together during the
reconstruction process. Each step of the Kalman
filter algorithm is identified by one of these TPC
clusters
T. Junk, DUNE-Doc-13933
https://docs.dunescience.org/cgi-bin/private/ShowDocument?docid=13933
FEDERICOBATTISTI
KALMAN FILTER: COORDINATE SYSTEM
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β’ Note that in our coordinate sytem z is the flux direction, y is the vertical direction and x is the drift direction (i.e.
the magnetic field direction)
y
z
x
FEDERICOBATTISTI
KALMAN FILTER APPLICATION: INITIAL ESTIMATES
β’ Before the Kalman filter algorithm can be applied, we need an initial estimate for the state vector, which in our case
includes π¦, π§, Ξ€1 π , π and π and the covariance matrix
π₯0π = (π¦0 π§0 1/π0 π0 π0) = 0 0 0.1 0 0
π0 =
12 0 00 12 0000
000
0.52
00
000
0.52
0
0000
0.52
STATE VECTOR
COVARIANCE
MATRIX
38
β’ The estimated quantities from the state vector can be used to estimate the particleβs momentum
ΰ΅
ππ₯ = ππ tan πππ¦ = ππ sinπ
ππ§ = ππ cosπ
ππ Ξ€GeV π = 0.3 Γ π΅ π Γ π(π)
FEDERICOBATTISTI
KALMAN FILTER APPLICATION: PREDICTION AND MEASUREMENT
39
β’ From the equation of motion we obtain the prediction function for our state vector. Note that the drift direction π₯ is
used as our free parameter
ΰ·π₯πβ = π
ΰ·π¦πβ1+
ΖΈπ§πβ1+
1/ ΖΈππβ1+
ππβ1+
αππβ1+
=
ΰ·π¦πβ1+ + dxk Γ cot αππβ1 Γ sin ππβ1
+
ΖΈπ§πβ1+ + dxk Γ cot αππβ1 Γ cos ππβ1
+
1/ ΖΈππβ1+
ππβ1+ + dxk Γ cot αππβ1 Γ1/ ΖΈππβ1
+
αππβ1+
β’ The only measured quantities in our case are π¦ and π§, and are set at the center of the TPC cluster correspondent to
the present step.
π πβ =
π¦πβ
π§πβ
Note: the prediction model
does not account for dE/dx
energy loss
FEDERICOBATTISTI
KALMAN FILTER APPLICATION: COVARIANCE MATRIX PREDICTION
40
β’ First step to make the prediction for the covariance matrix is to calculate the Jacobian
πΉπβ1 =ππ ΰ·π₯πβ1
+
π ΰ·π₯πβ1+ =
1 00000
1000
001
ππ₯π cot αππβ1+
0
ππ₯π cot αππβ1+ cos ππβ1
+
βππ₯π cot αππβ1+ sin ππβ1
+
010
ππ₯π sin ππβ1+ (β1 β cot2 αππβ1
+ )
ππ₯π cos ππβ1+ (β1 β cot2 αππβ1
+ )0
Ξ€ππ₯π ΖΈππβ1+ (β1 β cot2 αππβ1
+ )1
β’ The step uncertainty matrix is also needed
π =
0 00000
0000
00
πΞ1/π00
000πΞπ0
0000πΞπ
β’ The prediction is then:
ππβ = πΉπβ1ππβ1πΉπβ1
π + π
FEDERICOBATTISTI
KALMAN FILTER APPLICATION: EVALUATE THE RESIDUAL
β’ We now evaluate the residual and Kalman Gain
π¦π = π πβ βπ» ΰ·π₯π
β =π¦πβ β ΰ·π¦π
β
π§πβ β ΖΈπ§π
β
πΎπ = ππβπ» π + π»ππ
βπ»π β1
π» =1 0 00 1 0
00
00
π =Ξ£π¦π§2 0
0 Ξ£π¦π§2
KALMAN GAIN
RESIDUAL With:
CONVERSION MATRIX
MEASUREMENT NOISE COVARIANCE
Note: The uncertainties in R are fixed, before the Kalman filter is applied, as external parameters: R is not updated.
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FEDERICOBATTISTI
KALMAN FILTER APPLICATION: PREDICTION UPDATE
β’ We are now finally able to update our estimates using both the a priori prediction and the measurement
ΰ·π₯π+ = ΰ·π₯π
β + πΎπ π¦π
ππ+ = 1 β π»ππΎπ ππ
β
STATE VECTOR
COVARIANCE MATRIX
42
FEDERICOBATTISTI
TOY MONTE CARLO
β’ Plots describing the evolution of the estimate of the non measured quantities (1
π, π, π) as a function of the free parameter π₯ for the
perfect helix
43
FILTER PROPAGATION DIRECTION
43
ππ₯, ππ¦π§ = (0.5ππ, 1ππ)
FEDERICOBATTISTI
TOY MONTE CARLO
β’ Plots comparing the reconstructed tansverse momentum ππ and the reconstructed total momentum ππ from the Kalman fitter
algorithm to the MC truth as a function of the free parameter π₯
44FORWARD FITππ₯, ππ¦π§ = (0.5ππ, 1ππ)
FEDERICOBATTISTI
RANDOMIZED X STEP
β’ Plots describing the evolution of the estimate of the non measured quantities (1
π, π, π) as a function of the free parameter π₯ for the
perfect helix
45
FILTER PROPAGATION DIRECTION
45
ππ₯, ππ¦π§ = (0.5ππ, 1ππ)
FEDERICOBATTISTI
UNDERSTANDING STEP DETERMINATION
β’ Reapplying the fit with the new ππ¦π§ = 4cm we see that the 3D predictions are more in line with the Montecarlo truth, past the
first few steps
46
FITTER
PROPAGATION
DIRECTION
ππ¦π§ = 2ππ ππ¦π§ = 4ππ
FEDERICOBATTISTI
UNDERSTANDING STEP DETERMINATION
β’ Reapplying the fit with the new ππ¦π§ = 4cm we see that the 3D predictions are more in line with the Montecarlo truth, past the
first few steps
47
ππ¦π§ = 2ππ
FITTER
PROPAGATION
DIRECTION
ππ¦π§ = 4ππ