Bayesian Statistics to calibrate the LISA Pathfinder
experimentNikolaos Karnesis for the LTPDA team
!!
LISA Symposium X 20/05/2014
http://gwart.ice.cat/
• LPF System Overview
• LPF System Identification Experiments
• Data Analysis framework
• Applications to Simulated Data
• The Pipeline Design for on-line analysis
Outline
eLISA to LISA Pathfinder
* Squeeze two eLISA SCs into one SC.• Prove geodesic motion by
monitoring the relative acceleration of the two test masses.
• Characterise all noise sources of the instrument, build accurate noise models.
• Test all key technologies for the future space-based gravitational-wave detectors.
> Science Mode: TM2 is following TM1 through capacitive actuators.
(simplified 1D case)
> The LPF noise budget*:
*Antonucci et al, CQG, 29-124014
• “Kick” the system, measure the response, get the system parameters.
• Two main dynamics sys-id experiment families:
• X-axis
• cross-talk
System Identification
Data Analysis & Parameter Estimation
• LTPDA toolbox!
• It’s there: http://www.lisa.aei-hannover.de/ltpda/
• All the data analysis tools presented here are available in the toolbox with the proper documentation!
• Command along the “sensitive” x-axis between the two test-masses
• Large signal-to-noise ratio, satisfactory recovery of the parameters.
• Three experiments:
1. “fake displacement”, unmatched stiffness.
2. “fake displacement”, matched stiffness.
3. Out-of-loop forces injections to the three bodies of the system.
Sensitive x-axis system identification
• For the parameter estimation, the standard approach:
A. Assume that
B. thenwhere, and
C. Perform the fit using MCMC* methods.
D. Also use linear•, or non-linear† methods.
Data Analysis & Parameter Estimation
~d = ~h+ ~n
< ~a|~b >= 2
1Z
0
ha⇤(f)b(f) + a(f)b⇤(f)
i/Sn(f)
�2 ⌘< ~d� ~h(~✓)|~d� ~h(~✓) >
⇡(~d|~✓) = C ⇥ exp[�1/2⇥ <
~d� ~h(~✓)|~d� ~h(~✓) >]
* PRD82, 122002, (2010), •CQG, 28 094006 (2011), †PRD85, 122004, (2012)
• Perform the fit in the “acceleration” domain.
• The model now, looks like:
• And in particular, for the differential acceleration (x-axis):
System identification along the x-axis
�a =
NgX
j=1
�gj
(~✓) +�gnoise
�a =
d
2
dt
2+ !
22
�x12(t� ⌧) +
�!
22 � !
21
�x1(t� ⌧)�AFcmd,TM2
• Given this equation we can now minimise the log-likelihood by following the following recipes:
A. Iterative chi2 minimisation:
1. define initial set of parameters
2. estimate
3. minimise the log-likelihood
4. get an estimate of
5. set , repeat from 2, until equilibrium.
System identification along the x-axis: -iterative chi^2 method
~✓new
Sn(~✓)
~✓0 = ~✓new
ms−
2 H−1/2
Frequency [Hz]
10−14
10−13
10−12
10−11
10−10
0.0001 0.001 0.01 0.1
initial
loop 1
loop 2
loop 3
loop 4
loop 5
~✓0
log(⇡(~d|~✓)) = ��
2/2
System identification along the x-axis: - By modelling the noise
* Littenberg et al, PRD80, 063007, 2009
m2s−4 H
z−1
10−30
10−25
10−20
Frequency [Hz]
η
0.8
1
1.2
0.001 0.01
Sn,i ! ⌘jSn,i, ij < i ij+1
log(⇡(~d|✓)) = �1/2
0
@�
2 +X
j
Nj log(⌘j)
1
A+ C
B. Assume that the noise can be written as*
1. then define the log-likelihood function as
2. assign priors, sample the posterior.
i ! bin, j ! segment
comparison of the iterative chi^2 and the noise modelled log-likelihood resulting parameter estimates.
τ
0.4007 0.40075 0.4008
Asus
1.0499 1.05
ω1 × 10−6-1.36 -1.35
ω2 × 10−6-2.262 -2.26 -2.258 -2.256 -2.254
Real Pole (Hz)
0.199 0.1995
χ2
noise fit
• the log-likelihood then turns into where, Q is the set of DFT coefficients and the residuals.
System identification along the x-axis: - Assuming unknown and unmodeled noise
† Vitale et al, arXiv:1404.4792, submitted to PRD
C. Assume that all noise sources zero-mean and Gaussian. Also taking into account the spectral window properties, one can marginalise over the noise parameters†.
*** See next talk from D. Vetrungo, for a more detailed explanation!
log(⇡(
~
✓|~d)) = �Ns
X
k2Q
log
✓|n
hk,
~
✓
i|2◆
n
Cross-talk system identification
• Command forces and torques in different degrees of freedom (φ1, φ2, y1, y2, Φ).
• measure with the sensitive differential channel (o12).
• estimate cross-talk/cross-coupling coefficients.
• Lower resulting SNR.
• The parameters to estimate in this case are:
1. system parameters (gains, delays)
2. cross-talk terms (piston effects, mechanical imperfections, cross-stiffness, secondary effects)
Cross-talk System Identification
din = 4 mm
ΣCin = 4.40 pF
dy = 2.9 mm
Cy = .83 pF
dx = 4 mm
Cx = 1.15 pF
dz = 3.5 mm
Cz = .61 pF
Ctot = 25.6 pF
y
z
x
θ
φ
η
* See also, next talk by D. Vetrungo for the physics behind
0 1 2 3 4 5 6 7 8
x 104
−1
−0.5
0
0.5
1
x 10−3
Origin: 2013−11−16 08:00:01.000 UTC − [s]
Va
lue
[ra
d]
−2
−1
0
1
2
x 10−8
Va
lue
[m
]
iy1
iy2
iΦ
iφ1
iφ2
• Analysis of each experiment separately, or
• define a model that describes all the cross-coupling terms for all the cross-talk experiments
Cross-talk System Identification
• Analyse each experiment separately: good understanding of physics for each injection case.
• Analyse the joint experiments: verify that all cross-talk terms are included in the dynamics model subtract total produced acceleration and reach the noise level.
• An example of the joint analysis model could be:
Cross-talk System Identification
a12,ct =� �
�1�1 � �
�1�1 + �
�
21�
21
+ ��N�
�N
cmd,�1(t� ⌧)�N
cmd,�2(t� ⌧)�
� �
�2�2 � �
�2�2 + �
�
22�
22
� �
y1 y1 � �
y1y1 � �
y2 y2 � �
y2y2.
+ ��N✓
�N
cmd,✓1(t� ⌧)�N
cmd,✓2(t� ⌧)�
+ ��N⌘
�N
cmd,⌘1(t� ⌧)�N
cmd,⌘2(t� ⌧)�
� !
22(x12 + x1) + !
21x1 +A
sus
F
cmd,x2(t� ⌧).
Cross-talk System Identification
Sample the posterior with MCMC methods. Extract covariance/correlation matrices from the chains.
Cross-talk System Identification
Sample the posterior with MCMC methods. Extract covariance/correlation matrices from the chains.
ms−2 H
z−1/2
Frequency (Hz)
0.0001 0.001 0.01 0.1
10−14
10−13
10−12
10−11
10−10
noise level acceleration
experiment acceleration
estimated residual acceleration
• For the given simulated data-set, the results are quite satisfying!
Cross-talk System Identification
• Since the cross-talk experiment requires a high dimensionality model,
• and many physical effects contribute with very low SNR…
• we can apply other Bayesian techniques like the Reversible Jump MCMC to perform model selection*.
Cross-talk System Identification
* Karnesis et al, PRD89, 062001, 2014
‣ A generalised MCMC: allows transdimensional moves.
‣ Directly calculates the Bayes factor (ratio of the “evidences” of the models)
‣ Will most probably be used off-line. Other approximations (like the Laplace) can be put to use during operations.
Frequency
16pδ y
δηδθ
16pδ y
η 215p
16p
¨δy
1
10
100
103
104
105
106
107
• Since the cross-talk experiment requires a high dimensionality model,
• and many physical effects contribute with very low SNR…
• we can apply other Bayesian techniques like the Reversible Jump MCMC to perform model selection*.
Cross-talk System Identification
* Karnesis et al, PRD89, 062001, 2014
‣ A generalised MCMC: allows transdimensional moves.
‣ Directly calculates the Bayes factor (ratio of the “evidences” of the models)
‣ Will most probably be used off-line. Other approximations (like the Laplace) can be put to use during operations.
• This work presented here, is integrated in data analysis pipelines.
• The pipeline includes all analysis steps, from downloading the telemetry, to the submission of the analysis results.
• Can be modified/tuned/configured by the user.
• Flexibility to adjust the model symbolic equation on the spot.
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• We have developed a Bayesian tool to perform system identification/Model Selection for the LPF experiments.
• It has been integrated to data analysis pipelines.
• Already being tested systematically in numerous Simulations.
• Always improving/enhancing
• Getting ready for the launch!!!
Summary
Thank you! Questions?
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