Bayesian Statistics to calibrate the LISA Pathfinder ...• Analyse each experiment separately: good...

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!

http://www.lisa.aei-hannover.de/ltpda/

• 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


• 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:


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� ⌧).


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!


• 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*.


* 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*.


* 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.

The Sys-ID Pipeline for on-line analysis��

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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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