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The extended Global Sky Model (eGSM)
Adrian Liu, UC Berkeley/McGill
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The extended Global Sky Model (eGSM) project
AL, UC Berkeley/McGillDoyeon “Avery” Kim, UC BerkeleyEric Switzer, NASA Goddard Haoxuan “Jeff” Zheng, MIT/Intel
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What does the sky look like in all directions at “all” frequencies?
10 MHz
408 MHz
85 MHz
??????
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My motivation
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Cosmic Microwave Background
21cm Cosmology
(See AL,et al. 2013 PRD 87, 043002 for more details)
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How does one model the sky?
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Global Sky Model v1 (de Oliveira-Costa et al. 2008, MNRAS 388, 247)
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Take a wide selection of survey data…
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…identify common regions…
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…which are then used to train three principal component
spectral templates…
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…that are used to fit the spectra in every pixel of the sky…
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…that are used to fit the spectra in every pixel of the sky…
T (r̂, ⌫) = m1(r̂)c1(⌫) +m2(r̂)c2(⌫) + . . .
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…that are used to fit the spectra in every pixel of the sky…
T (r̂, ⌫) = m1(r̂)c1(⌫) +m2(r̂)c2(⌫) + . . .
Given these
Solve for these
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…that are used to fit the spectra in every pixel of the sky…
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…and are interpolated to produces maps of the sky at
“any” frequency
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Global Sky Model v2 (Zheng… Kim, AL… et al. 2017, MNRAS 464, 3486)
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Take an even wider selection of updated maps…
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…now using six spectral components…
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…and iteratively fitting for spectral and spatial information across the whole sky
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…and iteratively fitting for spectral and spatial information across the whole sky
T (r̂, ⌫) = m1(r̂)c1(⌫) +m2(r̂)c2(⌫) + . . .
Given these
Solve for these
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…and iteratively fitting for spectral and spatial information across the whole sky
T (r̂, ⌫) = m1(r̂)c1(⌫) +m2(r̂)c2(⌫) + . . .
Solve for these
Given these
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…and iteratively fitting for spectral and spatial information across the whole sky
T (r̂, ⌫) = m1(r̂)c1(⌫) +m2(r̂)c2(⌫) + . . .
Given these
Solve for these
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…to derive even even better fits to the data.
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Global Sky Model v3 (Kim, AL, Switzer 2017, in prep.)
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Why three components? Why six components?Too few components: inadequate fits to data Too many components: overfitting of data
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Computing the Bayesian Evidence provides a way to determine the optimal
number of principal components to fitIm
age
cred
it:
Zoub
in G
hahr
aman
i
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Computing the Bayesian Evidence provides a way to determine the optimal
number of principal components to fit
Maximum likelihood
Greatest evidence
Imag
e cr
edit:
Zo
ubin
Gha
hram
ani
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Optimal number of principal components
2 13
TFrequency [GHz]
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Optimal number of principal components
2 13
Evid
ence
3 4 5 6 7
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Optimal number of principal components
2 13
Evid
ence
3 4 5 6 7
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Optimal number of principal components
2 13
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But why even commit to a model? Use evidence as a weight for constructing hybrid models that are noncommittal
to the number of componentsEv
iden
ce
3 4 5 6 7
Wei
ghts
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2 13
Effective number of principal components
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The old versions of the GSM had no error bars!
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Solution: construct models for the errors in the input data, and Monte Carlo to
get final errors in our predictions
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Solution: construct models for the errors in the input data, and Monte Carlo to
get final errors in our predictions• Where available, use provided estimates of errors
and covariances
LWA 74 MHz, Dowell et al. (2017)
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Solution: construct models for the errors in the input data, and Monte Carlo to
get final errors in our predictions• Where available, use provided estimates of errors
and covariances
• Errors in the model itself modelled empirically
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Run model again with an input map removed, making a prediction for the missing map
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Subtract the new predicted map from the observed data
Prediction Data
—
=
“Error”
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“Error”
Ansatz for errors in image space:
proportional to error map
Ansatz for harmonic space:determined by Cl of whitened error map
Error model
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An example 408 MHz prediction
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Errors on the 408 MHz prediction
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Solution: construct models for the errors in the input data, and Monte Carlo to
get final errors in our predictions• Where available, use provided estimates of errors
and covariances
• Errors in the model itself modelled empirically
• Interpolation errors accounted for using Gaussian Process regression.
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Lots more coming soon to a Github repo near you!
• Position-dependent number of components. • Error bars in output maps. • Framework for incorporating monopole
measurements. • Inclusion of new map data.
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Lots more coming soon to a Github repo near you!
• Position-dependent number of components. • Error bars in output maps. • Framework for incorporating monopole
measurements. • Inclusion of new map data.
End goal: a publicly hosted, self-updating, best-guess
model of the sky