power spectrum estimation in theory and in practice
DESCRIPTION
Power Spectrum Estimation in Theory and in Practice. Adrian Liu, MIT. What we would like to do. Inverse noise and foreground covariance matrix. Vector containing measurement. What we would like to do. “Geometry” -- Fourier transform, binning. Bandpower at k . - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/1.jpg)
Power Spectrum Estimation in Theory and
in Practice
Adrian Liu, MIT
![Page 2: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/2.jpg)
What we would like to do
Inverse noise and foreground covariance
matrix
Vector containing
measurement
![Page 3: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/3.jpg)
What we would like to do
Bandpower at k“Geometry” -- Fourier
transform, binningNoise/residual
foreground bias removal
![Page 4: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/4.jpg)
Why we like this method• Lossless
CleanedData
RawDataCleaning
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Why we like this method• Lossless• Smaller “vertical” error bars
![Page 6: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/6.jpg)
Why we like this method• Lossless• Smaller “vertical” error bars
100
0.02 0.04 0.060.08
101
10 mK
1 K100 mK
3.0
2.5
2.0
1.5
1Log10 T(in mK)
Errors using Line of Sight Method
AL, Tegmark, Phys. Rev. D 83, 103006 (2011)
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Why we like this method• Lossless• Smaller “vertical” error bars
100
0.02 0.04 0.060.08
101
<10 mK
130 mK
3.0
2.5
2.0
1.5
1Log10 T(in mK)
Errors using Inverse Variance Method
30 mK
AL, Tegmark, Phys. Rev. D 83, 103006 (2011)
![Page 8: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/8.jpg)
Why we like this method• Lossless• Smaller “vertical” error bars• Smaller “horizontal” error bars
![Page 9: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/9.jpg)
Why we like this method• Lossless• Smaller “vertical” error bars• Smaller “horizontal” error bars
100
101
10-210-110010-1
1.0
0.60.50.40.30.20.1
0.70.80.9
AL, Tegm
ark, Phys. Rev. D
83, 103006 (2011)
![Page 10: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/10.jpg)
Why we like this method• Lossless• Smaller “vertical” error bars• Smaller “horizontal” error bars
100
101
10-210-110010-1
1.0
0.60.50.40.30.20.1
0.70.80.9
AL, Tegm
ark, Phys. Rev. D
83, 103006 (2011)
![Page 11: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/11.jpg)
Why we like this method• Lossless• Smaller “vertical” error bars• Smaller “horizontal” error bars• No additive noise/foreground bias
![Page 12: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/12.jpg)
Why we like this method• Lossless• Smaller “vertical” error bars• Smaller “horizontal” error bars• No additive noise/foreground bias• A systematic framework for evaluating
error statistics
![Page 13: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/13.jpg)
Why we like this method• Lossless• Smaller “vertical” error bars• Smaller “horizontal” error bars• No additive noise/foreground bias• A systematic framework for evaluating
error statistics
BUT
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Why we like this method• Lossless• Smaller “vertical” error bars• Smaller “horizontal” error bars• No additive noise/foreground bias• A systematic framework for evaluating
error statistics
BUT• Computationally expensive because
matrix inverse scales as O(n3). [Recall C-1x]
• Error statistics for 16 by 16 by 30 dataset takes CPU-months
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Quicker alternatives
Full inverse variance
AL, Tegmark 2011
O(n log n) versionDillon, AL, Tegmark (in
prep.)
FFT + FKPWilliams, AL,
Hewitt, Tegmark
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Quicker alternatives
Full inverse variance
AL, Tegmark 2011
O(n log n) versionDillon, AL, Tegmark (in
prep.)
FFT + FKPWilliams, AL,
Hewitt, Tegmark
![Page 17: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/17.jpg)
O(n log n) version• Finding the matrix inverse C-1 is the
slowest step.
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O(n log n) version• Finding the matrix inverse C-1 is the
slowest step.• Use the conjugate gradient method for
finding C-1x, which only requires being able to multiply by Cx.
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O(n log n) version• Finding the matrix inverse C-1 is the
slowest step.• Use the conjugate gradient method for
finding C-1, which only requires being able to multiply by C.
• Multiplication is quick in basis where matrices are diagonal.
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O(n log n) version• Finding the matrix inverse C-1 is the
slowest step.• Use the conjugate gradient method for
finding C-1, which only requires being able to multiply by C.
• Multiplication is quick in basis where matrices are diagonal.
• Need to multiply by C = Cnoise + Csync + Cps + …
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Different components are diagonal in different
combinations of Fourier space
C = Cps + Csync + Cnoise + …
Real spatialFourier spectral
Fourier spatialFourier spectral
Real spatialReal
spectral
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Comparison of Foreground Models
GSMOur
model
Eig
enva
lue
AL, Pritchard, Loeb, Tegmark, in prep.
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Quicker alternatives
Full inverse variance
AL, Tegmark 2011
O(n log n) versionDillon, AL, Tegmark (in
prep.)
FFT + FKPWilliams, AL,
Hewitt, Tegmark
![Page 24: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/24.jpg)
FKP + FFT version
Bandpower at k“Geometry” -- Fourier
transform, binningNoise/residual
foreground bias removal
![Page 25: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/25.jpg)
FKP + FFT version• Foreground avoidance instead of
foreground subtraction.
100
0.02 0.04 0.060.08
101
10 mK
1 K100 mK
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FKP + FFT version• Foreground avoidance instead of
foreground subtraction.• Use FFTs to get O(n log n) scaling,
adjusting for non-cubic geometry using weightings.
![Page 27: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/27.jpg)
FKP + FFT version• Foreground avoidance instead of
foreground subtraction.• Use FFTs to get O(n log n) scaling,
adjusting for non-cubic geometry using weightings.
• Use Feldman-Kaiser-Peacock (FKP) approximation– Power estimates from neighboring k-cells
perfectly correlated and therefore redundant.– Power estimates from far away k-cells
uncorrelated.– Approximation encapsulated by FKP
weighting.– Optimal (same as full inverse variance
method) on scales much smaller than survey volume.
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FKP + FFT version
100
0.02 0.04 0.060.08
101
10 mK
1 K100 mK
![Page 29: Power Spectrum Estimation in Theory and in Practice](https://reader035.vdocuments.site/reader035/viewer/2022081420/5681620b550346895dd2369f/html5/thumbnails/29.jpg)
Summary
Full inverse variance
AL, Tegmark 2011
O(n log n) versionDillon, AL, Tegmark (in
prep.)
FFT + FKPWilliams, AL,
Hewitt, Tegmark