Post on 22-Feb-2016
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
Bandpower at k“Geometry” -- Fourier
transform, binningNoise/residual
foreground bias removal
Why we like this method• Lossless
CleanedData
RawDataCleaning
Why we like this method• Lossless• Smaller “vertical” error bars
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)
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)
Why we like this method• Lossless• Smaller “vertical” error bars• Smaller “horizontal” error bars
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)
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)
Why we like this method• Lossless• Smaller “vertical” error bars• Smaller “horizontal” error bars• No additive noise/foreground bias
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
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
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
Quicker alternatives
Full inverse variance
AL, Tegmark 2011
O(n log n) versionDillon, AL, Tegmark (in
prep.)
FFT + FKPWilliams, AL,
Hewitt, Tegmark
Quicker alternatives
Full inverse variance
AL, Tegmark 2011
O(n log n) versionDillon, AL, Tegmark (in
prep.)
FFT + FKPWilliams, AL,
Hewitt, Tegmark
O(n log n) version• Finding the matrix inverse C-1 is the
slowest step.
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.
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.
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 + …
Different components are diagonal in different
combinations of Fourier space
C = Cps + Csync + Cnoise + …
Real spatialFourier spectral
Fourier spatialFourier spectral
Real spatialReal
spectral
Comparison of Foreground Models
GSMOur
model
Eig
enva
lue
AL, Pritchard, Loeb, Tegmark, in prep.
Quicker alternatives
Full inverse variance
AL, Tegmark 2011
O(n log n) versionDillon, AL, Tegmark (in
prep.)
FFT + FKPWilliams, AL,
Hewitt, Tegmark
FKP + FFT version
Bandpower at k“Geometry” -- Fourier
transform, binningNoise/residual
foreground bias removal
FKP + FFT version• Foreground avoidance instead of
foreground subtraction.
100
0.02 0.04 0.060.08
101
10 mK
1 K100 mK
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.
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.
FKP + FFT version
100
0.02 0.04 0.060.08
101
10 mK
1 K100 mK
Summary
Full inverse variance
AL, Tegmark 2011
O(n log n) versionDillon, AL, Tegmark (in
prep.)
FFT + FKPWilliams, AL,
Hewitt, Tegmark