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