1 / 29

Power Spectrum Estimation in Theory and in Practice

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 .

brilliant
Download Presentation

Power Spectrum Estimation in Theory and in Practice

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Power Spectrum Estimation in Theory and in Practice Adrian Liu, MIT

  2. What we would like to do Inverse noise and foreground covariance matrix Vector containing measurement

  3. What we would like to do “Geometry” -- Fourier transform, binning Bandpower at k Noise/residual foreground bias removal

  4. Why we like this method • Lossless Raw Data Cleaning Cleaned Data

  5. Why we like this method • Lossless • Smaller “vertical” error bars

  6. Why we like this method Errors using Line of Sight Method 3.0 101 • Lossless • Smaller “vertical” error bars 2.5 10 mK 2.0 100 1.5 100 mK 1 K 1 0.02 0.04 0.06 0.08 Log10 T (in mK) AL, Tegmark, Phys. Rev. D 83, 103006 (2011)

  7. Why we like this method Errors using Inverse Variance Method 3.0 101 30 mK • Lossless • Smaller “vertical” error bars 2.5 <10 mK 2.0 100 1.5 130 mK 1 0.02 0.04 0.06 0.08 Log10 T (in mK) AL, Tegmark, Phys. Rev. D 83, 103006 (2011)

  8. Why we like this method • Lossless • Smaller “vertical” error bars • Smaller “horizontal” error bars

  9. Why we like this method 1.0 101 0.9 • Lossless • Smaller “vertical” error bars • Smaller “horizontal” error bars 0.8 0.7 0.6 100 0.5 0.4 AL, Tegmark, Phys. Rev. D 83, 103006 (2011) 0.3 0.2 10-1 0.1 10-2 10-1 100

  10. Why we like this method 1.0 101 0.9 • Lossless • Smaller “vertical” error bars • Smaller “horizontal” error bars 0.8 0.7 0.6 100 0.5 0.4 AL, Tegmark, Phys. Rev. D 83, 103006 (2011) 0.3 0.2 10-1 0.1 10-2 10-1 100

  11. Why we like this method • Lossless • Smaller “vertical” error bars • Smaller “horizontal” error bars • No additive noise/foreground bias

  12. 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

  13. 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

  14. 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

  15. Quicker alternatives Full inverse variance AL, Tegmark 2011 O(n log n) version Dillon, AL, Tegmark (in prep.) FFT + FKP Williams,AL, Hewitt, Tegmark

  16. Quicker alternatives Full inverse variance AL, Tegmark 2011 O(n log n) version Dillon, AL, Tegmark (in prep.) FFT + FKP Williams,AL, Hewitt, Tegmark

  17. O(n log n) version • Finding the matrix inverse C-1 is the slowest step.

  18. 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.

  19. 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.

  20. 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 + …

  21. Different components are diagonal in different combinations of Fourier space C = Cps + Csync + Cnoise + … Real spatial Fourier spectral Fourier spatial Fourier spectral Real spatial Real spectral

  22. Comparison of Foreground Models Our model Eigenvalue GSM AL, Pritchard, Loeb, Tegmark, in prep.

  23. Quicker alternatives Full inverse variance AL, Tegmark 2011 O(n log n) version Dillon, AL, Tegmark (in prep.) FFT + FKP Williams,AL, Hewitt, Tegmark

  24. FKP + FFT version “Geometry” -- Fourier transform, binning Bandpower at k Noise/residual foreground bias removal

  25. FKP + FFT version 101 • Foreground avoidance instead of foreground subtraction. 10 mK 100 100 mK 1 K 0.02 0.04 0.06 0.08

  26. 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.

  27. 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.

  28. FKP + FFT version 101 10 mK 100 100 mK 1 K 0.02 0.04 0.06 0.08

  29. Summary Full inverse variance AL, Tegmark 2011 O(n log n) version Dillon, AL, Tegmark (in prep.) FFT + FKP Williams,AL, Hewitt, Tegmark

More Related