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- 1. Using Principal Component Analysis to Remove Correlated Signal from Astronomical Images Kim Scott National Radio Astronomy Observatory Data Science Meet-up February 18, 2014
- 2. Galaxy Evolution in One Slide...
- 3. Galaxy Evolution in One Slide...
- 4. Galaxy Evolution in One Slide... ?
- 5. Galaxy Surveys – What Are We Missing?
- 6. Galaxy Surveys – What Are We Missing? Optical surveys miss ~50% of star formation in galaxies Optical surveys are biased Dust reemits stellar radiation at infrared to millimeter wavelengths (λ ~ 20 – 2000 μm)
- 7. Galaxy Surveys at (Sub)mm Wavelengths Atmospheric emission 1000× stronger than signal from galaxies Extragalactic emission: Transmitted Absorbed
- 8. Removing the Atmosphere by Modulating the Signal in Time Detector array Galaxy
- 9. Removing the Atmosphere by Modulating the Signal in Time Detector array i=1 i=2 i=3 Galaxy xij: power measured for time sample i on detector j
- 10. Surveys at λ=1.1mm with AzTEC ASTE Telescope AzTEC Dewar AzTEC Array (117 detectors)
- 11. Raw Time-stream Data Sample rate = 1∕(15.625 ms)
- 12. Raw Time-stream Data Sample rate = 1∕(15.625 ms) (20 s = 1280 samples)
- 13. Principal Component Analysis (PCA) [Used in supervised learning to compress data - ﬁt to fewer number of features] • xij: power measured for time sample i on detector j • n = number of detectors; m = number of time samples • X = [ x1 x2 ... xm ] → n × m matrix *Only input needed for PCA*
- 14. Principal Component Analysis (PCA) Step 1: Mean normalization (and feature scaling) • Compute μj = (1∕m) Σi=1,m xij for each detector • Compute σ2j = (1∕(m-1)) Σi=1,m (xij - μj)2 for each detector • Set xij (xij − μj) ∕ σj • X = [ x1 x2 ... xm ] → n × m matrix
- 15. Principal Component Analysis (PCA) Step 1: Mean normalization (and feature scaling) • Compute μj = (1∕m) Σi=1,m xij for each detector • Compute σ2j = (1∕(m-1)) Σi=1,m (xij - μj)2 for each detector • Set xij (xij − μj) ∕ σj • X = [ x1 x2 ... xm ] → n × m matrix
- 16. Principal Component Analysis (PCA) Step 1: Mean normalization (and feature scaling) • Compute μj = (1∕m) Σi=1,m xij for each detector • Compute σ2j = (1∕(m-1)) Σi=1,m (xij - μj)2 for each detector • Set xij (xij − μj) ∕ σj • X = [ x1 x2 ... xm ] → n × m matrix 1mV *PCA can identify lower level correlations among subsets of the detectors*
- 17. Principal Component Analysis (PCA) Step 2: Calculate covariance matrix • C = (1∕m) X XT (recall m = # time samples) • C → n × n symmetric matrix (recall n = 117 detectors) Step 3: Eigen decomposition • C = Q Λ Q-1 (*solve using SVD*) • Q = [ q1 q2 ... qn ] → n × n matrix containing eigenvectors qi • Λ → n × n diagonal matrix containing eigenvalues λi = Λii • Principal components = uncorrelated variables
- 18. Principal Component Analysis (PCA) Step 4: Choose number of components to remove • Goal: choose fewest number of components (k) to REMOVE most of the observed variance in the data • QR = [ qk+1 qk+2 ... qn ] → n × k matrix, k < n • Z = [ z1 z2 ... zm ] = QRT X → k x m matrix • To derive model of galaxy intensities on sky, use Z instead of X (but...) Choosing k: Variance after PCA (given k) < 0.05 Variance with average subtraction only
- 19. Principal Component Analysis (PCA) Step 5: Reconstruct data without correlated signal • Know RA/Dec for each detector: need to reconstruct approximation for data to make image • XR = QR Z → n × m matrix with correlated signal removed! 1mV
- 20. Principal Component Analysis (PCA) Step 5: Reconstruct data without correlated signal • Know RA/Dec for each detector: need to reconstruct approximation for data to make image • XR = QR Z → n × m matrix with correlated signal removed! 20μV *Variance reduced by factor of 50*
- 21. Image of PKS J1127-1857 Make the map: • Use information on sky position for each detector at each time sample (RAij, Decij) and bin data onto image grid • Set the intensity of each image pixel to the average of the xRij values that fall into that bin • Smooth image by telescope point-spread response function (Gaussian with FWHM=30’’) Average Subtraction PCA Cleaned • raw data = 30 MB • ttot = 4 min • 16640 samples/detector
- 22. An Extragalactic Survey at λ=1.1 mm • Most galaxies are 100× fainter than PKS J1127-1857 • raw data ~ 25 GB • ttot ~ 80 hrs • ~ 2×107 samples/detector • AzTEC/COSMOS survey • 0.7 deg2 • 500× area of HUDF • 160 hrs versus 11 days for HUDF • 130 mm-bright galaxies Aretxaga et al. 2011
- 23. An Extragalactic Survey at λ=1.1 mm • AzTEC/COSMOS survey • 0.7 deg2 • 500× area of HUDF • 160 hrs versus 270 hrs for HUDF • 130 mm-bright galaxies
- 24. An Extragalactic Survey at λ=1.1 mm • AzTEC/COSMOS survey • 0.7 deg2 • 500× area of HUDF • 160 hrs versus 270 hrs for HUDF • 130 mm-bright galaxies
- 25. An Extragalactic Survey at λ=1.1 mm • AzTEC-3 • Observed 1 Gyr after Big Bang • Starburst galaxy (SFR~1000 Msun/yr) Capak et al. 2011 • AzTEC/COSMOS survey • 0.7 deg2 • 500× area of HUDF • 160 hrs versus 270 hrs for HUDF • 130 mm-bright galaxies Aretxaga et al. 2011

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