This document discusses using Principal Component Analysis (PCA) to remove correlated atmospheric noise from astronomical images taken by radio telescopes. PCA identifies lower-level correlations between subsets of detectors in the raw time-stream data. It decomposes the data into orthogonal principal components, allowing selection of components correlated with the atmosphere to remove. Reconstructing the data without these components significantly reduces noise and allows efficient extragalactic surveys over large areas compared to traditional methods. As an example, the AzTEC survey mapped 0.7 square degrees in 160 hours using PCA, compared to 11 days required for an area by traditional methods.

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