Using Principal Component Analysis to Remove Correlated Signal from Astronomical Images
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Transcript of Using Principal Component Analysis to Remove Correlated Signal from Astronomical Images
Using Principal Component Analysis to Remove Correlated
Signal from Astronomical Images
Kim ScottNational Radio Astronomy Observatory
Data Science Meet-upFebruary 18, 2014
Galaxy Evolution in One Slide...
Galaxy Evolution in One Slide...
Galaxy Evolution in One Slide...
?
Galaxy Surveys – What Are We Missing?
Galaxy Surveys – What Are We Missing?
Optical surveysare biased
Optical surveys miss ~50% of star formation in galaxies
Dust reemits stellar radiation at infrared to millimeter wavelengths (λ ~ 20 – 2000 μm)
Galaxy Surveys at (Sub)mm Wavelengths
Extragalactic emission:Transmitted
Absorbed
Atmospheric emission1000× stronger than signal from galaxies
Removing the Atmosphere by Modulating the Signal in Time
Galaxy
Detector array
Removing the Atmosphere by Modulating the Signal in Time
xij: power measured for
time sample i on detector j
i = 3
i = 2i = 1
Galaxy
Detector array
Surveys at λ=1.1mm with AzTEC
ASTE Telescope
AzTEC DewarAzTEC Array
(117 detectors)
Raw Time-stream Data
Sample rate = 1∕(15.625 ms)
Raw Time-stream Data
Sample rate = 1∕(15.625 ms)(20 s = 1280 samples)
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*
Principal Component Analysis (PCA)
Step 1: Mean normalization (and feature scaling)
• Compute μj = (1∕m) Σi=1,m xij for each detector• Compute σ2
j = (1∕(m-1)) Σi=1,m (xij - μj)2 for each detector• Set xij ⇒ (xij − μj) ∕ σj• X = [ x1 x2 ... xm ] → n × m matrix
Principal Component Analysis (PCA)
Step 1: Mean normalization (and feature scaling)
• Compute μj = (1∕m) Σi=1,m xij for each detector• Compute σ2
j = (1∕(m-1)) Σi=1,m (xij - μj)2 for each detector• Set xij ⇒ (xij − μj) ∕ σj• X = [ x1 x2 ... xm ] → n × m matrix
Principal Component Analysis (PCA)
Step 1: Mean normalization (and feature scaling)
• Compute μj = (1∕m) Σi=1,m xij for each detector• Compute σ2
j = (1∕(m-1)) Σi=1,m (xij - μj)2 for each detector• Set xij ⇒ (xij − μj) ∕ σj• X = [ x1 x2 ... xm ] → n × m matrix
*PCA can identify lower levelcorrelations among subsets of the detectors*
1mV
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
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)Variance with average subtraction only
< 0.05
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
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*
Image of PKS J1127-1857Make the map:
• Use information on sky position for each detector at each time sample (RAi
j, 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’’)
PCA CleanedAverage Subtraction
• raw data = 30 MB• ttot = 4 min• 16640 samples/detector
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
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
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
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
Aretxaga et al. 2011
Capak et al. 2011
• AzTEC-3• Observed 1 Gyr after Big Bang• Starburst galaxy (SFR~1000 Msun/yr)