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...

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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)