Principal Component Analysis of Cas A :Principal Component Analysis of Cas A Jessica S. Warren John P. Hughes Rutgers University


PCA Technique :PCA Technique Goal: To look at spectral variations in SNRs Unbiased technique for identifying spectral variations in a statistically quantifiable way Specific to Cas A: Look for Si, Fe, non-thermal regions Use to quantify differences in Si/Fe regions Use to find most Fe-rich regions


Cas A – 3 Color :Cas A – 3 Color Hughes et al. 2000 – 5 ks image Red: 0.6 – 1.65 keV Green: 1.65 –2.25 keV Blue: 2.25 – 7.50 keV


Cas A – Line-to-Continuum Ratio :Cas A – Line-to-Continuum Ratio Hwang & Laming 2003 Fe L image overlaid with contours of Fe L line-to-continuum ratios


What is PCA? :What is PCA? PCA is a statistical technique often used to reduce the dimensionality of a dataset. Represent the data in matrix form Rows: spatial regions; Columns: spectral channels PCA finds the eigenvalues & eigenvectors of the data matrix Eigenvectors: axes which maximize the variance of the data Eigenvalues: quantify amount of variation accounted for by that eigenvector


PCA - Picture :PCA - Picture Data: n objects (spatial regions), each with m attributes (spectral channels) Each object represented by a vector in space of m attributes Eigenvectors are new axes in m-space that: Maximize variances of all objects (C) Equivalent to: Minimize distances of all objects (B)


How do we use PCA for SNRs? :How do we use PCA for SNRs? Divide SNR into many spatial regions Each region has at least a minimum number of user-determined counts 50 ks observation 1000 minimum counts Spatial binning: 4”x4” to 16”x16”


Using PCA :Using PCA Get spectrum of each region Bin each spectrum in energy in the same way Normalize each to the total number of counts in that region Input to PCA code (Murtagh & Heck 1987) Outputs will be eigenvectors and eigenvalues


Energy Binning – First Try! :Energy Binning – First Try!


Results: A Work in Progress :Results: A Work in Progress Output from PCA are eigenvectors, or principal components As many components as spectral channels Rank the components based on eigenvalues Consider only 1st two here Represent 38% and 31% of variation in spectra Still to do: quantify significance of remaining components


Eigenvectors :Eigenvectors Soft vs. hard spectra – perhaps column density Si-rich vs. Si-poor spectra Si


Projections of Spectra in PC1-PC2 Plane :Projections of Spectra in PC1-PC2 Plane Project each spectrum onto the new axis (eigenvector). The features in this graph are indications of different types of spectra.


Map of Projections – PC1 :Map of Projections – PC1


Map of Projections – PC1 :Map of Projections – PC1 But dark regions don’t just represent non-thermal emission . . .


Projections :Projections Look at regions in pc1-pc2 plane: degeneracy is broken Thermal hard spectra Softer spectra Non-thermal hard spectra


Map of Projections – PC2 :Map of Projections – PC2


Map of Projections – PC2 :Map of Projections – PC2 But light regions don’t just represent iron-rich material . . .


Projections :Projections Si rich Fe rich Non-thermal Look at regions in pc1-pc2 plane: degeneracy is broken


Conclusions . . . :Conclusions . . . Able to separate out major differences in spectra Need to: Go from pc1-pc2 plot to the spectra to ID regions of interest Is 3rd component useful? Optimize spectral binning vs. spatial binning for 1 Ms observation Thermal hard Si rich Soft Fe rich Non-thermal


Simulated Data :Simulated Data Need to determine significance of results Calculate mean spectrum Input to Poisson random number generator Only variations then due to Poisson noise Input simulations to PCA Tells which eigenvectors are significant


SCREE Test :SCREE Test 2 significant eigenvalues This is the variance explained by a particular eigenvector.