Principal Component Analysis of Cas A

Principal Component Analysis of Cas A Jessica S. Warren John P. Hughes Rutgers University 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     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   Hwang & Laming 2003 Fe L image overlaid with contours of Fe L line-to-continuum ratios 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  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?  Divide SNR into many spatial regions     Each region has at least a minimum number of userdetermined counts 50 ks observation 1000 minimum counts Spatial binning: 4”x4” to 16”x16” 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! Bin 1 2 3 4 5 6 7 8 9 10 Name Absorbed Oxygen Iron L Magnesium Continuum/Mg Silicon Sulphur Argon Calcium Iron K/Continuum Energy (keV) 0.2 – 0.45 0.45 – 0.7 0.7 – 1.2 1.2 – 1.4 1.4 – 1.6 1.6 – 2.2 2.2 – 2.6 2.6 – 3.4 3.4 – 5.0 5.0 – 9.0 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 Soft vs. hard spectra – perhaps column density Si-rich vs. Si-poor spectra Si 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 But dark regions don’t just represent nonthermal emission . . . Projections Look at regions in pc1-pc2 plane: degeneracy is broken Non-thermal hard spectra Softer spectra Thermal hard spectra Map of Projections – PC2 Map of Projections – PC2 But light regions don’t just represent iron-rich material . . . Projections Look at regions in pc1-pc2 plane: degeneracy is broken Non-thermal Fe rich Si rich Conclusions . . .   Able to separate out major differences in spectra Need to:  Non-thermal Fe rich   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 Soft Si rich 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  2 significant eigenvalues This is the variance explained by a particular eigenvector.