deletions | additions       diff --git a/subsubsection_Principal_Component_Analysis_Principle__.tex b/subsubsection_Principal_Component_Analysis_Principle__.tex  index c5ca23c..c392a25 100644  --- a/subsubsection_Principal_Component_Analysis_Principle__.tex  +++ b/subsubsection_Principal_Component_Analysis_Principle__.tex 
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Principle component analysis (PCA) determines a set of orthogonal axes that maximize the variance of the data. As applied to spectral data cubes, it identifies differences between the line profiles, and thus, is a useful tool for distinguishing between kinematic changes and noise \citep{heyer97,heyer02}. Subsequent work established an empirical and analytic formalism connecting PCA to the underlying turbulent velocity fluctuations, including the spectral slope \citep{brunt03,bunt13}. In PCA analysis, the first step involves constructing , a 2-D covariance matrix from the spectra. Next, the eigenvalues and eigenvectors of this matrix are determined. Here, we use the magnitude of the eigenvalues to assess the degree of difference between two datasets.  Figure 3 shows the velocity channel covariance matrices of runs W1T2t0.2 and T2t0. Both runs show a signal for velocities $|v| \lessim \lesssim  2 \kms$, which roughly encompasses the range of turbulent gas velocities. However, W1T2t0.2 exhibits multiple strong covariance peaks around a few $\kms$. These features exist to a lesser degree for T2t0, but feedback augments and further separates the peaks. The strongest covariance corresponds to the typical expansion rate of the wind shells. Because the eigenvectors provide a measure of the strength of different eigenvectors, they also serve as a proxy of the amount of power on different scales \citep{brunt08}. We find a clear difference in the eigenvalues for the cases with and without feedback. %Ask Mark about PCA and feedback -- Brunt et al. 2003 paper analyzed simualtions with point expostions? but no break