PRELIMINARY PCA RESULTS

Principal Component Analysis (PCA) is a multivariate statistical technique that finds an optimum set of variables for describing a given system, in terms of linear combinations of the original statistical variables. The optimal variables are selected by finding a set of orthogonal (eigen-)vectors in the hyperspace of the original variables, each of which corresponds to the direction of maximum variance in the subspace orthogonal to the previous vector.

Heyer & Schloerb (1997, hereafter HS97) and Heyer & Brunt (1999, hereafter HB99) have presented PCA studies of the Sh 155 molecular cloud complex and the W3 cloud, respectively. Ghazzali et al. (1998) have also applied this technique to a sample of objects in the HI Canadian Galactic Plane Survey. In these works, the ``random variables'' are taken to be the intensity values at each velocity channel, and the ``sample'' is provided by all of the positions in the map. Eigenimages can be constructed for each eigenvector by showing in each pixel the magnitude of the projection of the velocity distribution onto the eigenvector. The eigenvectors effectively give velocity ``filters'', selected by the structures present in the PV space. Note that these filters may consist of both positive or negative entries (amplitudes at a given velocity) of the eigenvectors.

In what follows we present preliminary results of applying PCA in a similar fashion to our ISM simulation and compare them with those of HS97 and HB99 for the CO data. Although this simulation contains parameters most suited for the atomic and ionized components of the ISM, there are reasons to believe that the dynamics should not be too different from that of the molecular gas (Ballesteros-Paredes et al. 1999). A comparison with PCA results for HI gas will be presented elsewhere.

Figure 2 shows the resulting eigenvectors (a) and eigenimages (b) for this simulation. The following points can be noticed:

• The first few (low-order) eigenvectors exhibit wide filters, corresponding to broad velocity-difference intervals, while the high-order eigenvectors show narrow filters. In the corresponding eigenimages, the structures highlighted by these filters are seen to decrease in characteristic size as successively higher orders are considered. This is similar to the behavior found in the study of HB99 (fig. 3), supporting their conclusion that this behavior is a manifestation of the turbulence in the clouds.
• While the first eigenvector in Heyer's analysis has positive-definite entries (fig. 3), for our simulation data this is not true in general. This is due to the fact that the observational CO data contains a number of pixels with no signal (Heyer, private communication), causing the direction of maximum scatter of the data points in the hyperspace to be from the origin to the cluster of points with signal. The simulation data have no pixels with zero integrated signal. We have tested that simulation data with pixels with zero integrated signal do produce eigenvectors with positive-definite entries.