## 2. PCA inversionLet the spectral signal be which
could represent the flux for a stellar spectral line, or any one of
the Stokes parameters for a solar
spectral line. For simplicity imagine for the moment that we are
dealing with just a flux or intensity profile; generalisation to the
other Stokes parameters will be discussed at the end of this section.
Suppose the line is sampled at where For a particular atmospheric model one can generate a synthetic model profile . Inversion of an observed spectral line involves finding the model which minimises the Euclidean distance This is a non-linear least squares problem which is usually solved by iterative adjustment of the model parameters (eg. del Toro Iniesta & Ruiz Cobo 1996b. Our approach sidesteps the numerical difficulties of this method. Assume that we compute a database of model profiles for models with parameters
ranging over all likely values. Here
is the signal for the wavelength and
model. We use PCA to achieve a
compact linear reconstruction of these model signals in terms of an
orthonormal set of principal components, or In Sect. 3 we use the form where is the mean signal vector. In this case the eigenprofiles are the
eigenvectors of the signal Ignoring a constant factor, the covariance estimate is Then we have which can be solved by singular value decomposition (Press et al., 1988). In Sects. 4.1 we use the alternative form, where the eigenprofiles are eigenvectors of the signal
an matrix composed of the synthetic signal vectors . This form is also used in Sect. 4.2, but the training data are the observations themselves, rather than the synthetic signals. Note that PCA achieves significant data compression. The principal components are indexed in decreasing order of significance in a least-squares sense, and good signal reconstruction can be obtained using only a small number of them, ie. . The appropriate cut-off can be estimated from a study of the eigenvalues (eg. Murtagh & Heck 1987). Thus using PCA we can map the high-dimensional signal vector
onto a low-dimensional
Using the fact that the eigenprofiles are orthonormal, it follows that Eq. 2 is, to a good approximation, equivalent to The eigenfeature vectors derived
from the model signal vectors can be
regarded as
Consider now the problem of inverting polarised spectra. Let be the profiles, or signal vectors, of the Stokes parameters. Stokes inversion involves solving for the model (magnetic field and other atmospheric parameters) which minimises the composite Euclidean distance, where the are constant weights. Applying PCA as above to each of the Stokes parameters one can readily show that where the dimensions of the Stokes eigenfeature vectors may differ. We now need to search four databases to find the closest points (with consistent model parameters) on four model manifolds . Any observed spectra will produce a model solution using this approach, and one must be sure that the solution is physically meaningful. One way to test this would be to reject solutions where the distance to each model manifold exceeds some prescribed threshold. There is, of course, the risk that the model itself is inadequate, but that is a problem faced by any inversion method. An interesting possibility might be to use the solution found by PCA inversion to initialise a non-linear least squares inversion procedure. Assuming the PCA solution is near optimality, one might expect rapid convergence. However this is not guaranteed, given the unknown topology of the error surface. As shown in Sects. 3 and 4, the © European Southern Observatory (ESO) 2000 Online publication: March 9, 2000 |