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Astron. Astrophys. 343, 760-774 (1999)

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Appendix A: non parametric analysis

The non parametric solutions of Eq. (3) and (4) are described by their projection onto a complete basis of functions

[EQUATION]

of finite support, which are chosen here to be cubic B-splines (i.e. the unique [FORMULA] function which is defined to be a cubic over 4 adjacent intervals and zero outside, with the extra property that it integrates to unity over that interval):

[EQUATION]

[EQUATION]

The parameters to fit are the weights [FORMULA] and [FORMULA]. Calling [FORMULA] or [FORMULA] (the parameters) and [FORMULA] or [FORMULA] (the measurements) Eq. (3) and (4) then become formally

[EQUATION]

where [FORMULA] is a [FORMULA] matrix with entries given by

[EQUATION]

[FIGURE] Fig. A1. Sketch of deprojection: concentric B-spline shells of increasing logarithmic radii are projected onto the sky; their corresponding light distribution is calculated analytically and represents the basis over which the data is expanded.

[FIGURE] Fig. A2. The B-spline basis functions and their transform as a function of radius. Top panel: the B-spline projection; middle panel: corresponding density distribution; bottom panel: corresponding self consistent potential.

The projection and the self consistent potential of B-splines can be computed analytically. Their knots can be placed arbitrarily in order to resolve high frequencies in the profiles which are believed to be signal rather than noise (this is a requirement when using a penalty function which operates on the spline coefficient since imposing a correlation between these coefficients would truncate the high frequency). The analytic properties of B-splines and their transform turns out handy in particular since Taylor expansions are available when dealing with exponential profile where the dynamical range is large. Another useful property of B-spline is extrapolation: the correlation of the spline coefficient induced by the penalty function yields an estimate for the behaviour of the profile beyond the last measured point; since the Abel transform requires integration to infinity, this estimate corrects in part for the truncation. Note that an explicit analytic continuation of the model can be added to the spline basis if required. Finally here the requirement is that [FORMULA] is smooth, which is more strigent than requiring that [FORMULA] (or [FORMULA]) are smooth.

Assuming that we have access to discrete measurements of [FORMULA] and [FORMULA] (via binning as discussed above), and that the noise in [FORMULA] and [FORMULA] can be considered to be Normal, we can estimate the error between the measured profiles and the non parametric B-spline model as

[EQUATION]

where the weight matrix [FORMULA] is the inverse of the covariance matrix of the data (which is diagonal for uncorrelated noise with diagonal elements equal to one over the data variance). Linear penalty functions obey

[EQUATION]

where [FORMULA] is a positive definite matrix. In practice, we use [FORMULA] where [FORMULA] is a finite difference second order operator

[EQUATION]

In short, the solution of Eq. (3) (or Eq. (4)) is found by minimizing the quantity [FORMULA] where [FORMULA] and [FORMULA] are respectively the likelihood and regularization terms given by Eq. (A5) and (A6), [FORMULA] are the (large number) of parameters, and where the Lagrange multiplier [FORMULA] allows us to tune the level of regularization. The introduction of the Lagrange multiplier µ is formally justified by the fact that we want to minimize [FORMULA], subject to the constraint that [FORMULA] should be equal to some value. For instance, with [FORMULA] the problem is to minimize [FORMULA] subject to the constraints that [FORMULA] is in the range [FORMULA]). In practice, the minimum of

[EQUATION]

is:

[EQUATION]

The last remaining issue involves setting the level of regularization. The so-called cross-validation method (Wahba 1990) adjusts the value of µ so as to minimise residuals between the data and the prediction derived from the data. Let us define

[EQUATION]

We make use of the value for µ given by Generalised Cross Validation (GCV) (Wahba & Wendelberger 1979) estimator corresponding to the mimimum of

[EQUATION]

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© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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