Considerations about some problems on functional parametrical models implementation from a discrete set of data

The least squares method is widely used in Fundamental Astronomy in the determination of some parameters that are usually coefficients of functional developments based on certain regularity hypothesis about the developed function. This hypothesis of regularity in the working domain, together with the spatial distribution of the discrete data and their statistical properties should be carefully treated if we want to obtain reliable results. The use of a kernel based method is shown as a robust procedure and allows to generalize, in numerical terms, the usual least-squares statistical treatment. 1. PROBLEMS INVOLVING PARAMETRICAL ADJUSTMENT FOR BIASED DATA Given αi, δi, Zi be a set of n discrete values on the celestial sphere. A parametrical adjustment is given by the search of such ck that Zi = ∑ k ckΦk (αi, δi) for functions Φk. Some aspects of the problem are: The points αi, δi, are homogeneously distributed on the celestial sphere (other case can be similarly considered using an estimation of the density function from the discrete data), Zi are the random variable values of Z. Z is also the function, searched as an adjustment, this will require some hypothesis (usually, the supposition of a integrable square on the sphere) The discrete distribution of Zi values is determined by its mean and variance. The residual R = Z − ∑ k ckΦk is also understood as random variable as well as a function. With two random variables Zi and Yj over the same set αi, δi, the search of the Zi = ∑ k ckΦk (αi, δi), Yi = ∑ k dkΨk (αi, δi) developments can be done with the same considerations that before. Another possibility is to suppose ck = dk for a given reordenation of the Φk, Ψk basis. This case does not exclude the individual studies, but the basis functions will be different in each case. We shall use pairs ∆αcosδ, ∆δ or ∆μαcosδ, ∆μδ where ∆ represents the differences in α, δ or in the proper motions μαcosδ, μδ for two different catalogues. Provided the usual regularity hypothesis, there are spherical harmonics developments for each of the individual variablesfunctions. Analogously, it is possible to suppose a priori that the variables are related in pairs by means of infinitesimal rotations or spins, respectively. We use the same notation for the random variables and the function. The existence of infinitesimal rotations (or spins) should be compatible with the analytical and statistical properties for each variable-function considered. We are going now to deal with the functional and the statistical adjustment. They both are related to the most general least squares method: Given a function Z, which is defined in the unity sphere and has integrable square, then it may be developed in spherical harmonics. In practice, the series should be truncated up to an order m through the minimization problem ∥