ABSTRACT
Let O be a sequence of n observational data in a domain X ⊆ R2 in the form O = {(x1, y1, Z1), . . . , (xi , yi , Zi ), . . . , (xn, yn, Zn)} (2.1)
with (xi , yi ) ∈ X Zi = {zi,1, . . . , zi,mi } (2.2)
where zi, j ∈ R represents the j th observation at the point (xi , yi ). As previously discussed in the introduction, a first approach to data reduction
is to model the observation set Zi with the interval Ii defined by the minimum and maximum values of the observations at the point (xi , yi ) [40]:
IO = {(x1, y1, I1), . . . , (xi , yi , Ii ), . . . , (xn, yn, In)} (2.3) where Ii is the interval
For some applications, this reduction is too coarse because it does not take into account the distribution and the quality of the data Zi at (xi , yi ). If the data set Zi comprises a large number of observations zi, j , a representation of Zi in terms of a probability density can be achieved and then a stochastic model can be fitted to the data in the form of a stochastic surface. However, in general, this approach requires special assumptions on the joint probability density of the data distribution, which are difficult to check from the data themselves. Furthermore, the interrogation of the stochastic surface model can be done, in general, only through Monte Carlo simulation, which is both noisy and computationally very expensive. Finally, this approach cannot be pursued when the data set contains only few data and the data quality is judged subjectively from the observer.
