the
same
ance
is
|
|
|
e of |
|
|
|
|
|
|
the
re no
where e = s- s
p P
At the check points, if any, discrepances can be
computed by:
where Ww are a small set of data use for the
crossvalidation.
2.3 Finite element method (e.g. tricubic spline
interpolation)
A tricubic spline function is given by the product
of three orthogonal cubic spline functions:
S(x,y,z) 9» S(x) Sly) S(z)
The choice for the number of cells and the number
of knots depends on the number of observations m
and the interpolation step ô.
The number of cells is the product of the number
of classes in three directions x,y and z:
x uy az
where p int (AX/8) +.1
um z int (AY/8) * 1
v - int (42/8) + |
z
being AX, AY and AZ the dimensions of the space
region in three directions and & the chosen
interpolation step. Consequently the number of
knots is:
n-n5n n n sp 43)4(» 43] (y^ € 3)
X-:y--Z x y zZ
The tricubic spline interpolation is performed, as
a classical least squares problem, by writing a
system of observation equations:
Hays k = 1,m
k
4
4
Lo. e pau Ji j L' $ 1 (5 1, Sy)
s =
k
D» ="
i=1
and associating it with the least squares norm:
The weights are mostly assumed equal one; however
more complex stochastic model should be defined
including correlations between the observations,
but they are usually omitted in sake of
brevity.
The following formulas are the legenda of the
functional model; indeed for the x direction, the
coordinate of the k-th knot respect the initial
corner is splitted in two parts:
Ax =1 8 + 8x
k k
where the number of the preceding knots is:
I = int (Ax /8)
and the position inside the class is:
Er = 8x /ó
being 8x = Ax - IS
k k
analogously, for the y direction:
Ay, =J 3d + y.
where J = int (Ay, /8)
and "n ay /9
being ôy, = Ay, — I8
and for the z direction:
Az L6 t 62
k k
where L - int (Az 78)
and C - óz /ó
k k
being óz, - Az - Ió
k k
Note that suitable constraints for the knots
should be introduced at the border and in empty
regions.
3. THE SYSTEM OF PROGRAMS
(Because of the modularity of the system of
programs, there is a high degree of similarity
between this system and those dedicated to 2D
problems and time series, see: Crippa/Mussio,
1987. )
The system consists of a set of programs, which
allow for the following operations: