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dependent and independent variables is deterministic
instead of stochastic; b) the error measurements are
random, following the normal distribution, average
zero and constant variance; and c) the explanatory
variables don't show correlation among themselves
(Seber and Wild, 2005). This technique of landform
attributes representation has some advantages over
other techniques, basically by representing the
landform attributes by mathematical equations
instead of images, latitude, longitude and elevation
coordinates, or map of elevation levels. One of the
advantages is the significant reduction of the amount
of necessary information to represent the landform
attributes of a certain area, since, with the regression
technique, the landform attributes of an entire area
can just be represented through the coefficients of
two-dimensional polynomial. By the polynomial
representation it is possible to generate images with
different resolution levels because, with a
polynomial function, we may generate as many
points as needed. Besides these advantages, there is
a possibility to apply, on the polynomial, several
mathematical operations, such as finding the
maximal and minimal point, derivation, etc.
However, the technique of polynomial regression
has some disadvantages too, since the complexity
and the computational power demanded in obtaining
such polynomial is very high and sometimes
impractical.
For that reason the objective of this article is to
present a methodology designed to make possible
the distribution of the necessary processing to
compute a two-dimensional polynomial that
represents the landform attributes of an area and, as
an example, we use the area of the state of Minas
Gerais, in Brazil, located in the Southeastern area of
the country. Some estimative calculations, presented
in this article, show that the necessary time of
centralized processing to estimate such a polynomial
is prohibitive, being in order of dozens of
uninterrupted years of processing.
2 MATHEMATICAL METHOD
In this work we present a methodology for landform
attributes representation using the method of
nonlinear regression to adjust a two-dimensional
polynomial. The regression analysis is a statistical
instrument very used in science. Its frequent use is
due to the fact of making possible the description of
phenomena through mathematical models from a
data sample. Graphically, it is equal to identify the
curve or mathematical surface that best adjusts to the
points in a dispersion diagram. The mathematical
models of regression are based on three statistical
facts: a) the relationship among the dependent and
independent variables is deterministic instead of
stochastic; b) the errors are random with normal
distribution, average zero and constant variance; and
c) the explanatory variables don't present correlation
among themselves (Seber and Wild, 2005).
When a mathematical model of regression is
used, the most used method of estimating the
parameters is the Least Squares Method which
consists of estimating a function to represent a group
of points, minimizing the square of the deviations
(Nobel and Daniel, 1986). Considering a group of
geographic coordinates (x, y, z), representing
longitude, latitude, and elevation of each point,
respectively, we may take an estimate elevation
function
),(
ˆ
yxfz
of these points. A
polynomial of degree r in x and degree s in y can be
given, according to Equation 1, and the estimated
error ε
ij
is given by Equation 2 where 0 ≤ i ≤ m and
0 ≤ j ≤ n.
∑∑
==
==
r
k
s
l
l
j
k
iklji
yxayxfz
00
),(
ˆ
(1)
ijijij
zz
ˆ
(2)
The coefficients a
kl
(k = 0, 1, ..., r, l = 0, 1, ..., s)
that minimize the errors of the estimated function
f (x, y) can be obtained by solving Equation 3 for c =
0, 1, ..., r and d = 0, 1, ..., s.
0=
∂
cd
a
(3)
where
∑∑∑∑
====
−==
m
i
n
j
jiij
m
i
n
j
ij
zz
11
2
11
2
)
ˆ
(
εξ
(4)
and
x
i
longitude i of DEM column, for 1 ≤ i ≤ k
y
j
latitude j of DEM line, for 1 ≤ j ≤ l,
zij elevation of point (xi, yi)
r polynomial degree in x,
s polynomial degree in y,
akl coefficients which minimize the error of the
estimated function f (x, y)
A MATHEMATICAL FORMULATION OF A MODEL FOR LANDFORM ATTRIBUTES REPRESENTATION FOR
APPLICATION IN DISTRIBUTED SYSTEMS
185