Geoscience Reference
In-Depth Information
regressions are then carried out inside of each cluster. The most obvious example is
the one developed by. Douguédroit and de Saintignon [DOU 70].
Knowing that
mountainous areas make it difficult to obtain high correlation coefficients between
temperature and altitude, Douguédroit and de Saintignon started their research by
separating the mountains areas into two sub-areas. On one hand they had the south-
facing slopes, and on the other, they had the north-facing slopes and the bottoms of
valleys. Each of the both sub-areas was subject to regressions, which improved the
relationship that was being studied. The main difficulty using this method was
linked to distinguishing boundaries, for example, when does a south-facing slope
end and a north-facing slope begin? Where does the bottom of a valley begin?
Another problem was linked to how the border and limits between two different
areas could be managed. This is an excellent example of some of the problems that
exist when we want to draw out geographical information.
In the 1980s [CAR 82; LAB 84; JOL 87], the development of multiple
regression software makes possible to add descriptive geographical variables. In
addition to altitude, other variables could be used, such as slope, exposure, position
of the area in relation to a local level used as a basis, distance from the sea (if there
is one), latitude, and longitude (if necessary), etc.
With multiple regression software a hyper-plan is used where the number of
dimensions used is equal to the total number of variables that are used. Remember
that the point cloud equation is written as:
Y= a
0
+ a
1
x
1
+ a
2
x
2
+.....+ a
n
x
n
+ H
whilst the regression equation that summarizes a cloud in a hyper-plan is written as:
Y'= a
0
+ a
1
x
1
+ a
2
x
2
+.....+ a
n
x
n
WhereH ҏrepresents the set of residuals, the sum of the squares of the distances
that exist between the points of the cloud and the hyper-plan. The performance of
the adjustment increases as the value of H is weaker:
- Y= the observed value of the dependent variable,
- Y'= the value that is estimated by the equation,
- x
1
, x
2
...x
n
are independent variables (regressors),
- a
1
, a
2
...an are regression coefficients,
- a
0
is a constant.
The multiple determination coefficient (R
2
) refers to the part of the variance that
is considered by the regression, with V being the standard deviation for the series:
V
2
Y = V
2
Y. R
2
+ V
2
Y (1- R
2
)
with:
-V
2
Y = total variance of Y,
