Geology Reference
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Fig. 6.13 Variations of a
paleolatitude regression
spline with smoothing
5(Fig. 6.12 ) about a
spline regression plot
having smoothing
parameter “
D
D
300
being analyzed. The possibility to perform such
operation is quite intuitive. If we know the Euler
pole that restores the position of a continent
with respect to a reference plate at time t ,for
example the Brazilian craton with respect to
central Africa at 130 Ma, then applying this
rotation to its paleopoles of age t gives a new set
of paleomagnetic poles that are fully compatible
with those obtained by sampling directly on the
reference continent. We say that the paleopoles
have been rotated into the coordinate system of
the reference continent. Such paleopole transfer
technique has been applied by several authors
to fill gaps in the paleomagnetic record of some
continents (especially Africa) and to build global
APW paths (e.g., Besse and Courtillot 1991 ).
The next step in the technique of Schettino
and Scotese ( 2005 ) was to select an arbitrary
reference point on each continent and to build pa-
leolatitude and declination plots relative to these
points. Then, the two time series were analyzed
independently through a non-parametric spline
regression technique. Parametric regression mod-
els assume that the form of the regression func-
tion, , is known except for a finite number of
parameters. Non-parametric regression models,
on the other hand, only require some qualitative
properties for , for example that belongs to
some functional space . An overall measure of
performance of an estimator can be expressed
by a combination of goodness of the fit and
smoothness of the curve D ( t ). For example,
n
X
D .1 q/ 1
n
w i Œx i .t i / 2
¦ 2
iD1
Z
d m
dt m
2
b
C q
dt
(6.56)
a
where x i D x ( t i )arethe n observed values with
positive weights w i , m is a positive integer, and
the parameter 0 < q < 1isusedasabalancing
factor between goodness-of-fit and smoothing of
the estimator. If we set: “ q /(1 - q ), then the
functional to be minimized, 2 , can be rewritten
as:
Z
d m
dt m
2
b
n
X
1
n
w i Œx i .t i / 2
¦ 2
D
C
dt
iD1
a
(6.57)
The minimizing function, ,isa natural
smoothing spline estimator (Eubank 1999 )
and the parameter “, which controls balancing
between goodness-of-fit and smoothness, is
called the smoothing parameter .When“ is
large ( q 1) smoothness is favoured, whereas
estimators having large m -th derivative are
penalized. Conversely, small values of “ ( q 0)
tend to select classic least squares estimators and
privilege goodness-of-fit. It can be shown that
smoothing spline estimators are natural splines ,
that
is,
piecewise
polynomials
subject
to
a
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