Quantitative Stratigraphy, Splining and Geologic Time Scales - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 325

y y

y i = a+bx i +cx i 2 +dx i 3

e i

X i ≤ x i < X i+1

y i = Y i
y i = y i-1
y

At x=X i :

ʣ e i

SF=

Y i − 1 Y i

Y i + 1

i = y i-1

a b

X i-1 X i X i + 1 x x

Fig. 9.12 Schematic diagrams of cubic interpolation spline and cubic smoothing spline. The cubic
polynomials between successive knots have continuous first and second derivatives at the knots.
The smoothing factor (SF) is zero for interpolation splines. In most applications, the abscissae of
the knots coincide with those of the data points (Source: Agterberg 1990 , Fig. 3.8)

data points. A separate cubic polynomial curve with four coefficients is computed
for each interval between two successive data points. These cubics must have
continuous first and second derivatives. After setting the second derivative equal
to zero at the first and last data points, the continuity constraints yield so many
conditions that all (4 n

4) coefficients can be computed. Smoothing splines have
the same properties as interpolation splines except that they do not pass through the
data points. Instead of this, they deviate from the observed values by an amount that
can be regulated by means of the smoothing factor (SF) based on the average mean
squared deviation. For each specific value of SF, which can be set in advance, or
estimated by cross-validation, a single smoothing spline is obtained. Various
methods of estimating the smoothing factor were discussed by Wahba ( 1990 ).

9.2.2 Irregularly Spaced Data Points

In his topic on spline smoothing and non-parametric regression, Eubank ( 1988 ,
p. 153) discusses that unequally spaced data points may give poor results for
smoothing splines. De Boor ( 1978 ) pointed this out for interpolation splines. In
order to avoid poor results of this type for constructing age-depth curves from
biostratigraphic data, Agterberg et al. ( 1985 ) proposed the following simple “indi-
rect” method. The following experiment with interpolation splines illustrates how
the problem of unrealistic oscillations can be avoided, using this indirect method. It
should be kept in mind that the problem of oscillations in data gaps becomes even
more serious if the data are subject to much “noise” as in microfossil applications in
biostratigraphy.
Figure 9.13 is based on an example of De Boor ( 1978 , Fig. 8.1, p. 224). In total,
49 observations were available for a property of titanium ( Y ) as a function of
temperature ( X ). These data have regular spacing along the X -axis. Irregular
spacing was simulated by De Boor by selecting n ¼ 12 data points which are closer