Selected Topics for Further Research - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 464

of the objects of study. Practical examples are spatial distribution of acidic and
mafic volcanics in the Bathurst area, New Brunswick, and in the Abitibi volcanic
belt on the Canadian Shield in east-central Ontario and western Quebec.

Keywords Incomplete information • Jackknife method • Compositional data anal-
ysis • Non-linear process modeling • Random cut model • Accelerated dispersion
• Universal multifractals • Cell composition modeling • Permanent frequency
distributions • Probnormal model • Star Kimberlite • South Saskatchewan till
• Bathurst acidic volcanics • Abitibi acidic volcanics

12.1 Bias and Grouped Jackknife

In Sect. 7.2.1 elevations of the top of the Arbuckle Formation in Kansas were
analyzed as in Agterberg ( 1970 ) in order to compare various trend surface and
kriging applications with one another. The data set was randomly divided into three
subsets: two of these subsets were used for control and results derived for the two
control sets were applied to the third “blind” subset in order to see how well results
for the control subsets could predict the values in the third subset ( cf . Sect. 7.2.1 ). In
his comments on this approach Tukey ( 1970 ) stated that this form of cross-
validation could be used but a better technique would be the Jackknife proposed
by Mosteller and Tukey ( 1968 ).
Efron ( 1982 ) describes cross-validation in the following terms. In original appli-
cations of cross-validation, a data set was randomly divided into two halves and one
of these halves used for model testing. For example, a regression model fitted to the
first half is used to predict the second half. Normally, the first half predictions do less
well in predicting the second half than they did for the first half. After computers
became widely available, cross-validation could be generalized. It is more common
now to leave one data point out at a time and fit the model to the remaining points to
see howwell it does on the excluded point. The average of the prediction errors, each
point being left out once, then is the cross-validated measure of prediction error. This
“leave-off-one” cross-validation technique was used in Sect. 9.5.3 to find the best
smoothing factor (SF) used for cubic spline-curve fitting of age determinations
plotted along a relative geologic timescale. Cross-validation, the jackknife and
bootstrap are three techniques that are closely related. Efron ( 1982 , Chap. 7) dis-
cusses their relationships in a regression context pointing out that, although the three
methods are close in theory, they can yield different results in practical applications.
In Chap. 2 it was pointed out that for a set of n independent and identically
distributed (iid) data the standard deviation of the sample mean ðÞ x satisfies

r

X n
i

2

1 x i

ð x Þ

¼

σ . Although this is a good result it cannot be extended to

other estimators such as the median. However, the jac kk nife and bootstrap can be
used to make this type of extension. Suppose x i ¼

ðÞ ¼ x

nn 1 ð Þ

nx x i

n 1 represents the sample
average of the same data set b ut with the data point x i deleted. Let x JK represent
the mean of the n new values x i . The jackknife estimate of the standard deviation is