Page 84 - MATLAB Recipes for Earth Sciences

76 4 Bivariate Statistics

4.7 Jackknife Estimates of the Regression Coeffi cients

The jackknife method is a resampling technique that is similar to the boot-
strap method. However, from a sample with n data points, n subsets with
n-1 data points are taken. Subsequently, the parameters of interest are cal-

culated, such as the regression coefficients. The mean and dispersion of the

coefficients are computed. The disadvantage of this method is the limited
number of n samples. The jackknife estimate of the regression coeffi cients
is therefore less precise in comparison to the bootstrap results.
MATLAB does not provide a jackknife routine. However, the correspond-
ing code is easy to generate:
for i = 1 : 30
% Define two temporary variables j_meters and j_age
j_meters = meters;
j_age = age;
% Eliminate the i-th data point
j_meters(i) = [];
j_age(i) = [];
% Compute regression line from the n-1 data points
p(i,:) = polyfit(j_meters,j_age,1);
end
The jackknife for n-1=29 data points can be obtained by a simple for loop.
Within each iteration, the i-th element is deleted and the regression coef-

ficients are calculated for the i-th sample. The mean of the i samples gives
an improved estimate of the coefficients. Similar to the bootstrap result, the

slope of the regression line (fi rst coefficient) is clearly defined, whereas the

intercept with the y-axis (second coefficient) has a large uncertainty,
mean(p(:,1))
ans =
5.6382
compared to 5.6023+/-0.4421 and

mean(p(:,2))

ans =
1.0100
compared to 1.3366+/-4.4079 as calculated by the bootstrap method. The
true values are 5.6 and 1.2, respectively. The histogram of the jackknife
results from 30 subsamples
hist(p(:,1));
figure
hist(p(:,2));