96
T. Basu et al.
Fig. 3.7 Relative frequency of occurrence of variables, for refit-LASSO applied on the Gaia
dataset
3.4.1.1 Example: Gaia Dataset
We applied the refit-LASSO on the Gaia dataset. We have taken 100 simulation runs
for the selection of important variables. The result is displayed in Fig. 3.7. We set
the desired proportion of inclusion at 50% as indicated by a horizontal line. Then we
have applied OLS fit on the important variables; in Table 3.1 we show the standard
error of our prediction with its “t-value” and corresponding probability. We also give
a comparison between the refit-LASSO estimates and the original cross-validated
LASSO estimates in the last two columns.
We notice from the Fig. 3.7 that the third variable appeared to be important in
several runs. However, it is not important in most of the runs.
3.4.2 Bootstrap Method
Bootstrap is a general frequentist method to quantify statistical accuracy, where one
randomly draws samples from a given training dataset with replacement, the sample
size being equal to that of the original training dataset. This is done for B times
(often multiples of 1000). Then one fits the model to each of these B datasets and
examines the empirical distributions of the estimated parameters.