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Robust geomechanical characterization Chapter 5 137

balances the importance between the SSE term and the regularization term,
which is the L 1 norm of the coefficient vector.
When α increases, the regularization term punishes the coefficient matrix to
be sparser. The α term is the hyperparameter of the LASSO model that is
optimized by testing a range of values for α. We select α equals to 4.83 for
which the LASSO model achieves the best generalization performance.
2
R values for the DTC and DTS predictions are 0.79 and 0.75, respectively.
For α equals to 4.83, the LASSO model learns the value of parameter
(coefficient) for each feature (input log), as listed in Table 5.1. LASSO
model-derived coefficients for 6 out of the 13 features are 0. Those logs with
a coefficient values (Table 5.1) close to zero are either well correlated with
other logs or less important for the desired log synthesis as compared with
other logs with nonzero coefficients. A similar redundancy in features was
noticed with the PLS model. Table 5.1 indicates that shallow-resistivity logs,
deep-resistivity logs, and density porosity log are not essential for the desired
DTC and DTS logs (Fig. 5.2). Shallow resistivity (RLA0 and RLA1) and
deep resistivity (RLA4 and RLA5) logs are correlated with the medium
sensing RLA2 and RLA3 logs. Further, RLA2 and RLA3 being medium
sensing have similar depth of investigation as the sonic DTC and DTS logs;
consequently, RLA2 and RLA3 are used by the LASSO model, while the
other resistivity logs are not used for the synthesis of DTC and DTS logs.

2.4.4 ElasticNet model
Similar to LASSO, the ElasticNet algorithm uses a regularization term to
penalize the coefficients of correlated and nonessential features. ElasticNet
model learns linear relationship between features and targets using a
regularization term that is a weighted sum of L1 norm and L2 norm of the
coefficients. Unlike the LASSO model, ElasticNet model preserves certain
groups of correlated features that improve the precision and repeatability of
the predictions. ElasticNet model does not penalize correlated features as
severely as LASSO model. ElasticNet algorithm generates more unique
model as compared to the LASSO model for high-dimensional data with
highly correlated variables. The objective function of the ElasticNet model is
formulated as
1 2 2
kk + α 2 w
min k Xw yk + α 1 w 1 kk 2 (5.8)
2
w 2n
where the penalty parameters α 1 and α 2 are the hyperparameters of the
ElasticNet model and determined through optimization to be equal to 4.8
and 0.1, respectively. This is aligned with the findings of the LASSO model
because α 2 is a small value and α 1 is almost equal to α of the LASSO
model. Dataset used in our study is not a high-dimensional dataset, and the
benefits of ElasticNet model in comparison to the LASSO model are only
observed for high-dimensional datasets.