Description Usage Arguments Details Value
An effective grid search for the lasso variants is based on starting with a maximal value for the penalty parameter λ. This is due to ability of the lasso variants to select features. The idea is to determine a value of λ such that any further increase of this value simply results in a solution with no selection (apart from fixed effects).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29  lambda_max_fitted_group_lasso(
DELTA,
VECTOR_Y,
VECTOR_GROUPS,
VECTOR_WEIGHTS_FEATURES,
VECTOR_WEIGHTS_GROUPS,
VECTOR_FULL_COLUMN_RANK,
VECTOR_BETA,
MATRIX_X
)
lambda_max_group_lasso(
VECTOR_Y,
VECTOR_GROUPS,
VECTOR_WEIGHTS_FEATURES,
VECTOR_BETA,
MATRIX_X
)
lambda_max_lasso(VECTOR_Y, VECTOR_WEIGHTS_FEATURES, VECTOR_BETA, MATRIX_X)
lambda_max_sparse_group_lasso(
ALPHA,
VECTOR_Y,
VECTOR_GROUPS,
VECTOR_WEIGHTS_FEATURES,
VECTOR_BETA,
MATRIX_X
)

DELTA 
numeric value, which is squared and added to the main diagonal of Z^{(l)T} Z^{(l)} for group l, if this matrix is not invertible. 
VECTOR_Y 
numeric vector of observations. 
VECTOR_GROUPS 
integer vector specifying which effect (fixed and random) belongs to which group. 
VECTOR_WEIGHTS_FEATURES 
numeric vector of weights for the vectors of fixed and random effects [b^T, u^T]^T. 
VECTOR_WEIGHTS_GROUPS 
numeric vector of precalculated weights for each group. 
VECTOR_FULL_COLUMN_RANK 
Boolean vector, which harbors the information of whether or not the groupwise parts of the filtered matrix Z, i.e., Z^{(l)} for each group l, have full column rank. 
VECTOR_BETA 
numeric vector of features. At the end of this function, the random effects are initialized with zero, but the fixed effects are initialized via a least squares procedure. 
MATRIX_X 
numeric design matrix relating y to fixed and random effects [X Z]. 
ALPHA 
mixing parameter of the penalty terms. Satisfies: 0 < α < 1. The penalty term looks as follows: α * "lasso penalty" + (1α) * "group lasso penalty". 
The value is calculated under the following prerequisites: The algorithm shall converge after a single iteration and the solution shall be equal to the initial solution. Additionally, the converged solution shall be zero for all random effects (which corresponds to "no selection".) The estimates for fixed effects shall simply remain unchanged after one iteration. Due to the explicit formulas of the proximal gradient descent algorithm, this naturally leads to a set of values for λ which guarantee to meet all the mentioned requirements. The lower bound of this set is then λ_{max}.
Particularly for the sparsegroup lasso, the calculation involves to find
the positive root of a nontrivial polynomial of second degree. In order to
solve this, an additional bisection algorithm is implemented. (See:
seagull_bisection
.)
the maximum value for the penalty parameter λ.
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