# lambda_max: Maximal lambda In seagull: Lasso, Group Lasso, and Sparse-Group Lasso for Mixed Models

## Description

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).

## Usage

 ``` 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 ) ```

## Arguments

 `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 pre-calculated weights for each group. `VECTOR_FULL_COLUMN_RANK` Boolean vector, which harbors the information of whether or not the group-wise 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" + (1-α) * "group lasso penalty".

## Details

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 sparse-group lasso, the calculation involves to find the positive root of a non-trivial polynomial of second degree. In order to solve this, an additional bisection algorithm is implemented. (See: `seagull_bisection`.)

## Value

the maximum value for the penalty parameter λ.

seagull documentation built on April 20, 2021, 5:06 p.m.