Solves the weighted aggregation problem using the CLASSO module in Python. The optimization problem is
trac(Z, y, A, fraclist = NULL, nlam = 20, min_frac = 1e-04, w = NULL)
| Z | n by p matrix containing log(X) |
|---|---|
| y | n vector (response) |
| A | p by (t_size-1) binary matrix giving tree structure (t_size is the total number of nodes and the -1 is because we do not include the root) |
| fraclist | (optional) vector of tuning parameter multipliers. Or a list of length num_w of such vectors. Should be in (0, 1]. |
| nlam | number of tuning parameters (ignored if fraclist non-NULL) |
| min_frac | smallest value of tuning parameter multiplier (ignored if fraclist non-NULL) |
| w | vector of positive weights of length t_size - 1 (default: all equal to 1). Or a list of num_w such vectors. |
a list of length num_w, where each list element corresponds to the solution for that choice of w. Note that the fraclist depends on the choice of w. beta0 is the intercept; beta is the coefficient vector on the scale of leaves of the tree; gamma is the coefficient vector on nodes of the tree where the features are sums of logs of the leaf-features within that node's subtree; alpha is the coefficient vector on nodes of the tree where the features are log of the geometric mean of leaf-features within that node's subtree.
minimize_beta, beta0, gamma 1/(2n) || y - beta0 1_n - Z_clr beta ||^2 + lamda_max * frac || W * gamma ||_1 subject to beta = A gamma, 1_p^T beta = 0 where W = diag(w) with w_u > 0 for all u
Observe that the tuning parameter is specified through "frac", the fraction of lamda_max (which is the smallest value for which gamma is nonzero).