Project study-level functions onto the simplex spanned by basis functions

For each study i, solves the constrained projection $$\hat\pi_i = \arg\min_{\pi \in \Delta_{K-1}} \left\| \hat f^{(i)} - \sum_{k=1}^K \pi_k \hat g_k \right\|_{L^2(\mu)}$$ where the norm is the weighted $L^2$ norm defined by grid_weights. This is Equation (3) of the paper and yields the study-specific weights hat pi_i used downstream for weight-model fitting and prediction.

Usage

project_to_simplex(F_hat, bases, grid_weights = NULL, ridge = 1e-10)

Arguments

F_hat: An m-by-G_grid numeric matrix; row i is the study function $\hat f^{(i)}$ evaluated on the shared grid.
bases: A K-by-G_grid numeric matrix of basis functions on the same grid, typically the bases slot of a dfspa() result.
grid_weights: Optional length-G_grid non-negative numeric vector of grid weights defining the $L^2(\mu)$ inner product. Defaults to uniform weights 1 / G_grid.
ridge: Small non-negative scalar added to the diagonal of the QP Hessian for numerical stability. Defaults to 1e-10.

Value

An m-by-K numeric matrix of simplex weights; rows sum to 1 and entries are non-negative.

Details

The projection reduces to the quadratic program $$\min_{\pi \in \mathbb R^K}\ \pi^\top D\,\pi - 2\,d^\top \pi \quad \text{s.t.} \quad \mathbf 1^\top \pi = 1,\ \pi \ge 0$$ with $D = G W G^\top$ and $d = G W f^{(i)}$, where $G$ is the K-by-G_grid basis matrix, $W = \mathrm{diag}($grid_weights$)$, and $f^{(i)}$ is the i-th row of F_hat. Solved via quadprog::solve.QP(). A tiny ridge is added to D for numerical stability.

Examples

set.seed(1)
G <- 40
x <- seq(0, 1, length.out = G)
basis <- rbind(sin(pi * x), cos(pi * x), x)
true_pi <- rbind(diag(3), c(0.4, 0.3, 0.3), c(0.1, 0.7, 0.2))
F_hat <- true_pi %*% basis
fit <- dfspa(F_hat, K = 3, denoise = FALSE)
pi_hat <- project_to_simplex(F_hat, fit$bases)
round(pi_hat, 3)
#>      [,1] [,2] [,3]
#> [1,]  0.0  1.0  0.0
#> [2,]  1.0  0.0  0.0
#> [3,]  0.0  0.0  1.0
#> [4,]  0.3  0.4  0.3
#> [5,]  0.7  0.1  0.2