Denoised functional Successive Projection Algorithm (d-fSPA)

Recovers a set of K latent basis functions from a collection of study-level function estimates under the low-rank cross-study heterogeneity assumption of Shi, Imai, and Zhang. Implements Algorithm 1 of the paper ("The d-fSPA Algorithm for basis hunting").

Usage

dfspa(F_hat, K, grid_weights = NULL, N = NULL, Delta = NULL, denoise = TRUE)

Arguments

F_hat

An m-by-G numeric matrix where row i is the evaluation of the estimated function \(\hat f^{(i)}\) at G grid points.

K

Integer number of basis functions to recover. Must satisfy 1 <= K <= m after denoising.

grid_weights

Optional length-G non-negative numeric vector of grid weights defining the \(L^2(\mu)\) inner product. Defaults to uniform weights 1 / G.

N, Delta

Optional numeric tuning parameters controlling denoising. See Details.

denoise

Logical; if FALSE, the denoising step is skipped and plain fSPA is run. Defaults to TRUE.

Value

An object of class "dfspa": a list containing

bases

A K-by-G matrix whose rows are the recovered basis functions evaluated on the grid (denoised, if applicable).

selected

Length-K integer vector of the selected row indices into the post-denoising function matrix.

original_indices

Length-K integer vector of the selected study indices in the original input F_hat (before any rows were dropped by denoising).

kept

Integer vector of row indices of F_hat that survived denoising.

F_denoised

The post-denoising function matrix (length(kept)-by-G).

grid_weights

Grid weights used.

N, Delta

Tuning parameters actually used (or NA when denoise = FALSE).

K

Number of bases requested.

call

The matched call.

Details

Each study-level function is represented by its evaluations on a shared grid of G points. The (weighted) \(L^2(\mu)\) inner product is \(\langle f,g\rangle = \sum_{j=1}^G w_j f(x_j) g(x_j)\), where the grid_weights w_j are proportional to the measure \(\mu\). If not supplied, uniform weights 1 / G are used.

Denoising follows Jin (2024): for each study i, let \(B_\Delta(\hat f^{(i)}) = \{j : \|\hat f^{(j)} - \hat f^{(i)}\| \le \Delta\}\). If \(|B_\Delta(\hat f^{(i)})| < N\), study i is discarded; otherwise \(\hat f^{(i)}\) is replaced by the average of the functions in \(B_\Delta\). After denoising, the functional SPA step iteratively selects, at each of the K iterations, the remaining function with the largest norm after projecting out the span of previously selected bases.

Default tuning parameters follow the heuristics of the paper: N = 0.5 * log(m) and \(\Delta = \max_{ij} \|\hat f^{(i)} - \hat f^{(j)}\| / 10\).

Examples

set.seed(1)
G <- 50
x <- seq(0, 1, length.out = G)
basis <- rbind(sin(pi * x), cos(pi * x), x)          # 3 true bases
pi_mat <- rbind(diag(3),                              # 3 pure studies
                c(0.5, 0.3, 0.2),
                c(0.2, 0.5, 0.3),
                c(0.3, 0.3, 0.4))
F_hat <- pi_mat %*% basis                             # m = 6, G = 50
fit <- dfspa(F_hat, K = 3, denoise = FALSE)
fit$original_indices    # should be a permutation of 1, 2, 3
#> [1] 2 1 3