Extract the Linear-Smoother Matrix of a Fixed-Configuration LPS Fit

For a fixed configuration (singleton support.grid, degree.grid, and kernel.grid, with an explicit numeric chart dimension) the LPS fitted vector is linear in the response: fitted = S %*% y with S depending on X, the kernel weights, and the configuration, but not on y. This function reconstructs S analytically, one evaluation row at a time, by rebuilding each local fit's support, kernel weights, local chart, and design through the same internal routines fit.lps() used, and reading off the influence row of the local weighted least-squares solve.

Usage

# S3 method for class 'lps'
smoother.matrix(object, check.tol = 1e-10, ...)

Arguments

object

A fitted "lps" object from fit.lps() at a fixed configuration (see Details).

check.tol

Positive scalar: maximum allowed absolute discrepancy of the self-guard identity S %*% y == fitted.values.raw. Default 1e-10, the program's algebraic tolerance.

...

Additional arguments are not currently supported.

Value

A numeric matrix S with nrow(object$X.eval) rows and nrow(object$X) columns: row i holds the weights through which the training responses enter the prediction at evaluation point i.

Details

The extraction refuses configurations where the linearity premise fails or is unsupported: it requires outcome.family = "gaussian", the R backend, design.basis = "orthogonal.polynomial.drop", singleton grids (no data-driven selection: the CV-selected pipeline is not a linear smoother), and coordinate.method = "coordinates" or "local.pca" with an explicit numeric chart.dim (never "auto" or "local.auto").

Self-guard: before returning, the function verifies max(abs(S %*% y - fitted.values.raw)) <= check.tol against the fit it was given (and that the NA patterns agree), so any divergence between the reconstruction and the estimator is a hard error, never a silent drift.

Local fits that fell back are represented honestly: a weighted-mean fallback (unstable.action = "mean") contributes its exact linear row w / sum(w); an unstable.action = "na" non-fit contributes an all-NA row.

See also

lps.pointwise.band() for pointwise variances and confidence bands derived from S.

smoother.matrix for supported fitted objects and migration.

Examples

set.seed(1)
n <- 30
X <- matrix(runif(2 * n, -1, 1), ncol = 2)
y <- sin(pi * X[, 1]) + 0.1 * rnorm(n)
fit <- fit.lps(X, y, foldid = rep(1:2, length.out = n),
               support.grid = 12L, degree.grid = 1L,
               kernel.grid = "tricube", backend = "R",
               design.basis = "orthogonal.polynomial.drop",
               ridge.multiplier.grid = 0, ridge.condition.max = Inf,
               unstable.action = "na")
S <- smoother.matrix(fit)
max(abs(S %*% y - fit$fitted.values))   # ~1e-15: the linear identity
#> [1] 4.440892e-16
sum(diag(S))                            # effective degrees of freedom
#> [1] 10.75683