Pointwise Variance and Confidence Band for a Fixed-Configuration LPS Fit

Computes, from the analytically extracted linear-smoother matrix S of a fixed-configuration LPS fit (see smoother.matrix()), the pointwise variance Var(fitted_i) = sigma^2 * sum_j S_ij^2, the effective degrees of freedom df = tr(S), the plug-in noise estimate sigma.hat^2 = RSS / (n - 2*tr(S) + sum(S^2)), and the pointwise confidence band fitted_i +/- z * sigma * ||S_i.||_2 with z = qnorm(1 - (1 - level) / 2).

Usage

lps.pointwise.band(
  object,
  sigma = NULL,
  level = 0.95,
  check.tol = 1e-10,
  variance.method = c("residual", "legacy")
)

Arguments

object

A fitted "lps" object from fit.lps() at a fixed configuration, with X.eval identical to X.

sigma

Known noise standard deviation (positive scalar), or NULL (default) to use the plug-in sigma.hat.

level

Confidence level of the band, a single number strictly between 0 and 1. Default 0.95. For independent errors with common variance and a fixed smoother, the expected residual sum of squares is sigma^2 * sum((I - S)^2) + ||(I - S) mu||^2. The default denominator is therefore the residual noise degrees of freedom, n - 2*tr(S) + sum(S^2), computed as sum((I - S)^2) for stability. This corrects the variance contribution; it cannot remove signal bias. Normal bands target the smoothed mean S mu. Their interpretation as bands for mu requires negligible smoothing bias. Plug-in bands are approximate, pointwise, and conditional on the specified configuration; they do not account for tuning on the same responses, dependent errors, or simultaneous coverage.

check.tol

Passed to smoother.matrix()'s self-guard.

variance.method

Noise-variance denominator: "residual" (default) uses sum((I - S)^2); "legacy" reproduces the former n - tr(S) rule. Neither removes smoothing bias. Added after check.tol to preserve positional calls. The choice affects sigma.hat even when a known sigma is supplied.

Value

A list of class "lps.pointwise.band" with named fields: fitted (the fit's raw fitted values), se, variance, lower, upper, level, z, sigma (the supplied known sigma, or NA in plug-in mode), sigma.hat, sigma.source ("known" or "plug.in"), df (tr(S)), residual.df, variance.method, rss, n.train, smoother.row.norm (||S_i.||_2), and configuration (the pinned fit configuration).

Details

With sigma supplied (known noise standard deviation), the band uses it directly (sigma.source = "known"); with sigma = NULL the plug-in sigma.hat is used (sigma.source = "plug.in"). sigma.hat is reported in both modes. The variance is purely the linear-smoother sampling variance: the band makes no bias correction, so where bias dominates (boundary, high curvature) undercoverage is expected and must be reported, not masked.

Requires X.eval identical to X (the square training-fit smoother, on which tr(S) and RSS are defined), in addition to all restrictions of smoother.matrix(). If any evaluation point's local fit returned NA, its variance/band entries are NA, and df, rss, and sigma.hat are NA as well; supplying a known sigma still yields bands at the unaffected points.

See also

smoother.matrix()

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")
band.known <- lps.pointwise.band(fit, sigma = 0.1)
band.plugin <- lps.pointwise.band(fit)
band.known$df
#> [1] 10.75683
band.plugin$sigma.hat
#> [1] 0.2206855