Nonparametric Functional Regression

Fits a functional regression model using kernel smoothing (Nadaraya-Watson). Supports fixed bandwidth (h), or k-nearest neighbors with global (kNN.gCV) or local (kNN.lCV) cross-validation.

Usage

fregre.np(
  fdataobj,
  y,
  h = NULL,
  knn = NULL,
  type.S = c("S.NW", "kNN.gCV", "kNN.lCV"),
  Ker = "norm",
  metric = metric.lp,
  ...
)

Arguments

fdataobj: An object of class 'fdata' (functional covariate).
y: Response vector.
h: Bandwidth parameter. If NULL and knn is NULL, computed automatically.
knn: Number of nearest neighbors to consider for bandwidth selection. Only used when type.S is "kNN.gCV" or "kNN.lCV".
type.S: Type of smoother: "S.NW" for Nadaraya-Watson with fixed h (default), "kNN.gCV" for k-NN with global CV (single k for all observations), "kNN.lCV" for k-NN with local CV (different k per observation).
Ker: Kernel type for smoothing. Default is "norm" (Gaussian).
metric: Distance metric function. Default is metric.lp.
...: Additional arguments passed to metric function.

Value

A fitted regression object of class 'fregre.np' with components:

fitted.values: Fitted values
residuals: Residuals
h.opt: Optimal/used bandwidth (for type.S = "S.NW")
knn: Number of neighbors used (for kNN methods)
k.opt: Optimal k value(s) - scalar for global, vector for local
type.S: Type of smoother used
Ker: Kernel type used
fdataobj: Original functional data
y: Response vector
mdist: Distance matrix
sr2: Residual variance
metric: Metric function used
call: The function call

Details

Three smoothing approaches are available:

Fixed bandwidth (type.S = "S.NW"): Uses a single bandwidth h for all predictions. If h is not provided, it is set to the median of non-zero pairwise distances.

k-NN Global CV (type.S = "kNN.gCV"): Selects a single optimal k for all observations using leave-one-out cross-validation. The bandwidth at each point is set to include k neighbors.

k-NN Local CV (type.S = "kNN.lCV"): Selects an optimal k_i for each observation i, allowing adaptive smoothing. Useful when the data has varying density across the functional space.

Examples

# Create functional data
t <- seq(0, 1, length.out = 50)
n <- 50
X <- matrix(0, n, 50)
for (i in 1:n) X[i, ] <- sin(2*pi*t) * i/n + rnorm(50, sd = 0.1)
y <- rowMeans(X) + rnorm(n, sd = 0.1)
fd <- fdata(X, argvals = t)

# Fixed bandwidth
fit1 <- fregre.np(fd, y, h = 0.5)

# k-NN with global CV
fit2 <- fregre.np(fd, y, type.S = "kNN.gCV", knn = 20)

# k-NN with local CV
fit3 <- fregre.np(fd, y, type.S = "kNN.lCV", knn = 20)