Fits the Nadaraya–Watson smoother for manifold-valued predictors and finite real scalar responses. For bandwidth \(h>0\), weights at \(x\) are proportional to \(\exp\{-d(x,X_i)^2/(2h^2)\}\). The selected distance and training observations are retained for prediction. Training fitted values include the observation's own weight; they are not cross-validated predictions.
Arguments
- riemobj
A
riemdataobject containing the training predictors.- y
A finite numeric response vector with one entry per predictor.
- bandwidth
One finite, strictly positive bandwidth.
- geometry
Geometry name or specification.
NULLresolves the input's geometry, defaulting to its intrinsic geometry when unspecified. Legacy"intrinsic"and"extrinsic"aliases are accepted.
Value
An object of class m2skreg. Legacy fields ypred,
bandwidth, and inputs are retained. Additional fields include
resolved geometry, call, input dimensions, schema and package
versions, and training_diagnostics. No distance matrix is retained.
Details
Weights are normalized after subtracting the smallest squared distance in the exponent. The difference is evaluated without first squaring the distances, avoiding all-zero weights for small bandwidths or distant predictions. Numerically negligible relative weights can still underflow to zero. This is floating-point evaluation of the Gaussian smoother, not a change to a nearest-neighbor model.
The effective weight count and nearest training distance divided by the bandwidth describe the weights and distance scale. They are not confidence intervals or assurances of adequate statistical support.
Examples
theta <- seq(0, pi / 2, length.out = 8)
X <- wrap.sphere(cbind(cos(theta), sin(theta)))
fit <- riem.m2skreg(X, sin(2 * theta), bandwidth = 0.3)
predict(fit, X)
#> [1] 0.3097147 0.4882286 0.6854487 0.8194095 0.8194095 0.6854487 0.4882286
#> [8] 0.3097147
summary(fit)
#> Manifold-to-scalar Gaussian kernel regression
#> Observations: 8 Bandwidth: 0.3
#> Geometry: round
#> Training RMSE (includes self-weights): 0.1819012