Lorentz-Hyperbolic Weighted Penalized Regression

Fit local Gaussian or binomial penalized regressions whose observation weights are defined by Lorentz-hyperbolic neighborhoods.

Usage

lhwpr(
  formula,
  data,
  Z,
  bw,
  family = c("gaussian", "binomial"),
  penalty = c("lasso", "mcp", "scad"),
  alpha = 1,
  lambda = NULL,
  nlambda = 100L,
  lambda_min_ratio = NULL,
  gamma = NULL,
  penalty_factor = NULL,
  case_weights = NULL,
  standardize = TRUE,
  kernel = hgwr_kernel_names(),
  adaptive = TRUE,
  bandwidth_type = c("knn", "volume"),
  geometry = c("lorentz", "tangent"),
  tangent_ref = NULL,
  dMat = NULL,
  engine = c("dense", "knn"),
  keep_path = c("auto", "all", "selected"),
  control = lhwpr_control()
)

# S3 method for class 'lhwpr'
print(x, ...)

# S3 method for class 'lhwpr'
summary(object, ...)

# S3 method for class 'summary.lhwpr'
print(x, ...)

# S3 method for class 'lhwpr'
coef(object, s = NULL, ...)

# S3 method for class 'lhwpr'
fitted(object, ...)

# S3 method for class 'lhwpr'
residuals(object, ...)

# S3 method for class 'lhwpr'
predict(
  object,
  newdata = NULL,
  Znew = NULL,
  type = c("response", "link", "coefficients"),
  s = NULL,
  ...
)

# S3 method for class 'lhwpr'
plot(
  x,
  focus = 1L,
  xvar = c("lambda", "lambda_ratio"),
  ...
)

Arguments

formula: A model formula with one response and one or more predictors.
data: A data frame containing variables in formula.
Z: An n by 3 matrix of Lorentz-hyperboloid coordinates.
bw: Kernel bandwidth, or neighbor count for adaptive kNN fitting.
family: Gaussian or binomial response family.
penalty: Structural penalty: lasso, MCP, or SCAD. It is ignored when alpha = 0.
alpha: Mixing value in [0, 1]. Zero is ridge; a value strictly between zero and one is elastic net and requires the lasso penalty; one gives lasso, MCP, or SCAD.
lambda: Optional strictly decreasing common absolute lambda path.
nlambda: Number of automatically generated lambda values.
lambda_min_ratio: Ratio of the final to initial automatic lambda.
gamma: MCP or SCAD concavity parameter.
penalty_factor: Optional non-negative multiplier for each non-intercept coefficient.
case_weights: Optional non-negative case weights.
standardize: Whether predictors are centered and scaled once on the full analysis sample.
kernel: Lorentz-distance kernel.
adaptive: Whether bw is adaptive.
bandwidth_type: Neighbor-count or hyperbolic-volume bandwidth scale.
geometry: Intrinsic Lorentz or tangent-space distance geometry.
tangent_ref: Optional Lorentz point defining the tangent plane.
dMat: Optional precomputed distance matrix.
engine: Dense or exact compact-kernel kNN computation.
keep_path: Whether to retain all local coefficient paths.
control: A control object from lhwpr_control().
x, object: A fitted lhwpr object, or its summary.
s: A lambda value exactly represented in the fitted path.
newdata: Optional predictor data for new focal locations.
Znew: Lorentz coordinates corresponding to newdata.
type: Prediction scale or local coefficients.
focus: One-based focal-location index for a path plot.
xvar: Horizontal path scale.
...: Additional arguments passed to the plotting method.

Details

At every focal location, kernel and case weights are multiplied and normalized before fitting. Predictor standardization is global rather than focal. For a given bandwidth, every focal model uses the same absolute lambda values.

Lasso and elastic-net paths use warm starts, sequential strong screening, and a full KKT repair pass. MCP and SCAD use curvature-aware scalar minimization and full stationarity scans. Binomial models use proximal Newton/IRLS.

The intercept is never penalized. Group penalties are not part of LHWPR. Classical unpenalized LHWR inference is not attached to penalized fits.

With keep_path = "auto", a full dense coefficient cube is retained only when it contains at most ten million values. Selected coefficients and all fitted-value paths are retained regardless. Use keep_path = "all" to override this memory guard.

Value

lhwpr() returns an object of class "lhwpr" with selected local coefficients and predictions, lambda metadata, optional paths, convergence and KKT diagnostics, preprocessing metadata, and geometry information. Extractor methods return the corresponding component.

Examples

set.seed(1)
H <- simulate_H2(40, r_max = 2)
dat <- data.frame(
  y = rnorm(40),
  x1 = rnorm(40),
  x2 = rnorm(40)
)
fit <- lhwpr(
  y ~ x1 + x2,
  data = dat,
  Z = H$Z,
  bw = 15,
  alpha = 0.5,
  nlambda = 10
)
coef(fit)