hfr {hfr}

Fit a hierarchical feature regression

Description

HFR is a regularized regression estimator that decomposes a least squares regression along a supervised hierarchical graph, and shrinks the edges of the estimated graph to regularize parameters. The algorithm leads to group shrinkage in the regression parameters and a reduction in the effective model degrees of freedom.

Usage

hfr(
  x,
  y,
  weights = NULL,
  kappa = 1,
  q = NULL,
  intercept = TRUE,
  standardize = TRUE,
  partial_method = c("pairwise", "shrinkage"),
  l2_penalty = 0,
  ...
)

Arguments

x

Input matrix or data.frame, of dimension (N\times p); each row is an observation vector.

y

Response variable.

weights

an optional vector of weights to be used in the fitting process. Should be NULL or a numeric vector. If non-NULL, weighted least squares is used for the level-specific regressions.

kappa

The target effective degrees of freedom of the regression as a percentage of p.

q

Thinning parameter representing the quantile cut-off (in terms of contributed variance) above which to consider levels in the hierarchy. This can used to reduce the number of levels in high-dimensional problems. Default is no thinning.

intercept

Should intercept be fitted. Default is intercept=TRUE.

standardize

Logical flag for x variable standardization prior to fitting the model. The coefficients are always returned on the original scale. Default is standardize=TRUE.

partial_method

Indicate whether to use pairwise partial correlations, or shrinkage partial correlations.

l2_penalty

Optional penalty for level-specific regressions (useful in high-dimensional case)

...

Additional arguments passed to hclust.

Details

Shrinkage can be imposed by targeting an explicit effective degrees of freedom. Setting the argument kappa to a value between 0 and 1 controls the effective degrees of freedom of the fitted object as a percentage of p. When kappa is 1 the result is equivalent to the result from an ordinary least squares regression (no shrinkage). Conversely, kappa set to 0 represents maximum shrinkage.

When p > N kappa is a percentage of (N - 2).

If no kappa is set, a linear regression with kappa = 1 is estimated.

Hierarchical clustering is performed using hclust. The default is set to ward.D2 clustering but can be overridden by passing a method argument to ....

For high-dimensional problems, the hierarchy becomes very large. Setting q to a value below 1 reduces the number of levels used in the hierarchy. q represents a quantile-cutoff of the amount of variation contributed by the levels. The default (q = NULL) considers all levels.

When data exhibits multicollinearity it can be useful to include a penalty on the l2 norm in the level-specific regressions. This can be achieved by setting the l2_penalty parameter.

Value

An 'hfr' regression object.

Author(s)

Johann Pfitzinger

References

Pfitzinger, Johann (2024). Cluster Regularization via a Hierarchical Feature Regression. _Econometrics and Statistics_ (in press). URL https://doi.org/10.1016/j.ecosta.2024.01.003.

See Also

cv.hfr, se.avg, coef, plot and predict methods

Examples

x = matrix(rnorm(100 * 20), 100, 20)
y = rnorm(100)
fit = hfr(x, y, kappa = 0.5)
coef(fit)