Cross-validated HAL Conditional Density Estimation

haldensify(
A,
W,
wts = rep(1, length(A)),
grid_type = "equal_range",
n_bins = round(c(0.5, 1, 1.5, 2) * sqrt(length(A))),
cv_folds = 5L,
lambda_seq = exp(seq(-1, -13, length = 1000L)),
smoothness_orders = 0L,
hal_basis_list = NULL,
...
)

## Arguments

A

The numeric vector observed values.

W

A data.frame, matrix, or similar giving the values of baseline covariates (potential confounders) for the observed units. These make up the conditioning set for the density estimate. For estimation of a marginal density, specify a constant numeric vector or NULL.

wts

A numeric vector of observation-level weights. The default is to weight all observations equally.

grid_type

A character indicating the strategy to be used in creating bins along the observed support of A. For bins of equal range, use "equal_range"; consult the documentation of cut_interval for more information. To ensure each bin has the same number of observations, use "equal_mass"; consult the documentation of cut_number for details. The default is "equal_range" since this has been found to provide better performance in simulation experiments; however, both types may be specified (i.e., c("equal_range", "equal_mass")) together, in which case cross-validation will be used to select the optimal binning strategy.

n_bins

This numeric value indicates the number(s) of bins into which the support of A is to be divided. As with grid_type, multiple values may be specified, in which case cross-validation will be used to choose the optimal number of bins. The default sets the candidate choices of the number of bins based on heuristics tested in simulation.

cv_folds

A numeric indicating the number of cross-validation folds to be used in fitting the sequence of HAL conditional density models.

lambda_seq

A numeric sequence of values of the regularization parameter of Lasso regression; passed to fit_hal via its argument lambda.

smoothness_orders

A integer indicating the smoothness of the HAL basis functions; passed to fit_hal. The default is set to zero, for indicator basis functions.

hal_basis_list

A list consisting of a preconstructed set of HAL basis functions, as produced by fit_hal. The default of NULL results in creating such a set of basis functions. When specified, this is passed directly to the HAL model fitted upon the augmented (repeated measures) data structure, resulting in a much lowered computational cost. This is useful, for example, in fitting HAL conditional density estimates with external cross-validation or bootstrap samples.

...

Additional (optional) arguments of fit_hal that may be used to control fitting of the HAL regression model. Possible choices include use_min, reduce_basis, return_lasso, and return_x_basis, but this list is not exhaustive. Consult the documentation of fit_hal for complete details.

## Value

Object of class haldensify, containing a fitted

hal9001 object; a vector of break points used in binning A

over its support W; sizes of the bins used in each fit; the tuning parameters selected by cross-validation; the full sequence (in lambda) of HAL models for the CV-selected number of bins and binning strategy; and the range of A.

## Details

Estimation of the conditional density A|W through using the highly adaptive lasso to estimate the conditional hazard of failure in a given bin over the support of A. Cross-validation is used to select the optimal value of the penalization parameters, based on minimization of the weighted log-likelihood loss for a density.

## Note

Parallel evaluation of the cross-validation procedure to select tuning parameters for density estimation may be invoked via the framework exposed in the future ecosystem. Specifically, set plan for future_mapply to be used internally.

## Examples

# simulate data: W ~ U[-4, 4] and A|W ~ N(mu = W, sd = 0.5)
set.seed(429153)
n_train <- 50
w <- runif(n_train, -4, 4)
a <- rnorm(n_train, w, 0.5)
# learn relationship A|W using HAL-based density estimation procedure
haldensify_fit <- haldensify(
A = a, W = w, n_bins = 10L, lambda_seq = exp(seq(-1, -10, length = 100)),
# the following arguments are passed to hal9001::fit_hal()
max_degree = 3, reduce_basis = 1 / sqrt(length(a))
)