WH_1d {WH}

1D Whittaker-Henderson Smoothing

Description

Main package function to apply Whittaker-Henderson smoothing in a one-dimensional survival analysis framework. It takes as input a vector of observed events and a vector of associated central exposure, both depending on a single covariate, and build a smooth version of the log-hazard rate. Smoothing parameters may be supplied or automatically chosen according to an adequate criterion such as "REML" (the default), "AIC", "BIC" or "GCV". Whittaker-Henderson may be applied in a full maximum likelihood framework (the default) or an approximate gaussian framework (the original).

Usage

WH_1d(
  d,
  ec,
  lambda,
  criterion,
  method,
  q = 2,
  framework,
  y,
  wt,
  quiet = FALSE,
  ...
)

Arguments

d

Vector of observed events, should have named elements.

ec

Vector of central exposure. The central exposure corresponds to the sum of the exposure duration over the insured population. An individual experiencing an event of interest during the year will no longer be exposed afterward and the exposure should be computed accordingly.

lambda

Smoothing parameter. If missing, an optimization procedure will be used to find the optimal smoothing parameter. If supplied, no optimal smoothing parameter search will take place unless the method argument is also supplied, in which case lambda will be used as the starting parameter for the optimization procedure.

criterion

Criterion to be used for the selection of the optimal smoothing parameter. Default is "REML" which stands for restricted maximum likelihood. Other options include "AIC", "BIC" and "GCV".

method

Method to be used to find the optimal smoothing parameter. Default to "fixed_lambda" if lambda is supplied, meaning no optimization is performed. Otherwise, default to the most reliable "outer" method based on the optimize function from package stats.

q

Order of penalization. Polynoms of degrees q - 1 are considered completely smooth and are therefore unpenalized. Should be left to the default of 2 for most practical applications.

framework

Default framework is "ml" which stands for maximum likelihood unless the y argument is also provided, in which case an "reg" or (approximate) regression framework is used.

y

Optional vector of observations whose elements should be named. Used only in the regression framework and if the d and ec arguments are missing (otherwise y is automatically computed from d and ec). May be useful when using Whittaker-Henderson smoothing outside of the survival analysis framework.

wt

Optional vector of weights. As for the observation vector y, used only in the regression framework and if the d and ec arguments are missing (otherwise wt is automatically set to d). May be useful when using Whittaker-Henderson smoothing outside of the survival analysis framework.

quiet

Should messages and warnings be silenced ? Default to FALSE, may be set to TRUE is the function is called repeatedly.

...

Additional parameters passed to the smoothing function called.

Value

An object of class WH_1d i.e. a list containing :

• d The inputed vector of observed events (if supplied as input)

• ec The inputed vector of central exposure (if supplied as input)

• y The observation vector, either supplied or computed as y = log(d) - log(ec)

• wt The inputed vector of weights, either supplied or computed as d

• y_hat The vector of values fitted using Whittaker-Henderson smoothing

• std_y_hat The vector of standard deviation associated with the fit

• res The vector of deviance residuals associated with the fit

• edf_obs The vector of effective degrees of freedom associated with each observation

• edf_par The vector of effective degrees of freedom associated with each eigenvector

• diagnosis A data.frame with one line containing the sum of effective degrees of freedom for the model, the deviance of the fit as well as the AIC, BIC, GCV and REML criteria

• Psi The variance-covariance matrix associated with the fit, which is required for the extrapolation step.

• lambda The smoothing parameter used.

• p The number of eigenvectors kept on each dimension if the rank reduction method is used (it should not in the one-dimensional case).

• q The supplied order for the penalization matrix.

Examples

d <- portfolio_mort$d
ec <- portfolio_mort$ec

y <- log(d / ec)
y[d == 0] <- - 20
wt <- d

# Maximum likelihood
WH_1d(d, ec, lambda = 1e2)
WH_1d(d, ec) # default outer iteration method based on the optimize function
WH_1d(d, ec, criterion = "GCV")
# alternative optimization criterion for smoothing parameter selection

# Regression
WH_1d(y = y, wt = wt, lambda = 1e2) # regression framework is default when y is supplied
WH_1d(d, ec, framework = "reg", lambda = 1e2)
# setting framework = "reg" forces computation of y from d and ec