Revisiting Whittaker-Henderson Smoothing


What is Whittaker-Henderson smoothing ?

Whittaker-Henderson (WH) smoothing is a gradation method aimed at correcting the effect of sampling fluctuations on an observation vector. It is applied to evenly-spaced discrete observations. Initially proposed by Whittaker (1922) for constructing mortality tables and further developed by the works of Henderson (1924), it remains one of the most popular methods among actuaries for constructing experience tables in life insurance. Extending to two-dimensional tables, it can be used for studying various risks, including but not limited to: mortality, disability, long-term care, lapse, mortgage default, and unemployment.

How to use the package?

The WH package may easily be installed from CRAN by running the code install.packages("WH") in the \(\mathsf{R}\) Console.

It features two main functions WH_1d and WH_2d corresponding to the one-dimensional and two-dimensional cases respectively. Two arguments are mandatory for those functions:

Additional arguments may be supplied, whose description is given in the documentation of the functions.

The package also embed two fictive agregated datasets to illustrate how to use it:

# One-dimensional case
d <- portfolio_mort$d
ec <- portfolio_mort$ec

WH_1d_fit <- WH_1d(d, ec)
Using outer iteration / Brent method
# Two-dimensional case
keep_age <- which(rowSums(portfolio_LTC$ec) > 5e2)
keep_duration <- which(colSums(portfolio_LTC$ec) > 1e3)

d  <- portfolio_LTC$d[keep_age, keep_duration]
ec <- portfolio_LTC$ec[keep_age, keep_duration]

WH_2d_fit <- WH_2d(d, ec)
Using performance iteration / Nelder-Mead method

Functions WH_1d and WH_2d output objects of class "WH_1d" and "WH_2d" to which additional functions (including generic S3 methods) may be applied:

An object fitted using the WH_1D function
Initial data contains 74 data points:
  Observation positions:  19  to  92 
Smoothing parameter selected: 13454 
Associated degrees of freedom: 7.5 
An object fitted using the WH_2D function
Initial data contains 90 data points:
  First dimension:  75  to  89 
  Second dimension:  0  to  5 
Smoothing parameters selected: 301   2 
Associated degrees of freedom: 13 

plot(WH_1d_fit, "res")

plot(WH_1d_fit, "edf")


plot(WH_2d_fit, "std_y_hat")

WH_1d_fit |> predict(newdata = 18:99) |> plot()

WH_2d_fit |> predict(newdata = list(age = 50:99,
                                    duration = 0:19)) |> plot()

WH_1d_df <- WH_1d_fit |> output_to_df()
WH_2d_df <- WH_2d_fit |> output_to_df()

Further WH smoothing theory

See the package vignette or the upcoming paper available here


Carballo, Alba, Maria Durban, and Dae-Jin Lee. 2021. “Out-of-Sample Prediction in Multidimensional p-Spline Models.” Mathematics 9 (15): 1761.

Henderson, Robert. 1924. “A New Method of Graduation.” Transactions of the Actuarial Society of America 25: 29–40.

Whittaker, Edmund T. 1922. “On a New Method of Graduation.” Proceedings of the Edinburgh Mathematical Society 41: 63–75.