whittaker.mortalityTable {MortalityTables}

Smooth a life table using the Whittaker-Henderson method, intepolation possibly missing values

Description

whittaker.mortalityTable uses the Whittaker-Henderson graduation method to smooth a table of raw observed death probabilities, optionally using the exposures stored in the table as weights (if no exposures are given, equal weights are applied). The weights (either explicitly given, implicitly taken from the exposures or implicit equal weights) will be normalized to sum 1. The parameter lambda indicates the importance of smootheness. A lower value of lambda will put more emphasis on reproducing the observation as good as possible at the cost of less smoothness. In turn, a higher value of lambda will force the smoothed result to be as smooth as possible with possibly larger deviation from the input data. All ages with a death probability of NA will be interpolated in the Whittaker-Henderson method (see e.g. Lowrie)

Usage

whittaker.mortalityTable(
  table,
  lambda = 10,
  d = 2,
  name.postfix = ", smoothed",
  ...,
  weights = NULL,
  log = TRUE
)

Arguments

table

Mortality table to be graduated. Must be an instance of a mortalityTable-derived class.

lambda

Smoothing parameter (default 10)

d

order of differences (default 2)

name.postfix

Postfix appended to the name of the graduated table

...

additional arguments (currently unused)

weights

Vector of weights used for graduation. Entries with weight 0 will be interpolated. If not given, the exposures of the table or equal weights are used. Weight 0 for a certain age indicates that the observation will not be used for smoothing at all, and will rather be interpolated from the smoothing of all other values.

log

Whether the smoothing should be applied to the logarithms of the table values or the values itself

References

Walter B. Lowrie: An Extension of the Whittaker-Henderson Method of Graduation, Transactions of Society of Actuaries, 1982, Vol. 34, pp. 329–372

See Also

whittaker

Examples

# A sample observation table with exposures and raw probabilities
obsTable = mortalityTable.period(
    name = "trivial observed table",
    ages = 0:15,
    deathProbs = c(
        0.0072, 0.00212, 0.00081, 0.0005, 0.0013,
        0.001, 0.00122, 0.00142, 0.007, 0.0043,
        0.0058, 0.0067, 0.0082, 0.0091, 0.0075, 0.01),
    exposures = c(
        150, 222, 350, 362, 542,
        682, 1022, 1053, 1103, 1037,
        968, 736, 822, 701, 653, 438))

# Effect of the different parameters
obsTable.smooth = whittaker.mortalityTable(obsTable,
    lambda = 1/10, d = 2, name.postfix = " smoothed (d=2, lambda=1/10)")
obsTable.smooth1 = whittaker.mortalityTable(obsTable,
    lambda = 1, d = 2, name.postfix = " smoothed (d=2, lambda=1)")
obsTable.smooth2 = whittaker.mortalityTable(obsTable,
    lambda = 1/10, d = 3, name.postfix = " smoothed (d=3, lambda=1/10)")
plot(obsTable, obsTable.smooth, obsTable.smooth1, obsTable.smooth2,
    title = "Observed death probabilities")

# Missing values are interpolated from the Whittaker Henderson
obsTable.missing = obsTable
obsTable.missing@deathProbs[c(6,10,11,12)] = NA_real_
obsTable.interpolated = whittaker.mortalityTable(obsTable,
    lambda = 1/10, d = 2, name.postfix = " missing values interpolated")
plot(obsTable.missing, obsTable.interpolated,
    title = "Missing values are automatically interpolated") + geom_point(size = 3)