STARS: Spatial-temporal autoregressive model

In demofit: Parametric Mortality Curve Fitting and Mortality Forecasting Tools

Spatial-temporal autoregressive model

Description

Fits and forecasts mortality rates using spatial-temporal autoregressive model.

Usage

STARS(
  x,
  M,
  curve = c("gompertz", "makeham", "oppermann", "thiele", "wittsteinbumsted", "perks",
    "weibull", "vandermaen", "beard", "heligmanpollard", "rogersplanck", "siler",
    "martinelle", "thatcher", "gompertz2", "makeham2", "oppermann2", "thiele2",
    "wittsteinbumsted2", "perks2", "weibull2", "vandermaen2", "beard2",
    "heligmanpollard2", "rogersplanck2", "siler2", "martinelle2", "thatcher2"),
  h = 10
)

Arguments

x	vector of ages.
M	matrix of mortality rates (rows as years and columns as ages).
curve	name of mortality curve for smoothing forecasted mortality rates (including gompertz, makeham, oppermann, thiele, wittsteinbumsted, perks, weibull, vandermaen, beard, heligmanpollard, rogersplanck, siler, martinelle, thatcher, gompertz2, makeham2, oppermann2, thiele2, wittsteinbumsted2, perks2, weibull2, vandermaen2, beard2, heligmanpollard2, rogersplanck2, siler2, martinelle2, thatcher2, where first 14 curves' parameters are unconstrained and last 14 curves' parameters are generally restricted to be positive).
h	forecast horizon (default = 10).

Details

The spatial-temporal autoregressive (STAR) model is specified as

ln(M_t) = \mu + R ln(M_{t-1}) + E_t,

where \mu contains intercept parameters and R is banded matrix with non-zero entries in main diagonal and two subdiagonals below that.

The model is estimated by constrained regression and is forecasted via its autoregressive nature. Constraints in R include: all non-zero entries are between zero and one and sum of each row is one. It can be applied to whole age range.

Value

An object of class STARS with associated S3 methods coef, forecast, plot, and residuals.

References

Li, H. and Lu, Y. (2017). Coherent forecasting of mortality rates: A sparse vector-autoregression approach. ASTIN Bulletin, 47(2), 563-600.

Examples

x <- 60:89
a <- c(-4.8499,-4.7676,-4.6719,-4.5722,-4.4847,-4.3841,-4.2813,-4.1863,-4.0861,-3.9962,
-3.8885,-3.7896,-3.6853,-3.5737,-3.4728,-3.3718,-3.2586,-3.1474,-3.0371,-2.9206,
-2.7998,-2.6845,-2.5653,-2.4581,-2.3367,-2.2159,-2.1017,-1.9941,-1.8821, -1.7697)
b <- c(0.0283,0.0321,0.0335,0.0336,0.0341,0.0358,0.0368,0.0403,0.0392,0.0395,
0.0396,0.0399,0.0397,0.0386,0.039,0.0375,0.0367,0.0368,0.035,0.0354,
0.0336,0.0323,0.0313,0.0295,0.0282,0.0265,0.024,0.0226,0.0219,0.0183)
k <- c(12.11,10.69,11.18,9.64,9.35,8.21,6.89,5.74,4.56,3.6,
3.27,2.04,1.11,-0.44,-1.05,-1.03,-1.84,-2.9,-4.03,-4.12,
-5.18,-5.64,-6,-6.51,-6.91,-6.9,-8.32,-8.53,-9.69,-9.31)
set.seed(123)
M <- exp(outer(k,b)+matrix(a,nrow=30,ncol=30,byrow=TRUE)+rnorm(900,0,0.035))
fit <- STARS(x=x,M=M,curve="makeham",h=30)
coef(fit)
forecast::forecast(fit)
plot(fit)
residuals(fit)

demofit documentation built on April 9, 2026, 5:09 p.m.