Mort1Dsmooth which is a P-splines
smooth of the input data of degree and order fixed by the
user. Specifically tailored to mortality data.
Mort1Dsmooth(x, y, offset, w, overdispersion=FALSE, ndx = floor(length(x)/5), deg = 3, pord = 2, lambda = NULL, df = NULL, method = 1, coefstart = NULL, control = list())2 ndx + 1 of them. y must be a
vector of the same length as x. NULL or a numeric vector of length either one or equal to the
number of cases. NULL or a numeric vector of length
equal to the number of cases. Details. Default: FALSE. floor(length(x)/5). method = 1 (default) adjusts the smoothing parameter
so that the BIC is minimized. method = 2 adjusts lambda
so that the AIC is minimized. method = 3 uses the value
supplied for lambda. method = 4 adjusts lambda so
that the degrees of freedom is equal to the supplied df.Details. Mort1Dsmooth with components:x. The response variables must be Poisson distributed counts,
though overdisperion can be accounted. Offset can be provided,
otherwise the default is that all weights are one. The function is specifically tailored to smooth mortality data in
one-dimensional setting. In such case the argument x would be
either the ages or the years under study. Death counts will be the
argument y. In a Poisson regression setting applied to actual
death counts the offset will be the logarithm of the exposure
population. See example below.
The function can obviously account for zero counts and definite
offset. In a mortality context, the user can apply the function to
data with zero deaths, but it has to take care that no exposures are equal
to zero, i.e. offset equal to minus infinitive. In this last case, the
argument w can help. The user would need to set weights equal
to zero when exposures are equal to zero leading to interpolation of
the data. See example below.
Regardless the presence of exposures equal to zero, the argument
w can also be used for extrapolation and interpolation of the
data. Nevertheless see the function
predict.Mort1Dsmooth for a more comprehensive way to
forecast mortality rates.
The method produces results similar to function smooth.spline,
but the smoothing function is a B-spline smooth with discrete
penalization directly on the differences of the B-splines
coefficients. The user can set the order of difference, the degree of
the B-splines and number of them. Nevertheless, the smoothing
parameter lambda is mainly used to tune the smoothness/model
fidelity of the fitted values.
The range in which lambda is searched is given in control -
RANGE. Though it can be modified, the default values are
suitable for most of the application.
There are print.Mort1Dsmooth,
summary.Mort1Dsmooth, plot.Mort1Dsmooth,
predict.Mort1Dsmooth and
residuals.Mort1Dsmooth methods available for this
function.
Four methods for optimizing the smoothing parameter are available. The BIC
is set as default. Minimization of the AIC is also possible. BIC will
give always smoother outcomes with respect to AIC, especially for
large sample size. Alternatively the user can directly provide the
smoothing parameter (method=3) or the degree of freedom to be
used in the model (method=4). Note that Mort1Dsmooth
uses approximated degree of freedom, therefore method=4 will
produce fitted values with degree of freedom only similar to the one
provided in df. The tolerance level can be set via
control - TOL2.
Note that the 'ultimate' smoothing with very large lambda will
approach to a polynomial of degree pord.
Starting coeffients for the B-spline basis can be provided by the
user. This feature can be useful when a grid-search is manually
performed by the user.
The argument overdispersion can be set to TRUE when possible presence of
over(under)dispersion needs to be considered in the selection of the
smoothing parameter. Mortality data often present overdispersion
also known, in demography, as heterogeneity. Duplicates in insurance
data can lead to overdispersed data, too. Smoothing parameter
selection may be affected by this phenomenon. When
overdispersion=TRUE, the function uses a penalized
quasi-likelihood method for including an overdisperion parameter
(psi2) in the fitting procedure. With this approach expected
values are assumed equal to the variance multiplied by the parameter
psi2. See reference. Note that the inclusion of the
overdisperion parameter within the estimation might lead to select
higher lambda, leading to smoother outcomes. When
overdispersion=FALSE (default value) or method=3 or
method=4, psi2 is estimated after the smoothing
parameter have been employed. Overdispersion parameter considerably larger
(smaller) than 1 may be a sign of overdispersion (underdispersion).
The control argument is a list that can supply any of the
following components:
MON: Logical. If TRUE tracing information on the
progress of the fitting is produced. Default: FALSE.
TOL1: The absolute convergence tolerance for each completed
scoring algorithm. Default: 1e-06.
TOL2: Difference between two adjacent smoothing parameters in
the (pseudo) grid search, log-scale. Useful only when method is
equal to 1, 2 or 4. Default: 0.5.
RANGE: Range of smoothing parameters in which the grid-search
is applied, commonly taken in log-scale. Default: [10^-4 ; 10^6].
MAX.IT: The maximum number of iterations for each completed
scoring algorithm. Default: 50.
The arguments MON, TOL1 and MAX.IT are kept
during all the grid search when method is equal to 1,
2 or 4. Function cleversearch from package
svcm is employed to speed the grid search. See
Mort1Dsmooth_optimize for details.
Camarda, C. G. (2012). MortalitySmooth: An R Package for Smoothing Poisson Counts with P-Splines. Journal of Statistical Software. 50, 1-24. http://www.jstatsoft.org/v50/i01/.
predict.Mort1Dsmooth,
plot.Mort1Dsmooth, Mort1Dsmooth_optimize. ## selected data
years <- 1950:2006
death <- selectHMDdata("Japan", "Deaths", "Females",
ages = 80, years = years)
exposure <- selectHMDdata("Japan", "Exposures", "Females",
ages = 80, years = years)
## various fits
## default using Bayesian Information Criterion
fitBIC <- Mort1Dsmooth(x=years, y=death,
offset=log(exposure))
fitBIC
summary(fitBIC)
## subjective choice of the smoothing parameter lambda
fitLAM <- Mort1Dsmooth(x=years, y=death,
offset=log(exposure),
method=3, lambda=10000)
## plot
plot(years, log(death/exposure),
main="Mortality rates, log-scale.
Japanese females, age 80, 1950:2006")
lines(years, fitBIC$logmortality, col=2, lwd=2)
lines(years, fitLAM$logmortality, col=3, lwd=2)
legend("topright", c("Actual", "BIC", "lambda=10000"),
col=1:3, lwd=c(1,2,2), lty=c(-1,1,1),
pch=c(1,-1,-1))
## see vignettes for examples on
## - Extra-Poisson variation
## - interpolation
## - extrapolation
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