Mort2Dsmooth which is a
two-dimensional P-splines smooth of the input data of degree and order
fixed by the user. Specifically tailored to mortality data.
Mort2Dsmooth(x, y, Z, offset, W, overdispersion=FALSE, ndx = c(floor(length(x)/5), floor(length(y)/5)), deg = c(3, 3), pord = c(2, 2), lambdas = NULL, df = NULL, method = 1, coefstart = NULL, control = list())2 ndx[1] + 1 of them. 2 ndx[2] + 1 of them. Z should
correspond to the length of x and y, respectively. NULL or
a numeric matrix of dimensions of Z or a single numeric
value. NULL or a numeric matrix of dimensions
of Z or a single numeric value. Details. Default: FALSE. floor(length(x)/5),
floor(length(y)/5)]. method = 1 (default) adjusts the two smoothing
parameters so that the BIC is minimized. method = 2 adjusts
lambdas so that the AIC is minimized. method = 3 uses
the values supplied for lambdas. Isotropic smoothing is allowed
in this method. method = 4 adjusts lambdas so that the
degrees of freedom is equal to the supplied df. Details. Mort2Dsmooth with components:x. y. x. y. x and y. The response variables must be
a matrix of Poisson distributed counts. 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
the ages and the argument y the years under study. The
matrix of death counts will be the argument Z. In a Poisson
regression setting applied to actual death counts the offset
will be the logarithm of the matrix of 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.Mort2Dsmooth for a more comprehensive way to
forecast mortality rates over ages and years.
The method produces results from a smoothing function which is the
Kronecker product of B-spline basis over the two axes and include a
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 for each of the axis. Nevertheless,
the smoothing parameters lambdas are mainly used to tune the
smoothness/model fidelity of the fitted values.
The ranges in which lambda is searched is given in control -
RANGEx and RANGEy. Though they can be modified, the
default values are suitable for most of the application.
There are print.Mort2Dsmooth,
summary.Mort2Dsmooth, plot.Mort2Dsmooth
predict.Mort2Dsmooth and
residuals.Mort2Dsmooth methods available for this
function.
Four methods for optimizing the smoothing parameters 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
parameters (method=3) or the degrees of freedom to be used in
the model (method=4). In this last case isotropic smoothing
(same smoothing parameter over x and y) is employed. If
the user provides only a single value for the argument lambdas,
isotropic smoothing is applied (with warning). Note that
Mort2Dsmooth uses approximated degrees 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 two-dimensional 'ultimate' smoothing with very large
lambda will approach to a surface which is a product of two polynomial
of degree pord[1] and pord[2], respectively. In
particular, when pord=c(2,2) the 'ultimate' smoothing is a
bi-linear surface over x and y.
The argument overdispersion can be set to TRUE when
smoothing parameters selection has to account for possible presence of
over(under)dispersion. Mortality data often present overdispersion
also known, in demography, as heterogeneity. Duplicates in insurance
data can lead to overdispersed data, too. Smoothing parameters
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 references. Note that with overdispersed data both BIC
and AIC might select higher lambdas, leading to smoother
outcomes. When overdispersion=FALSE (default value) or
method=3 or method=4, psi2 is estimated after the
smoothing parameters have been employed. Overdispersion parameter
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.
RANGEx: Range of smoothing parameters over x in which
the grid-search is applied, commonly taken in log-scale.
Default: [10^-4 ; 10^6].
RANGEy: Range of smoothing parameters over y 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 (pseudo) grid search when method is equal to 1,
2 or 4. Function cleversearch from package
svcm is employed to speed the grid search. See
Mort2Dsmooth_optimize for details.
The inner functions work using an arithmetic of arrays defined as
Generalized Linear Array Model (GLAM) (see references). In order to
avoid construction of large Kronecker product basis from the large
number of B-splines along the axes, the function profits of the
special structure of both the data as rectangular array and the model
matrix as tensor product. It uses sequence of nested matrix
operations and this leads to low storage and high speed computation
within the IWLS algorithm. Moreover, the function do not vectorize the
whole system keeping the actual two-dimensional array structure within
the scoring algorithm.
Currie, I. D., M. Durban, and P. H. C. Eilers (2006). Generalized linear array models with applications to multidimentional smoothing. Journal of the Royal Statistical Society. Series B. 68, 259-280.
Eilers, P. H. C., I. D. Currie, and M. Durban (2006). Fast and compact smoothing on large multidimensional grids. Computational Statistics & Data Analysis. 50, 61-76.
predict.Mort2Dsmooth,
plot.Mort2Dsmooth. ## selected data
ages <- 50:100
years <- 1950:2006
death <- selectHMDdata("Sweden", "Deaths", "Females",
ages = ages, years = years)
exposure <- selectHMDdata("Sweden", "Exposures", "Females",
ages = ages, years = years)
## fit with BIC
fitBIC <- Mort2Dsmooth(x=ages, y=years, Z=death,
offset=log(exposure))
fitBIC
summary(fitBIC)
## plot age 50 log death rates (1st row)
plot(years, log(death[1,]/exposure[1,]),
main="Mortality rates, log-scale.
Swedish females, age 50, 1950:2006")
lines(years, fitBIC$logmortality[1,], col=2, lwd=2)
## plot over age and years
## fitted log death rates from fitBIC
grid. <- expand.grid(list(ages=ages, years=years))
grid.$lmx <- c(fitBIC$logmortality)
levelplot(lmx ~ years * ages , grid.,
at=quantile(grid.$lmx, seq(0,1,length=10)),
col.regions=rainbow(9))
## see vignettes for examples on
## - Extra-Poisson variation
## - interpolation
## - extrapolation
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