Use components from a Singular Value Decomposition (SVD) to model an interaction involving age and time, or age, sex/gender and time, where the coefficients evolve over time.
SVD_AR(
ssvd,
n_comp = NULL,
indep = TRUE,
n_coef = 2,
s = 1,
shape1 = 5,
shape2 = 5,
con = c("none", "by")
)SVD_AR1(
ssvd,
n_comp = NULL,
indep = TRUE,
min = 0.8,
max = 0.98,
s = 1,
shape1 = 5,
shape2 = 5,
con = c("none", "by")
)
SVD_RW(ssvd, n_comp = NULL, indep = TRUE, s = 1, sd = 1, con = c("none", "by"))
SVD_RW2(
ssvd,
n_comp = NULL,
indep = TRUE,
s = 1,
sd = 1,
sd_slope = 1,
con = c("none", "by")
)
An object of class "bage_prior_svd_ar",
"bage_prior_svd_rw", or "bage_prior_svd_rw2".
Object of class "bage_ssvd"
holding a scaled SVD. See below for scaled SVDs
of databases currently available in bage.
Number of components from scaled SVD
to use in modelling. The default is half
the number of components of ssvd.
Whether to use separate or
combined SVDs in terms involving sex or gender.
Default is TRUE.
See below for details.
Number of AR coefficients in SVD_RW().
Scale for standard deviations terms.
Parameters for prior
for coefficients in SVD_AR().
Defaults are 5 and 5.
Constraints on parameters.
Current choices are "none" and "by".
Default is "none". See below for details.
Minimum and maximum values
for autocorrelation coefficient in SVD_AR1().
Defaults are 0.8 and 0.98.
Standard deviation
of initial value for random walks. Default is 1.
Can be 0.
Standard deviation in prior
for initial slope. Default is 1.
When the interaction being modelled only involves age and time, or age, sex/gender, and time
$$\pmb{\beta}_t = \pmb{F} \pmb{\alpha}_t + \pmb{g},$$
and when it involves other variables besides age, sex/gender, and time,
$$\pmb{\beta}_{u,t} = \pmb{F} \pmb{\alpha}_{u,t} + \pmb{g},$$
where
\(\pmb{\beta}\) is an interaction involving age, time, possibly sex/gender, and possibly other variables;
\(\pmb{\beta}_t\) is a subvector of \(\pmb{\beta}\) holding values for period \(t\);
\(\pmb{\beta}_{u,t}\) is a subvector of \(\pmb{\beta}_t\) holding values for the \(u\)th combination of the non-age, non-time, non-sex/gender variables for period \(t\);
\(\pmb{F}\) is a known matrix; and
\(\pmb{g}\) is a known vector.
\(\pmb{F}\) and \(\pmb{g}\) are constructed from a large database of age-specific demographic estimates by applying the singular value decomposition, and then standardizing.
With SVD_AR(), the prior for the \(k\)th element
of \(\pmb{\alpha}_t\) or \(\pmb{\alpha}_{u,t}\) is
$$\alpha_{k,t} = \phi_1 \alpha_{k,t-1} + \cdots + \phi_n \beta_{k,t-n} + \epsilon_{k,t}$$
or
$$\alpha_{k,u,t} = \phi_1 \alpha_{k,u,t-1} + \cdots + \phi_n \beta_{k,u,t-n} + \epsilon_{k,u,t};$$
with SVD_AR1(), it is
$$\alpha_{k,t} = \phi \alpha_{k,t-1} + \epsilon_{k,t}$$
or
$$\alpha_{k,u,t} = \phi \alpha_{k,u,t-1} + \epsilon_{k,u,t};$$
with SVD_RW(), it is
$$\alpha_{k,t} = \alpha_{k,t-1} + \epsilon_{k,t}$$
or
$$\alpha_{k,u,t} = \alpha_{k,u,t-1} + \epsilon_{k,u,t};$$
and with SVD_RW2(), it is
$$\alpha_{k,t} = 2 \alpha_{k,t-1} - \alpha_{k,t-2} + \epsilon_{k,t}$$
or
$$\alpha_{k,u,t} = 2 \alpha_{k,u,t-1} - \alpha_{k,u,t-2} + \epsilon_{k,u,t}.$$
For details, see AR(), AR1(),
RW(), and RW2().
With some combinations of terms and priors, the values of the intercept, main effects, and interactions are are only weakly identified. For instance, it may be possible to increase the value of the intercept and reduce the value of the remaining terms in the model with no effect on predicted rates and only a tiny effect on prior probabilities. This weak identifiability is typically harmless. However, in some applications, such as forecasting, or when trying to obtain interpretable values for main effects and interactions, it can be helpful to increase identifiability through the use of constraints.
Current options for constraints are:
"none" No constraints. The default.
"by" Only used in interaction terms that include 'along' and
'by' dimensions. Within each value of the 'along'
dimension, terms across each 'by' dimension are constrained
to sum to 0.
HMD Mortality rates from the
Human Mortality Database.
HFD Fertility rates from the
Human Fertility Database.
LFP Labor forcce participation
rates from the OECD.
SVD_AR(), SVD_AR1(), SVD_RW(), and SVD_RW2()
priors assume that, in any given period,
the age profiles or age-sex profiles for the quantity
being modelled looks like they were drawn at random
from an external demographic database. For instance,
the SVD_AR() prior obtained via
SVD_AR(HMD)
assumes that profiles look like they were obtained from the Human Mortality Database.
For details of the construction of scaled SVDS see the vignette here.
SVD() SVD prior for non-time-varying terms
RW() Smoothing via random walk
RW2() Smoothing via second-order random walk
Sp() Smoothing via splines
priors Overview of priors implemented in bage
set_prior() Specify prior for intercept,
main effect, or interaction
set_var_sexgender() Identify sex or gender variable in data
SVD_AR1(HMD)
SVD_RW(HMD, n_comp = 3)
SVD_RW2(HMD, indep = FALSE)
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