Create the Blending and Distilling Object

Based on model and output partitions, create a mixing partition and associated weights. That table of mixing values can be used to properly discretize a continuously varying (or otherwise high resolution) parameter to a relatively low resolution compartmental stratification, and then subsequently allocate the low-resolution model outcomes into the most likely high-resolution output partitions.

Usage

alembic(
  f_param,
  f_pop,
  model_partition,
  output_partition,
  pars_interp_opts = interpolate_opts(fun = stats::splinefun, kind = "point", method =
    "natural"),
  pop_interp_opts = interpolate_opts(fun = stats::approxfun, kind = "integral", method =
    "constant", yleft = 0, yright = 0)
)

Arguments

f_param: a function, f(x) which transforms the feature (e.g. age), to yield the parameter values. Alternatively, a data.frame where the first column is the feature and the second is the parameter; see xy.coords() for details. If the latter, combined with pars_interp_opts to create a parameter function.
f_pop: like f_param, either a density function (though it does not have to integrate to 1 like a pdf) or a data.frame of values. If the latter, it is treated as a series of populations within intervals, and then interpolated with pop_interp_opts to create a density function.
model_partition: a numeric vector of cut points, which define the partitioning that will be used in the model; must be length > 1
output_partition: the partition of the underlying feature; must be length > 1
pars_interp_opts: a list, minimally with an element fun, corresponding to an interpolation function. Defaults to splinefun() "natural" interpolation.
pop_interp_opts: like pars_interp_opts, but for density. Defaults to approxfun() "constant" interpolation.

Value

a data.table with columns: model_partition, output_partition, weight and relpop. The first two columns identify partition lower bounds, for both the model and output, the other values are associated with; the combination of model_partition and output_partition forms a unique identifier, but individually they may appear multiple times. Generally, this object is only useful as an input to the blend() and distill() tools.

Details

The alembic function creates a mixing table, which governs the conversion between model and output partitions. The mixing table a data.table::data.table() where each row corresponds to a mixing partition $c_i$, which is the union of the model and output partitions - i.e. each unique boundary is included. Within each row, there is a weight and relpop entry, corresponding to

$$ \textrm{weight}_i = \int_{c_i} f(x)\rho(x)\text{d}x $$ $$ \textrm{relpop}_i = \int_{c_i} \rho(x)\text{d}x $$

where $f(x)$ corresponds to the f_param argument and $\rho(x)$ corresponds to the f_pop argument.

This mixing table is used in the blend() and distill() functions.

When blending, the appropriately weighted parameter for a model partition is the sum of $\textrm{weight}_i$ divided by the $\textrm{relpop}_i$ associated with mixing partition(s) in that model partition. This corresponds to the properly, population weighted average of that parameter over the partition.

When distilling, model outcomes associated with weighted parameter from partition $j$ are distributed to the output partition $i$ by the sum of weights in mixing partitions in both $j$ and $i$ divided by the total weight in $j$. This corresponds to proportional allocation according to Bayes rule: the outcome in the model partition was relatively more likely in the higher weight mixing partition.

Examples

ifr_levin <- function(age_in_years) {
  (10^(-3.27 + 0.0524 * age_in_years))/100
}
age_limits <- c(seq(0, 69, by = 5), 70, 80, 101)
age_pyramid <- data.frame(
  from = 0:101, weight = ifelse(0:101 < 65, 1, .99^(0:101-64))
)
age_pyramid$weight[102] <- 0
# flat age distribution, then 1% annual deaths, no one lives past 101
ifr_alembic <- alembic(ifr_levin, age_pyramid, age_limits, 0:101)