Introduction to the vital package

vital objects

The basic data object is a vital, which is a time-indexed tibble that contains vital statistics such as births, deaths, population counts, and mortality and fertility rates.

We will use Norwegian data in the following examples. First, let’s remove the “Total” Sex category and collapse the upper ages into a final age group of 100+.

nor <- norway_mortality |>
  filter(Sex != "Total") |>
  collapse_ages(max_age = 100)
nor
#> # A vital: 25,048 x 7 [1Y]
#> # Key:     Age x Sex [101 x 2]
#>     Year   Age OpenInterval Sex    Population Deaths Mortality
#>    <int> <int> <lgl>        <chr>       <dbl>  <dbl>     <dbl>
#>  1  1900     0 FALSE        Female      30070 2376.    0.0778 
#>  2  1900     1 FALSE        Female      28960  842     0.0290 
#>  3  1900     2 FALSE        Female      28043  348     0.0123 
#>  4  1900     3 FALSE        Female      27019  216.    0.00786
#>  5  1900     4 FALSE        Female      26854  168.    0.00624
#>  6  1900     5 FALSE        Female      25569  140.    0.00538
#>  7  1900     6 FALSE        Female      25534  108.    0.00422
#>  8  1900     7 FALSE        Female      24314   93.5   0.00376
#>  9  1900     8 FALSE        Female      24979   93.5   0.00380
#> 10  1900     9 FALSE        Female      24428   90     0.00365
#> # ℹ 25,038 more rows

This example contains data from 1900 to 2023. There are 101 age groups and 2 Sex categories. A vital must have a time “index” variable, and optionally other categorical variables known as “key” variables. Each row must have a unique combination of the index and key variables. Some columns are “vital” variables, such as “Age” and “Sex”.

We can use functions to see which variables are index, key or vital:

index_var(nor)
key_vars(nor)
vital_vars(nor)

Plots

There are autoplot() functions for plotting vital objects. These produce rainbow plots (Hyndman and Shang 2010) where each line represents data for one year, and the variable is plotted against age.

nor |>
  autoplot(Mortality) +
  scale_y_log10()

We can use standard ggplot functions to modify the plot as desired. For example, here are population pyramids for all years.

nor |>
  mutate(Population = if_else(Sex == "Female", -Population, Population)) |>
  autoplot(Population) +
  coord_flip() +
  facet_grid(. ~ Sex, scales = "free_x")

Life tables and life expectancy

Life tables (Chiang 1984) can be produced using the life_table() function. It will produce life tables for each unique combination of the index and key variables other than age.

# Life tables for males and females in Norway in 2000
nor |>
  filter(Year == 2000) |>
  life_table()

Life expectancy (\(e_x\) with \(x=0\) by default) is computed using life_expectancy():

# Life expectancy for males and females in Norway
nor |>
  life_expectancy() |>
  ggplot(aes(x = Year, y = ex, color = Sex)) +
  geom_line()

Smoothing

Several smoothing functions are provided: smooth_spline(), smooth_mortality(), smooth_fertility(), and smooth_loess(), each smoothing across the age variable for each year. The methods used in smooth_mortality() and smooth_fertility() are described in Hyndman and Ullah (2007).

# Smoothed data
nor |>
  filter(Year == 1967) |>
  smooth_mortality(Mortality) |>
  autoplot(Mortality) +
  geom_line(aes(y = .smooth), col = "#0072B2") +
  ylab("Mortality rate") +
  scale_y_log10()

Lee-Carter models

Lee-Carter models (Lee and Carter 1992) are estimated using the LC() function which must be called within a model() function:

# Lee-Carter model
lc <- nor |>
  model(lee_carter = LC(log(Mortality)))
lc

Models are fitted for all combinations of key variables excluding age. To see the details for a specific model, use the report() function.

lc |>
  filter(Sex == "Female") |>
  report()

The results can be plotted.

autoplot(lc)

The components can be extracted.

age_components(lc)
time_components(lc)

Forecasts are obtained using the forecast() function

# Forecasts from Lee-Carter model
lc |>
  forecast(h = 20)

The forecasts are returned as a distribution column (here transformed normal because of the log transformation used in the model). The .mean column gives the point forecasts equal to the mean of the distribution column.

Functional data models

Functional data models (Hyndman and Ullah 2007) can be estimated in a similar way to Lee-Carter models, but with an additional smoothing step, then modelling with LC replaced by FDM.

# FDM model
fdm <- nor |>
  smooth_mortality(Mortality) |>
  model(hu = FDM(log(.smooth)))
fc_fdm <- fdm |>
  forecast(h = 20)
autoplot(fc_fdm) +
  scale_y_log10()

Functional data models have multiple principal components, rather than the single factor used in Lee-Carter models.

fdm |>
  autoplot(show_order = 3)

By default, six factors are estimated using FDM(). Here we have chosen to plot only the first three.

The components can be extracted.

age_components(fdm)
time_components(fdm)

Coherent functional data models

A coherent functional data model (Hyndman, Booth, and Yasmeen 2013), is obtained by first computing the sex-products and sex-ratios of the smoothed mortality data. Then a functional data model is fitted to the smoothed data, forecasts are obtained, and the product/ratio transformation is reversed. The following code shows the steps.

fdm_coherent <- nor |>
  smooth_mortality(Mortality) |>
  make_pr(.smooth) |>
  model(hby = FDM(log(.smooth), coherent = TRUE))
fc_coherent <- fdm_coherent |>
  forecast(h = 20) |>
  undo_pr(.smooth)
fc_coherent

Here, make_pr() makes the product-ratios, while undo_pr() undoes them.

The argument coherent = TRUE in FDM() ensures that the ARIMA models fitted to the coefficients are stationary when applied to the sex-ratios.