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The goal of vital is to allow analysis of demographic data using tidy tools.

Installation

You can install the stable version from CRAN:

pak::pak("vital")

You can install the development version from Github:

pak::pak("robjhyndman/vital")

Examples

First load the necessary packages.

vital objects

The basic data object is a vital, which is 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: 35,754 x 6 [1Y]
#> # Key:     Age x Sex [101 x 2]
#>     Year   Age OpenInterval Sex    Population Mortality
#>    <int> <int> <lgl>        <chr>       <dbl>     <dbl>
#>  1  1846     0 FALSE        Female      17990   0.109  
#>  2  1846     1 FALSE        Female      16132   0.0498 
#>  3  1846     2 FALSE        Female      16404   0.0279 
#>  4  1846     3 FALSE        Female      17564   0.0205 
#>  5  1846     4 FALSE        Female      16352   0.0140 
#>  6  1846     5 FALSE        Female      14538   0.00995
#>  7  1846     6 FALSE        Female      13426   0.00861
#>  8  1846     7 FALSE        Female      13531   0.00735
#>  9  1846     8 FALSE        Female      13664   0.00628
#> 10  1846     9 FALSE        Female      13746   0.00457
#> # ℹ 35,744 more rows

This example contains data from 1846 to 2022. It 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)
#> [1] "Year"
key_vars(nor)
#> [1] "Age" "Sex"
vital_vars(nor)
#>          age          sex   population 
#>        "Age"        "Sex" "Population"

Plots

There are autoplot() functions for plotting vital objects. These produce rainbow plots 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

Lifetables can be produced using the life_table() function. It will produce lifetables for each unique combination of the index and key variables other than age.

# Lifetable in 2000
nor |>
  filter(Year == 2000) |>
  life_table()
#> # A vital: 202 x 13 [?]
#> # Key:     Age x Sex [101 x 2]
#>     Year   Age Sex        mx      qx    lx      dx    Lx    Tx    ex    rx    nx
#>    <int> <int> <chr>   <dbl>   <dbl> <dbl>   <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#>  1  2000     0 Fema… 3.28e-3 3.27e-3 1     3.27e-3 0.997  81.4  81.4 0.997     1
#>  2  2000     1 Fema… 2.77e-4 2.77e-4 0.997 2.76e-4 0.997  80.4  80.6 1.00      1
#>  3  2000     2 Fema… 3.09e-4 3.09e-4 0.996 3.08e-4 0.996  79.4  79.7 1.00      1
#>  4  2000     3 Fema… 1.33e-4 1.33e-4 0.996 1.32e-4 0.996  78.4  78.7 1.00      1
#>  5  2000     4 Fema… 1.68e-4 1.68e-4 0.996 1.67e-4 0.996  77.4  77.7 1.00      1
#>  6  2000     5 Fema… 3.30e-5 3.30e-5 0.996 3.29e-5 0.996  76.4  76.7 1.00      1
#>  7  2000     6 Fema… 1.35e-4 1.35e-4 0.996 1.34e-4 0.996  75.4  75.7 1.00      1
#>  8  2000     7 Fema… 1.68e-4 1.68e-4 0.996 1.67e-4 0.996  74.4  74.7 1.00      1
#>  9  2000     8 Fema… 6.70e-5 6.70e-5 0.996 6.67e-5 0.995  73.4  73.7 1.00      1
#> 10  2000     9 Fema… 1.97e-4 1.97e-4 0.995 1.96e-4 0.995  72.4  72.7 1.00      1
#> # ℹ 192 more rows
#> # ℹ 1 more variable: ax <dbl>

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

# Life expectancy
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.

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

Lee-Carter models

Lee-Carter models (Lee & Carter, JASA, 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
#> # A mable: 2 x 2
#> # Key:     Sex [2]
#>   Sex    lee_carter
#>   <chr>     <model>
#> 1 Female       <LC>
#> 2 Male         <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()
#> Series: Mortality 
#> Model: LC 
#> Transformation: log(Mortality) 
#> 
#> Options:
#>   Adjust method: dt
#>   Jump choice: fit
#> 
#> Age functions
#> # A tibble: 101 × 3
#>     Age    ax     bx
#>   <int> <dbl>  <dbl>
#> 1     0 -3.73 0.0156
#> 2     1 -5.25 0.0232
#> 3     2 -5.86 0.0216
#> 4     3 -6.22 0.0214
#> 5     4 -6.33 0.0193
#> # ℹ 96 more rows
#> 
#> Time coefficients
#> # A tsibble: 177 x 2 [1Y]
#>    Year    kt
#>   <int> <dbl>
#> 1  1846  83.2
#> 2  1847  95.7
#> 3  1848  93.8
#> 4  1849  88.5
#> 5  1850  86.1
#> # ℹ 172 more rows
#> 
#> Time series model: RW w/ drift 
#> 
#> Variance explained: 78.83%

The results can be plotted.

The components can be extracted.

age_components(lc)
#> # A tibble: 202 × 4
#>    Sex      Age    ax     bx
#>    <chr>  <int> <dbl>  <dbl>
#>  1 Female     0 -3.73 0.0156
#>  2 Female     1 -5.25 0.0232
#>  3 Female     2 -5.86 0.0216
#>  4 Female     3 -6.22 0.0214
#>  5 Female     4 -6.33 0.0193
#>  6 Female     5 -6.58 0.0197
#>  7 Female     6 -6.66 0.0186
#>  8 Female     7 -6.72 0.0176
#>  9 Female     8 -6.69 0.0152
#> 10 Female     9 -6.74 0.0147
#> # ℹ 192 more rows
time_components(lc)
#> # A tsibble: 354 x 3 [1Y]
#> # Key:       Sex [2]
#>    Sex     Year    kt
#>    <chr>  <int> <dbl>
#>  1 Female  1846  83.2
#>  2 Female  1847  95.7
#>  3 Female  1848  93.8
#>  4 Female  1849  88.5
#>  5 Female  1850  86.1
#>  6 Female  1851  85.3
#>  7 Female  1852  87.2
#>  8 Female  1853  94.0
#>  9 Female  1854  79.3
#> 10 Female  1855  80.8
#> # ℹ 344 more rows

Forecasts are obtained using the forecast() function

# Forecasts from Lee-Carter model
lc |>
  forecast(h = 20) 
#> # A vital fable: 4,040 x 6 [1Y]
#> # Key:           Age x (Sex, .model) [101 x 2]
#>    Sex    .model      Year   Age        Mortality   .mean
#>    <chr>  <chr>      <dbl> <int>           <dist>   <dbl>
#>  1 Female lee_carter  2023     0 t(N(-5.1, 0.03)) 0.00614
#>  2 Female lee_carter  2024     0 t(N(-5.1, 0.06)) 0.00613
#>  3 Female lee_carter  2025     0 t(N(-5.1, 0.09)) 0.00613
#>  4 Female lee_carter  2026     0 t(N(-5.2, 0.12)) 0.00613
#>  5 Female lee_carter  2027     0 t(N(-5.2, 0.15)) 0.00612
#>  6 Female lee_carter  2028     0 t(N(-5.2, 0.18)) 0.00612
#>  7 Female lee_carter  2029     0 t(N(-5.2, 0.21)) 0.00612
#>  8 Female lee_carter  2030     0 t(N(-5.2, 0.25)) 0.00611
#>  9 Female lee_carter  2031     0 t(N(-5.2, 0.28)) 0.00611
#> 10 Female lee_carter  2032     0 t(N(-5.2, 0.31)) 0.00610
#> # ℹ 4,030 more rows

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 & Ullah, CSDA, 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
fit <- nor |> 
  smooth_mortality(Mortality) |> 
  model(hu = FDM(log(.smooth)))
fc <- fit |>
  forecast(h = 20) 
autoplot(fc) +
  scale_y_log10()

Coherent functional data models

A coherent functional data model (Hyndman, Booth & Yasmeen, Demography, 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.

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

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.