bayesPop: Probabilistic Population Projections

Abstract

We describe bayesPop, an R package for producing probabilistic population projections for all countries. This uses probabilistic projections of total fertility and life expectancy generated by Bayesian hierarchical models. It produces a sample from the joint posterior predictive distribution of future age- and sex-specific population counts, fertility rates and mortality rates, as well as future numbers of births and deaths. It provides graphical ways of summarizing this information, including trajectory plots and various kinds of probabilistic population pyramids. An expression language is introduced which allows the user to produce the predictive distribution of a wide variety of derived population quantities, such as the median age or the old age dependency ratio. The package produces aggregated projections for sets of countries, such as UN regions or trading blocs. The methodology has been used by the United Nations to produce their most recent official population projections for all countries, published in the World Population Prospects.

1. Introduction

Projections of countries’ future populations, broken down by age and sex, are widely used by governments at all levels for planning purposes, by international organizations for monitoring development and other goals, such as the Millennium Development Goals, by social and health researchers, and the private sector for strategic and marketing decisions.

Most population projections are currently done deterministically using the cohort component method. This is an age- and sex-structured version of the basic demographic identity that the population of a country at the next time point is equal to the current population, plus the number of births, minus the deaths, plus the immigrants, minus the emigrants (Leslie 1945; Preston et al. 2001).

Standard projections are deterministic, meaning that they yield a single value for each projected future population quantity of interest. However, probabilistic projections are widely desired because they are useful for decision-making when one wants to be reasonably sure of not under- or overpredicting a number, for assessing changes and deviations of population quantities from expectations, and for providing a general assessment of uncertainty.

A systematic framework for producing probabilistic population projections for all countries, both developed and developing, has recently been proposed by Raftery et al. (2012). It consists of probabilistically projecting total fertility rate and life expectancy using Bayesian hierarchical models (Alkema et al. 2011; Raftery et al. 2013b), converting the results to age-specific rates, and projecting the population forward using the cohort component method applied to each trajectory simulated from their predictive distributions (Ševčíková et al. 2015). The median projection from the method has been used as the official medium projection of the United Nations (UN) in the 2012 revision of the World Population Prospects (United Nations 2013). In July 2014 for the first time, the United Nations released official probabilistic population projections for all countries (Gerland et al. 2014; United Nations 2014).

Here we describe a package called bayesPop that was used by the UN to produce the official probabilistic projections. It is implemented in R (Ihaka and Gentleman 1996; R Core Team 2013), and it was developed to allow users beyond the UN to implement the methodology. The package allows an analyst to reproduce the UN projections, to generate variations on them corresponding to different inputs or modeling assumptions, or to use their own data. We also introduce a flexible expression language which allows probabilistic results to be summarized and visualized in graphs, maps or population pyramids. The software can be conveniently controlled from a graphical user interface.

The paper is organized as follows. In Section 2 we review the basic probabilistic population projection methodology underlying the package. In Section 3 we describe the bayesPop package and how to generate and view the probabilistic population projections using it. In Section 4, we show how to display probabilistic population pyramids for visualizing the age-specific results. In Section 5 we describe our expression language for generating probabilistic projections of user-defined derived population quantities. Examples include the median age of the population or the ratio of the population of one country to that of another. In Section 6 we indicate how to produce probabilistic projections of aggregated population quantities for groups of countries, such as UN regions or trading blocs. We conclude in Section 7 with a brief discussion of some related R packages.

2. Methodology

Most methods for predicting population P in country c at time period t are based on the demographic balancing equation, namely

$$P_{c,t}=P_{c,t-1}+B_{c,t}-D_{c,t}+M_{c,t},$$ (1)

where B denotes the number of births, D denotes the number of deaths and M denotes net international migration. In most applications this equation is solved deterministically using the cohort component method (Whelpton 1928, 1936), which decomposes it into age- and sex-specific components.

As it has traditionally been implemented by the United Nations Population Division (United Nations 1956, 1989), the cohort component method for projecting a country’s population by age and sex in future time periods t > 0 is deterministic, and requires the following inputs:

• sex- and age-specific population estimates at the initial time t = 0,

• projections of future total fertility rates (TFR),

• projections of sex ratio at birth,

• projections of female and male life expectancies (e₀),

• historical data on sex- and age-specific death rates (for t ≤ 0),

• historical data on fertility distribution by age (for t ≤ 0), and

• projections of future sex- and age-specific net international migration.

In each time period t, a projection of the fertility distribution by age is obtained using historical data (Ševčíková et al. 2015). Then this distribution is used to convert the TFR to age-specific fertility rates in time period t. Using the historical data on death rates, life expectancy is converted to age-specific mortality rates using a variant of the Lee-Carter method (Lee and Carter 1992). See Ševčíková et al. (2015) for more detail on these conversion steps. Finally the cohort component method is applied.

To communicate uncertainty in the context of this deterministic approach, until recently the UN used three scenarios, the high, medium and low variants. The medium projection is the main one. The high and low variants are generated by adding half a child to or subtracting half a chile from the TFR, respectively, and then applying the method above. Such an approach suffers from not having a probabilistic basis and can lead to inconsistencies (Lee and Tuljapurkar 1994; National Research Council 2000).

Methods for probabilistic projection of the two most important inputs have recently been developed, namely TFR (Alkema et al. 2011) and life expectancy (Raftery et al. 2013b). Raftery et al. (2012) describes a way to combine these components into overall probabilistic population projections. The idea is to simulate a large set of trajectories of future values of TFR (as implemented in bayesTFR, see Ševčíková et al. (2011)), then to simulate an equal number of trajectories of life expectancy (as implemented in bayesLife, see Ševčíková and Raftery (2013a)), and finally to convert each of the trajectories into a future trajectory of all sex- and age-specific populations, using the current UN methodology as described above. The resulting set of values is viewed as a sample from the predictive distribution of population numbers. This approach is implemented in the R package bayesPop (Ševčíková and Raftery 2013b). A graphical user interface (GUI) for the three packages, bayesTFR, bayesLife and bayesPop, is provided by the R package bayesDem (Ševčíková 2013a). Together, these packages allow one to generate probabilistic projections of TFR and life expectancy, and combine those results into probabilistic population projections from a single interface; see Figure 1. In addition to the CRAN repository, all of these packages are hosted on https://github.com/PPgp. Furthermore, a mailing list accessible from bayespop.csss.washington.edu is available to interested users.

Figure 1.

Structure of packages supported by bayesDem. Boxes shown on the left-hand side (connected by blue arrows) depict the main steps needed for generating probabilistic population projections. Boxes on the right-hand side show supporting functionality of the packages. The packages operate on UN datasets included in one of the wpp packages which can be explored and visualized using the wppExplorer package.

3. Using bayesPop

3.1. Generating population projections

The main function for generating probabilistic population projection is called pop.predict. It can be run for a single country or for a given set of countries. By default, projections are generated for all countries for which inputs are available. The data packages wpp2008 (Ševčíková and Gerland 2013a), wpp2010 (Ševčíková and Gerland 2013b), and wpp2012 (Ševčíková et al. 2013) provide such data for most countries. An argument wpp.year (which can be one of 2008, 2010, or 2012) determines the default dataset used in the projections. We will refer to the corresponding data package simply as a wpp package.

Inputs

The projection inputs listed in Section 2 are given in the argument inputs which is a list containing the various input components. The deterministic components include:

• popM, popF: Initial male, female age-specific population counts.

• mxM, mxF: Estimates of historical male and female age-specific death rates.

• pasfr: Estimates of historical age-specific fertility rates as percentages of TFR.

• srb: Projection of future sex ratio at birth.

• mig.type: Migration base year, and an argument determining whether migration is assumed to occur at the end of the five-year interval or to be evenly distributed over the interval.

• migM, migF: Projection of future male and female age-specific net international migration.

If any of these inputs is not specified, the default dataset from the given wpp package is used. If the user wishes to overwrite a default dataset, the corresponding component can be given as a tab-delimited text file (the manual for pop.predict describes the structure of these files). Note that the pop and mig components must be on the same scale which becomes the final scale of the projections. The wpp packages maintain their datasets on the scale of thousands.

Each of the probabilistic components of the inputs argument, namely TFR and e₀, can be specified in several ways, either as a directory, or as a file. In both cases, a set of trajectories is passed to the prediction function. The pop.predict function is designed to operate on the results of tfr.predict and e0.predict from bayesTFR and bayesLife, respectively. Thus one would specify here the directories in which the resulting TFR and e₀ trajectories are stored:

• tfr.sim.dir: Simulation directory used to store results of tfr.predict.

• e0F.sim.dir: Simulation directory used to store results of e0.predict for projections of female life expectancy.

• e0M.sim.dir: This can be a directory with a simulation of male life expectancy that is independent of the simulation in e0F.sim.dir. Preferably, however, it can be the keyword ‘joint ’ in which case it is assumed that male and female e₀ were generated jointly using the gap model (Raftery et al. 2014), and thus, they are both extracted from e0F.sim.dir.

For convenience, these probabilistic inputs can also be specified as text files, which can be useful, for example, if generated with software other than bayesTFR and bayesLife.

If neither of the probabilistic components of the inputs argument is given, the function creates three trajectories at most which are extracted from the wpp package, namely the projection median, and the low and high variants if available.

For experimental purposes, the function allows the user to enter multiple trajectories of net migration which can be used for example to test different migration scenarios. These components are called migMtraj and migFtraj and if used they replace the deterministic components migM and migF.

Other arguments

Besides specifying countries ( argument countries), one can define how many trajectories to generate ( nr.traj), the end year of the projection ( end.year), the initial year ( start.year), or the present year ( present.year). The initial year defines the first year of the historical death rates to be used to estimate the parameters of the Lee-Carter method. The present year defines the last observed time period, i.e. the initial time t = 0 from which projections start.

By default, vital events, such as births and deaths, used during the computation in the cohort component method are discarded. However, if the argument keep.vital.events is set to TRUE, they are stored together with the projection trajectories and can be used later for an analysis. It should be noted that storing vital events more than doubles the amount of data stored per country.

The argument output.dir specifies the location on disk where the results are to be stored.

Outputs

The pop.predict function applies the cohort component method to each set of trajectories (a set meaning one trajectory for each of TFR, female e₀ and male e₀), using the deterministic components for the remaining input data. As a result, we have a set of sex- and age-specific population trajectories which can be used to construct posterior distributions of various population quantities of interest.

The trajectories are stored in the “ output.dir/predictions” directory, one file per country, called “totpop_countryx.rda,” where x is the numerical country code. If storing vital events is requested, trajectories for number of births by age of mother and sex of child, trajectories for sex- and age-specific numbers of deaths, net migration, and trajectories for sex- and age-specific fertility and mortality rates are stored in files called “vital events countryx.rda”. Thus in such a case, there are two files per country. The resulting file structure is depicted in Figure 2. The approximate file sizes in this figure correspond to a simulation with 100 trajectories, such as in the example below. The optional aggregations sub-directory will be described in Section 6.

Figure 2.

Structure of folders and files created when generating projections with bayesPop. The approximate sizes correspond to a simulation with 100 trajectories, 162 countries and 6 aggregated areas.

An object returned by the pop.predict function is of class bayesPop.prediction. It contains pre-computed quantiles for the main population quantities, including total population, total population by sex and total population by sex and age. It also contains a pointer to the disk location where the country-specific trajectories are stored. The package contains various functions that help the user to view and analyze those results; these will be described in the following sections.

3.2. Example

We first show an example of how to generate probabilistic population projection from scratch, including generating all the probabilistic components. The blue arrows in Figure 1 show the workflow in this process. Note that estimating and projecting TFR and e₀ (step 1 and 2 below) use packages bayesTFR and bayesLife, respectively, and can be therefore considered as prerequisites for generating population projection with bayesPop. The data used in this example are taken from the wpp2012 package. At the time of writing, 2012 is the default value for the wpp.year argument in all three packages involved, but we expect this to change to 2015 some time after the 2015 Revision of the World Population Prospects has been published.

A word of caution about this example is in order. We will show an example that can be reproduced in a time-efficient manner and thus the Markov Chain Monte Carlo chains (MCMC) of the TFR and e₀ models might not converge. Thus results that will be shown throughout this paper are not UN official projections and may not even be realistic. For a real simulation, increase the number of iterations (argument iter), set it to “auto,” or use default values. Using multiple chains and setting an argument parallel to TRUE in steps 1 and 2 below could improve estimation results in less run time. Finally, steps 1 and 2 can be carried out independently of one another. If all the steps are processed sequentially, allow about an half an hour processing time on a current standard laptop.

1. Estimate TFR parameters for the phase II and phase III models (Raftery et al. 2013a) and generate TFR projections (about 15 minutes):

   R> sim.dir.tfr <- file.path(getwd(), "TFRprojections")
   R> run.tfr.mcmc(iter = 1000, nr.chains = 1, thin = 1,
   + output.dir = sim.dir.tfr, seed = 1)
   R> run.tfr3.mcmc(sim.dir = sim.dir.tfr, iter = 1000,
   + nr.chains = 1, thin = 1, seed = 1)
   R> tfr.predict(sim.dir = sim.dir.tfr, nr.traj = 100,
   + burnin = 500, burnin3 = 500, seed = 1)
   

   For details on the estimation, MCMC diagnostics, TFR projections etc, see Ševčíková et al. (2011).

2. Estimate e₀ parameters using female data and generate joint female and male projections of life expectancy (about 10 minutes):

   R> sim.dir.e0 <- file.path(getwd(), "e0_projections")
   R> run.e0.mcmc(sex = "Female", iter = 1000, nr.chains = 1, thin = 1,
   + output.dir = sim.dir.e0, seed = 1)
   R> e0.predict(sim.dir = sim.dir.e0, nr.traj = 100, burnin = 500, seed = 1)
   

3. Generate probabilistic population projections (about 10 minutes):

   R> sim.dir.pop <- file.path(getwd(), "pop_projections")
   R> pop.pred <- pop.predict(output.dir = sim.dir.pop,
   + inputs = list(
   +  tfr.sim.dir = sim.dir.tfr,
   +  e0F.sim.dir = sim.dir.e0,
   +  e0M.sim.dir = "joint_"
   + ),
   + keep.vital.events = TRUE, verbose = TRUE)
   

   The last call generates 100 trajectories, one for each trajectory of TFR and e₀. (In practice far more trajectories would be needed, but 100 can be run relatively quickly for illustrative purposes.)

At the end of this command sequence the user has three new directories in the working directory, for TFR, e₀ and population, respectively. One can now use functions from the bayesTFR, bayesLife, and bayesPop, respectively, to analyze and visualize results.

To access population projections in later sessions, issue the command:

R> pop.pred <- get.pop.prediction(sim.dir.pop)

3.3. Population trajectories

Population trajectories can be viewed on a country-specific basis. A simple summary function gives one a quick look at quantiles of a country’s projections:

R> country <- "Netherlands"
R> summary(pop.pred, country)

Projections: 18 (2013 – 2098)
Initial time point: 2008
Observed time points: 12 (1953 – 2008)
Trajectories: 100
Number of countries: 162

Country: Netherlands

Projected Population for both:

	mean	SD	2.5%	5%	10%	25%	50%	75%	90%	95%	97.5%
2008	16615	0.000	16615	16615	16615	16615	16615	16615	16615	16615	16615
2013	16838	63.066	16722	16742	16758	16791	16833	16882	16916	16935	16940
2018	17019	127.435	16774	16797	16848	16938	17020	17097	17166	17262	17277
2023	17163	196.703	16779	16844	16902	17051	17153	17278	17388	17474	17606
2028	17254	266.896	16713	16777	16891	17098	17262	17414	17592	17681	17752
2033	17266	350.300	16577	16699	16816	17047	17261	17489	17704	17830	17917
2038	17200	434.620	16415	16471	16620	16938	17210	17494	17666	17951	18012
2043	17081	537.395	16146	16195	16367	16752	17086	17412	17668	18011	18119
2048	16931	645.696	15845	15936	16064	16494	16936	17337	17669	17959	18174
2053	16778	769.594	15469	15612	15776	16276	16807	17283	17654	17942	18165
2058	16652	898.871	15010	15361	15495	16057	16716	17234	17661	18137	18233
2063	16570	1042.949	14553	15082	15247	15856	16571	17325	17759	18213	18502
2068	16523	1203.044	14158	14830	15122	15762	16495	17342	17882	18496	18810
2073	16503	1359.145	13784	14501	14848	15600	16481	17292	18038	18714	19067
2078	16481	1527.195	13365	14218	14680	15504	16433	17367	18233	18915	19388
2083	16450	1695.052	12977	13941	14494	15332	16489	17418	18446	19288	19685
2088	16411	1863.701	12570	13638	14326	15121	16462	17384	18528	19485	19983
2093	16381	2039.013	12217	13284	14069	14965	16394	17449	18686	19979	20278
2098	16345	2218.959	11882	12956	13907	14841	16353	17468	18929	20268	20624

The unit of the projections corresponds to the unit of the initial population, which is in this case a thousand.

Trajectories can be plotted using:

R> pop.trajectories.plot(pop.pred, country = country, sum.over.ages = TRUE)

The resulting plot is shown in the left panel of Figure 3. The pop.trajectories.plot accepts arguments for specifying sex and age. For example, the right panel of Figure 3 shows the projection for male population up to age 14:

R> pop.trajectories.plot(pop.pred, country = country, sex = "male",
+ age = 1:3, sum.over.ages = TRUE)

Figure 3.

Projected trajectories by time for the Netherlands for total population (left panel) and for males aged 0–14 (right panel).

If sum.over.ages is FALSE, separate plots for each age group are generated. Regarding the age argument, see below about defining ages in bayesPop. An optional argument nr.traj can be used to control how many trajectories are plotted. It defaults to the total number of available trajectories, which is 100 in our example.

In addition to plotting trajectories by time, one can view them by age, see the left panel of Figure 4:

R> pop.byage.plot(pop.pred, country = country, year = 2100)

Figure 4.

Projected trajectories by age for the Netherlands: in 2100 (left panel) and in 2060 (right panel). In the right panel, the main projections (red lines) are compared to 1960 (blue line).

The argument year can be either a projected year or a past time point, i.e., any year from the x-axis of Figure 3. To compare the age structure from multiple years, the right panel of Figure 4 shows an analogous plot for 2060 (with 50 trajectories) and 1960 in the same graph:

R> pop.byage.plot(pop.pred, country = country, year = 2060,
+ pi = 80, nr.traj = 50)
R> pop.byage.plot(pop.pred, country = country, year = 1960, add = TRUE,
+ col = "blue", show.legend = FALSE)

Tabular analogues to the trajectory plots are implemented in the functions pop.trajectories.table and pop.byage.table, respectively. All four functions understand an argument pi specifying the probability intervals to be viewed. The functions pop.trajectories.plotAll and pop.byage.plotAll can be used to plot population trajectories for all countries at once.

Cohort projections

It is often of interest to obtain projections for specific cohorts. Simply running

R> pop.cohorts.plot(pop.pred, country = country)

will show projections of population born in ten different cohorts, starting with the present cohort. To extract the underlying data containing all cohorts (including the ones born in the past), one can use

R> cohort.data <- cohorts(pop.pred, country = country)

The result is a list of matrices where each list item corresponds to one cohort:

R> head(names(cohort.data))

[1] "1880-1885" "1885-1890" "1890-1895" "1895-1900" "1900-1905" "1905-1910"

Each of the elements is a matrix where the first row corresponds to the median and the following rows correspond to quantiles which are configurable via the optional argument pi. The cohort.data object can be also passed to the plotting function above. For example, the following code produces a plot of projections for two already born cohorts and one future cohort, as shown in Figure 5:

R> pop.cohorts.plot(pop.pred, cohort.data = cohort.data,
+ cohorts = c(1980, 2000, 2020))

Figure 5.

Projections of three different cohorts for the Netherlands.

Finally, all functions described in this section accept an argument called expression for exploring trajectories of other population quantities which will be described in Section 5.

Defining age

Many functions in the package accept an argument called age. It refers to an index of an ordered array of five-year age groups, as shown in Table 1. Functions that handle observed data accept values of the age index up to 21 (in which case 21 corresponds to the age group 100+), whereas functions that deal with projections accept values of the age index up to 27. This distinction is due to the low availability of observed data for high ages. However, considering increasing longevity, it is useful to generate projections for those high ages. The package extrapolates mortality to higher ages using the Kannisto method. Other measures can be derived by using life tables and assuming that the initial population at those ages is zero. Ševčíková et al. (2015) describes this in more detail.

Table 1.

Definition of the argument age in various functions of the package. It is an index of an ordered array of age groups.

Age group	0–4	5–9	10–14	15–19	20–24	25–29	30–34	35–39	40–44
age	1	2	3	4	5	6	7	8	9
Age group	45–49	50–54	55–59	60–64	65–69	70–74	75–79	80–84	85–89
age	10	11	12	13	14	15	16	17	18
Age group	90–94	95–99	100–104	105–109	110–114	115–119	120–124	125–129	130+
age	19	20	21	22	23	24	25	26	27

A bayesPop.prediction object contains a component called ages which is an array of the starting ages of each age group. Thus it can be used to determine the correspondence between index and age, for example the one used in Figure 3 for ages 0–14:

R> which(pop.pred$ages < 15)
[1] 1 2 3
R> pop.pred$ages[1:3]
[1] 0 5 10

4. Probabilistic population pyramids

The bayesPop package supports plotting probabilistic population pyramids for any given country and year. In addition, multiple years can be plotted in one pyramid. There are two different kinds of pyramids – a classic pyramid consisting of boxes, and a so-called trajectory pyramid which is created using age trajectories. The classic pyramid can display projections for up to two years in one pyramid with one set of probability intervals; the trajectory pyramid can include any number of years and any number of probability intervals.

A classic pyramid can be created using the function pop.pyramid:

R> pop.pyramid(pop.pred, country, year = c(2100, 2010), age = 1:23)

Here we are comparing the end year of the projections with the current year (see Figure 6 on the left). An optional argument pi for defining the probability intervals shown can be given. In addition, the function accepts various arguments for controlling the appearance of the pyramid, such as colors, height and thickness of the boxes etc.; see below for an example. Using the optional argument indicator, births and deaths can be also shown in a pyramid, if vital events were stored during the prediction.

Figure 6.

Probabilistic population pyramid for the Netherlands. The classic type on the left compares the end year of the projections with the current year. The trajectory type on the right compares three different years on the scale of proportions.

The following code creates a trajectory pyramid with three years (the end year, the second prediction year, and the first observed year) with 95% probability intervals around the two prediction years:

R> pop.trajectories.pyramid(pop.pred, country, year = c(2100, 2020, 1950),
+ age = 1:23, pi = 95, nr.traj = 0, proportion = TRUE)

It results in the right graph of Figure 6. Here the argument proportion is used, which switches the x-axis to a proportional scale. Note that the order of the values in the year argument matters, especially in the classic pyramid case. The first value is used to create the main pyramid (i.e., with boxes of probability intervals in classic pyramid and using red color in case of trajectory pyramid), whereas the remaining ones are used for the additional pyramids in the graph.

The functions pop.pyramidAll and pop.trajectories.pyramidAll can be used to produce pyramids for all countries and for a set of years at once. The year argument is then expected to be a list where each element is a vector to be passed to the underlying function, i.e., pop.pyramid or pop.trajectories.pyramid. For example, to create the pyramid on the left-hand side of Figure 6 and a similar pyramid comparing 2050 to the current year for all countries, one can do:

R> pop.pyramidAll(pop.pred, year = list(c(2100, 2010), c(2050, 2010)),
+ age = 1:23, output.dir = "mypyramids")

This will create a png file for all combinations of countries and year elements, in this case two files per country, and place it into the directory “mypyramids”. Alternatively, one can set the optional argument one.file to TRUE in which case all pyramids are put into a single file.

Pyramids for user-defined data

So far, we have shown how to create probabilistic pyramids for an object of class bayesPop.prediction. However, both pyramid functions are S3 methods that can be also applied to an object of class bayesPop.pyramid. This is a structure containing all the data necessary for a pyramid graph; they do not need to be created using bayesPop. Thus any data fitting into a pyramid structure can be used. An S3 method called get.bPop.pyramid can convert a matrix, a data frame, or a list of matrices or data frames into the bayesPop.pyramid structure. In addition, it can be also applied to a bayesPop.prediction object. One advantage of the latter conversion is that it gives the user a finer control over the plot.

The main element of the bayesPop.pyramid list is called pyramid. It is a list of data frames, each having two columns containing data for the left and right side of one pyramid and row names determining the age labels. Consider an example dataset containing population estimates for Washington State and King County in 2011:

R> data <- read.table(
+ file.path(find.package("bayesPop"), "ex-data", "popestimates_WAKing.txt"),
+ header = TRUE, row.names = 1
+ )
R> head(data)

	WA.male	WA.female	King.male	King.female
0–4	224883	214629	61537	58672
5–9	219450	209870	57814	55298
10–14	224828	213562	56848	53958
15–19	234336	221463	59294	56604
20–24	238228	223712	65651	64287
25–29	244381	233876	81481	78426

In order to show the two pyramids in one graph, we create two data frames with the same column names:

R> WA <- data[, c("WA.male", "WA.female")]; colnames(WA) <- c("M", "F")
R> Ki <- data[, c("King.male", "King.female")]; colnames(Ki) <- c("M", "F")

Now, one can create a bayesPop.pyramid object, specifying which columns contain data for the left and right part of the pyramid, respectively:

R> pyr <- get.bPop.pyramid(list(WA, Ki), LRcolnames = c("M", "F"),
+ legend = c("Washington", "King County"))

Simply using plot and optionally specifying some aesthetics will create the pyramid shown on the left-hand side of Figure 7:

R> plot(pyr, main = "Population in 2011",
+ pyr2.par = list(height = 0.7, col = "violet", border = "violet"))

Figure 7.

Population pyramids for user-defined data. The pyramid on the right is on the scale of proportions.

It can also be useful to compare such data on the scale of proportions, where what is plotted is not the actual numbers of people in each age group, but the numbers as a proportion of the total population. The following code creates such an object and its graph (shown in Figure 7 on the right):

R> pyr.prop <- get.bPop.pyramid(list(WA, Ki), is.proportion = NA,
+ LRcolnames = c("M", "F"), legend = c("Washington", "King County"))
R> pop.pyramid(pyr.prop, main = "Population in 2011 (proportions)",
+ pyr1.par = list(col = "lightgreen", border = "lightgreen",
+  density = 30),
+ pyr2.par = list(col = "darkred", border = "darkred",
+  density = 50, height = 0.3))

If the argument is.proportion has a logical value, it determines whether the data are on a proportional scale. An NA value means that the data frames, here WA and Ki, are not on a proportional scale but that such a scale is desired and thus, should be computed on the fly. Using the aesthetic arguments pyr1.par (for the main pyramid) and pyr2.par (for the secondary pyramid) allows the user to create a wide variety of different pyramid graphs.

The plot function is an alias for pop.pyramid. If the pop.trajectories.pyramid function is to be used, it should be called explicitly.

Apart from the pyramid element, the bayesPop.pyramid object also contains an element for storing the probability intervals, called CI, which can be passed directly to the get.bPop.pyramid function. Thus uncertainty can be included in the visualization of user-defined pyramid data. The pop.trajectories.pyramid can display any number of pyramids and probability interval sets into one graph, whereas pop.pyramid uses only the first two elements of pyramid and only one element of CI. See the manual for more details on the structure of bayesPop.pyramid.

5. bayesPop expressions

As mentioned in Section 3.1, a bayesPop.prediction object contains information about sex-and age-specific population projections. It is often of interest, however, to analyze quantities derived from the basic population counts and vital rates, such as potential support ratio, mean age at child-bearing, median age, and so on. For this purpose, the package implements a simple expression language that allows one to compute such quantities on the fly.

A bayesPop expression is a collection of basic components connected via usual arithmetic operators and combined using parentheses. Standard R functions and pre-defined functions can be also used within expressions.

Basic component

A basic component of an expression is a character string that consists of four sub-strings, the first two of which are mandatory. They must be in the following order:

1. Measure identification. The following upper-case characters are currently allowed and one of them must be provided:

   P	Population
   D	Deaths
   B	Births
   S	Survival ratio
   F	Fertility rate
   R	Percent age-specific fertility
   M	Mortality rate
   Q	Probability of dying
   G	Net migration

   Note that all but the P and G indicators can be used only if keep.vital.events was switched on during the prediction. P and G are always available. Since net migration, G, is currently a deterministic input to the projections, it results in only one trajectory, namely the input.

2. Country part. This mandatory part can be a numerical country code, or a two- or three-character ISO 3166 code (International Organization for Standardization 2013), or characters “XXX” which serve as a wildcard for a country code. For example, “P528”, “PNL”, and “PNLD” are all expressions for the total population of the Netherlands. The use of “XXX” is limited to specific functions and will be discussed later in this section.

3. Sex sub-string. The country part can be optionally followed by either “_F” or “_M”, specifying female or male indicator, respectively. An expression consisting of two basic components “P528_F/P528” gives the ratio of female to total population in the Netherlands.

4. Age sub-string. If the age sub-string is used, the basic component is concluded by an array of age indices (as defined in Table 1). Such an array is delimited by either brackets (“[ ]”) or curly braces (“{ }”). The former invokes a summation of counts over the given ages, while the latter is used when no summation is desired. Note that if the age sub-string is missing, the counts are automatically summed over all ages. To use all ages without summing, empty curly braces can be used. For example, the number of females in childbearing age in France can be calculated as “PFR_F[4:10]”. As another example, the potential support ratio can be defined as “PFR[5:13]/PFR[14:27]”.

   In addition to the age index in Table 1, the indicators S, M and Q also allow an index -1 which corresponds to the age group 0–1, and an index 0 which corresponds to the age group 1–4.

Not all combinations of the four parts above make sense. For example, fertility rate can be combined only with female sex and a subset of the age groups, namely the childbearing ages (indices 4 to 10). Births are also restricted to those age groups. The rate-like indicators S, M, and Q should include all four components, since summing over sexes or age groups is meaningless for this type of measure.

Connecting components

When an expression is evaluated, each basic component is replaced by the corresponding data in the form of a four-dimensional array with the following dimensions:

1. Country dimension: It is equal to one if a specific country code is given. If “XXX” is used in the country sub-string, this dimension equals the number of countries in the prediction object.

2. Age dimension: It is equal to one if the age sub-string is missing or is defined within square brackets. If the age is defined within curly braces, this dimension corresponds to the length of the age array.

3. Time dimension: Depending on the context in which the expression is evaluated, this dimension corresponds to either the number of projection periods or the number of observation periods.

4. Trajectory dimension: It corresponds to the number of trajectories in the prediction object, or if the expression is evaluated on observed data, it is equal to one.

This array is returned by the function get.pop, which is evaluated either on projections or observed data, controlled by the logical argument observed.

Various arithmetic operations, such as +,−, *,/, ˆ, %%, %/%, and R functions can be performed on these four-dimensional arrays. An expression should be constructed in such a way that the age dimension is eliminated, for example by using the apply function or one of the pre-defined functions described below. An exception to this rule is when an expression is used in the context of functions plot.byage.plot, pop.byage.table, or the cohort functions, as illustrated below.

There are a few aspects to keep in mind when combining basic components. They are rooted in the fact that the combined arrays must have the same dimensions. For example, the deterministic indicator G cannot simply be combined with the probabilistic components, unless it is on observed data. In such cases, a special function, pop.combine (see below), needs to be used. Furthermore, if using curly braces, the age index of the basic components must have the same length. For B, F and R, only age indices between 4 and 10 are allowed, so that “BNL{}” has length 7 on its age axis whereas “PNL{}” has length 27 for predictions and 21 for observed data. Therefore, if B, F and R are combined with other indicators, the age index specified must be of the correct length, e.g., “BNL{}/PNL{4:10}”. For debugging purposes, one can use the get.pop function to check dimensions of basic components:

R> B <- get.pop("BNL{}", pop.pred, observed = FALSE)
R> P <- get.pop("PNL{4:10}", pop.pred, observed = FALSE)
R> all(dim(B) == dim(P))

[1] TRUE

Pre-defined functions

There are a few pre-defined functions implemented in the package for convenience. The most commonly used are:

• gmedian and gmean for computing the median and mean of grouped data;

• pop.apply is an apply wrapper for applying a function along the age dimension;

• pop.combine for combining basic components of different shapes;

• age.index01 for an index to age groups 0, 1–4, 5–9, … (allowed for S, M, and Q);

• age.index05 for an index to age groups 0–4, 5–9, …

Furthermore, to generate an expression for the mean age of childbearing, one can use the function mac.expression(…), see below for an example.

Using expressions in bayesPop functions

All functions that accept expressions have an argument called expression.

Expressions can be used to view projection trajectories by time using functions pop.trajectories.plot and pop.trajectories.table, as well as trajectories by age using functions plot.byage.plot and pop.byage.table. The former two evaluate expressions on both, observed and projected data. Each of the latter two accepts an argument year that specifies on which data it is evaluated. In the case of projected data, the functions use the trajectory dimension of the resulting array to compute desired quantiles, and possibly to show trajectories in the plot. For example, using the pop.pred object created in Section 3.2, we can plot the median age of women in childbearing ages in Germany by

R> expr <- "pop.apply(PDE_F{4:10}, gmedian, cats=seq(15, by=5, length=8))"
R> pop.trajectories.plot(pop.pred, nr.traj = 20, expression = expr,
+ main = "Median age of women in childbearing ages")

which results in the left graph of Figure 8. The cats argument in the expression is passed to gmedian and it is a categories definition of the grouped data.

Figure 8.

Results of evaluating expressions. Left panel: Median age of women in childbearing ages by time in Germany. Right panel: The mean age of childbearing in Germany.

An expression for the mean age of childbearing can be obtain an plotted as follows:

R> expr <- mac.expression(country="DE")
R> expr

[1] "(17.5*RDE[4] + 22.5*RDE[5] + 27.5*RDE[6] + 32.5*RDE[7] + 
37.5*RDE[8] + 42.5*RDE[9] + 47.5*RDE[10])/100"

R> pop.trajectories.plot(pop.pred, nr.traj = 20, expression = expr,
+ main = "Mean age of childbearing")

This results in the right graph of Figure 8.

For an age-specific plot, the number of births by mother’s age per women of the same age can be viewed using

R> pop.byage.plot(pop.pred, expression = "BDE{}/PDE_F{4:10}", year = 2030,
+ nr.traj = 20, main = "Births by mother’s age per woman – 2030")

which is shown on the left-hand side of Figure 9. This output can be obtained in tabular rather than graphical format using the function pop.byage.table. Both of these age-specific functions require the use of curly braces in the expressions, as the age axis of the resulting array must not be eliminated.

Figure 9.

Results of evaluating expressions for Germany. Left panel: Births by mother’s age per women of the same age in an age-specific plot. Right panel: Total number of births for women of the cohort 2015–2020 as a function of time.

Expression can be also used in the cohort functions described in Section 3.3. For example, the total number of children born to women of the next future cohort in Germany can be obtained by

R> pop.cohorts.plot(pop.pred, expression="BDE{}", cohorts=2015,
+ legend.pos="topleft", main="Births for female cohort 2015-2020")

which can be seen in the right panel of Figure 9. Note that expressions used in the cohort functions must contain curly braces.

In all functions mentioned so far in this section, the expressions must be country specific. However, basic components from different countries can be combined. For example, one could use “PDE/PFR” to view projection of the ratio of German to French population.

Expressions can also be used in maps. For that purpose, the “XXX” wildcard should be used. To generate a map with infant mortality in 2050, do

R> pop.map(pop.pred, expression = "MXXX[−1]", year = 2050)

The pop.map function is built on top of the rworldmap package (South et al. 2012). Alternatively, one can use the pop.map.gvis function which builds on the googleVis package (Gesmann and de Castillo 2011). Creating a map via an expression involves, for each country, evaluating an expression in which the wildcard is replaced by its country code. This means loading trajectories of all countries from the disk one by one, evaluating the expression and obtaining results, which can be time-expensive. For that reason, the package has a simple caching mechanism, in which results of an expression evaluation for all time periods are stored on disk (in a file called “cache.rda” located in the prediction directory). Next time the same expression is used, for example with a different time point, the cached data are re-used. Even though creating a map with a new expression is processed in parallel if possible (depending on the number of cores in the user’s computer) it still can take substantially more time than using a previously used expression. The cache is removed every time a new projection is generated. Alternatively, the function pop.cleanup.cache can be used for a manual removal of the data.

In addition to maps, expressions with the “XXX” wildcard can be passed to other functions that involve all countries, such as pop.trajectories.plotAll, plot.byage.plotAll, or write.pop.projection.summary.

More examples of bayesPop expressions can be found by typing ?pop.expressions.

6. Aggregations

In addition to producing population estimates at the country level, the UN also provides projections for population quantities aggregated over many different sets of countries, such as geographic regions and trading blocs. bayesPop offers two methods for producing probabilistic projections of aggregated quantities:

• Country-based method: This combines the posterior samples from the different countries trajectory by trajectory: aggregation is done by simply summing the population counts in each trajectory across the countries of the regions in question.

  If the input TFR and life expectancy trajectories came from the original Bayesian hierarchical models (BHM), this corresponds to the conditional independence assumptions in the BHM. If there is between-country correlation beyond that, the resulting posterior distributions may underestimate uncertainty.

  However, this method can be made to take account of correlations between the forecast errors for different countries. Fosdick and Raftery (2014) proposed a method for taking the between-country correlation in TFR forecast errors into account. This is implemented in bayesTFR (controlled by the logical argument use.correlation in the tfr.predict function). If their method is used, the output from bayesTFR that is fed into bayesPop will take account of between-country correlations in TFR forecast errors.

• Region-based method: Here aggregations are generated using a cohort component method, similarly to pop.predict, but the function operates on aggregated input components. While deterministic input components are aggregated on the fly, the method requires that aggregations of all probabilistic input components described in Section 3.1 exist. This can be achieved using the functions run.tfr.mcmc.extra and tfr.predict.extra from bayesTFR for TFR, and run.e0.mcmc.extra and e0.predict.extra from bayesLife for life expectancy.

  In practice we have found that, when projecting aggregates of countries whose demographic histories are not well aligned, the regional method tends to overestimate uncertainty, often giving predictive intervals that are too wide.

Here is an example of aggregating over continents and over the whole world using the country-based method:

R> pop.aggr <- pop.aggregate(pop.pred, input.type = "country",
+ # World, Africa, Europe, Northern America, Asia, Latin Am.
+ regions = c(900, 903, 908, 905, 935, 904))

The region codes must correspond to the column “area_code” of the UNlocations dataset in the wpp package. Alternatively, user-defined aggregations are also supported (see the function help file for more information).

The function pop.aggregate accepts an optional argument input.type which defaults to the method name and is used for labeling the aggregation. Thus one can create several aggregations for the same prediction object. They are stored in the main simulation directory, here sim.dir.pop from Section 3.2, under the given name. In later R sessions, the object can be retrieved using

R> pop.aggr <- get.pop.aggregation(sim.dir.pop, name = "country")

The stored data have the same structure as in the case of the prediction object, as described in Section 3.1. Indeed, the object that results from the two calls above is again of class bayesPop.prediction and thus can be used in any of the summarizing and plotting function described in the previous sections, including in combination with expressions:

R> par(mfrow = c(2, 2))
R> for(country in c(903, 908, 905, 935))
+ pop.trajectories.plot(pop.aggr, country, sum.over.ages = TRUE)

R> pop.pyramid(pop.aggr, 900, year = c(2100, 2010), proportion = TRUE)

R> pop.byage.plot(pop.aggr, expression = "P908{}/P900{}", year = 2100,
+ main = "Proportion of European population to world population",
+ pi = 80, ylim = c(0,0.35), show.legend = FALSE)
R> pop.byage.plot(pop.aggr, expression = "P908{}/P900{}", year = 2010,
+ add = TRUE, show.legend = FALSE, col = "blue")
R> legend("topleft", legend = c(2100, 2010), col = c("red", "blue"), lty = 1)

The resulting graphs are shown in Figures 10 and 11. Note that the regions are aggregated only from countries that are available in the underlying pop.pred object. These do not yet include a large part of Africa, because we do not yet have probabilistic projections of life expectancy for the countries with generalized HIV/AIDS epidemics, many of which are in Africa. Thus the African projections are only illustrative.

Figure 10.

Population trajectories for aggregated regions obtained via the country-based method.

Figure 11.

Visualizing results of aggregation. Left panel: Population pyramid for the world in 2010 and 2100. Right panel: Number of Europeans as a proportion of world population for each age group in 2100 and the same indicator in 2010, expressed as “P908{}/P900{}”.

The expression used in the last call of pop.byage.plot (right graph of Figure 11) combines indicators from two regions. It is also possible to combine non-aggregated indicators with aggregated ones:

R> pop.trajectories.plot(pop.pred, expression = "PIND/P900",
+ main = "Proportion of population of India to the world population")

In such a call, the original (non-aggregated) prediction object, here pop.pred, should be passed as the first argument. The function then tries to find the aggregated object automatically by iterating over the available aggregation objects until the region’s code is found. In this process, an aggregation called “country” has priority above objects with other names.

7. Discussion

We have described an R package called bayesPop to produce and display probabilistic population projections, using a methodology which is being used by the United Nations Population Division as part of the process for producing its official population projections for all countries. The package produces a sample from a joint posterior predictive distribution of population quantities.

It allows the user to visualize the probabilistic projections in various ways, including different kinds of probabilistic population pyramids. It also includes an expression language that yields probabilistic projections of arbitrary user-defined derived future population quantities, such as the median age of the population, the potential support ratio or the ratio of the population of one country to that of another. Finally, it gives probabilistic projections of population quantities that are aggregated over an arbitrary set of countries, such as UN regions or trading blocs.

bayesPop is a command line package. However, there is also a graphical user interface, implemented in the bayesDem R package, for controlling all three packages bayesPop, bayesTFR and bayesLife, and visualizing their results. The UN’s most recent official historical population estimates and population projections at the time of writing are contained in the data package wpp2012. The previous revisions of the UN’s official World Population Prospects are available in the data packages wpp2010 and wpp2008. Data in all three wpp packages can be visualized in a browser using the R package wppExplorer (Ševčíková 2013b), or viewed online at https://rstudio.stat.washington.edu/shiny/wppExplorer/. An additional new data package may be added after the 2015 Revision of the UN’s World Population Prospects is published.

There is now a wealth of R packages that do demographic analysis in some form, but relatively few oriented to human populations. Apart from bayesPop, the only one that we know of that does probabilistic projections of human populations is demography (Hyndman 2013), which does stochastic population forecasting using the functional data approach of Hyndman and Ullah (2007).

Several packages use statistical models to estimate and forecast age-specific mortality rates. The YourCast package is based on the methods of Girosi and King (2008). MortalityS-mooth (Camarda 2012) uses P-splines to smooth and forecast age-specific mortality rates. The HPbayes package (Sharrow 2012a; Sharrow et al. 2013) estimates the Heligman-Pollard model for age-specific mortality from mortality data using Bayesian Melding with incremental mixture importance sampling (IMIS; Raftery and Bao (2010)). The LifeTables package (Sharrow 2012b; Clark and Sharrow 2011) produces model life tables by applying model-based clustering (Fraley and Raftery 2002) to the Human Mortality Database.

The popReconstruct package (Wheldon 2013) does probabilistic reconstruction of past population quantities rather than forecasting of the future; it is based on the methods of Wheldon et al. (2013). Giza (Striessnig 2012) is a graphics package that constructs panels of population pyramid plots.

There are several packages that provide tools for the construction and analysis of deterministic matrix population models, often oriented more to animal than to human populations. These include popbio (Stubben et al. 2012), based on the work of Caswell (2001), popdemo (Stott et al. 2012), and primer (Stevens 2012). The IPMpack package (Metcalf et al. 2013) builds and analyzes integral projection models; these are also deterministic and take demographic rates as fixed inputs.

There are also several packages that provide tools for analyzing the interaction between demography and population genetics, again typically in the context of animal populations. These also usually treat demographic rates as fixed inputs. These include AlleleRetain (Welser et al. 2012), which analyzes the effect of demography on allele retention, and Biodem (Boattini and Calboli 2012), which provides biodemographic functions, with an emphasis on kinship and inbreeding. Another such package is lmf (Kvaines 2013), which provides methods for inference about genetic selection in age-structured populations; it is based on the methods of Engen et al. (2012).