Large-scale spatial population synthesis for Denmark

Population synthesis has been approached from different methodological perspectives [19, 25], including reweighting approaches, matrix fitting approaches and simulation-based approaches.

Approaches based on reweighting typically aim at estimating weights or “expansion factors” which can be used to expand surveys or smaller samples such that these are representative for the populations. Reweighting based on quadratic optimisation has been proposed in Daly [10], whereas others have used combinatorial optimisation [1, 30, 35] and maximum cross-entropy [3, 17, 22]. It should be noted that reweighting and matrix fitting are closely connected as a matrix fitting indirectly generates weights relative to a starting solution. Matrix fitting approaches typically apply different variants of Iterative Proportional Fitting [13], which may include multi-level fitting [27] and intermediate stages to circumvent missing values in the starting solution. The correspondence between cross-entropy and IPF under convex constraints has been covered in McDougall [23], Dykstra [12] and Darroch and Ratcliff [11] who showed that IPF throughout the iteration scheme increases entropy monotonously. Applications of IPF have been presented in Beckman et al. [5], Arentze et al. [2], and Simpson and Tranmer [31]. Other slightly more advanced applications can be found in Pritchard and Miller [27] in which Hierarchical IPF procedures were proposed for fitting household and individuals jointly. Another application is in Rich and Mulalic [28] which proposed a pre-harmonisation procedure in order to account for inconsistent targets. Recently there has been an increased interest for using simulation based approaches to generate synthetic populations. Mostly these approaches has been concerned with the problem of generating synthetic ‘pools’ of individuals by sampling from an original data source. Farooq et al. [16] present an application of a Gibbs sampler whereas Borysov et al. [4] suggest using deep generative modelling in order to address scalability issues and sparsity in the origin data. However, simulation-based approaches are not new to population synthesis [7] and have often been used in post allocation stages, e.g. to generate agents and household entities from aggregated population matrices. In this sense, most applied frameworks for population synthesis are “hybrids” that combine matrix fitting with simulation stages although these combined applications are rarely described in the literature. Even pure simulation-based population synthesis as presented in Farooq et al. [16] and Borysov et al. [4] typically require a re-sampling stage where individuals are ‘importance-sampled’ such that the resulting population are consistent with population targets.

While referring to the literature and the many different approaches that have been applied, it should be stressed that choice of strategy for the population synthesis is closely connected with the amount, the type and quality of data at hand. Generally speaking, if detailed high quality data are available as a basis for the starting solution it is important to utilise this information as much as possible. In that case, IPF/Cross Entropy or Markov Chain Monte Carlo approaches are natural choices because they maintain the correlation structure captured in the starting solution well [8] by maintaining odd ratios. Another determining criterion is whether ‘hard’ or ‘soft’ targets are required. Certain methods, such as quadratic optimisation [10], do not facilitate ‘hard targets’ and may not be acceptable from an application perspective or at least require a subsequent quota-based sampling stage.

The paper provides a detailed description of a large-scale population synthesis framework. The model is implemented as part of the Danish National Transport Model and is used for population synthesis for the entire Danish population. Main contributions to the existing literature are as follows. First, to our knowledge, almost no population synthesis frameworks have been presented in the scholarly literature describing all stages from target harmonisation, matrix fitting to the final allocation stage. We have deliberately included a complete and coherent description in order to provide a standing example of a self-contained model for population synthesis for an entire country. Secondly, on the more technical side, the paper proposes a simulation-based household allocation procedure which involves the concept of “spouse matching” and “kids matching”. Thirdly, we introduce a pre-harmonisation stage of population targets combined with a two-stage IPF fitting procedure to enable adjustments of the fitted matrix and to enforce consistency among targets. Finally, we provide some validation insight by analysing the degree of accumulated household simulation noise and by comparing a prediction from 2010 to 2015 with observed data.

In Section 2 the model framework is presented, which includes two fitting stages, a harmonisation stage and a simulation stage. Section 3 is concerned with model application and provides an overview of data and notation and considers model validation. Finally in Section 4, we offer a conclusion, including a research outlook.

The harmonisation stage is a pre-processing stage with the sole objective of ensuring consistency between input target constraints. The issue has not received much attention in the scholarly literature where it is common to assume that inputs are correct and consistent. It is also a principle discussion whether we should either correct inputs (via harmonisation) or require inputs to be correct before running the model.

However, from a user and application perspective the auto-harmonisation of targets makes the process of generating scenarios much easier and the application of the model less prone to errors. As an example, if a scenario is concerned with a generic income increase then the user can focus on shifting the conditional income distribution without worrying about the actual population counts. Another advantage of pre-harmonised targets is that it solves the problem of convergence problems that may occur as a result of rounding errors when defining the targets. As we typically have a convergence threshold below E-6 in the matrix fitting stage this can easily turn into a problem. The harmonisation stage prevents this problem by ensuring consistency at the level of the machine precision. As the harmonisation process has not received much attention in the literature we discuss this in more details below.

Firstly, why is it a problem? When having many targets constraints of which some are cross-linked in that they share common attributes, it is not trivial to make sure that these are consistent across all dimensions. This is a problem that can easily arise when users of the model are editing targets for a given scenario. As a result, summing different targets across attributes that are similar (across targets) may render sums that are different. This in turn will lead to inconsistent input data, which will affect convergence and bias outputs. This is not just a problem that relates to the IPF algorithm but for any approach which is supposed to align a population with future targets one way or the other. Clearly, the simple solution to the problem is to reduce the number of targets and their complexity. However, this generally conflict with the increasing need for more detailed long-term forecasts. Having few and simple targets will make it difficult to capture social processes and trends in the population synthesis stage.

A solution was proposed in Rich and Mulalic [28] in which constraints were harmonised according to a ranking procedure and this procedure has been applied in the Danish National Transport Model. The idea is that the different targets are ranked according to reliability and subsequently adjusted such that lower ranked constraints always comply with higher ranked constraints. The ranking that is used in the National Transport Model is presented in Table 1 below.

The idea is that rather than using targets directly, a harmonised version of the targets is used instead. As a result, it is the harmonised targets that are passed on to the IPF algorithm in order to ensure consistency across inputs and convergence of the algorithm.

Per definition, the target with rank 1 is the ‘ground truth’. This target will therefore not be harmonised but is used as a baseline target. The other targets are harmonised based on the levels of the highest ranked target. Below \( {\overset{\sim }{T}}_{ai}\left({z}_0,a,i\right) \) represent the harmonised targets for T_ai(z₀, a, i) and similarly for the other targets.

As can be seen, we apply only the relative distribution of T_ai(z₀, a, i), T_al(z₀, a, l) and T_af(z₀, a, f) by i, l and f. As a result, summing over any dimension in all targets will reproduce the same sum. In the baseline all targets are naturally harmonised as they are drawn directly from harmonised register data. However, the harmonisation is mainly supposed to support users of the model who alter the targets for scenario analysis.

In the case described in Table 1, the cross-linking of targets is relative simple in that at most two dimensions are shared. Moreover, both of these two dimensions are included in the highest ranked target which makes it straightforward to harmonise subsequent targets as described. However, if the cross-linking is more complex the harmonisation stage will be equally complicated. Sometimes it may involve running separate IPF steps in the harmonisation. To illustrate this potential problem consider a slightly modified set of targets in Table 2.

Although only three targets are considered with only two attributes it is not possible to apply the ranking procedure introduced above. The problem arise in the harmonisation of T_il(i, l) as this target shares attributes with T_ai(a, i) and T_al(a, l). In other words, we cannot represent T_il(i, l) as scaled marginal probabilities as before. However, in this particular case potential inconsistencies that may arise from T_il(i, l) can be solved by running a pre-stage IPF.

More specifically, the harmonised target \( {\overset{\sim }{T}}_{il}\left(i,l\right) \) could be calculated using an IPF algorithm with starting value T_il(i, l) and constraints formed by \( {T}_1(i)={\sum}_a{T}_{il}\left(a,i\right) \) and \( {T}_2(l)={\sum}_a{\overset{\sim }{T}}_{il}\left(a,l\right) \). The harmonised \( {\overset{\sim }{T}}_{il}\left(a,l\right) \) is constructed in a similar way as before. That is, \( {\overset{\sim }{T}}_{il}\left(a,l\right)=P\left(l|a\right){\sum}_i{T}_{ai}\left(a,i\right) \).

The intuition behind this approach is straightforward. The best estimate for \( {\overset{\sim }{T}}_{il}\left(i,l\right) \) is T_il(i, l) except that the matrix might fail to be consistent with T_il(a, i) and the previously harmonised target \( {\overset{\sim }{T}}_{il}\left(a,l\right) \). Hence, we would like to preserve the correlation structure in T_il(i, l) as closely as possible while adjusting the row and column sums to be consistent. This minimal deconstruction of the probability distribution is preserved under iterative proportional fitting in that odds log ratios are preserved [8].

A final remark that relates to the definition of targets and the harmonisation of these is that it is sometimes observed that the spanned target space includes cells that are not included in the starting solution or the other way around. This essentially corresponds to a situation where targets are inconsistent and will generally lead to convergence problems. It is therefore important to align the dimensionality of the target space and the solution space before starting the iteration process. Typically this is solved as part of a pre-processing stage where the input space and target space are aligned. In practise it may lead to an extended input space to allow for new cell-entries.

The fitting of the master table translates into finding the maximum cross-entropy of T(a, g, i, l, f, c, z) provided we have a starting solution T₀(a, g, i, l, f, c, z) and subjected to a set of pre-defined constraints.

The corresponding maximum cross-entropy for the problem at hand is provided in (9) below.

The constraints are provided in (10)–(13).

As seen, the right-hand side is the harmonised targets from (1)–(7) in the previous section. The matrix fitting is solved using an IPF algorithm which is carried out in two stages (refer to Appendix 2 for a more elaborate description of the fitting stage). In a first stage the problem that corresponds to the above cross entropy problem is solved. This problem returns the fitted matrix T^∗(a, g, i, l, f, c, z). In the second stage fitting we allow users to add additional constraints at a more detailed zone level. This introduces a possibility of ‘aligning’ the fitted solution to additional information that may be available as local projections of zone population. The outcome of the second-stage fitting is a T^∗∗(a, g, i, l, f, c, z) matrix.

The outcome of the population fitting (as we considered in Section 2.2) is a new master table, which conforms to the new targets and to the structure of the master table for individuals (refer to Table 3 in Section 3).

The next stage of the population synthesis is to convert prototypical individuals into micro agents and to allocate these individuals to households. This task is necessary because the master table is not a micro representation of the population but simply a weighted list of prototypical individuals grouped into socio-groups. Although the socio-economic groups are very detailed, certain entries in the master table may represent as much as 30–50 prototypical individuals and others very small fractions of individuals. If we apply micro-simulation directly to the master table it would mean that all of these 30–50 individuals would be treated in a similar way as regards the household sampling, which is not desirable. We therefore create an enumerated list of all individuals in the population and process those in a micro-simulation algorithm where individuals are grouped into households.

Many have considered the challenge of how to translate a numerical representation of individuals into an integer representation [18, 32]. In the Danish National Transport Model this is accomplished by allowing fractions of individuals and fractions of households to be processed. Hence, in the demand model that precedes the population synthesis framework an internal numerical weighting of each individual and household is facilitated. Hence, the enumeration of a households and individuals is allowed to be strictly different from 1. So, if in a cell, there are 11.3 households or individuals, we process the decimal part as a 0.3 fraction of a household and introduce a reweighting in the demand model. In case the demand model cannot process fractions a truncation or rounding process is required where the residual fractions are recirculated in the simulation algorithm.

The micro-simulation scheme is based on the following overall steps;

The most important stage in the simulation scheme is Step 5. It involves several stages as illustrated in Fig. 3 below.

From the matrix fitting stage the type of household is known. Hence, for households classified as ‘singles’ the only decisions to consider in the simulation stage is related to the number of kids and the characteristics of these. For households with two adults we start by selecting a first adult for the household. Subsequently, we select a partner based on a spouse-match model (Step 5b). The probability of selecting a given spouse depends on the income, age, labour market association and gender of both. The conditional “spouse matching” probability is defined on the basis of register data, e.g.

The conditional sampling from this joint distribution takes account of the strong correlation when matching people into households. Although the marginal distribution in (14) excludes the spatial dimension its dimensionality is very large and consists of more than 3 million potential cells. To circumvent the problem of sparsity, less detailed sampling schemes are enforced depending on the “sparsity” of a given sampling (refer to Table 9).

Another important allocation in the simulation stage is the allocation of kids and the type of these. Whereas the spouse-matching model were concerned with the choice of individuals, the models for allocating kids is a household based model. The allocation of kids (Step 5d) involves two sequential steps. In a first stage the number of kids c_h is sampled as a function of household income, age, gender and labour market association of each of the adults in the household. That is;

After having sampled the number of kids we sample a type classification for each of these. This is based on household income and age and labour market association for the two adults as seen below.

The sampling of type of kids is somewhat simplified in that we do not condition the probability of selecting one kid with other kids. This would significantly complicate the sampling scheme and require very large tables as input data. For the same reason, the gender classification has been excluded as well.

In the current implementation we sample from marginal tables which have been produced on the basis of register data and constitute one of many input tables. However, it is possible to construct any mathematical model for the spouse matching or the sampling of kids to be able to reflect possible future fluctuations in the way households are formed. These external prediction models could be modelled using time-series analysis, however a detailed discussion of this is beyond the scope of the paper.

Although the simulation stage solves the important allocation problem of grouping individuals into households it introduces a potential inconsistency problem. The final list of individuals, after joining these into households, may not be entirely consistent with the master table when aggregating over the different dimensions. This is because the household simulation stage it is based on random draws, which although consistent at the aggregate level due to the law of large numbers may not be entirely consistent with the targets from Tables 4, 5, 6, 7 and 8. There are different solutions to this problem. The simplest solution is to introduce a re-scaling of individuals. It means that for a household, one person might have a weight of e.g. 1.02 when used in the demand model. In case of the Danish National Transport Model, this is the approach taken and the accumulated trip matrices are calculated as a weighted matrix of trips across all households and individuals at the micro level. In that case it is perfectly fine to have weighted individuals.

However, if the model following the population synthesis requires complete agent-based inputs, which at all times needs to be consistent with targets, this is a substantial complication. Although there is a relative developed literature on joint hierarchical matrix fitting of households and individuals (Muller and Axhausen, [26]) it is less clear how to ensure consistency in the micro-simulation stage. One possible approach is to direct the micro simulation to the pool of individuals listed from the IPF. Then join people into households without replacement and essentially continue until the pool is used up. However, from a computational (combinatorial) and book-keeping point of view this is a relative cumbersome process.

The synthesis is based on Danish register data. This is essentially a micro database for all Danish citizens with a wide range of attributes related to demography, income, social class, job, family structure and location. Most importantly, however, the data is generally of a very high quality as it is the basis for tax payments and income transfers. The data are reported directly from firms and public authorities to Statistic Denmark.

In the fitting of the master table for individuals (refer to Table 3 below) we consider a matrix T(a, g, i, l, f, c, z) that is spanned by seven dimensions. The master table represents the fundamental input and output format for the IPF. When fully spanned it represents approximately 17.4 million matrix entries or 19,200 potential socio-economic groups for each of the 907 zones. A more detailed description of the variable definitions is provided in Table 14 in Appendix 1.

The constraints for the IPF is shown in Tables 4, 5, 6, 7 and 8. Most of these operate at the municipality level because this is the lowest level for which official demographic forecasts exists. In all, there are more than 22,000 constraints and these are cross-linked in the sense that they share common variables. In order to account for consistency issues the ranking harmonisation procedure as described in Section 2.1 is used.

The choice of targets is a balance between the precision of the final matrix and what variables we can actually forecast with a reasonable precision and amount of effort. Clearly, if more dimensions and more variables had been introduced, the final matrix would have been more “heavily” constrained, and given that the constraints proved correct, the final matrix would then be more correct. However, it is not trivial to establish spatial forecasts for e.g. 2020 and 2030 for very detailed variables, and the uncertainty of these forecasts is likely to be high. As a result, we have chosen a rather simple set of constraints in which the fundamental constraint, T_ga(z0, g, a) in Table 4 is provided as an official forecast.

Whereas the T_ga(z0, g, a) constraint can be based on official forecasts, this is not in general the case for income forecasts. Hence, if we are to predict a change in population per income category as a result of an overall increase in income this is non-trivial as the underlying distribution is asymmetric and right skewed. In order to generate future income targets micro-simulation is used. In a first stage, we generate numerical income measures for each individual by random sampling from the income intervals of the targets. These incomes can then be subjected to possible transformations of which a simple uniform upscaling of the average income level is the simplest scenario. Finally, after the income vector has been modified it is converted back to the categorical representation of the target. In practise, it means that, as people gets richer the frequency table representing the income target is shifted to the right as it should.

The resampling stage where individuals are matched in households is based on conditional sampling from tables generated on the basis of register data. The spouse matching table is shown below in Table 9 and further illustrated in a ‘heat map’ type of plot in Fig. 4.

As commented in Section 2.3 we implement different sampling schemes depending on the “sparseness” in the matching. Figure 4 is intended as an illustration of the correlation between the ages of the two adult persons in the household. Although in reality it is a categorical grid it is illustrated in the form of a smoothened heat-map to illustrate the correlation density. Similar correlation patterns can be plotted for income and for the labour market association.

In addition to the sampling of the household composition, the model uses sampling as a mean to allocate kids to households. This is conditional on the spouse matching and considers both the number of kids and a classification of which type of kid to allocate. Tables 10 and 11 represent the marginal probability tables for the sampling of kids and for illustration of the correlation between the number of kids and the age of the adult female in the household, we offer a contour type of plot in Fig. 5.

The validation of population synthesis models is non-trivial. It compares to the validation of model fit for high dimensional probability distributions which is an active research area in statistics. The challenge is that usual norms such as root-mean-square (RMSE) or Wasserstein metrics offers little value when evaluated across many dimensions. At least it provides no information at the level of the cells. Also, because of the many dimensions simple illustrations of the “performance” cannot be carried out. As a result, the paper will not represent a complete validation of the model framework but provide important validation insights by focusing on two aspects: i) Uncertainty that result from the household simulation stage and propagate to the final output, and ii) Prediction performance when evaluated at the level of zones and when considering a prediction horizon between 2010 and 2015.

The model framework is deterministic in the sense that if we select a specific random seed for the household simulation stage, the same population will be replicated given the inputs are the same. However, it is relevant to ask how sensible the final results are with respect to this seed. In other words, how much sampling noise is generated and propagated from the household simulation stage to the final list of agents. An even more important and relevant question however, is to what extent this noise will affect the final output of the transport demand model. To analyse this specific question the entire model framework has been run 20 times with different seed numbers. Hence, for each of these runs, the composition of the households is different due to different seed numbers. In Fig. 6 the overall percentage deviation is illustrated for trips and mileage for 6 different transport modes.

As can be seen, the variation at the aggregated level is small and in most cases comfortably below 0.03%. Clearly, this is the result of the “law of large numbers” and suggests that we do not have to worry about the sampling noise when results are aggregated over many households. However, if we consider more detailed outputs, noise caused by the sampling will increase. In Tables 12 and 13 the share of origin zones where the percentage deviation is below certain thresholds is illustrated.

The column representing the “weighted” distribution has a better profile in that fewer entries are above the 1% deviation. In particularly very few “weighted entries” are above the 5% threshold. Again, this is because of the law of large numbers as smaller zones will have fewer agents which in turn will drive up the relative sampling noise. From an application point of view, what matters is the weighted measurement of trips and mileage and the results suggest that although sampling noise exists, it is at a low level.

It is also relevant to compare the population synthesis with observed register data. More specifically we predict the 2015 population based on a starting matrix form 2010 and by using correct targets for 2015. Hence, per assumption there are no biases in the constraints at the municipality level. This gives an indication of a “best-case” benchmark for the synthesizer. Others have looked at various sensitivity investigations when targets and the starting solutions have been varied [24]. However, this in itself involves a rather comprehensive Monte-Carlo scheme which may not be of any particular interest in general and is certainly outside the scope of the present paper.

Firstly, in Fig. 7 the weighted percentage deviation at the sub-zones level is presented. This test essentially looks into how well the model recovers the population at more detailed spatial levels (levels that are not supported by the constraints). As can be seen, over the 5-year period there is an average deviation of around 3.5% which equals approximately to 0.7% deviation per year. However, for certain zones it can be substantially higher. The results suggest that there is definitely variation over the years and for some zones this variation may be substantial. This may reflect the development of new urban settlements not reflected in the first-stage fitting. Not surprisingly, the prediction error is significantly smaller per year than has been observed in Krishnamurthy S, Kockelman KM [21] and in McCray et al. [24]. However, these studies considered the total observed error and also included uncertainty in the population targets.

In Fig. 8 we add additional age groups in order to assess the deviation at the subzone level when combined with age groups. As expected, the variation gets bigger and especially for younger generations which are much more mobile compared to older generations. Still, the annual deviation across age groups is in most cases below a 1% deviation per year.

While these results suggest that the synthesis framework give rise to a sizable deviation, two important elements should be considered. First, in Denmark most large cities are experiencing a large inflow of people and to give room for these people new settlement areas are continuously developed. Some of these areas are quite large and will even for a 5 year period represent substantial relative changes to the existing population (examples for Copenhagen are Nordhavn and Sydhavn). These fluctuations are not captured by the synthesis framework as these processes happen inside the municipality. To some extent this uncertainty can be captured in forecasts by aligning the sub-zone population with projected expectations at this level (this is accomplished in the second stage IPF). Such projections typically exist and would substantially improve the performance. The second issue relates to the socio-dynamics of the population over time. As an example, while for Copenhagen there has been a general increase in the population between 2010 and 2015 of approximately 10% the 20–29 years old have increased 17% and kids between the age of 3 and 6 as much as 20%. Hence, predicting this socio-dynamics (which is even more extreme at the level of detailed zones) is quite a challenge.

Future research seen from an applied perspective may focus on the following directions.

In general, any improvement to the population synthesis stage will benefit not only the prediction of the population in itself, but all subsequent modelling steps. This is particularly important for transport models as local transport demand is a mirror of the residing population.