Boyam Fabrice Yaméogo

Pascal Gastineau
email: pascal.gastineau@univ-eiffel.fr

Pierre Hankach
email: pierre.hankach@univ-eiffel.fr

Pierre-Olivier Vandanjon
email: pierre-olivier.vandanjon@univ-eiffel.fr

Fabrice Boyam
email: boyam-fabrice.yameogo@univ-eiffel.fr

Yaméogo

Comparing Methods for Generating a Two-Layered Synthetic Population

Keywords: Synthetic Population Generation, Multi-level, Microsimulation, Simultaneous Control

Synthetic population is used in many transport models ranging from trip-based, hybrid trip, tourbased and activity-based models. As mobility decision depends on both individual's characteristics and family situation, generating a two-layered population that take into account not only the individuallevel but also household-level is essential.

In the literature, three main categories of methods for two-layered population generation have been proposed. These categories are Synthetic Reconstruction (SR), Combinatorial Optimization (CO), and Statistical Learning (SL). SR and CO methods produce synthetic populations by means of replicating individuals, while SL methods generate a population following a joint probability estimation. However, selecting a generation process is not straightforward as it depends on input data and synthetic population characteristics. To the best of our knowledge, no clear methodology for selecting between these methods exists.

The main objectives of this paper is to provide 1) a detailed description of the available methods 2) a comparison between these methods and 3) a decision-making procedure for selecting between these methods. The description and comparison of the methods relies on different criteria: marginals availability, sample size, number of potential attributes that can be handled, population size to generate, possibility of zero-cell problem, etc. The advantages and shortcomings of each method are illustrated, and method performance is assessed. The decision-making procedure is carried out through the proposal of a decision tree.

Researchers and practitioners have now access to a comprehensive and unified framework to select the appropriate method depending on available data and features of their modeling purposes.

Introduction

In recent years, Agent-Based Models (ABMs) have become more popular for simulating transport systems given their powerful capacity to simulate complex systems with a large number of agents (1; 2). For this reason, ABMs have also been applied to other fields, e.g. health (3; 4), economic policy evaluation (5; 6), geography [START_REF] O'sullivan | Geographical information science: agent-based models[END_REF].

In transport applications, disaggregated population information (individuals, households, etc.) is used as ABM input data. However, for privacy reasons, no comprehensive dataset containing the sociodemographic characteristics of individuals and households at a small geographical scale exists. Only the characteristics of a sample and aggregated statistics (marginal distributions) of socio-demographic variables of the actual population are known. In order to perform agent-based simulation, a necessary intermediate step therefore is the generation of a "synthetic population" representative of the actual population derived from available data. The resulting synthetic population constitutes a simplified microscopic representation of the actual population since only those variables of interest are to be reproduced [START_REF] Chapuis | A brief review of synthetic population generation practices in agentbased social simulation[END_REF].

Most of the approaches developed to generate a synthetic population (for transport or other applications) have emphasized generating either individual-centered or household-centered populations. For example, Iterative Proportional Fitting (IPF), which is by far the most widely used technique for generating a synthetic population, does not enable generating populations linking households with individuals (and introducing controls at both levels). In using this technique, a population of individuals can be generated but without linking any individual characteristics to household information. For transport modeling, this absence of a link is clearly a shortcoming, since an individual's mobility decision depends on both his/her characteristics and family situation (9; 10; 11). Highlighted herein is the need to generate synthetic populations that take into account not only the individual level but also household-level information. Therefore, only synthetic population generation methods that are capable of taking these two levels into account are considered in this paper.

The specific case of generating a two-layered synthetic population must satisfy certain conditions, namely:

• maintaining the hierarchical structure of the data by associating individual and household variables in the most optimal way possible;

• reflecting the heterogeneity of the distribution of households and individuals between geographical areas [START_REF] Münnich | On the simulation of complex universes in the case of applying the german microcensus[END_REF];

• reproducing the interdependencies among agents in the same household [START_REF] Sun | A hierarchical mixture modeling framework for population synthesis[END_REF];

• demonstrating the ability to fit with marginals.

To tackle this two-layered (or multi-level) synthetic population generation issue, various approaches have recently been developed. Different classifications of these methods have been proposed (8; 13; 14). Our categorization is derived from Sun et al. [START_REF] Sun | A hierarchical mixture modeling framework for population synthesis[END_REF]. Three main categories of methods are thus considered: Synthetic Reconstruction (SR), Combinatorial Optimization (CO) and Statistical Learning (SL). SR and CO methods produce synthetic populations by means of replicating individuals, while SL methods draw a population following a joint probability estimation. SR methods are deterministic whereas CO and SL are both stochastic.

Figure 1 provides an overview of these three categories, along with their primary associated methods for generating a synthetic population of individuals and households. Some of these methods have been implemented as commercial or open source software. All the implemented population synthesizers fall under the SR category. Transport practitioners most often use these population synthesizers without necessarily being aware of their suitability to their applications (input data, population size, sample size, etc.) and available alternatives.

The objective of this paper is to introduce and assess the various approaches based on the criteria listed above. A few studies have already carried out a review of these methods. Hermes and Poulsen [START_REF] Hermes | A review of current methods to generate synthetic spatial microdata using reweighting and future directions[END_REF] analysed generation methods but only at a single level: either household or individual; and do not take into account SL methods. Müller and Axhausen (16;[START_REF] Müller | Multi-level fitting algorithms for population synthesis[END_REF] reviewed two-layered methods but only covered methods from the SR family. Chapuis and Taillandier [START_REF] Chapuis | A brief review of synthetic population generation practices in agentbased social simulation[END_REF] analysed two-layered generation methods classifying them in two main categories (grouping SR and SL methods into one category and maintaining CO methods in a distinct one). However the review was limited to articles published in one journal (Journal of Artificial Societies and Social Simulation) over a five-years period. None of the above reviews proposed neither a detailed comparison on identified criteria nor a decision-making procedure to help practitioners select between them.

The main contribution of this paper is threefold. First, we provide a detailed description of the characteristics of the three approaches and the main methods of each approach. Second, we compare the methods using a set of criteria that illustrates the advantages and shortcomings of each one, as well as their applicability depending on available input data. Third, we propose a decision-making procedure, in the form of a decision tree, for selecting among these methods. The decision criteria include: sample representativeness, availability of marginal distribution, number of considered attributes, and population size.

The remainder of this paper is organized as follows. First, the approaches will be described according to their classification into three main categories: Synthetic Reconstruction (Section 2), Combinatorial Optimization (Section 3), and Statistical Learning (Section 4). Section 5 will discuss the three previous sections and provide both a comparison table and decision tree for selecting among the methods. The final section offers a conclusion and outlook.

Synthetic Reconstruction

This category of methods is the most widely used to generate synthetic populations and relies on a two-step procedure: fitting and allocation. The fitting step entails assigning positive weights to the individuals and households in the sample in such a way that the sum of these weights corresponds to the marginal sums. The resulting weights are typically non-integers (i.e. fractions of individuals). During the allocation step, these non-integer weights are converted into integer weights, and individuals are replicated in proportion to their weights. For example, an individual assigned a weight of 8 will be replicated 8 times in the synthetic population. Once this procedure has been completed, each member of the synthetic population will have clearly defined attributes. These two steps form the foundation of the synthetic reconstruction approach [START_REF] Templ | Simulation of synthetic complex data: The r package simpop[END_REF]. Figure 2 shows a simplified flowchart of how these steps are implemented. SR methods are deterministic, which means that depending on the sample used, the weights obtained during the fitting stage never vary subsequently.

Most SR methods require having both a sample and aggregate data. The underlying assumption here is that the sample represents the true correlation structure among the attributes [START_REF] Farooq | Simulation based population synthesis[END_REF], and the interactions present in the sample are, to a great extent, preserved for the synthetic agents [START_REF] Müller | Population synthesis for microsimulation: State of the art[END_REF]. The sample therefore needs to be consistent and representative while containing at least one observation for each type of individual in the actual population. This last constraint avoids the so-called "zero-cell problem", whereby in the absence of an observation in the sample of a specific individual present in the actual population, then SR methods cannot generate this specific individual. Barthelemy and Toint (19) developed an alternative SR method that does not use a sample (and hence is called a sample-free method). Although this approach provides more flexibility in terms of data needs (only marginals are required), it ignores the dependence between household and individual levels without maintaining consistency between these two layers (20; 21). Therefore this method fails to satisfy the conditions we specified in the introduction for two-layered synthetic population generation and has not been considered in this paper.

Iterative Proportional Fitting (IPF) approach and its adaptations to the two-layered problem

A commonly used SR technique is Iterative Proportional Fitting (IPF). IPF adjusts a contingency table constructed from the sample in order to match marginal distributions; it minimizes discrimination information, and the contingency table is derived by iteratively adjusting the cell values so as to minimize deviation from the marginals of the attributes. Interested readers are referred to Beckman et al. [START_REF] Beckman | Creating synthetic baseline populations[END_REF] and Pritchard and Miller (23) for a complete description of this technique. The IPF technique has several advantages, the first being its guarantee that marginals will be fit and the correlation structure will be preserved [START_REF] Rich | Generating synthetic baseline populations from register data[END_REF]. As a second advantage, the distribution of variables not used to match marginal distributions are reasonably estimated [START_REF] Beckman | Creating synthetic baseline populations[END_REF]. IPF is simple, fast, accurate [START_REF] Lovelace | truncate, replicate, sample': A method for creating integer weights for spatial microsimulation[END_REF] and moreover offers a pervasive method to estimate socioeconomic variables [START_REF] Choupani | Population synthesis using iterative proportional fitting (ipf): A review and future research[END_REF]. In its original formulation however, IPF cannot simultaneously estimate both household and individual level attributes. Some IPF-based approaches have sought to deal with both household and individual attributes. Arentze et al. [START_REF] Arentze | Creating synthetic household populations: problems and approach[END_REF] proposed a two-step procedure: in the first step, marginal distributions of individuals are converted into a marginal distribution of households by using a relation matrix; in the second step, the resulting household marginal distribution serves to constrain a multiway table of household counts. Guo and Bhat [START_REF] Guo | Population synthesis for microsimulating travel behavior[END_REF] and Auld and Mohammadian (29) estimated the joint distributions of attributes for households and persons separately by means of IPF techniques. Household samples can then be drawn into the synthetic population iteratively depending on how well the types of households and their members fit to the estimated joint distributions at both the household and person levels [START_REF] Zhu | Synthetic population generation at disaggregated spatial scales for land use and transportation microsimulation[END_REF]. According to Pritchard and Miller's generation process [START_REF] Pritchard | Advances in population synthesis: fitting many attributes per agent and fitting to household and person margins simultaneously[END_REF], the person population is first fitted with IPF. These authors selected a set of shared attributes between the person and household levels and performed a second IPF at the household level in considering the shared attributes as marginals. At the generation stage, households and persons are matched through these shared attributes, and a conditional Monte Carlo method is employed to assign persons to households [START_REF] Zhu | Synthetic population generation at disaggregated spatial scales for land use and transportation microsimulation[END_REF].

In all these studies, the joint distribution of household and individual-level attributes is fitted either separately or sequentially, which does not guarantee consistency between these two levels.

Two-layered generation methods

Iterative Proportional Update (IPU), Hierarchical Iterative Proportional Fitting (HIPF), entropy maximization (ent) and Generalized Raking (GR) all generate an intrinsically two-layered population. As such, they generate populations of persons grouped into households by computing household-level weights that satisfy the marginals at both the household and person levels.

Iterative Proportional Update (IPU) [START_REF] Ye | A methodology to match distributions of both household and person attributes in the generation of synthetic populations[END_REF] and Hierarchical IPF (HIPF) (17; 31) are intended to simultaneously estimate both household and individual-level attributes. These methods respectively compute the factors and weights linked to individual and household records using an iterative approach with a cross-categorization of individual types into household types [START_REF] Chapuis | A brief review of synthetic population generation practices in agentbased social simulation[END_REF]. The HIPF approach constantly switches between the household domain and the person domain, through employing an entropyoptimizing adjustment step [START_REF] Müller | Hierarchical ipf: Generating a synthetic population for switzerland[END_REF]. In the IPU approach, weights are adjusted to satisfy household-level constraints first and then updated to satisfy person-level constraints.

The population synthesis problem can also be formulated as a constrained optimization problem. The goal of this formulation is to find a weight for each household such that the distributions of characteristics in the weighted sample match the exogenously given distributions in the population, for both household and person characteristics [START_REF] Bar-Gera | Estimating survey weights with multiple constraints using entropy optimization methods[END_REF]. This optimization problem can be solved using two approaches: the entropy-optimizing procedure, and the calibration estimation.

The entropy-optimizing procedure is based on the thermodynamic concept of entropy and seeks the most likely configuration of elements within a constrained situation [START_REF] Lee | Cross-entropy optimization model for population synthesis in activity-based microsimulation models[END_REF]. This method directly optimizes an information-based similarity metric of the weights and therefore introduces the least possible amount of new information, just like IPF [START_REF] Müller | A generalized approach to population synthesis[END_REF]. The technique can offer an alternative to IPU and HIPF since it also generates weights associated with individual and household records.

The aim of a calibration estimation is to weight a sample of individuals using information available on certain variables, called calibration variables, which are the control variables or marginals obtained from the aggregated data. Deville et al. [START_REF] Deville | Generalized raking procedures in survey sampling[END_REF] called these methods "generalized raking" (GR) as they generalize the classical statistical raking ratio method. This is done by minimizing a distance measure between initial weights and final weights, subject to calibration equations. For example, for a qualitative variable, the weighted counts of the categories (attributes) of this variable in the sample, after calibration, will be equal to the known counts of the population.

Different distance functions can be used to solve this optimization problem: linear method, multiplicative method (or raking ratio method), logit method, and truncated linear method. Generalized raking directly adjusts weights to satisfy both individual and household-level constraints and is a viable alternative to the multi-level fitting methods like HIPF or IPU.

Almost all population synthesizers fall under the SR category. For example, PopulationSim [START_REF] Paul | Multi-level population synthesis using entropy maximizationbased simultaneous list balancing[END_REF] and PopSynIII [START_REF] Vovsha | New features of population synthesis[END_REF] are both based on the entropy maximization procedure, TransCAD [START_REF] Balakrishnaa | An enhanced and efficient population synthesis approach to support advanced travel demand models[END_REF] uses an enhanced approach of IPU. For a more in-depth knowledge of the characteristics of some synthetic population generators, the interested readers are referred to (31) (chapter 2).

Combinatorial Optimization

The second category of approaches refers to Combinatorial Optimization (CO) techniques. CO-based techniques are two-layered since they can directly generate a list of households and persons [START_REF] Ma | Synthetic population generation with multilevel controls: A fitness-based synthesis approach and validations[END_REF].

Similar to SR methods, CO requires information on both the sample and marginal level, and the synthetic population is obtained by a replication of individuals (without explicitly determining the joint distribution across all controlled attributes). But unlike SR methods, CO is a stochastic process. Another difference is that the CO methods result in entire individuals or households being allocated to each zone, whereas SR methods yield fractions of individuals or households being allocated to each zone [START_REF] Lovelace | Spatial microsimulation with R. Chapman and Hall/CRC[END_REF]. One key advantage of CO methods is that data requirements remain less restrictive than those for SR methods [START_REF] Templ | Simulation of synthetic complex data: The r package simpop[END_REF]. However, a major drawback of CO methods is their computational complexity for large population sizes. As population size grows, CO methods cannot guarantee the optimal solution and require too much computational time [START_REF] Lee | Cross-entropy optimization model for population synthesis in activity-based microsimulation models[END_REF]. For this reason, CO-based methods are not widely used.

A description of the method has been given by ( 41) and [START_REF] Templ | Simulation of synthetic complex data: The r package simpop[END_REF]. A simplified flowchart of how CO techniques function is presented in Figure 3. Starting from a sample of households and individuals, the sample is first divided into exclusive groups (e.g. small geographic areas), for which some control variables (marginals) are available. The process then becomes iterative:

• an initial number of households is randomly chosen from the sample in each small geographic area in order to form an appropriately sized group; next, the fitness of the selected population to the marginals for the newly formed group is estimated;

• •one household is either added, replaced or swapped with another from the sample, and the goodness of fit is recalculated. If the replacement improves the fit, this household is kept; otherwise, this household will be removed from the zone. The process is repeated until a certain goodness-of-fit threshold or an arbitrarily fixed number of iterations has been reached.

An example of a statistical measure of fit that could be used is the overall relative sum of squared Z scores (RSSZ) (42; 43). Various optimization procedures have been proposed to estimate the goodnessof-fit measure, i.e.: genetic algorithm [START_REF] Birkin | A synthetic demographic model of the uk population: Methods, progress and problems[END_REF], greedy heuristic [START_REF] Srinivasan | Procedure for forecasting household characteristics for input to travel-demand models. project report of university of florida, gainesville; florida department of transportation[END_REF], hill-climbing [START_REF] Abraham | Population synthesis using combinatorial optimization at multiple levels[END_REF], and simulated annealing [START_REF] Harland | Creating realistic synthetic populations at varying spatial scales: A comparative critique of population synthesis techniques[END_REF].

Most studies using CO methods have focused on synthesizing a population with a handful of attributes, typically either individual or household. For example, Ryan et al. [START_REF] Ryan | Population synthesis: Comparing the major techniques using a small, complete population of firms[END_REF]) showed that CO methods produce more accurate populations than IPF, although their study was conducted on a relatively small population with just three individual level variables. Few studies have sought to simultaneously synthesize populations with controls at both the household and person levels (46; 39).

The former uses a hill-climbing algorithm to resolve the combinatory problem, while the latter employs its own algorithm, called the fitness-based synthesis (FBS).

Statistical Learning

The third approach that may be used to generate a two-layered synthetic population is statistical learning (SL), also known as a simulation-based approach. SL focuses on the joint distribution of all attributes in the sample by directly estimating a probability for each combination, including those not observed in the sample [START_REF] Sun | A hierarchical mixture modeling framework for population synthesis[END_REF]. In fact, in the actual population, individuals are characterized by discrete or continuous attributes, which feature a unique joint distribution. Due to privacy concerns and data availability, only partial views (samples, marginals or conditional-marginal distribution) of this unique joint distribution are available. The aim of SL methods is to use these partial views to construct a synthetic population, whereby the empirical distribution of this synthetic population is as close as possible to the unique joint distribution in the actual population. The basic premise behind this approach is given by Farooq et al. [START_REF] Farooq | Simulation based population synthesis[END_REF] and Sun et al. [START_REF] Sun | A hierarchical mixture modeling framework for population synthesis[END_REF].

SL methods offer greater flexibility in terms of data requirements and data sources; in general, they demonstrate good performance in dealing both with the lack of heterogeneity problem raised in both SR and CO [START_REF] Sun | A hierarchical mixture modeling framework for population synthesis[END_REF] and with small sample sizes. Also, these methods provide a more systematic way of imputing or interpolating data and are capable of reproducing agents that do not exist in the sample. Another advantage with SL is that these methods can generate a synthetic population from sample data only, when marginal information at the zonal level is not available. However, a major drawback of SL methods is that they fail to satisfy the conditional distributions while satisfying the marginal distributions of all variables simultaneously. During the population synthesis process, when marginals are available, it is indeed necessary to precisely match the observed marginal distributions with the generated population at the zonal level. Sun and Erath (21) and Sun et al. [START_REF] Sun | A hierarchical mixture modeling framework for population synthesis[END_REF] both recommend applying SR methods as a post-processing step (when marginals are available) on a synthetic sample of the population generated from SL methods so as to create a population matching the marginals. In some configurations, combining SL and SR methods actually proves to be the most relevant means for generating a synthetic population. If sample data are not comprehensive (do not contain at least one observation for each type of individual in the actual population) and representative (valid description of the study area's population [START_REF] Müller | Multi-level fitting algorithms for population synthesis[END_REF]) of population data, yet marginals are available, then SL methods can be applied first to construct a suitable sample. Afterwards, SR methods can be introduced to create a population matching the marginals. Figure 4 Many SL-based methods for synthetic population generation have been proposed. Farooq et al. [START_REF] Farooq | Simulation based population synthesis[END_REF] suggested implementing Gibbs sampling and a Markov Chain Monte Carlo method (MCMC) to generate the synthetic population. This method uses the conditional-marginal distribution to simulate drawing from the joint distribution in the actual population. Farooq et al. [START_REF] Farooq | Simulation based population synthesis[END_REF] used discrete choice models to construct these conditionals. Once the conditionals are obtained, the Gibbs sampler is run and a synthetic population obtained by drawing the number of individuals/households corresponding to the required population size.

Saadi et al. [START_REF] Saadi | Hidden markov model-based population synthesis[END_REF] presented a Hidden Markov Model (HMM) to generate a synthetic population. These authors considered that each attribute represents a state. The number of states is known and each state of attribute i is connected to all states of the subsequent attribute i + 1. An HMM is a Markov process characterized by two parameters, i.e.: the symbol emission describing the emission probabilities of each state, and the transition probabilities of changing to another state. The basic idea behind this process is to position the attributes in a certain sequence (by descending order of the number of attributes) and then estimate these probabilities. However, both Markov process-based methods (MCMC and HMM) only pay slight attention to the hierarchical household structure (48; 13). In order to generate a two-layered synthetic population, Casati et al. [START_REF] Casati | Synthetic population generation by combining a hierarchical, simulation-based approach with reweighting by generalized raking[END_REF] proposed an extension of Farooq et al.'s MCMC model [START_REF] Farooq | Simulation based population synthesis[END_REF] to account for the hierarchical structure of households and individuals; they named this process hierarchical MCMC (or hMCMC). hMCMC entails ordering the agents living in the same household according to their household roles. Three roles are thus defined by a rule-based approach: owners (persons with the highest income in their household or chosen by virtue of another selection criterion), intermediate types (spouses and children), and others. To generate a household, all owners are first generated according to the methodology used in [START_REF] Farooq | Simulation based population synthesis[END_REF], thus taking into account the variables characterizing the household (as all members of a household share the same household attributes). If the household size is greater than 1, then the other household members are generated as follows:

1. the conditionals of the spouse are generated by incorporating certain variables relating to the owner;

2. the conditionals of a child are generated by incorporating certain variables relating to both the owner and spouse;

3. the conditionals of others solely depend on their owners and intermediate types.

In this manner, individual and household attributes are simultaneously combined, and the association between individuals and households is respected. However, as noted by the authors, the main flaw in this method pertains to the fact that the number of categories of an attribute characterizing a certain agent type may differ when this variable is involved in the generation of its agent type and when it influences the generation of subsequent types. It may often be unclear which of the owner's variables should be used to generate intermediate types or other person types; another uncertainty lies in the implications of choosing one particular set over another.

Another SL-based method applied to population synthesis is the Bayesian Network (BN) (21; 50). A BN is a probabilistic graphical representation of the factorization of the joint probability distribution into conditional distributions, presented by means of a directed acyclic graph (DAG) [START_REF] Borysov | How to generate micro-agents? a deep generative modeling approach to population synthesis[END_REF]. Sun and Erath [START_REF] Sun | A bayesian network approach for population synthesis[END_REF] argued that a BN, by abstracting the structure of population systems using a DAG and local conditional probabilities, is able to capture and reproduce the complex dependence and higher-order interactions among a large set of variables. According to the DAG, nodes represent random variables, while arrows show the probabilistic dependencies between these variables. From the definition given by Zhang et al. [START_REF] Zhang | Connected population synthesis for transportation simulation[END_REF], BN consists of two main parts: (1) structural learning to define the network structure that describes the conditional independence of the random variables through a scoring approach, and (2) parameter estimation to learn a conditional distribution of random variables given this DAG structure. The synthetic population can then be generated accordingly, by sampling from the obtained Bayesian Network. With this BN method, the hierarchical structure of the data (households and individuals) is respected. Nevertheless, learning a Bayesian structure is a computationally challenging task, particularly when a large number of attributes for each agent must be considered. As pointed out by Sun and Erath (21), a network with six nodes (where each node represents an attribute) contains some 3 million possible DAGs. Zhang et al. [START_REF] Zhang | Connected population synthesis for transportation simulation[END_REF] suggested keeping the Bayesian Network as simple as possible and creating a "whitelist" defining a set of relationships in the structural learning procedure that are to be preserved and guaranteed present in the final graph. This procedure involves building the network based on expert knowledge; yet despite this, mistakes can still be made.

Sun et al. [START_REF] Sun | A hierarchical mixture modeling framework for population synthesis[END_REF] proposed an SL method that relies on a hierarchical mixture (HM) modeling framework. According to this approach, the authors considered the existence of latent classes at the household level.

For each household-level latent class, latent classes also exist at the individual level. The interaction between these two levels can thus be captured by using a conditional distribution on the latent class labels. The proposed framework integrates three component models (probabilistic tensor factorization, multi-level latent class model, rejection sampling) to reproduce the underlying statistical distribution between the two levels. However, as the number of attributes increases, the HM method becomes very challenging to apply. Another challenge with this approach pertains to its robustness with respect to the selection of latent variables [START_REF] Borysov | How to generate micro-agents? a deep generative modeling approach to population synthesis[END_REF].

More recently, Borysov et al. [START_REF] Borysov | How to generate micro-agents? a deep generative modeling approach to population synthesis[END_REF] proposed a deep generative modeling (DGM) approach based on a Variational Autoencoder (VAE). According to these authors, each person is represented by a vector of random variables. The objective of such a VAE (which is an unsupervised generative model based on a deep artificial neural network) is to learn the joint distribution of all individuals in the sample. A consistent synthetic population of households and individuals with a large number of attributes can thus be generated. Implementation of this method however requires fairly advanced computer science skills.

Comparison of methods

The methods that serve to generate a representative synthetic population of households and individuals differ depending on their underlying assumptions, the approach employed or the data required. According to Sun et al. ( 13), let's sort these methods into three categories: synthetic reconstruction (SR), combinatorial optimization (CO), and statistical learning (SL).

The SR approach combines information from sample data and aggregate statistics (marginals) to compute the weights that reflect each sample agent's representativeness in a given zone [START_REF] Chapuis | A brief review of synthetic population generation practices in agentbased social simulation[END_REF]. The CO methods also use the sample and aggregate data to select an appropriate combination of households that best fit the marginals. SL methods solely consider the sample and focus on the joint distribution of all attributes by estimating a probability for each combination.

The main characteristics, advantages and shortcomings of these approaches are summarized in Table 1, with a discussion of these characteristics given below:

• In order to operate, CO and SR methods require both sample data and aggregate statistics (marginals), while SL methods can produce a population from sample data only, as needed.

• The output from CO and SR methods directly fits to marginals, whereas the SL method necessitates a post-processing step in order to achieve this step.

• SL methods can produce consistent results even if the sample size is small. In contrast, sample size is critical for both CO and SR. In fact, controlling the marginal statistics of attributes with infrequent categories requires larger reference samples. For example, in their case study, Sun and Erath [START_REF] Sun | A bayesian network approach for population synthesis[END_REF] showed that for IPF results to be similar to BN results, the sample size needs to be greater than 20%.

• SR methods have been proven to be scalable. Many papers report on SR applications for generating large synthetic populations with a large number of attributes and categories (30; 31; 33; 34).

CO methods are hindered by their computational complexity, which winds up compromising their scalability when generating large synthetic populations. Most applications of this method have been restricted to small population sizes (46; 39). Some SL methods are hindered by the scalability problem when the number of attributes is large. For Bayesian Networks and Hierarchical Mixtures, it is preferable for the number of attributes to be small, as pointed out by Borysov et al ( 51) and Zhang et al. [START_REF] Zhang | Connected population synthesis for transportation simulation[END_REF]. In fact, a Bayesian network with six nodes (corresponding to attributes) contains some 3 million possible DAGs, and this number becomes 1.1 billion with a network of seven nodes [START_REF] Sun | A bayesian network approach for population synthesis[END_REF].

• SR and CO methods generate a synthetic population by reproducing a reference sample. Consequently, combinations of attributes that are not part of the sample cannot be generated; this restriction is known as the "zero-cell problem" and can occur more frequently when dealing with small geographic zones. The zero-cell problem is inherently present in both SR and CO methods.

A pre-processing procedure to eliminate the zero cell problem is often necessary. SL methods, by virtue of estimating a joint population distribution from the sample, effectively treat this issue.

Conclusion

For transport modeling, the ability to generate a two-layered population (i.e., persons and households) is essential since an individual's mobility decisions depend on both his/her characteristics and family situation. Generating such a population is one of the most challenging problems in population synthesis [START_REF] Fournier | Integrated population synthesis and workplace assignment using an efficient optimization-based person-household matching method[END_REF]. This paper has provided a state of practice of the main methods available for generating a twolayered synthetic population. It has also proposed a decision tool (in the form of a decision tree) for selecting between these methods. To the best of our knowledge, no other study has provided such a detailed analysis of these methods.

The various methods employed to tackle the two-layered (or multi-level) synthetic population generation problem have been reviewed. A classification of these methods, into three categories (Synthetic Reconstruction (SR), Combinatorial Optimization (CO) and Statistical Learning (SL)) based on Sun et al. [START_REF] Sun | A hierarchical mixture modeling framework for population synthesis[END_REF], has been provided, along with a detailed description of the characteristics of methods found in of the three categories. Different criteria have been used to draw a method comparison, namely: processing approach, output format, solution quality (fitting marginals), input data characteristics, mandatory sample size, number of attributes potentially handled, data preparation, the zero-cell problem, and dissemination. This comparison illustrates the advantages and shortcomings of each method, as well as its applicability depending on available input data. However, the paper does not deal with the cases where no initial sample is available. In order to facilitate the choice of an appropriate method for generating a two-layered synthetic population, a decision tree is proposed. This choice is driven by the amount, type and quality (representativeness and comprehensiveness) of available data and synthetic population characteristics. Decision criteria include: marginals availability, number of attributes to be considered, and synthetic population size. Facing the problem of generating a synthetic two-layered population, transport modelers have now access to a comprehensive and unified framework to select the appropriate method depending on available data and features of their modeling purposes. We here only consider the generation of synthetic population based on census or surveys data. We are nevertheless witnessing the emergence of heterogeneous data sources derived from big data that may be exploited in the context of synthetic population generation. The next step would be to look for a methodology to integrate this new type of data. The deep learning methods seem to be a promising way to do it.

Figure 1 :

 1 Figure 1: Different approaches to create a two-layered synthetic population (households and individuals)

Figure 2 :

 2 Figure 2: Simplified flowchart of Synthetic Reconstruction methods to generate a two-layered synthetic population (households and individuals)

Figure 3 :

 3 Figure 3: Simplified flowchart of Combinatorial Optimization methods to generate a two-layered synthetic population (adapted from (15))

Figure 4 :

 4 Figure 4: Simplified flowchart using Statistical Learning methods to generate a two-layered synthetic population (households and individuals)

  -processing step using SR methods like GR to fit marginals. these methods have been implemented recently and only tested by their designers

Table 1 :

 1 A brief summary of different methods for generating a two-layered synthetic population.

	Statistical Learning		hierarchical Bayesian Network Markov Chain Hierarchical Mixture Monte Carlo Deep Generative Modeling	Definition of Learning both A framework A method using	different groups model structure integrating a Variational	of agent types and model three models Autoencoder	and computation parameter to to estimate the (VAE) to	of conditional estimate the joint probability estimate the	distribution joint probability distribution. joint probability	from a Gibbs distribution. distribution.	sampling for each	type of agent	to estimate the	joint probability	distribution.	+ Generation of a pool of individuals respecting the joint distribution	Only a sample (both the individual and household levels).
	Combinatorial	Optimization				A stochastic	algorithm based	on a combination	of individuals and	households to	satisfy a fitness	criterion.	Sample and aggre-	gate data on both	the individual and	household levels.
	Synthetic	Reconstruction	IPU, HIPF, GR,	entropy	optimization	A deterministic	algorithm using	a two step	procedure: fitting	and allocation.	Sample and aggre-	gate data on both	the individual and	household levels.
			Characteristics		Main processing	algorithms						Data preparation

Decision-making procedure

The choice of appropriate method for generating a two-layered synthetic population is critical to transport modelers. This choice is partially driven by the amount, type and quality (representativeness and comprehensiveness) of available data [START_REF] Rich | Large-scale spatial population synthesis for denmark[END_REF]. In order to facilitate this process and in addition to the method comparison presented herein, the decision tree shown in Figure 5 identifies the most suitable methods for generating a two-layered synthetic population, based on the following parameters: available input data (sample and marginals), number of attributes to be considered, and synthetic population size. Our starting assumption is that there is a sample with both household and person characteristics (census or household travel surveys data) as it is the case in most studies.

When marginal data are available and the sample is representative (at least 5% of the population), SR methods are preferred for generating a large population size (millions of persons and households with a high number of attributes as in (32; 37; 11)).

CO methods are limited to the generation of small size populations. This is due to the computational complexity of these methods. For example, Hafezi and Habib (53) and Ma and Srinivasan (39) generate populations with fewer than 60.000 persons and 30.000 households.

When the sample size is very small (less than 5%) but marginal data are available, combining the SL and SR methods is more relevant. SL methods can be applied first to construct a sample of suitable size, afterwards SR methods can be applied to create a population matching the marginals. Following this line, Borysov et al. [START_REF] Borysov | How to generate micro-agents? a deep generative modeling approach to population synthesis[END_REF] in Denmark and Sun et al. [START_REF] Sun | A hierarchical mixture modeling framework for population synthesis[END_REF] in Singapore generated a population from household travel survey samples corresponding to 2.5% and 1% of the total population, respectively.

If marginals are not available, only SL methods can be applied to generate the population.


Florian Beaugnon

Sara Quiligotti

Sylviane Chevreux

Gilles Wallez
email: gilles.wallez@sorbonne-universite.fr

On the monoclinic distortion of β-anhydrite CaSO4

Keywords: anhydrite, gypsum, crystal structure, phase transition, X-ray diffraction, Rietveld method

β-anhydrite, the stable form of CaSO4, is a common mineral and a key phase in gypsum's dehydration mechanism. An in-depth multi-scale investigation of its structure by powder X-ray diffraction revealed a very slight distortion of the crystal cell at room temperature (β = 90.092(9) °), that led to rule out the commonly admitted orthorhombic model (space group Cmcm) in favor of monoclinic C2/c subgroup. According to high-temperature XRD, the structure resumes however the orthorhombic symmetry around 400 °C.

Introduction

Anhydrite (CaSO4), that results from the dehydration of gypsum, exists in three forms: the metastable γ-anhydrite, the stable and naturally occuring β-anhydrite, and α-anhydrite which only exists at temperatures higher than 1200 °C. Little is known about the latter, but the structures of both γ-and βanhydrite have been described at length in previous works. The crystal structure of γ-anhydrite (Fig. 1) is somewhat similar to that of bassanite (CaSO4•½H2O), also obtained by dehydration of gypsum, except that the channels containing the water molecules are empty in the former structure. It crystallizes in the orthorhombic system (space group C222) [START_REF] Bezou | Investigation of the crystal structure of γ-CaSO4, CaSO4·0.5H2O and CaSO4·0.6H2O by Powder Diffraction Methods[END_REF] but shows a strong pseudo-hexagonal symmetry around the Ca-SO4-chains which has led to its description in various space groups. βanhydrite was first identified as "anhydrous sulfated lime" (chaux sulfatée anhydre) in 1801 by R.-J. Haüy who wrote "Its mechanical division, similarly perfect in all the directions, results in integrant molecules with a cubic -or nearly -shape" [2], in agreement with its fairly equal cell parameters. The first radiocrystallographic analysis by Dickson & Binks in 1926 [3] concluded in the orthorhombic symmetry, as would all those that followed. However, notable progresses have been made in radiocrystallographic instrumentation since the last of these studies, published in the 1980s. The crystal structure is generally described in space group Amma [START_REF] Antao | Crystal-structure analysis of four mineral samples of anhydrite, CaSO4, using synchrotron high-resolution powder X-ray diffraction data[END_REF] or equivalent Bmmb [START_REF] Hartman | On the unit cell dimensions and bond lengths of anhydrite[END_REF] (non-standard settings for Cmcm, used in the following). It is noteworthy that the [001] Ca-SO4-chains of the γ-form remain unchanged in the β-form (along [010]), but they reorganize into a dense pseudo-tetragonal array in the latter form (a = 6.99 Å, c = 7.00 Å).

$

Fig. 1. [001] projection of the crystal structures of γ-(left) and β-anhydrite (right). The cornerconnected CaO8 (blue) and SO4 (yellow) polyhedra build chains following the projection axis.

As a mineral, β-anhydrite is of common occurence in evaporite deposits in association with gypsum and in other environments. Its stability, i.e. its low tendency to re-hydration, is due to its lower solubility in water compared to the γ form and bassanite. From an industrial point of view, the reactive plaster powder actually consists in a mixture of the two latter phases, obtained by heating gypsum at a temperature of 100-200 °C depending on water vapor partial pressure [START_REF] Preturlan | Comprehensive thermodynamic study of the calcium sulfate-water vapor system. Part 1: experimental measurements and phase equilibria[END_REF].

Two varieties of hemihydrate, often called "alpha" and "beta" plasters (not to be mistaken with the α and β forms of anhydrite) are commonly distinguished according to their microstructure, in link with their conditions of elaboration. Beta plaster, that results of the calcination of gypsum under dry atmosphere, is made of sub-micron sized crystals with high specific surface, while alpha plaster is prepared by calcination in a wet or hydrothermal environment, resulting in larger crystals with a low specific surface [START_REF] Follner | The setting behaviour of α-and β-CaSO4·0.5H2O as a function of crystal structure and morphology[END_REF][START_REF] Takahashi | SEM observation of β-calcium sulfate hemihydrate particles II. Effects of raw gypsum and condition of production on internal microstructure[END_REF].

If the material is further heated, γ-anhydrite gradually and irreversibly turns into inert βanhydrite, which behaves as a charge during the setting of plaster. It is therefore regarded as undesirable due to its misleading effect on the calculation of the water-plaster mixing ratio and its negative effect on the mechanical properties of the set plaster. Despite its importance, neither the precise mechanism nor the kinetics of the transformation are fully understood, but both the temperature range and the typical time of the transformation are known to be affected by the microstructure [START_REF] Jacques | Structural evolution during the dehydration of gypsum materials[END_REF][START_REF] Mitsuki | Transition of calcium sulfate hemihydrate to insoluble anhydrite[END_REF].

In the frame of a multiscale study on the structural mechanisms of the γ-β transition, the X-ray diffraction (XRD) diagram of the β form was carefully examined in order to determine its microstructure. The analysis led however to an unexpected discovery concerning the actual symmetry of this key phase.

Experimental

Samples

Sample 1 to be used for the structural analysis was prepared from a typical "alpha-type" hemihydrate powder supplied by Saint-Gobain Research Paris (hydrothermal process, grain size 2-20 µm), calcinated at 700 °C for two hours in a muffle furnace, in order to fully transform into β-anhydrite [START_REF] Jacques | Structural evolution during the dehydration of gypsum materials[END_REF]. Sample 2, from the same starting material, was heated 3 h at 820 °C, as well as Sample 3, prepared from a "beta-type" commercial plaster.

X-ray diffraction

X-ray powder diffraction was conducted on a Panalytical X'Pert Pro diffractometer in Bragg-Brentano geometry with a PIXcel1D detector. The X-ray source was a copper anode vacuum tube and the Cu Kα1 emission line was selected with a Ge111 monochromator. The instrument was fitted with an Anton Parr furnace for high-temperature measurements. The room-temperature (resp. hightemperature, every 100 °C up to 900 °C) patterns were recorded on a 20 ≤ 2θ ≤ 150 ° (20 ≤ 2θ ≤ 80 °) range, step 0.013 °, with a total 60 h (70 min) counting time.

Microstructural analysis

X-ray data for Sample 1 were first processed by Rietveld analysis using the Fullprof suite [START_REF] Rodriguez-Carvajal | Recent developments of the program FULLPROF[END_REF][START_REF] Rodriguez-Carvajal | FullProf 98 and WinPLOTR: New Windows 95/NT Applications for Diffraction[END_REF] in Le Bail (profile matching) mode, assuming the Cmcm symmetry and excluding the overlapping reflections, typically the hkl -lkh pairs with indexes of the same parity, due to the pseudotetragonal symmetry.

In a second step, the whole diagram was refined with cell parameters set at the previously refined values. The peaks FWHM were measured by implementing the Thompson-Cox-Hastings profile function with spherical harmonics expansion to take into account the anisotropy of the microstructure, then checked by single peak or doublet profile fitting. The global broadening β was determined for each peak by substracting the instrumental width measured on a LaB6 standard, then a Williamson-Hall analysis [START_REF] Williamson | X-ray line broadening from filed aluminum and tungsten[END_REF] was performed by plotting β.cosθ against sinθ. Least-squares regression lines were traced for several families including h00, 00l and h0l in order to calculate the mean crystal dimensions Lhkl = Kλ/β from the vertical intercept and the lattice strains εhkl from the slope.

Refinement of the crystal structure

The structure of β-anhydrite was determined by Rietveld refinement in space group Cmcm and in alternative monoclinic symmetry C2/c. Final reliability factors were (ortho / mono): χ² = 2.37 / 2.22, Rp = 0.072 / 0.070, Rwp = 0.095 / 0.092. For the sake of realism, the uncertainties measured by Fullprof were multiplied by 10 for the cell parameters and by Bérar's factor (2.5) [START_REF] Bérar | s and estimated probable error obtained in Rietveld refinements with local correlations[END_REF] for the intensitydependent variables.

Results and discussion

Williamson-Hall analysis

The Williamson-Hall plot for Sample 1 on Fig. 2 calls for a careful examination. The h00, 00l and other reflections with either h or l null (in the shaded area) exhibit a low strain factor (εhk0, ε0kl < 3.3 10 -4 ), as expected considering the temperature and duration of the thermal treatment. From the intercept of the h00 and 00l lines, the crystallites appear to be 200-300 nm wide in the (010) plane, meaning that the large alpha-plaster grains were heavily cracked during firing due to the shrinkage in the (010) plane. This feature is in agreement with the fibrous aspect often observed on natural anhydrite samples of various dimensions.

The other reflections (non-null h and l) however, are systematically beyond the h00 and 00l lines. Their slopes are all the higher that, on the one hand, h and l are close to each other, and on the other hand, k is low: the maximal value (13.5 10 -4 ) is reached for the h0h series. For example, a 0.082(5) ° broadening can be measured for the 202 peak in comparison with the narrowest ones of Sample 1. Samples 2 and 3, although prepared at higher temperature (and concerning the latter, from a different material) exhibit a similar (resp. 0.069(5) and 0.071(5) °) widening. The differences between the samples are probably due to microstructural effects.

While clearly inconsistent with a normal strain model, this phenomenon more probably stems from an homogeneous distortion of the crystal lattice. In line with the dependence of the slope against Miller's indexes, a monoclinic distortion in the (010) plane can be considered: the lifting of the degeneracy of the dhkl-d-hkl spacing pairs would actually have no effect should either h or l be zero, but would yield a maximal splitting if h = l (as a ≈ c) and if k is minimum.

The monoclinic structure of β-anhydrite

The examination of the extinction rules confirms the C-centered lattice and the c glide mirror normal to the b axis. On this basis, the two possible space groups are C2/c and Cc (n° 9 and 15), subgroups of Cmcm. Structure refinements with both groups lead to atomic positions (Table 1) indistinguishable from those in the Cmcm symmetry considering the standard uncertainties, therefore C2/c was retained as the highest possible symmetry (See Fig. 3 for the final Rietveld plot).

The cell parameters remained unchanged, except for β = 90.092(9) °. Note that β can be also infered from the strains measured following h0h and h00 (β ≈ π/2 + 2 (εh0h -εh00), details in Supplementary Data), giving β ≈ 90.12 °, in fair agreement with the Rietveld measurement. This near-90 value explains why the monoclinic distortion was not discovered until this work.

Phase transition

According to the literature, neither differential thermal analysis [START_REF] West | Thermography of Gypsum[END_REF] nor X-ray diffraction [START_REF] Flörke | Die Hochtemperaturmodifikationen von Kalzium-, Strontium-und Bariumsulfat[END_REF][START_REF] Evans | The thermal expansion of anhydrite to 1000 °C[END_REF][START_REF] Ballirano | Thermal behaviour of beta-anhydrite CaSO4 to 1263 K[END_REF] show any sign of a phase transition until the formation of the face-centered cubic α-anhydrite around 1200 °C. Most of these works are now ancient however, and did not benefit of the improvements in instrumentation that allowed to reveal the monoclinic distortion. The HTXRD data measured in this work can be used to monitor the evolution of the β angle, but the Rietveld analysis of the whole diagrams proved hazardous, as minor imperfections in the sample's surface orientation and the thermal expansion of the holder (despite corrected) do not allow to measure the peaks profile as precisely as with the ambient-temperature mounting. Besides, changes in the microstructure could occur on heating and modify the profile. Although no sintering occured during the experiment, a faint densification of the powder sample was observed afterwards. However, all these biases proved systematic for a given pattern, thus allowing to correct the measured FWHM of a ±hkl doublet by substracting the FWHM of a neighbor non-split peak. Fig. 4, obtained by applying this procedure to the sensitive ±202 doublet (2θ = 36.29 ° at RT), corrected by the 310 singlet (41.31 °), gives a clear evidence of the degeneracy of the doublet around 400 °C -a clue for a probable change to the orthorhombic symmetry. As the extinction rules remain unchanged, the space group of this new modification is probably the Cmcm supergroup once ascribed to the ambient form. This typical 2 nd order phase transition associated to an extremely faint displacive mechanism was not observed by DTA conducted in the frame of this work. Besides the modification of the peaks profile, the transition is too tenuous to be visible on the high-temperature XRD patterns (Fig. 5) or on the thermal expansion plots (Fig. 6) derived by Rietveld analysis. These plots are similar to those reported by Ballirano and Melis [START_REF] Ballirano | Thermal behaviour of beta-anhydrite CaSO4 to 1263 K[END_REF], who also noted that the strong expansion along [010] resulted from the stretching of the CaO8 polyhedron. To complete their statement, we can suggest that this phenomenon is due to Coulombic repulsions between Ca II and S VI , which share oxygen edges along the [010] chains.

Unlike the long-hidden pseudo-orthorhombic character of the ambient-temperature form, its pseudo-tetragonal aspect can be easily observed through the quasi-degeneracy of the hkl/lkh pairs. Actually, the [010] Ca-SO4-chains form a quasi-square-based array and both the SO4 and CaO8 polyhedra almost comply with a -4m2 point symmetry, but polyhedra of adjacent (010) layers are equivalent through the c-glide mirror in the (100) planes and through the C-translation in the (001) planes, thus ruling out any four-fold symmetry. Like those of the literature [START_REF] Evans | The thermal expansion of anhydrite to 1000 °C[END_REF][START_REF] Ballirano | Thermal behaviour of beta-anhydrite CaSO4 to 1263 K[END_REF], the present HTXRD measurements actually show an increase of the hkl/lkh splitting on heating due to differences in the mechanism of thermal expansion (Fig. 5). For these reasons, the existence of a tetragonal variety between the orthorhombic β and the cubic α ones is very improbable.

Conclusion

During the past decades, advances in XRD instrumentation have allowed to re-explore the symmetry of numerous inorganic compounds, such as γ-anhydrite, and now β-anhydrite. The present work was based upon an unconventional -and unexpected -implementation of the Williamson-Hall analysis, which appeared as a convenient and easy way to detect and even measure a lifting of degeneracy. Of course, synchrotron powder diffraction could be implemented in future works to observe this distortion in a more accurate and straightforward way. To this date, most of the crystallographic studies on βanhydrite were performed on single crystals obtained from natural samples, which can grow bigger than the sub-micron wide needles resulting from the calcination of plaster, like in the present case. Considering the progress achieved in the past decades in four-circle diffraction, re-visiting this approach with modern instruments could also be an excellent way to improve our knowledge of this phase.

The monoclinic, probably C2/c ambient structure of β-anhydrite is extremely similar to the previously assumed orthorhombic structure; it is therefore expected that most properties of anhydrite are not significantly affected. However, as slight as it is, this distortion is of importance for studies on the microstructures of β-anhydrite insofar as the lifting of degeneracy of the ±hkl pairs now allows to conduct fine characterizations based on diffraction peaks profiles. In addition, twin laws depend on the crystal symmetries and pseudo-symmetries. Indeed the main twinning direction of anhydrite is {101} [START_REF] Anthony | Handbook of Mineralogy[END_REF], reflecting the tetragonal pseudo-symmetry, but the lifting of the orthorhombic symmetry on cooling should also lead to {100} twins, investigated in a further study.

Alternative calculation of the β angle

Assuming that the excess in the strain measured for the h0h line (εh0h -εh00) results from the lifting of degeneracy between the dh0h and d-h0h spacings, one can write: $ In the monoclinic system (unique axis b) and assuming a ≈ c:

$ and $

Considering that cosβ ≈ 0 and assuming limited development

(1 + cosβ) -1/2 =

Structural data

Table S1. Comparison between the atomic positions refined in the Cmcm (upper) and C2/c (lower) groups.

Table S2. Selected bond lengths and angles calculated in the C2/c setting (s. u. multiplied by Bérar's factor = 2.5). da/a20 = 3.25 10 -9 T 2 + 9.63 10 -6 T -3.39 10 -4 db/b20 = 1.24 10 -8 T 2 + 2.19 10 -5 T -3.88 10 -4 dc/c20 = 7.17 10 -10 T 2 + 5.55 10 -6 T -3. 18 10 -4

Thermal expansion equations

(Fig. 2 .

 2 Fig. 2. Williamson-Hall analysis of β-anhydrite (orthorhombic indexing) showing the n th order reflections of the 200, 002, 111 and 202.

(Fig. 3 .

 3 Rietveld plot for the structure refinement in C2/c symmetry: observed (red dots), calculated (black line), difference (blue line) and Bragg positions (bars). Inset: high-angle x 20 vertical magnification of the 80-150 ° range.

!Fig. 4 .!Fig. 5 .Fig. 6 .

 456 Fig. 4. Thermal evolution of the broadening of the ±202 doublet (FWHM±202 -FWHM310) evidencing the change in symmetry.

Figure

  Figure S2. FWHM of the ±202 doublet (2θ = 36.29 °) in Samples 1, 2 & 3. The 022 -220 doublet (pseudo-tetragonal symmetry) at 2θ = 38.64 °, showned for comparison, is itself broader by about 0.01 ° than the singlet peaks.

Table 1

 1

	atom	symmetry	x	y	z	Beq (Ų)
	Ca	2	0	0.3478(4)	3/4	0.6(1)
	S	2	0	0.1560(6)	1/4	0.8(2)
	O1	1	0.1695(9)	0.0150(8)	0.252(2)	1.2(3)
	O2	1	0.003(3)	0.2979(9)	-0.5805(9)	0.9(3)

. Atomic positions and displacement parameters refined in space group C2/c, unit cell parameters a = 6.9924(3) Å, b = 6.2396(3) Å, c = 7.0001(3) Å and β = 90.092(9) °.

Acknowledgments

Financial support was received from the LabEx MATISSE and Saint-Gobain Research Paris. The authors also thank Saint-Gobain Research Paris for providing the alpha plaster sample, Doctor Pascal Loiseau of IRCP-Chimie ParisTech for his precious help in HTXRD experiments and the three anonymous reviewers whose relevant remarks helped improving the article.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A: Supplementary data

Detailed versions of the Williamson-Hall plots, an alternative calculation of the β angle, thermal expansion equations, interatomic distances and angles and the CIF file are available in the Supplementary data section.

Supplementary data 1. X-ray diffraction