dataesr/scientific-paragraphs-categorization huggingface dataset:
Conclusions The purpose of this paper was to investigate the behaviour of the multivariate DPM-G model when affine transformations are applied to the data. To this end we focused on the DPM-G model with independent normal and inverse-Wishart specification for the base measure. Our investigation covered both the finite sample size and the asymptotic framework. Specifically, in Proposition 1, given any affine transformation g, an explicit model specification, depending on g, was derived so to ensure coherence between posterior inferences carried out based on a dataset or its transformation via g. We then considered a different setting where the specification of the model is assumed independent of the specific transformation g. In this case, we formalised the notion of asymptotic robustness of a model under transformations of the data and identified mild conditions on the true data generating distributions which are sufficient to ensure that the DPM-G model features such a property.
Specifically, Theorem 1 shows that the posterior distributions obtained conditionally on a dataset or any affine transformation of it, become more and more similar as the sample size grows to infinity. Inference on densities and, as suggested by the simulation study, on the clustering structure underlying the data, thus becomes increasingly less dependent on the affine transformation applied to the data, as the sample size grows. As a special case, Theorem 1 implies that posterior inference based DPM-G models is asymptotically robust to data transformations commonly adopted for the sake of numerical efficiency, such as standardisation or normalisation. This observation is particularly relevant when dealing with the astronomical unsupervised clustering problem motivating this work. Due to the lack of prior information on the dimensional constants relating different physical units, we resorted to a standardisation of each component of the data and chose an arbitrary model specification.
Prior information was available in the form of the experts' prior opinion on the expected number of groups in the dataset and was used to elicit the hyperprior distribution for α, the precision parameter of the DP. × exp - 1 2 (Cx + b -m) S -1 (Cx + b -m) P (dm, dS; π g ) = | det(C)| fπg (g(x)). A simple reparametrisation leads to fπg = | det(C)| -1 fπ • g -1. All the identities in this proof are deterministic, that is they hold for every ω ∈ Ω.
order to test the robustness of the model under affine transformations of the data, we compressed or stretched the generated datasets by using five different constants, namely c = 1/5, c = 1/2, c = 1, c = 2 and c = 5. For each constant, we multiplied the simulated data by c, thus obtaining a transformed dataset X (n) c := cX(n). We then fitted a DPM-G model, specified as in (1) and (2), to each one of the 5 × 3 × 100 = 1 500 resulting datasets. In order to enhance the flexibility of the model, we completed its specification by setting a normal/inverse-Wishart prior distribution for the hyperparameters (m 0 , B 0 ) of the base measure (2). Namely, we set B 0 ∼ IW (4, diag(15)) and m 0 | B 0 ∼ N (0, B 0 ), specification chosen so that E[µ] = 0 and to guarantee a prior guess on the location component µ flat enough to cover the support of the non-transformed data.
Acknowledgements This work was developed in the framework of the  Ulysses Program for French-Irish collaborations  ( 43135ZK ) and the  Grenoble Alpes Data Institute. The authors wish to thank  Carlo Nipoti  for suggesting the motivating astronomical problem, and  Stéphane Girard  for helpful discussions on the set of assumptions of Theorem 1. The authors are also grateful to  Bob Franke  for the picture in Figure 1.
This enables MRIO analysis to trace increasingly fragmented international supply chains across primary, secondary and tertiary producers, to give a more complete picture of the impacts of final consumption of nations, in comparison to biophysical accounting methods purely based on physical data  (Bruckner et al. 2015). A drawback of MRIO analysis in environmental impact studies is the lacking resolution to trace specific products and/or materials  (Schaffartzik et al. 2015)  or differentiate production technologies in detail. In addition, the efforts to harmonize sectoral and regional data and satellite accounts may require additional aggregation that can compromise the accuracy of environmental and socioeconomic results  (Steen-Olsen et al. 2014; Lenzen 2011). Today several global MRIO databases exist, such as Eora  (Lenzen et al.
2013) , WIOD  (Timmer et al. 2015) , GTAP-MRIO  (Aguiar et al. 2016) , the OECD-ICIO  (Yamano and Webb 2018) , and EXIOBASE  (Tukker et al. 2013). Ideally, a global MRIO is as detailed as possible on both the product/industry resolution as well as on the number of explicitly represented countries. In addition, the ideal MRIO should be available as a consistent long and up-to-date time series and provide detailed socioeconomic and environmental extensions  (Tukker and Dietzenbacher 2013). In order to have a consistent database between different world regions, MRIO developers necessarily need to deal with aggregations of extensions, regions and sectors into a standardized classification system  (Lenzen 2011). Due to lack of easily available data for many countries, the approach sometimes used to reach global coverage is by estimating "rest-of-the world regions" (RoW), which typically consist of the remaining countries that are not explicitly covered in the database.
(2010)  find that around 40 sectors are sufficient to avoid large uncertainties in CO 2 emissions embodied in exports. Comparing the impacts embodied in exports by disaggregating the SUTs of EXIOBASE at a detail of 59 sectors versus 129 sectors,  Wood et al. (2014)  found differences in the order of maximum 5% for labor and compensation of employees, while CO 2 impacts differed up to 50%. Steen-Olsen et al. (2014)  further investigated the effect of sector aggregation on CO 2 multipliers (kg CO 2 /$) in different MRIO databases. Similar to  Wood et al. (2014) , they found that aggregating sectors of different MRIOs to 17 sectors significantly changed the CO 2 multipliers, and that the multiplier errors increased with increased sectoral detail in the original database. Similarity in economic input structures among sectors did not imply similarity in terms of emission profiles.
(2014)  showed that the RoW regions' share of global land use (33-44% of the global total) are much larger than the equivalent share of global warming potential (17-22%). Furthermore,  Stadler et al. (2014)  found that 38% of global land exports originate in the RoW regions, underlining the need for a higher country resolution to reduce uncertainties in estimating land use embodied in trade. In terms of available MRIO databases, EXIOBASE has the highest consistent sector resolution of the available MRIO databases, but is limited in regional resolution. Eora has high country coverage and higher sector detail for some counties, but as the level of detail varies from region to region, this complicates the between-region comparison of impacts on a sectoral level. For example, Eora has only one sector aggregating all agricultural, forestry and fishing activities for most countries in the world.
The GTAP-MRIO probably has the best compromise of sectoral resolution (57 sectors) and country (140 regions), but is currently not available as a time series, and has limited sectoral resolution outside the agricultural and food sectors. Ideally, there would be a MRIO database with high sector resolution, individual country coverage and a full time-series. The aim of this paper is to describe the steps towards such an improved MRIO, by increasing the country resolution of EXIOBASE 3 to explicitly including all domestic economies registered in the UN main aggregates database (214 countries, see below). We use this extended EXIOBASE (named EXIOBASE 3rx) to show the relevance of additional regional disaggregation to estimate land use embodied in trade. We study the degree of regional aggregation errors on both a regional and on a harmonized and detailed product level.
In the following method section, we describe the development of EXIOBASE 3rx and present its methodological building blocks, describe the processing of land use extensions, and the method for comparing the two databases with different regional resolution. In the result section, we present land footprints and explore the degree of regional aggregation errors for land use embodied in trade. To isolate the effect of regional aggregation on land use, we compare an EXIOBASE version where the MRIO structure is pre-aggregated (aggregation of IO data before calculation of coefficients and results), referred to from now on as the aggregated database, with EXIOBASE 3rx, where the land use results of the full detailed database are aggregated to 49 regions. The implication of this work is further picked up in the next section, where we discuss our results for both MRIO development and the use of MRIO for land use studies now and in the future.
Building EXIOBASE 3rx The approach to building the monetary supply-use tables for EXIOBASE 3rx (Fig. 1 ) closely follows previous approaches establishing EXIOBASE 3 and EXIOBASE 2  (Wood et al. 2015 , Stadler et al. 2018). Deviations from the EXIOBASE 3 workflow can be found in Additional file 1: S1. In EXIOBASE 3, the economic structures of 44 regions are available in the form of (aggregate) supply-use tables (SUTs). These SUTs are both disaggregated and balanced to product, industry, and trade data. From the SUTs, a trade-linking procedure  (Wood et al. 2015)  and application of an IO construct  (Majeau-Bettez et al. 2014 ) is applied to obtain square MRIO tables. In order to estimate the SUTs for the RoW regions in EXIOBASE2 and 3, global average coefficient data was reconciled with product output, industry and trade data (see  Stadler et al.
(2014)  for more information). EXIOBASE 3 adds top-level constraints of macroeconomic data to ensure consistency between regions and over time at a highly aggregate level. EXIOBASE 3 had a strong European focus (28 EU member states, 16 major economies) and 5 RoW regions (RoW Asia and Pacific, RoW Europe, RoW Africa, RoW America, RoW Middle East). In this work, we extend the procedure used in estimating RoW regions in EXIOBASE 3, but apply it to individual countries in order to expand the number of regions from originally 49 to 214 (Additional file 2). As SUT data are not commonly available for the countries in the RoW regions, we follow the regional approach where we use proxy data in the form of generic estimates of coefficients of the supply (i. e. , market share relationships) and use matrices (intermediate use and final demand coefficients) to give an initial estimate of the product/industry transactions.
The coefficients are then reconciled to globally balanced estimates of trade data, estimates of product outputs for every country and macroeconomic data on value added, taxes, exports, imports, final consumption and gross capital formation (for an overview of regional data sources, see Additional file 1: S2). The macroeconomic data serve as the top-level data towards which all the other data are balanced. The number of countries is based on the available macroeconomic data from the UN National Main Aggregates Database (United Nations 2018a). Additionally, we estimate land use extensions for all 214 countries (more info in Additional file 1: S11).
Trade estimates and reconciliation In order to process the country-specific trade data, we combine data from three data sources when compiling the trade estimates. The BACI database is the main data source (balanced product trade data based on the UN Comtrade database, for more information see  Gaulier and Zignago (2010) ), while the UN services trade database (United Nations. 2018b) and the IEA database (International Energy Agency 2018) provide data for services and energy products/services, respectively. Re-exports are estimated in the same way as EXIOBASE 2 and 3 (based on SUT data for re-exports where available, and extrapolated based on Comtrade data). After compiling the initial estimate of the trade data, this is reconciled against the top-level macroeconomic trade data in current price obtained from the UN National Accounts Main Aggregates Database.
Here, we replace the quadratic programming approach with an information theoretical approach. We minimize cross-entropy (CE), also known as Kullback-Leibler Divergence  (Kullback and Leibler 1951) , between the final trade flows of product i from country r to country s , p rs i , and their initial estimate q rs i , subject to constraints requiring that total export and import values from the UN National Main Aggregates Database, EX r and IM s , are met. In addition to the constraint that total exports by country and product are less than gross output, x r max i. For the general methodology, see  Golan and Vogel (2000). As in  Többen and Schröder (2018) , we implement the computationally much more efficient unconstrained dual of the minimal cross-entropy problem. In the dual version, the cross-entropy model takes the form where r 1 and s 2 are Lagrangian multipliers referring to the equality constraints.
Estimating product output Product output estimates were processed in EXIOBASE 3  (Stadler et al. 2018 ) and combines data from several national account databases, FAOSTAT (2014), IEA energy balances (IEA 2015) and product output from EXIOBASE 2 (for more information see Additional file 1: S1 and S9 in  Stadler et al. (2018) ). The main difference is that for EXIOBASE 3rx we process the raw data on an individual country level also for all former RoW countries. In the next step, these data sources served to disaggregate the UN macroeconomic industry output data (United Nations. 2018a), which consists of gross value added from seven aggregated industries. By applying a concordance matrix between the seven UN industries and the 163 EXIOBASE industries (Additional file 1: S3) and by assigning a quality index to the different data sources based on their closeness to raw data, the routine disaggregates the UN industry data.
The disaggregation is based on the values in the chosen raw data source. The result is product output at the level of the 163 industries and 200 products of EXIOBASE. In general, this procedure should give reasonable estimates for agricultural, food and energy products, whilst missing detailed country-specific data on manufactured products and services. (1) max D = r r 1 EX r + s s 2 IM s + r i r max i -r min i x r max i - rs i q rs i exp r 1 + s 2 + r max i -r min i - r i r max i x r max i - r i r min i x r min i , (2) p rs i = q rs i exp r 1 + s 2 + r max i -r min i rs i q rs i exp r 1 + s 2 + r max i -r min i.
Balancing supply-use tables The monetary SUT balancing routine applies an algorithm similar to the approach in Stadler et al. (  2018 ) using a quadratic programming target function. One important difference here is that, due to lack of data on a detailed country level, taxes, trade and transport margins are not estimated as explicit layers in our approach. Hence, our system is an MRIO in basic pricing only. The results are monetary SUTs estimated for every country and year independently for a time series from 1995 to 2015 for 214 countries. The balancing routine was unable to find a solution for a few countries, about 3. 3% of all cases through the time series. See an overview in Additional file 1: S5 of the unbalanced countries.
Using this approach, we avoid the problem of negative coefficients that could be faced when applying, e. g. , the commodity technology construct  (Jansen and Raa 1990). The choice of producing trade-linked IO tables rather than fully compiled MRIO tables (as per EXIOBASE3) was due to the significantly lower loading and running time, and does not constitute a loss of data (we had no additional data to inform the trade relationships). Normal desktop computers are not able to handle the memory requirements of a fully complied MRIO system of the size of EXIOBASE 3rx, but can easily handle the tradelinked system. Because of the trade proportionality assumption over the import use estimates, if a full MRIO system is desired, either the approach of Peters et al. (  2011 ) could be followed if no memory constraints exist, or topological transformation of the data could be applied as explained in  Rodrigues et al.
Compiling the land use data To obtain land use data at the sectoral resolution of EXIOBASE, we followed a two-step procedure: First, we created spatially explicit maps for major land cover types based on publicly available state-of-the-art datasets. The data were harmonized following a closed-budget mapping approach  (Erb et al. 2007) , i. e. , the sum of all layers will add up to 100% or the available land area for each specific grid cell. In a second step, we utilized information from census statistics (FAOSTAT) to further disaggregate the data to closely match the EXIOBASE sector classification (in table format). See Additional file 1: S11 for a detailed description of establishing the land use dataset. The land use extensions comprise 207 countries, which cover most of the countries in EXIOBASE 3rx.
For the remaining seven countries, mainly Island states like Palau and Nauru, we use the land area variable from FAOSTAT (2019) to estimate the land use accounts of the missing countries. We first choose a country (country A) with existing land use data and geographical proximity to the country with missing data (country B). Next, the land use extensions of country B are estimated by scaling the data of country A based on the land area variable of country B relative to that of country A. Next, we remap the land use data into EXIOBASE 3rx format. Here, we follow the same procedure as in EXIOBASE 3, and therefore refer the reader to S6 of  Stadler et al. (2018). The resulting 40 land use extensions consist of land used by the EXIOBASE 3rx production sectors (F) and land directly allocated to households (F_hh).
Estimating land footprints Due to the large size of EXIOBASE 3rx (e. g. , the coefficient matrix (A) has 42,800 × 42,800 data points), most of the arrays are saved in a sparse format in MATLAB to reduce disk storage requirements. The sparse format database for one year is approximately 60 megabytes. We used the emissions embodied in bilateral trade (EEBT) approach  (Peters 2007 (Peters , 2008 ) to do land use calculations using EXIOBASE 3rx rather than calculating impacts from the MRIO system directly. The main difference is that we do not account for intermediate demand of imports that go to industries to produce exports. Hence, a limitation is that imports that are used for intermediate production, that later end up as exported goods are not accounted for. However, as we are studying aggregate land embodied in trade, and not that resulting from a particular final demand, the EEBT approach is suitable as discussed in  Peters (2007).
Analyzing the effect of regional aggregation To enable comparison of the pre-aggregated database and EXIOBASE 3rx for land use results, we aggregate the inter-industry flow matrix (Z), the final demand matrix (Y), the total land use of production (F), and land directly allocated to households (F_hh) to 49 regions using a regional bridging (Additional file 2). Next, we calculate the coefficient matrix (A) and the land use multipliers (S) per monetary unit. We refer to this as the aggregated database from now on. Note that we do not compare land use results of EXIOBASE 3rx and EXIOBASE 3 directly as it would be difficult to distinguish the effect of regional disaggregation to effects arising from other changes (see Additional file 1: S1 for an overview of the differences in workflows between the databases).
Two of the most prominent changes to the workflow are the mentioned updated trade processing and reconciliation, and re-processed and more detailed land use extensions. In addition, the land use dataset was newly established specifically for EXIOBASE 3rx. For comparing the land embodied in trade between the EXIOBASE 3rx and the aggregated database, we define the aggregation error as the sum of the absolute difference of the traded land in question: where T is a three-dimensional array of land embodied in imports or exports with dimensions imports/exports ( q ) by trade partner ( r ) by product ( p ). s corresponds to the summed-over dimension(s) and n is the number of data points in the summed-over dimension(s). n varies according to the type of aggregation error in question. We exam- ine aggregation errors of imports and exports of products, between regions, and specific product-region combinations.
Hence, for, e. g. , the product aggregation error of imports, we sum over q, r-exporting and importing countries. Similarly, for the aggregation error of exports of specific goods originating in specific countries, we sum over r-importing countries. Note that we exclude intra-RoW trade in EXIOBASE 3rx aggregated to 49 regions for the sake of comparison with the aggregated database, where intra-RoW trade is part of domestic demand. "Aggregation error" refers to the difference in results between those from one inputoutput table and those from a pure aggregation of the same input-output table prior to calculations (as per literature, e. g. ,  Gibbons et al. (1982) ). It must be noted that inputoutput tables are always estimates of actual transactions and the more disaggregated an input-output table is (especially in the case at hand where there is very poor statistical coverage of some countries) the higher the level of uncertainty of these transactions.
Most literature (e. g. ,  Lenzen (2011) ) point to the benefit of disaggregation for reducing the uncertainty of footprint calculations, but we do not explore that here. As such, it must be remembered that uncertainty related to disaggregation, and the concept of aggregation error are related, but different concepts. We expect, but cannot measure whether the accuracy of our results will increase by disaggregating EXIOBASE3, whilst we can measure the aggregation error between the disaggregated database and a pure of aggregation of the same database. Using Eq. 3 we define the aggregation error score ∈ s as the aggregation error divided by the export/imports of the region, product or product-region combination in the 49 region version of EXIOBASE 3rx:.
Trade comparisons The added regional detail changes the trade structure of EXIOBASE 3rx compared to the aggregated database and EXIOBASE 3. In EXIOBASE 3, intra-RoW trade flows are treated as "domestic" flows, while they are treated as inter-country trade flows in EXIOBASE 3rx. In 2015 intra-RoW trade (as classified in EXIOBASE 3) is the largest or second largest export destination of each continental region (Table  1 ). This has relevance to the regional disaggregation of EXIOBASE 3 for footprint analyses both for the countries within the RoW region and for the trade partners importing from the RoW region. In the former case a footprint resulting from a demand for an imported good from, e. g. , Thailand to the Philippines would be treated as domestic in EXIOBASE 3 with the land use (or emission) intensity equal to the RoW region, while in EXIOBASE 3rx the footprint is treated as imports using the land use intensity of Thailand, which can lead to highly differing results as discussed in the introduction.