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--- scenario_simulation [2022/01/03 11:17] – [Price depending crop yields and input coefficients] himics
+++ scenario_simulation [2022/01/04 10:27] – [Annex: Land use modelling] himics
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-====Annex: Land use modelling ====
+====Annex: Land supply and land transitions in the supply part of CAPRI====
+**Introduction**
+This technical paper explains how the most aggregate level of the CAPRI area allocation in the context of the supply models has been re-specified in the TRUSTEE project((https:%%//%%www.trustee-project.eu/
+)) and subsequently adopted in the CAPRI trunk. The former specification for land supply and transformation functions focused on agricultural land use and the transformation of agricultural land between arable land and grass land((See https://svn1.agp.uni-bonn.de/svn/capri/trunk/doc/landSupplyCAPRI_v5.pdf (Torbjörn Jansson, Wolfgang Britz, Alan Renwick and Peter Verburg (2010) Modelling CAP reform and land abandonment in the European Union with a focus on Germany.)
+)).
+During the subsequent period, CAPRI was increasingly adapted to analyses of greenhouse gas (GHG) emission studies. Examples include CAPRI-ECC, GGELS, ECAMPA-X, AgCLim50-X, (European Commission, Joint Research Centre), ClipByFood (Swedish Energy Board), SUPREMA (H2020). This vein of research is very likely to gain in importance in the future.
+In order to improve land related climate gas modelling within CAPRI, it was deemed appropriate to (1) extend the land use modelled to //all// available land in the EU (i.e. not only agriculture), and (2) to explicitly model //transitions// between land use classes. The pioneering work was carried out within the TRUSTEE project((https:%%//%%www.trustee-project.eu/
+)), but as always, an operational version emerged only after integrating efforts by researchers in several projects working at various institutions.
+This paper focusses on the theory applied while data and technical implementation are only briefly covered.
+**A simple theory of land supply**
+Recall the dual methodological changes attempted in this paper:
+  - Extend land use modelling to the entire land area, and
+  - Explicitly model transitions between each pair of land uses
+In order to keep things as simple as possible, we opted for a theory where the decision of how much land to allocate to each use is independent of the explicit transitions between classes. This separation of decisions is simplifying the theoretical derivations, but also seem to have some support in theory: land use transitions show a good deal of stability over time. We would like to remind sceptics of this assumption that the converse is not implied: land transitions are certainly strongly depending on the land use requirements.
+The land supply and transformation model developed here is a bilevel optimization model. At the higher level (sometimes termed the //outer problem//), the land owner decides how much land to allocate to each aggregate land use based on the rents earned in each use and a set of parameters capturing the costs required in order to ensure that the land is available to the intended use. At the lower level (sometimes termed the //inner problem//), the transitions between land classes are modelled, with the condition that the total land needs of the outer problem are satisfied. The inner problem is modelled as a stochastic process involving no explicit economic model.
+For the outer problem, i.e. the land owner’s problem, we propose a quadratic objective function that maximizes the sum of land rents minus a dual cost function. The parameters of the dual cost function were specified in two steps:
+  - A matrix of land supply elasticities was estimated (by TRUSTEE partner Jean Saveur Ay, CESEAR, Dijon (JSA). This estimation might be updated in future work or replaced with other sources for elasticities.
+  - The parameters of the dual cost function are specified so that the supply behaviour replicates the estimated elasticities as closely as possible while exactly replicating observed/estimated land use and land rents.
+The model is somewhat complicated by the fact that land use classes in CAPRI are defined somewhat differently compared to the UNFCCC accounting and also in the land transition data set. Therefore, some of the land classes used in the land transitions are different from the ones used in the land supply model. In particular, “Other land”, “Wetlands” and “Pasture” are differently defined. To reconcile the differences, we assumed constant shares of the intersections of the different sets, as explained below.
+**Inner model – land transitions**
+A vector of supply of land of various types could result from a wide range of different transitions. The inner model determines the matrix of land transitions that is “most likely”. The concept of “most likely” is formalized by assuming a joint density function for the land transitions, based on the historically observed transitions. The model then is to find the transition matrix that maximizes the joint density function.
+Since each transition is non-negative, but in principle unlimited upwards, we opted for a gamma density function, that has the support $\lbrack 0,\infty\rbrack$. For those that cannot immediately recall what the gamma density function looks like, and as entertainment for those that can, Figure 1 shows the graph of the density function for different parameters, all derived from an assumed mode of “1” and different assumed ratios “mode/standard deviations” (that we called “acc” for “accuracy” in the figure).
+[CHART]
+Figure 1: Gamma density graph for mode=1 and various standard deviations. “acc”="mode/standard deviation".
+Let $i$ denote land use classes in CAPRI definition, whereas //l// and //k// are land uses in UNFCCC classification. Let $\text{LU}_{k}$ be total land use after transitions and $\text{LU}_{l}^{\text{initial}}$ be land use before transitions. Furthermore, let $T_{\text{lk}}$ denote the transition of land from use $l$ to use $k$. Noting that it is simpler and fully equivalent to maximize a sum of logged densities than a product of densities, the likelihood maximization problem can be written (with //f// being the gamma density function)
+$${\max_{T_{\text{lk}}}{\log{\prod_{\text{lk}}^{}{f\left( T_{\text{lk}}|\alpha_{\text{lk}},\beta_{\text{lk}} \right)}}}}{= \max_{T_{\text{lk}}}{\sum_{\text{lk}}^{}{\log{f\left( T_{\text{lk}}|\alpha_{\text{lk}},\beta_{\text{lk}} \right)}}}}$$
+$$\Rightarrow \max_{T_{\text{lk}}}\sum_{\text{lk}}^{}\left\lbrack \left( \alpha_{\text{lk}} - 1 \right)\log T_{\text{lk}} - \beta_{\text{lk}}T_{\text{lk}} \right\rbrack$$
+subject to
+$$\text{LU}_{k} - \sum_{l}^{}T_{\text{lk}} = 0\text{~~~~~~~~~}\left\lbrack \tau_{k} \right\rbrack$$
+$$\text{LU}_{l}^{\text{initial}} - \sum_{k}^{}T_{\text{lk}} = 0\text{~~~~~~~~~}\left\lbrack \tau_{l}^{\text{initial}} \right\rbrack$$
+$$\text{LU}_{k} - \sum_{i}^{}{\text{shar}e_{\text{ki}}\text{LEV}L_{i}} = 0$$
+The last equation is needed to convert land use in UNFCCC classification to land use in CAPRI classification, using a fixed linear transformation matrix $\text{shar}e_{\text{ki}}$. This discrepancy between land class accounts will be expanded on in a subsequent section. Forming the Lagrangian function and taking the derivatives with respect to land transitions gives the following first-order optimality conditions:
+$$\ \left( \alpha_{\text{lk}} - 1 \right)T_{\text{lk}}^{- 1} - \beta_{\text{lk}} + \tau_{k}^{} + \tau_{l}^{\text{initial}} = 0$$
+The parameters $\alpha$ and $\beta$ of the gamma density function were computed by assuming that (i) the observed transitions are the mode of the density, and (ii) the standard deviation equals the mode. Then the parameters are obtained by solving the following quadratic system:
+$$\text{mode} = \frac{\alpha - 1}{\beta}$$
+$$\text{variance} = \frac{\alpha}{\beta^{2}}$$
+**Land use transitions as implemented in CAPRI**
+The implementation in CAPRI differs from the above general framework in that it explicitly identifies the //annual// transitions in year t $T_{\text{lk}}^{t}$ from the initial $\text{LU}_{l}^{\text{initial}}$ land use to the final land use $\text{LU}_{k}$. This is necessary to identify the annual carbon effects occurring only in the final year in order to add them to the current GHG emissions, say from mineral fertiliser application in the final simulation year. If the initial year is the base year = 2008 and projection is for 2030, then the carbon effects related to the change from the 2008 $\text{LU}_{l}^{\text{initial}}$ to the final land use $\text{LU}_{k}$ (=$T_{\text{lk}}$in the above notation, without time index) refer to a period of 22 years that cannot reasonably be aggregated with the “running” non-CO2 effects from the final year 2030. Furthermore the historical time series used to determine the mode of the gamma density for the transitions also refer to annual transitions.
+Initially the problem to link total to annual transitions has been solved by assuming a linear time path from the initial to the final period, but this was criticised as being an inconsistent time path (by FW). Ultimately the time path has been computed therefore in the supply model in line with a static Markov chain with constant probabilities $P_{\text{lk}}$ such that both land use $\text{LU}_{l}^{t}$ as well as transitions $T_{\text{lk}}^{t}$ in absolute ha require a time index (e_luOverTime in supply_model.gms).
+$$\text{LU}_{k}^{t} - \sum_{l}^{}{P_{\text{lk}}\text{LU}_{l}^{t - 1}} = 0\ ,\ t = \{ 1,\ldots s\}$$
+Where $\text{LU}_{k}^{s}$ is the final land use in the simulation year s and $\text{LU}_{k}^{0} = \text{LU}_{k}^{\text{iniital}}$ is the initial land use. The transitions in ha in any year may be recovered from previous years land use and the annual (and constant) transition probabilities (e_LUCfromMatrix in supply_model.gms).
+$$T_{\text{lk}}^{t} = P_{\text{lk}}*\text{LU}_{l}^{t - 1}$$
+The absolute transitions may enter the carbon accounting (ignored here) and if we substitute the last period’s transitions we are back to the condition for consistent land balancing in the final period from above:
+$$\text{LU}_{k}^{s} = \sum_{l}^{}{P_{\text{lk}}\text{LU}_{l}^{s - 1}} = \sum_{l}^{}T_{\text{lk}}^{s}$$
+When using the transition probabilities in the consistency condition for initial land use we obtain
+$$\text{LU}_{l}^{\text{initial}} - \sum_{k}^{}T_{\text{lk}}^{1} = 0\text{~~~~~~~~}$$
+$$\Longleftrightarrow \text{LU}_{l}^{\text{initial}} = \sum_{k}^{}{P_{\text{lk}}^{}\text{LU}}_{l}^{\text{iniital}}\text{~~~~~~~~}$$
+$$\Leftrightarrow 1 = \sum_{k}^{}P_{\text{lk}}$$
+So the simple condition is that probabilities have to add up to one (e_addUpTransMatrix in supply_model.gms). In this form the model is currently implemented in CAPRI.
+**Outer model – land supply**
+The outer problem is defined as a maximization of the sum of land rents minus a quadratic cost term, subject to the first order optimality conditions of the inner problem:
+$$\max{\sum_{i}^{}{\text{LEV}L_{i}r_{i}} - \sum_{i}^{}{\text{LEV}L_{i}c_{i}} - \frac{1}{2}\sum_{\text{ij}}^{}{\text{LEV}L_{i}D_{\text{ij}}\text{LEV}L_{j}}}$$
+subject to,
+$$\text{LU}_{k} - \sum_{i}^{}{\text{shar}e_{\text{ki}}\text{LEV}L_{i}} = 0$$
+$$\text{LU}_{k} - \sum_{l}^{}T_{\text{lk}} = 0\text{~~~~~~~~~}\left\lbrack \tau_{k} \right\rbrack$$
+$$\text{LU}_{l}^{\text{initial}} - \sum_{k}^{}T_{\text{lk}} = 0\text{~~~~~~~~~}\left\lbrack \tau_{l}^{\text{initial}} \right\rbrack$$
+$$\ \left( \alpha_{\text{lk}} - 1 \right)T_{\text{lk}}^{- 1} - \beta_{\text{lk}} + \tau_{k}^{} + \tau_{l}^{\text{initial}} = 0$$
+The parameters of the inner model **α** and **β// //**may be determined as explained in the previous sections. For the outer model, we need to define the parameters **c** and **D**. We have a single data point of land use and land rent for each land use class. Since we have, for $N$ land classes, $N + N(N - 1)/2$ parameters, but only $N$ price-quantity pairs (one data point for each land class). This means that without any additional information, we could e.g. calibrate the model exactly by computing the **c** parameter, but have no information left for defining **D**//.// However, we have at our disposal prior estimates of the regional matrices of land supply elasticities that may be used to define prior densities for the elasticity matrix implied jointly by the **c** and **D** parameters and the inner problem. Another way of expressing this is that we compute a meta parameter matrix $\mathbf{\eta}\left( \mathbf{c},\mathbf{D},\mathbf{L}\mathbf{U}^{\text{initial}} \right)$ that is a function of the real parameters, and use the prior elasticity matrix as a prior for this meta parameter. If cast in this way, the problem becomes a Bayesian econometric estimation.
+There are a few methodological and numerical challenges to overcome. In particular, we need to (i) analytically derive $\mathbf{\eta}\left( \mathbf{c},\mathbf{D},\mathbf{L}\mathbf{U}^{\text{initial}} \right)$, and (ii) ensure that the resulting model has the appropriate curvature to ensure a unique interior solution – anything else would result in a rather useless model. We start by simplifying the problem by observing that all the constraints (the first order conditions of the inner problem) can be replaced with an ordinary land constraint:
+$$\sum_{i}^{}{\text{LEV}L_{i}} - \sum_{l}^{}{LU_{l}^{\text{initial}}} = 0$$
+Note that the second sum is a constant. This simplification is based on the observation that the land transitions don’t appear in the objective function of the outer problem, so that all solutions to the inner problems are equivalent from the perspective of the outer problem, and that any land use vector that preserves the initial land endowment is a feasible solution to the inner problem.
+Next, we formulate the first order condition (FOC) of the modified outer problem to obtain land use as an implicit function of the parameters, $F\left( LEVL,c,D,LU^{\text{initial}},r \right) = 0$. We can then use the implicit function theorem to compute the derivative of land supply $\text{LEV}L_{i}$ with respect to land rent $r_{j}$, which in turn can be used to define the elasticity matrix $\mathbf{\eta}$.
+The first order conditions, and the implicit function, become
+$$F\left( LEVL,\lambda,c,D,LU^{\text{initial}},r \right) = \begin{bmatrix}
+\frac{\partial\mathcal{L}}{\partial LEVL_{i}} = & r_{i} - c_{i} - \sum_{j}^{}{D_{\text{ij}}\text{LEV}L_{j}} - \lambda & = 0 \\
+\frac{\partial\mathcal{L}}{\partial\lambda} = & \sum_{i}^{}{\text{LEV}L_{i}} - \sum_{l}^{}{LU_{l}^{\text{initial}}} & = 0 \\
+\end{bmatrix}$$
+In order to apply the implicit function theorem((Recall that the implicit function theorem states that if F(x,p) = 0, then dx/dp = -[dF/dx]<sup>-1</sup>[dF/dp]
+)) we need to differentiate the FOC once w.r.t. the variables $\text{LEV}L_{i}$ and $\lambda$ and once with respect to the parameter of interest, $r_{j}$, invert the former and take the negative of the matrix product. If (currently) irrelevant parameter are omitted, the following matrix of $(N + 1) \times (N + 1)$ is obtained (the “+1” is the uninteresting derivative of total land rent $\lambda$ with respect to individual land class rent $r_{i}$)
+$$\left\lbrack \frac{\partial LEVL}{\partial r} \right\rbrack = - \left\lbrack D_{LEVL,\lambda}F(LEVL,\lambda,r) \right\rbrack^{- 1}D_{r}F(LEVL,\lambda,r)$$
+$$\begin{bmatrix}
+\frac{\partial LEVL}{\partial r} \\
+\frac{\partial\lambda}{\partial r} \\
+\end{bmatrix} = - \begin{bmatrix}
+\frac{\partial F}{\partial LEVL} & \frac{\partial F}{\partial\lambda} \\
+\end{bmatrix}\left\lbrack \frac{\partial F}{\partial r} \right\rbrack$$
+Carrying out the differentiation specifically for land rent //r<sub>j</sub>//, we obtain:
+$$\begin{bmatrix}
+\frac{\partial LEVL_{i}}{\partial r_{j}} \\
+\frac{\partial\lambda}{\partial r_{j}} \\
+\end{bmatrix} = - \begin{bmatrix}
+\left\lbrack {- D}_{\text{ij}} \right\rbrack & - 1 \\
+ - 1' & 0 \\
+\end{bmatrix}^{- 1}\begin{bmatrix}
+I \\
+\\
+\end{bmatrix}$$
+Discarding the last row of the resulting $(N + 1) \times N$ matrix finally lets us compute the elasticity as
+$$\left\lbrack \eta_{\text{ij}} \right\rbrack = \left\lbrack \frac{\partial LEVL_{i}}{\partial r_{j}} \right\rbrack\left\lbrack \frac{r_{j}}{\text{LEV}L_{i}} \right\rbrack$$
+In the estimation, we assumed that the prior elasticity matrix is the mode of a density where each entry were independently distributed. Furthermore, the off-diagonal or any diagonal elements with negative priors were normally distributed, whereas the diagonal elements with positive priors (as required for a well-behaved curvature) were gamma distributed. For the standard deviation of elasticities we used either information from the prior estimates or some fall-back assumptions on standard deviations relative to the mode of elasticities. Denoting the prior elasticities with $e_{\text{ij}}$, we solved the following optimization problem, where parameters $\alpha$ and $\beta$ were already estimates as explained in the sections on the inner problem.
+$$\max_{\eta,c,D}{\sum_{ij \in normal(i,j)}^{}{- {\frac{1}{s_{\text{ij}}^{2}}\left( \eta_{\text{ij}} - e_{\text{ij}}^{\text{jsa}} \right)}^{2}} + \sum_{ij \in gamma(i,j)}^{}\left\lbrack \left( \alpha_{\text{ij}} - 1 \right)\log\eta_{\text{ij}} - \beta_{\text{ij}}\eta_{\text{ij}} \right\rbrack}$$
+subject to
+$$\left\lbrack \frac{\partial LEVL_{i}}{\partial r_{j}} \right\rbrack = - \begin{bmatrix}
+\left\lbrack {- D}_{\text{ij}} \right\rbrack & - 1 \\
+ - 1' & 0 \\
+\end{bmatrix}^{- 1}\begin{bmatrix}
+I \\
+\\
+\end{bmatrix}$$
+$$\left\lbrack \eta_{\text{ij}} \right\rbrack = \left\lbrack \frac{\partial LEVL_{i}}{\partial r_{j}} \right\rbrack\left\lbrack \frac{r_{j}}{\text{LEV}L_{i}} \right\rbrack$$
+$$\begin{matrix}
+ & r_{i} - c_{i} - \sum_{j}^{}{D_{\text{ij}}\text{LEV}L_{j}} - \lambda & = 0 \\
+ & \sum_{i}^{}{\text{LEV}L_{i}} - \sum_{l}^{}{LU_{l}^{\text{initial}}} & = 0 \\
+\end{matrix}$$
+and the curvature constraint using a stricter variant of the Cholesky factorization
+$$D_{\text{ij}}\left( 1 - \delta I_{\text{ij}} \right) = \sum_{k}^{}{U_{\text{ki}}U_{\text{kj}}}$$
+where $\delta$ is a small positive number and $I_{\text{ij}}$ entries of the identity matrix such that the factor $(1 - \delta I_{\text{ij}})$ shrinks the diagonal of the D-matrix, ensuring //strict// positive definiteness instead of //semi-//definiteness. We used $\delta = 0.05$. Furthermore, the Lagrange multiplier of the total land constraint, $\lambda$, was fixed at the weighted average of the rents $r_{i}$, i.e. $\lambda = \frac{\sum_{i}^{}{\text{LEV}L_{i}r_{i}}}{\sum_{i}^{}{\text{LEV}L_{i}}}$. Without the latter assumption, the parameters c and D are not uniquely identified.
+**Prior elasticities and area mappings**
+The empirical evidence obtained in the TRUSTEE project applied to prior elasticities for land categories based on Corine Land Cover (CLC) data. These categories are also covered in the CAPRI database based on various sources (see the database section in the CAPRI documentation):
+The introduction has mentioned already three systems of area categories that need to be distinguished. The first one is the set of area aggregates with good coverage in statistics that has been investigated recently by JS Ay (2016), in the following “JSA”:
+$$\text{LEVL} = \left\{ \text{ARAC},\ \text{FRUN},\ \text{GRAS},\ \text{FORE},\ \text{ARTIF},\text{OLND} \right\}$$
+Where
+ARAC = arable crops
+FRUN = perennial crops
+GRAS = permanent grassland
+FORE = forest
+ARTIF = artificial surfaces (settlements, traffic or industrial)
+OLND = other land
+The above categories are matching reasonably well with the definitions in JSA. A mismatch exists in the classification of paddy (part of ARAC in CAPRI but in the perennial group in JSA) and terrestrial wetlands (part of OLND in CAPRI and a separate category in JSA). Inland waters are considered exogenous in CAPRI and hence not included in the above set LEVL.
+For carbon accounting we need to identify the six LU classes from IPCC recommendations and official UNFCCC reporting:
+$$LU = \left\{ \text{CROP},\ \text{GRS}\text{LND},\ \text{FORE},\ \text{ARTIF},WETLND,RESLND \right\}$$
+which is typically indexed below with “l” or “k” ∈ LU and where
+CROP = crop land (= sum of arable crops and perennial crops)
+GRSLND = grassland in IPCC definition (includes some shrub land and other “nature land”, hence GRSLND>GRAS)
+WETLND = wetland (includes inland waters but also terrestrial wetlands)
+RESLND = residual land is that part of OLND not allocated to grassland or wetland, hence RESLND<OLND
+FORE = forest
+ARTIF = artificial surfaces
+In the CAPRI database, in particular for its technical base year, we have estimated an allocation of other land OLND into its components attributable to the UNFCCC classes GRSLND,WETLND, and RESLND:
+$$\text{OLND}^{0} = {\text{OLND}G}^{0} + {\text{OLND}W}^{0} + {\text{OLND}R}^{0}$$
+Lacking better options to make the link between sets LEVL (activity level aggregates) and LU (UNFCCC classes, technically in CAPRI code: set “LUclass”) we will assume that these shares are fixed and may estimate the “mixed” LU areas from activity level aggregates as follows
+^//GRSLND//^=^//GRAS + OLND · OLNDG<sup>0</sup>/OLND<sup>0</sup>//^
+|WETLND    |=|//INLW + OLND · OLNDW<sup>0</sup>/OLND<sup>0</sup>//|
+|RESLND    |=|//OLND · OLNDR<sup>0</sup>/OLND<sup>0</sup>//       |
+which means that the mapping from set LEVL to set LU only uses some fixed shares of LEVL areas that are mapped to a certain LU:
+$$LU_k=\sum_i{\text{share}_{\text{i,k}}\text{LEVL}_i}$$
+where 0 ≤ //share<sub>i,k</sub>// ≤ 1.
+**Technical implementation**
+The key equations corresponding to the approach explained above are collected in file supply_model.gms or the included files supply\declare_calibration_models_for_luc.gms and supply/declare_calibration_models_for_land_supply.gms. The declarations of parameters, variables, equations, models and even some sets only used in the calibration given in these files are included by the “supply_model.gms” only if “BASELINE==ON” or if it was a CAPREG base year task that was carried out. Loading of priors, initialisation of parameters and variables for the calibration as well as the organisation of solve attempts are handled in new sections of file “cal_land_nests.gms”, in turn called by the gams file “prep_cal.gms”. This implies that the land supply and land use change calibrations were inserted before the ordinary calibration of the supply models.
+The new land supply specification is only activated if the global variable %trustee_land%==on which may be set via the CAPRI GUI. In order to store the results of the calibration in a compact way that is compatible with the existing code, the existing parameter files “pmppar_XX.gdx” was used. The parameters of the land supply functions, called “c” and “D” above, were stored on two parameters “p_pmpCnstLandTypes” and “p_pmpQuadLandTypes”. As a new symbol (p_pmpCnstLandTypes) is introduced in an existing file, the first run of CAPRI after setting %trustee_land%==on may give errors if the file exists already but has been used with the previous land supply specification before. In this case it helps to delete or rename the old pmppar files.
+At this point, it should also be explained that rents for non-agricultural land types were entirely based on assumptions (a certain ratio to agricultural rents). As there were no plans to run scenarios with modified non-agricultural rents, these land rents //r// used in calibration for those land types were subtracted from the “c-paramter”, so that it is implicitly stored in p_pmpCnstLandTypes and enters the objective function through the PMP terms. This requires changes if the rents shall be modified or if non-agricultural production shall be included in some simplified form.
+Furthermore, the class Inland Waters (INLW) was given a special treatment: it is supposed to be entirely exogenous. For this purpose the special acronym “exogenousLandSupply” was introduced, and stored on the p_pmpCnstLandTypes and used to trigger an equation “e_exogenousLand “ in the supply model setting the variable to a constant. In that way, the fixity of INLW (or any land type, should it happen) is stored in the pmp terms and cannot be “forgotten”.
+More detailed explanations on the technical implementation are covered elsewhere, for example in the “Training material” included in the EcAMPA-4 deliverable D5.
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