domain: spi info math phys

text:
R Charayron
email: remy.charayron@onera.fr

T Lefebvre
email: thierry.lefebvre@onera.fr

N Bartoli

J Morlier
email: joseph.morlier@isae-supaero.fr

Pareto optimal fidelity level selection for multi-fidelity Bayesian optimization applied to drone design

Optimizing expensive models is a challenging task in the aeronautical design community. Multi-fidelity optimization proposes to use different levels of fidelity to alleviate the overall computational cost of the optimization. How to know which level of fidelity will be useful for the optimization process ? In this work, we propose a methodology to decide which levels of fidelity are useful. We construct a Pareto front for the fidelity levels in terms of cost and accuracy, and then select only the Pareto optimal fidelity levels. All the models considered in this work were considered as black-boxes, so a gradient-free optimizer is used to solve the optimization problems. We chose to use a multi-fidelity Bayesian algorithm in order to reduce the number of function evaluations. We showed the advantage of selecting the optimal Pareto fidelity levels in the multi-fidelity process by comparing the cost to achieve convergence using different scenarios for level selection. To this end, we applied our methodology to several analytical test cases and to a simple drone design problem for which we obtained an improvement of 38 percent improvement over the mono-fidelity algorithm and a 24 percent improvement over the multi-fidelity algorithm with random selection.

Introduction

Multi-fidelity Bayesian optimization proved to overtake mono-fidelity Bayesian optimization when solving problems where the involved models are very expensive [START_REF]Multi-fidelity constrained Bayesian optimization, application to drone design[END_REF][START_REF] Joaquim | Multi-fidelity efficient global optimization: Methodology and application to airfoil shape design[END_REF][START_REF]Non-Myopic Multifidelity Method for Multi-regime Constrained Aerodynamic Optimization[END_REF]. Indeed, multifidelity Bayesian optimizer can query low-fidelity models in order to explore the design space but also high expensive fidelity models when exploitation of the surrogate is needed. Sometimes a lot of fidelity levels are available to approach the quantities of interest. In this paper, we address the problem of choosing how many and which levels of fidelity to use in the multi-fidelity process. We propose an efficient approach to select the right combination of fidelity levels among all the possible combinations. To make this selection, we first evaluate each fidelity level according to two objectives which are their cost and accuracy, and then keep only the non-dominant fidelity levels. The paper is divided as follows. Section 2 gives a state of the art on the multi-fidelity Kriging and multi-fidelity Bayesian optimization. Next, Section 3 explains the proposed Pareto optimal fidelity level (POFL) methodology. Some analytical test cases and the associated results are presented in Section 4. The methodology is then experienced on a drone design test case in Section 5. Finally, some conclusions and perspectives are given in Section 6.

State of the art

This section presents the multi-fidelity framework that is used in this paper. We want to solve an optimization problem with inequality or/and equality constraints:

min x∈Ω f (x) such that g(x) ≤ 0 h(x) = 0 (1)
where the objective function f and the constraints g, h are evaluated with a Multidisciplinary Design Analysis (MDA) costly to evaluate. In a multi-fidelity context, the f , g and h can be obtained by different MDAs associated to different fidelity levels. To solve this multidisciplinary optimization (MDO) problem with expensive black-box functions, Bayesian optimizers [START_REF] Archetti | Bayesian optimization and data science[END_REF][START_REF] Frazier | A tutorial on Bayesian optimization[END_REF] are used to minimize the number of function evaluations. First Section 2.1 shows how to construct a multi-fidelity Gaussian process surrogate model. Then, in Section 2.2, the current Bayesian optimization approach is adapted to multi-fidelity by defining a criterion to choose the fidelity level to query. We assume here that the reader is familiar both with Gaussian processes (GP) interpolation also denoted Kriging models [START_REF]Gaussian processes for machine learning[END_REF][START_REF] Krige | A statistical approach to some basic mine valuation problems on the Witwatersrand[END_REF][START_REF] Matheron | Mémoires du Bureau de Recherche Géologiques et Minières[END_REF] and with the classical mono-fidelity Kriging based Bayesian optimization methods like the Efficient Global Optimization (EGO) [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF] algorithm for unconstrained problems and the Super Efficient Global Optimization (SEGO) algorithm [START_REF]Flexibility and efficiency enhancements for constrained global design optimization with Kriging approximations[END_REF][START_REF] Bartoli | An adaptive optimization strategy based on mixture of experts for wing aerodynamic design optimization[END_REF][START_REF] Bartoli | Improvement of efficient global optimization with application to aircraft wing design[END_REF] for constrained problems.

Multi-fidelity Kriging

The first extension of the Kriging model for multivariate functions, called co-Kriging, was first developed in geostatistics (see [START_REF]Geostatistics: modeling spatial uncertainty[END_REF][START_REF]Multivariate geostatistics: an introduction with applications[END_REF]). Next, [START_REF] Kennedy | Bayesian calibration of computer models[END_REF] figured that making assumptions to relate different levels of fidelity is a way to simplify multi-fidelity problems. In this section, focus will be made on Le Gratiet's [START_REF]Multi-fidelity Gaussian process regression for computer experiments[END_REF] recursive formulation that we used to construct multi-fidelity Kriging models. Let's suppose that L + 1 fidelity levels f 0 , ..., f L sorted from the lowest to the highest are available to approach the function f (the same philosophy could be applied to the constraint functions) and that we dispose of L + 1 design of experiments (DoE), one for each fidelity level, denoted (D i ) i=0...L with D i = ((x 0 , f i (x 0 )), ..., (x j , f i (x j )), ..., (x N i , f i (x N i ))) T , N i being the number of points in D i . We make the nested DoE assumption which states that each point evaluated at a fidelity level is also evaluated at all the lowest fidelity levels:

D x L ⊆ D x L-1 ⊆ . . . ⊆ D x
0 where D x i = (x 0 , ..., x j , ..., x N i ) T corresponds to the inputs part of D i . The nested DOE provides some properties useful to express the surrogate model variance in closed form. Let's make the following assumption first introduced by [START_REF] Kennedy | Bayesian calibration of computer models[END_REF] in the bi-fidelity case to link the fidelity levels

f l+1 (x) = ρ l f l (x) + δ l+1 (x) such that f l ⊥ δ l+1 ∀l = 0, ..., L -1 (2)
where δ l+1 (x) is a discrepancy function that captures the difference between the l + 1-th and the l-th fidelity levels while ρ l is a scaling factor which is applied to f l . Le Gratiet [START_REF]Multi-fidelity Gaussian process regression for computer experiments[END_REF] proposed to add the lowest fidelity function to the basis function set (h i ) i=1...p used in the universal Kriging regression term to get:

µ(x) = i=1,...,p β i h i (x) + β ρ 0 f 0 (x) (3)
where β ρ 0 is an estimation of ρ 0 and (β i ) i=1...p is a set of unknown coefficients to be multiplied by the basis functions. Estimations of these coefficients are done at the hyperparameter estimation step performed by maximizing the likelihood [START_REF]Cross validation and maximum likelihood estimations of hyper-parameters of Gaussian processes with model misspecification, Bachoc, François[END_REF][START_REF]Computing the maximum likelihood estimates: concentrated likelihood, EM-algorithm[END_REF]. Since we used a nested DoE structure, the independence between fidelity levels of the surrogate model is assumed. Then the following recursive formulation for the mean (μ l+1 ) and variance (σ 2 l+1 ) of each fidelity level GP surrogate model can be written:

∀l = 0, ..., L -1 μl+1 = ρ l μl + μδ l+1 σ2 l+1 = ρ 2 l σ2 l + σ2 δ l+1 (4) 2
In this case, ρ l is considered as a constant but it can depend on x and we have ρ l : x → ρ l (x). The learning process is the following, first the lowest fidelity level is learnt, then the relationships (scaling factor ρ l and discrepancy function δ l+1 with l = 0, ..., L -1) between every successive fidelity level are consecutively learnt. Since the variances can be expressed in closed form, the contribution of each fidelity level to the total variance of the multi-fidelity model can be expressed too. It constitutes the more remarkable advantage of Le Gratiet's recursive formulation. Denoting σ 2 cont (l, x) the variance contribution of the l th fidelity level at the point x, with the notation σ 2 δ 0 = σ 2 0 we have:

∀l = 0, ..., L -1 σ 2 cont (l, x) = σ 2 δ l (x) L-1 j=l ρ 2 j and σ 2 cont (L, x) = σ 2 δ L (x) (5)

Multi-fidelity Bayesian optimization

With a multi-fidelity Bayesian optimization process, one must decide which point is most promising and the level of fidelity at which to evaluate it. It has been proposed in [START_REF]Multi-fidelity Gaussian process regression for computer experiments[END_REF] to solve the problem of finding the point and level of fidelity in two successive steps. First the point is found using a classical acquisition function as in the mono-fidelity Bayesian optimization. Then the variance contribution knowledge at each fidelity level gives some information to smartly decide the fidelity level to choose. Let c 0 , ..., c L be respectively the querying costs of all the fidelity levels f 0 , ..., f L . Let us denote σ 2 red (l, x * ) the variance reduction of the high fidelity model when the point x * is evaluated with all the fidelity levels ≤ l to ensure the nested DoE structure

σ 2 red (l, x * ) = l i=0 σ 2 δ i (x * ) L-1 j=i ρ 2 j (6)
A criterion to choose the level of enrichment can then be written as:

l * = arg max l∈0,...,L σ 2 red (l, x * ) ( l i=0 c i ) 2 (7)
One can remark that Eq [START_REF] Bartoli | Improvement of efficient global optimization with application to aircraft wing design[END_REF] is a trade off between variance reduction and computational cost.

Then the cost ratios between the different fidelity levels are important parameters to the method. This two step approach from [START_REF] Joaquim | Multi-fidelity efficient global optimization: Methodology and application to airfoil shape design[END_REF] is described in Fig. 3 of Appendix 7.2. It is refered in the following as the Multi-Fidelity Super Efficient Global Optimization (MFSEGO). Now that the multi-fidelity model and optimization technique have been described, the method for selecting the Pareto optimal fidelity level that we experimented with in this paper is explained.

Pareto optimal fidelity levels methodology

In this section, we present a preprocessing method whose goal is to select the Pareto optimal fidelity levels (POFL) of a function to be used in a multi-fidelity context. We consider two objectives which are the accuracy and the computational cost associated with each fidelity level. In a multi-objective context, the notion of optimality must be revised, and we speak of Pareto optimality [START_REF] Martins | [END_REF]. A point A dominates a point B if it is better in all objectives. For example,in our case, a fidelity level that is more accurate and less expensive than another fidelity level will dominate it. A point is said non-dominated and is called Pareto optimal if no other evaluated point dominate it. The Pareto set is the set of Pareto optimal points. The Pareto front is the set of images of the Pareto set (the objective values at the points of the Pareto set). Suppose we have L + 1 fidelity levels f 0 , ..., f l , ..., f L that evaluate a function with different accuracies. We consider here that the fidelity levels are not hierarchically ordered, except for the last fidelity level L which is the highest fidelity level. Typically, it is possible that some of these fidelity levels are not very interesting to use. For example, if one fidelity level is less accurate than another but still more computationally expensive, it is not interesting to use it in the multi-fidelity process we are building. The strategy we propose here is to evaluate each level of fidelity at points Q i of a DoE Q and estimate their accuracy and cost. The number of points is card(Q) = n pts . To approximate the computational cost c l associated to the l-th level of fidelity, we compute the mean value ĉl defined by

ĉl = 1 n pts npts i=1 c i,l (8)
where c i,l is the computational cost at point Q i with the level l. Assuming that the higher fidelity level f L is the reference, ∀l ∈ L = {0, .., L} we approximate e l the accuracy of the l-th fidelity level by êl = 1

1 npts p=1...P ||(f L (Q))p-(f l (Q))p|| 2 2 1 n pts n pts i=1 (f L (Q i ))p (9)
f l (Q) is the output of the evaluation of f l at all the points Q. Denoting P equals to the sum of the number of objectives and the number of constraints, ∀l, ∀i, f l (Q i ) is a vector of size P . Thus the estimated accuracy of the fidelity levels depends on the accuracy of the objective value but also on the constraints. Next, we construct the Pareto front in terms of cost and accuracy to select only the optimal Pareto fidelity levels. The dominated fidelity levels will not be considered in the process [START_REF] Wu | A Gradient-based Sequential Multifidelity Approach to Multidisciplinary Design Optimization[END_REF].

Analytical test cases

The POFL method is validated on a selection of analytical test cases of increasing complexity (unconstrained and constrained Rosenbrock in 2-dimension and the Borehole function in dimension 8) to show its utility.

Four fidelities for the 2D-Rosenbrock function

For the purpose of this test case, we extend the two-dimensional two-fidelity Rosenbrock function to a four-fidelity function using the method proposed by [START_REF]Comparison of generic multi-fidelity approaches for bound-constrained nonlinear optimization applied to adjoint-based CFD applications[END_REF]. The high fidelity (HF) Rosenbrock function is defined by:

f HF : [-2, 2] 2 -→ R (x 0 , x 1 ) -→ 100(x 1 -x 2 0 ) 2 + (x 0 -1) 2 (10)
The low fidelity (LF) Rosenbrock functions are of the following form

f LF,xs : [-2, 2] 2 -→ R (x 0 , x 1 ) -→ 100(x s x 1 -x 2 0 ) 2 -2((x 1 -x s ) 2 + (x 0 -x s ) 2 ) (11)
The three LF functions are built using the same equation, but are parameterized by x s . Table 1 summarizes the chosen parameter x s and the chosen computational cost ratio for each fidelity level. Knowing x s and the cost ratio for each fidelity level, the POFL method defined in Section 3

Four fidelities for the constrained 2D-Rosenbrock function

This test case uses the same functions as the ones defined in Section 4.1 with some different cost ratios. We also add the following constraint

c a (x 0 , x 1 ) = -x 2 0 -(x 1 -1) 1 2 + 1.1 + a sin(10x 0 + 5x 1 ) ≥ 0 ( 12
)
where a is an amplitude parameter for the sine. Table 2 sums up the value of this parameter a and the cost ratio for each fidelity level.

Three fidelities for the 8D-Borehole function

The Borehole function is an unconstrained 8-dimensional test case, derived here to have three fidelity levels denoted by LF, MF and HF. The form of the Borehole function is expressed as: Table 3 gives the values of the parameters b 0 and b 1 and the chosen cost ratio for each fidelity level. 1.5 0.5 1 cost ratio 10 100 1

f b 0 ,b 1 : Ω -→ R x -→ b 0 x 2 (x 3 -x 5 ) ln x 1 x 0 (b 1 + 2x 6 x 2 ln(x 1 /x 0 )x 2 0 x 7 + x 2 x 4 ) (13
The Pareto front obtained with n pts = 100 points is shown in Fig. 5 in Appendix 7.3. The LF function does not belong to the Pareto front and could be eliminated.

Optimization results of analytical test cases

In this section the optimization results obtained on the analytical test cases are described. We compared a mono-fidelity optimization with several multi-fidelity optimizations: one using all the fidelity levels available, one using only the Pareto optimal fidelity levels and one using fidelity levels randomly chosen. Thus four scenarios are evaluated and compared.

The multi-fidelity Kriging models are computed with SMT [START_REF] Bouhlel | Advances in Engineering Software, A Python surrogate modeling framework with derivatives[END_REF][START_REF] Bouhlel | Surrogate Modelling Toolbox[END_REF][START_REF] Bouhlel | A Python surrogate modeling framework with derivatives[END_REF]. SMT is an open source toolbox developed by ONERA, ISAE-SUPAERO, NASA Glenn and University of Michigan specialized on the fast development of tailored surrogate models. The optimizations are performed with SEGOMOE-toolbox [START_REF] Segomoe ; , Sylvain And Salaün | Super Efficient Global Optimization with Mixture of Experts[END_REF] a Bayesian optimization toolbox developed at ONERA and ISAE-SUPAERO including the proposed Multi-Fidelity strategy (MFSEGO). For the random fidelity selection case, the HF level is always selected and the number of levels is equal to the one chosen by the POFL strategy. For each scenario, 10 runs are performed using a squared exponential kernel and the WB2 acquisition function [START_REF] Watson | Infill sampling criteria to locate extremes[END_REF]. For each run, first we built an initial DoE for the optimization that uses all of the fidelity levels. Next, to construct the initial DoE for each other scenario, we examine the selected fidelity levels with the POFL method, and then select the parts corresponding to those fidelity levels from the full initial DoE. For all analytical test cases, the initial DoE size for all multi-fidelity optimizations was chosen as follows. First, we ranked all fidelity levels by accuracy, from k = 0 the least accurate to k = L the most accurate. Next, we chose the initial DoE size for the HF level with ndoe HF equal to twice the problem dimension, i.e., ndoe HF = 4 for the two Rosenbrock problems and ndoe HF = 16 for the Borehole problem. Finally, the number of points evaluated with the k-th level of fidelity is determined by the formula

ndoe f id k = (ndoe HF )2 L-k
As MFSEGO needs a nested DoE structure as mentioned is Section 2.1, the initial DoEs have then been created respecting this requirement. Reference solutions are known for the analytical problems. Given this reference solution ref sol , a convergence criterion for the mean (over the 10 runs) of the objective value at the best feasible point ȳfeasible best is defined by:

|ȳ f easible best -f HF (ref sol )| ≤ ϵ 0 |f HF (ref sol )| + ϵ 1 ( 14
)
where ϵ 0 and ϵ 1 are respectively the relative and absolute tolerances. Typically we chose ϵ 0 = ϵ 1 = 1e -3 for the analytical test cases. Figure 1b shows the evolution of the mean of the best feasible objective value against the budget for the unconstrained Rosenbrock test case. 4. The term "not reached" in this table means that the maximum allocated budget for the optimization has been reached before the convergence criterion Eq. ( 14) was satisfied. For all the analytical test cases we can see that the POFL method overtakes all other methods. Moreover these results highlight that using all of the fidelity levels available is not efficient, even compared to the mono-fidelity strategy. In Fig 1b and Fig 7 for unconstrained and constrained Rosenbrock cases, the curve for the random scenario quickly approaches the reference solution, which means that for some of the 10 runs the random strategy was effective in finding the optimum. Nevertheless, the convergence criterion (see Eq. ( 14)) on the average over the 10 runs is not even reached using the whole allowed budget. We can therefore deduce that the random strategy is not at all robust.

Table 4: Summary of the needed budget to reach the convergence criterion for the three analytical test cases with ϵ 0 = ϵ 1 = 1e -3 ("not reached" means that the maximum allocated budget for the optimization has been reached before the convergence criterion Eq. ( 14) was satisfied).

Drone design test case

The drone model

For the purpose of this test case, we developed a parametric drone model with the OpenMDAO framework [START_REF] Openmdao ; Gray | An open-source framework for multidisciplinary design, analysis, and optimization[END_REF]. The core of the model is the aerostructural part which relies on an aerostructural simulation code called OpenAeroStruct [START_REF] Jasa | Open-source coupled aerostructural optimization using Python[END_REF] [12] [START_REF] Chaudhuri | Multifidelity optimization under uncertainty for a tailless aircraft[END_REF] (OAS) developed by University of Michigan within OpenMDAO. The model has a propulsion group handled using the electrical extension of the Breguet range equation. This equation is valid in the steady level flight case and its expression is given in Appendix 7.6. In order for the drone to ensure longitudinal stability, the model also has a group that balances the root chord tail such that the tail surface over wing surface ratio is equal to 0.15. This model has been derived in 5 fidelity levels by combining different mesh dimensions and drone speed parameterizations. The different combinations are given in Table 5. Using different speeds allows us to simulate several codes of a priori similar levels. This is a way to emulate more levels of fidelity for our problem without using more tools and codes.

m.s -1 LF1 (3,5) (3,5) 20 m.s -1 LF0 (3,5) (3,5) 30 m.s -1
The MDO problem to be solved can be defined as:

max x∈R 19 SOC(x) such that    CL ≥ 0.4 wing failure ≤ 0 tail failure ≤ 0
where the SOC is the state of charge of the battery at the end of the mission and x is the vector of 19 design variables relative to wing and tail geometrical parameters. They are detailed in

Optimization results for the drone design

This section presents the optimization results obtained with the drone design test case. The size of the initial DoEs and the cost ratios are given in Table 6. The same convergence criterion, see Eq. ( 14), as for the analytical cases is now parameterized by ϵ 0 = 0.005 and ϵ 1 = 0.0025 and the reference solution has been found using the SNOPT algorithm [START_REF]User's Guide for SNOPT Version 7: Software for Large-Scale Nonlinear Programming[END_REF] which stands for Sparse Nonlinear OPTimizer. For this test case we also compared the results with an other type of fidelity selection strategy that we named dominated selection. Indeed, since the low and medium fidelity levels are mainly intended to explore the design space, we decided to check the impact on the optimization performance of selecting the dominated low fidelity models instead of the non-dominated low fidelity models. Then, the same methodology as for the analytical cases is used to obtain the results presented in Fig. 2. The representation is slightly different to show the budget. For each of the 5 strategies, the number of runs that found the optimum is given by the blue line and the number of executions that found a feasible point with respect to the budget by the green line. Table 7 sums up the required budget to reach the convergence criterion. In the end, the POFL method is the most efficient. The dominated selection method is less efficient than the POFL method but still better than all other strategies. The mono-fidelity strategy is not the most effective but it is quite robust. On the opposite the random strategy is very effective for most of the runs but is slow for one of them. It emphasizes that this strategy is not robust to the initial DoE as we started to see with the analytical test cases. Finally, the all fidelity level strategy seems to be the worst. Using a large number of fidelity levels can lead to redundancy and complicate surrogate modeling, which could explain this inefficiency.

Table 7: Summary of the needed budgets to reach the convergence criterion for the drone design test case with ϵ 0 = 5 × 1e -3 and ϵ 1 = 2.5 × 1e -3 .

All fidelity levels Mono-fidelity POFL dominated selection Random 24.87 14 8.71 10.59 11.41

Conclusion

In this work, we proposed a method to select the most useful fidelity levels to use in a multifidelity process among all available levels. We experimented with this POFL method on several analytical optimization problems and on a UAV design problem that we solved using single and multi-fidelity Bayesian optimizations. Overall, the POFL selection proves to be the most computationally efficient for solving these problems. The budget required to achieve convergence was divided by a factor between 1.2 and 2.2 using the POFL method compared to the monofidelity method. However, the POFL has some limitations. For example, if the cost and accuracy of each model is not known or has not yet been estimated, it must be done to obtain the Pareto front and this can be expensive. Nevertheless, once this has been done once, the cost and accuracy estimates can be reused.

. Others optimization results figures

Figure 7 shows the evolution of the mean over the 10 runs of objective value at the best HF feasible point along the optimization for the constrained 2D-Rosenbrock test case. Figure 8 shows the evolution of the mean over the 10 runs of objective value at the best HF feasible point along the optimization for the constrained 8D-Borehole test case.

Design variables of the drone design problem

Table 8 sums up the design variables of the drone design optimization problem, their lower and upper bound, their unit and their dimension.

Electric Breguet range equation demonstration

First lets notice that the range R can be expressed with the speed and the time flying at that speed: R = V × t. Under ideal conditions, the time needed to empty a battery is given by: t = m batt e b P batt [START_REF] Jasa | Open-source coupled aerostructural optimization using Python[END_REF] where e b is the specific energy density of the battery (unit: J.kg -1 ), m batt is the battery mass and P batt is the battery power. Lets define the overall propulsion system efficiency η tot = Pprop P batt with P prop is the required propulsive power to reach the equilibrium. η tot can be expressed as

Note that the L D ratio can be replaced by the ratio of lift and drag coefficient C L C D . Indeed, L = qSC L and D = qSC D with S being the lifting surface area and q being the dynamic pressure.

) where b 0 and b 1

 1 are two parameters and the input space Ω ⊂ R 8 is defined by Ω = [0.05, 0.15]×[100, 50000]×[63070, 115600]×[990, 1110]×[63.1, 116]×[700, 820]×[1120, 1680]×[9855, 12045]

Figures 7 and 8 in

 8 Appendix 7.4 show the mean value evolutions respectively for the constrained Rosenbrock and Borehole test cases. Overall results are summed up in Table

Figure 1 :

 1 Figure 1: Unconstrained Rosenbrock test cases fidelity level Pareto front and convergence curves.

Figure 2 :

 2 Figure 2: Evolution of the number of runs that have found a feasible point and the number that have found the optimum.

Figure 3 :

 3 Figure 3: MFSEGO methodology diagram.

Figure 4 : 12 Figure 5 :Figure 6 :

 41256 Figure 4: Fidelity level Pareto front for the constrained 2D-Rosenbrock test case

Figure 7 :Figure 8 :

 78 Figure 7: Constrained Rosenbrock case: mean and 1-sigma confidence interval over 10 runs of the objective value at the best HF feasible point against budget

  [0; 0.5; 1] [0.5; 1; 1.5] deg 3 Total number of variables 19 14 the product of 3 individual efficiencies: η p the propeller efficiency, η m the motor efficiency and η ESC the electric speed controller efficiency. P prop = T V with T the required thrust. Assuming the steady level flight hypothesis, we haveT = D L = W (16)where L is the lift force, D is the drag force and W is the weight. Equation (16) implies g being the gravity acceleration. A relation is now available for the speed: the final expression for the range is given by:R = V × t = m batte b P batt × P batt η tot mg L D = L D η p η m η ESC e b g m batt m

Table 1 :

 1 LF and HF Rosenbrock functions: x s parameter and associated cost ratio to define the four fidelity levels.

		LF1 LF2 LF3 HF
	x s	0.4 0.65 0.9	1
	cost ratio 1000	10	100	1

is applied to build the associated Pareto front in terms of cost and accuracy using the data set Q with n pts = 100 points. The Pareto front is depicted in Fig

1a

, we can clearly see that with the chosen cost ratios, the second low fidelity function LF2 is dominated and could be removed from the list of available fidelities.

Table 2 :

 2 Constrained Rosenbrock test case: amplitude parameter and cost ratio for each fidelity level The Pareto front obtained with n pts = 100 points is shown in Fig 4 in Appendix 7.3. One can observe that the first low fidelity function LF1 does not belong to the Pareto front and could be removed On these first 2D examples, the POFL method proposes to eliminate one level of fidelity and thus to decrease the number of levels of fidelity from 4 to 3.

		LF1 LF2 LF3 HF
	a	0.15 0.1 0.05	0
	cost ratio 100 1000	10	1

Table 3 :

 3 Borehole parameters and cost ratio for each fidelity level

		LF MF HF
	b 0	5	7	2π
	b 1

Table 5 :

 5 Parameterization of the OpenAeroStruct models

		wing mesh dim	tail mesh dim	Drone speed
		(chordwise, spanwise) (chordwise, spanwise)
	HF	(19,41)	(17,37)	20 m.s -1
	MF1	(9,15)	(7,11)	20 m.s -1
	MF0	(9,15)	(7,11)	30

Table 8 in

 8 Appendix 7.5. The Pareto front obtained with n pts = 100 points is shown in Fig 6 in Appendix 7.3. We can see that the two models MF0 and LFO that consider a 30 ms -1 speed are not Pareto optimal.

Table 6 :

 6 Size of initial DoEs and cost ratios for the drone design test case.

		LF0 LF1 MF0 MF1 HF
	Initial DoE size	20	20	10	10	5
	cost ratio	100 100	10	10	1

Table 8 :

 8 Design variables of the drone design problem

		lower bound	upper bound	unit dimension
	span wing	5	6	m	1
	root chord wing	0.9	1.2	m	1
	taper wing	0.6	1	∅	1
	sweep wing	0	5	deg	1
	sweep tail	0	5	deg	1
	thickness cp wing

Acknowledgments

The PhD is funded by the defense innovation agency (AID) and by the french Directorate General of Armaments (DGA) as part of the CONCORDE project. This work is part of the activities of ONERA -ISAE -ENAC joint research group.

Appendix

In this appendix, we first find a pseudo-code of the MFSEGO methodology in Section 7.1. Then, Section 7.2 presents this MFSEGO methodology in a diagrammatic form. Section 7.3 shows the fidelity level Pareto fronts for the constrained Rosenbrock, Borehole and drone design test cases. Section 7.4 shows the optimization convergence curves for the constrained Rosenbrock and Borehole test cases. A table summarizing the lower and upper bounds, the unit and the dimension of all of the design variables for the drone test case is provided in Section 8. Finally, Section 7.6 details how to find the expression of the electric Breguet range equation.

Pseudo-code of the multi-fidelity Bayesian optimization methodology based on Le Gratiet's recursive formulation.

The following pseudo-code explains the MFSEGO methodology.


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Introduction

Mahatma Gandhi referred to the harvest of local non-domesticated plant species as "an antidote to poverty" [START_REF] Gandhi | Speech at opening of the village industry exhibition in Brindawan Bihar[END_REF]. Even though Gandhi was specifically referring to Neera, the sap extracted from the palm species Borassus flabellifer L., his quote is true for many non-timber forest products (NTFPs). Indeed, harvesting local NTFPs plays a central role for human wellbeing, especially in rural communities across tropical developing countries as food, medicine and income generation [START_REF] Brussaard | Reconciling biodiversity conservation and food security: scientific challenges for a new agriculture[END_REF][START_REF] Sunderland | Food security: why is biodiversity important?[END_REF]. Palms rank as one of the most economically important plant families across the tropics [START_REF] Balick | Useful palms of the world. A synoptic bibliography[END_REF][START_REF] Balslev | Palm harvest impacts in North-Western South America[END_REF][START_REF] Brisson | Trade in palm products in North-Western South America[END_REF]. Continental Africa, excluding Madagascar, currently harbors 68 palm species in 18 genera [START_REF] Cosiaux | Low extinction risk for an important plant resource: Conservation assessments of continental African palms (Arecaceae/Palmae)[END_REF][START_REF] Stauffer | An update to the African palms (Arecaceae) floristic and taxonomic knowledge, with emphasis on the West African region[END_REF]. The genus Raphia P. Beauv. is the most species rich, with currently 22 accepted species [START_REF] Mogue Kamga | Two new species of Raphia (Palmae/Arecaceae) from Cameroon and Gabon[END_REF]). Raphia's are large palms with conspicuous trunk fibers, generally thriving in humid and swampy regions. In several African countries Raphia's are among the most important palms with a multiplicity of uses [START_REF] Forenet | Subregional workshop on harmonization of national reviews on "Non-timber forest products (NTFPs) in Central Africa[END_REF][START_REF] Ingram | Review of non timber forest products (NTFPs) in Central Africa: Cameroon. Establishment of a Forestry Research Network for ACP Countries[END_REF][START_REF] Stauffer | Palmae (Arecaceae)[END_REF]) and have been highlighted as high priority species in several central African countries [START_REF] Ingram | Non-timber forest products: contribution to national economy and strategies for sustainable management[END_REF]. Although the importance of Raphia for African populations is undisputed, we still lack fundamental baseline information for the development of more responsible management practices. This includes a basic understanding of the genus' biology, taxonomy, genetic diversity, ecology, ethno botany, applied harvesting practices and management techniques. Some studies do not even recognize or classify products derived from Raphia species as officially regulated NTFPs [START_REF] Ingram | Win-wins in forest product value chains? How governance impacts the sustainability of livelihoods based on non-timber forest products from Cameroon[END_REF]Schure 2010, Couvreur and[START_REF] Couvreur | A l'Echelle du Raphia / On Raphia and Man[END_REF]. Besides this lack of knowledge, species identification remains challenging, leading to much confusion in scientific literature (Beccari 1910a;[START_REF] Mogue Kamga | Raphia vinifera (Arecaceae; Calamoideae): Misidentified for far too long[END_REF]Tuley 1995;[START_REF] Tuley | The Raphia palms reviewed[END_REF].

Biology and Taxonomy of Raphia

Raphia species are omnipresent across tropical Africa, ranging from medium to massive palms reaching up to 30 m in height (Fig. 1A). Most species have above-ground trunks, which may be solitary or clustered, while two (R. regalis, Fig. 1B; R. vinifera, Fig. 1C) have very short or subterranean (acaulescent) trunks [START_REF] Mogue Kamga | Raphia vinifera (Arecaceae; Calamoideae): Misidentified for far too long[END_REF]Tuley 1995). When the stem is above aground, it is typically hidden by old leaf sheaths and curly or straight trunk fibers. The pinnate leaves are very long reaching in some cases up to 25 or even 30 m in R. regalis [START_REF] Dransfield | Genera Palmarum: The evolution and classification of palms[END_REF][START_REF] Hallé | The longest leaf in palms?[END_REF][START_REF] Jacques-Félix | Note sur le stipe hypogé du Raphia regalis Becc. (Arecaceae-Lepidocaryeae)[END_REF]) Raphia palms are hapaxanthic (monocarpic, semelparous): each stem flowers only once in their lifetime and subsequently die [START_REF] Dransfield | Genera Palmarum: The evolution and classification of palms[END_REF][START_REF] Russell | The Raphia palms of West Africa[END_REF]. They are monoecious, with male and female flowers on the same flowering branches (rachillae), the former at the apex and the latter at the base. The various shapes of the inflorescences constitute a key character for identification and Raphia classification [START_REF] Otedoh | A revision of the genus Raphia Beauv.(Palmae)[END_REF]Tuley 1995). The fruits are covered by conspicuous scales (Fig. 1D) resembling a snake skin, a common trait of the Calamoideae subfamily [START_REF] Dransfield | Genera Palmarum: The evolution and classification of palms[END_REF]. Due to the massive stature, most Raphia species are rarely collected in the field, leading to a relatively poor understanding of their taxonomy [START_REF] Mogue Kamga | Two new species of Raphia (Palmae/Arecaceae) from Cameroon and Gabon[END_REF], 2019). Despite efforts made by several authors (Beccari 1910b;[START_REF] Otedoh | A revision of the genus Raphia Beauv.(Palmae)[END_REF]Tuley 1995;[START_REF] Tuley | The Raphia palms reviewed[END_REF], taxonomy and species delimitation still remain insufficiently understood. Current treatments of Raphia accept 22 species [START_REF] Mogue Kamga | Two new species of Raphia (Palmae/Arecaceae) from Cameroon and Gabon[END_REF][START_REF] Otedoh | A revision of the genus Raphia Beauv.(Palmae)[END_REF] divided into five sections based on overall inflorescence morphology. See Table S1 for a list of accepted species, author names, associated synonyms and distribution.

Raphia has a disjunct distribution between Africa and the Neotropics. In Africa, it is widely distributed, occurring from west through to eastern and southern Africa and has one species in Madagascar (R. farinifera). A single species (R. taedigera) occurs in South and Central America [START_REF] Dransfield | Genera Palmarum: The evolution and classification of palms[END_REF]. Its center of diversity is Central Africa, with 19 species recorded, 10 or 11 species in Cameroon and eight in Gabon (Fig. 1) [START_REF] Couvreur | Flore du Gabon-Palmae[END_REF]. Most Raphia species are well adapted to humid habitats and commonly occur in swamps, near streams in rain forests, gallery forests, and savannas. Raphia species mainly occur in lowland regions, except for R. vinifera [START_REF] Mogue Kamga | Raphia vinifera (Arecaceae; Calamoideae): Misidentified for far too long[END_REF]) and R. ruwenzorica, occurring between 800 and 2000 m in Cameroon/Nigeria and eastern Democratic Republic of the Congo (DRC), respectively. Some Raphia species form dense, nearly monospecific populations stretching over large areas [START_REF] Tuley | The Raphia palms reviewed[END_REF], providing important ecosystem services, such as water purification, limitation of soil erosion [START_REF] Mphoweh | The Degradation Of Raffia Palms and its Socio-Economic and Ecological Repercursions: The Case Study of Bamunka, Ndop, North West Province Cameroon[END_REF], and carbon sequestration [START_REF] Dargie | Age, extent and carbon storage of the central Congo Basin peatland complex[END_REF].

Raphia Uses

Since most parts of Raphia species are exploited, we divided the uses according to the plant organ and (where possible) to the species used. We further identified and classified uses into eight ethnobotanical categories, adapted from the Economic Botany Data Collection Standard prepared for the International Working Group on Taxonomic Databases for Plant Sciences [START_REF] Cook | Economic botany data collection standard[END_REF]: construction (CO), cultural (CU), domestic utensils and tools (D), edible (E), tools for hunting and fishing (HF), fuel (F), medicinal (M), and other uses (O) (environmental, toxic etc), and a new category: wearing apparels (W). Available studies on use and trade of Raphia species focus on particular species (e.g., R. hookeri), and species names are not usually provided (Raphia sp., "raffia", or just "palm") or probably incorrect. Here, we also report the uses for Raphia independent of the species investigated and their corresponding use categories.

Just under a hundred different Raphia uses are reported in scientific literature and span a considerable number of different categories, ranging from food (especially beverages), construction, tools, handicrafts, and energy supply through to medicinal, cultural and ecological purposes (Table 1, Table S2, Figs. 2,3,[START_REF] Hallé | The longest leaf in palms?[END_REF]Figs. S1,S2). Although most parts of Raphia palms are useful (Tuley 1995), leaves have undoubtedly the greatest number of uses (Table S2). Indeed, leaves appear in eight of nine use categories. The petiole and rachis are commonly referred to as "bamboo" and employed in the manufacture of domestic utensils and in construction. The leaf sheaths are used as fuel wood. The leaflets are useful in two ways: fibers are extracted from young, still expanding (spear) leaflets, whereas older leaves are used for thatching material. Across Africa, leaflet midribs are used in manufacturing brooms. The least useful part, the root, only appears in two use categories (Table S2).

We show that 21 out of 22 Raphia species are useful in one or more categories (Table 1, Fig. S2), with 19 species used for construction purposes and 18 for food (Fig. S1). Few species are mentioned for the categories fuel (fuel wood), wearing apparels, and medicine. Clearly some species are exploited more extensively than others. The underlying factors determining the degree to which a species is used include product type and quality, available expertise, knowledge, and accessibility. The latter is associated with morphology and the distance to the resource. Providing raw materials that are applicable across all use categories, R. hookeri and R. vinifera are the most intensively exploited in Africa. For now, the recently described, endemic Gabonese species R. gabonica has no reported use as it is locally endemic with few individuals reported [START_REF] Mogue Kamga | Two new species of Raphia (Palmae/Arecaceae) from Cameroon and Gabon[END_REF].

Fig. 2. Illustration of the major categories of Raphia species uses.

A. Raphia palm. B, E. construction (CO), C. cultural (CU). D. domestic utensils and tools (D). H, J. edible (E), tools for hunting and fishing (HF), fuel (F), F. Medicinal (M) and G. other uses (O) (environmental, toxic etc); I, G. wearing apparels (W). Drawings by Nicolas Morales Arregui

Main Raphia Products and Methods of Extraction

Raphia sap and derived products

Sap is the most important resource and the derived palm wine probably the most studied products of Raphia [START_REF] Ayogu | Evaluation of the performance of a yeast isolate from Nigerian palm wine in wine production from pineapple fruits[END_REF]Obahiagbon 2009). Most Raphia species can be used for tapping (Table 1) (Tuley 1995). Raphia hookeri is the most important and frequently used for wine production in Nigeria (Obahiagbon 2009). When several species are available, the species tapped is typically based on accessibility, distance, abundance, availability, and amount of sap produced per unit of time. Raphia hookeri is preferred in the littoral and central regions of Cameroon and R. vinifera in the Western Highlands. In certain parts of DRC, R. matombe is the main species for tapping. Finally, in Madagascar R. farinifera is used for wine extraction (Perrier de la Bathie 1931). Morphology plays an important role in choosing a species: Raphia hookeri is easily tapped since it has single stems, while R. laurentii, R. monbuttorum, or R. zamiana, with clustering stems, forming dense clumps covered in straight sharp fibers that are harder to climb. Where only one species exists, it is tapped even though access is difficult (e.g, R. zamiana in South Cameroon).

Sap Extraction Methods

Two main methods exist for sap extraction: "down wine tapping" and "inflorescence tapping" (Tuley 1995). In the former, individuals are felled and a container is placed under the inflorescence cut to collect the sap. This is destructive and has been observed in R. hookeri and R. monbuttorum, but is safe for the tapper, limiting accidents. Indeed, accidents related to climbing individuals are quite common [START_REF] Mbuagbaw | The palm wine trade: Occupational and health hazards[END_REF]. Falling from a Raphia during tapping can lead to disgrace within village communities (Mogue Kamga pers. com.). In the latter method tappers climb to the base of the crown (Fig. 3A), apply an incision near the emergent inflorescence primordium allowing the sap to flow out. This method necessitates expertise and skill of the tapper to climb to the top of the palm.

Tapping occurs just before flowering, thus the emergent inflorescence primordium is scarified at an early stage of its development. Before the onset of flowering, individual palms develop leaves that are generally shorter than the previous ones, indicating the right moment to harvest (Tuley 1995). Tapping is conducted on a daily basis, early in the morning or in the late evening (Otedoh 1990). Information about sap yield remains scare in the literature. [START_REF] Tuley | The production of Raphia wine in Nigeria[END_REF] reports that R. hookeri in Nigeria can produce up to 118 l of sap per individual per year, while Obahiagbon et al. (2007) report averages between 6 and 20 l (more rarely up to 30) per induvial per week, suggesting a high variability in sap yield between individuals. More studies should be conducted to provide a more precise of the yield of Raphia wine across its distribution.

Sap Products

Three main products are derived from Raphia sap: a non-alcoholic beverage, alcoholic wine (palm wine, Fig. 3B) and a high alcoholic concentration spirit.

The freshly tapped sap, or juice, is a translucent liquid representing a sweet and nonalcoholic drink rich in vitamins and minerals (Fig. 3B). In Nigeria, it is reported that Raphia sap is consumed by over 50 million people [START_REF] Obahiagbon | Sugar and macrominerals composition of sap produced by Raphia hookeri palms[END_REF]. The fresh sap of R. hookeri is composed of sucrose (12-15%), protein, fats, organic acids, vitamins, minerals, and water [START_REF] Bassir | Observations on the fermentation of palm wine[END_REF][START_REF] Nwokeke | Palm wine preservation using traditional plant that have preservative bases[END_REF][START_REF] Obahiagbon | Raphia vinifera Sap: Some Physico-Chemical and Biochemical Profiles. A Comparative Study[END_REF][START_REF] Okafor | Preliminary Microbiological Studies on the Preservation of Palm Wine[END_REF] as well as several types of sugars (e.g., arabinose, cellobiose, fructose) [START_REF] Faparusi | Sugars identified in Raphia palm wine[END_REF]Obahiagbon 2009;[START_REF] Obahiagbon | Sugar and macrominerals composition of sap produced by Raphia hookeri palms[END_REF]. It contains elements such as Ca, Cl, Cu, Fe, K, Mg, Mn, N, Na, P, and Zn (Obahiagbon et al. 2007). Raphia vinifera sap contains both useful and harmful microorganisms, which change spontaneously during fermentation at room temperature [START_REF] Tiempa | Physicochemical changes of raffia sap (Raphia mambillensis) contents during spontaneous fermentation[END_REF].

Subsequent fermentation converts the sap into a cloudy, alcoholic beverage named Raphia wine [START_REF] Uzochukwuru | Standard pure culture inoculum of natural fermented Palm sap[END_REF][START_REF] Obahiagbon | Sugar and macrominerals composition of sap produced by Raphia hookeri palms[END_REF]. Fermentation is aided by adding bark and sometimes seeds of different species. For example, the bark of Garcinia lucida Vesque is added to accelerate fermentation, in addition to being medicinal (see below). The use of Garcinia in palm wine production occurs in Central and West Africa [START_REF] Guedje | Garcinia lucida Vesque (Clusiaceae): from traditional uses to pharmacopeic monograph for an emerging local plant-based drug development[END_REF][START_REF] Ayogu | Evaluation of the performance of a yeast isolate from Nigerian palm wine in wine production from pineapple fruits[END_REF][START_REF] Bouquet | Féticheurs et médecines traditionnelles du Congo (Brazzaville)[END_REF][START_REF] Irvine | Woody plants of Ghana with special reference to their uses[END_REF]. In Bipindi-Akom, Cameroon, a number of other plant additives from species are used. In the DRC, the bark of Bridelia scleroneura Müll. Arg. adds color, while the fruit pulp of Landolphia camptoloba (K.Schum.) Pichon and L. lanceolata (K.Schum.) Pichon enhances flavor [START_REF] Moyene | Valorisation des produits forestiers non ligneux des plateaux de Bateke en peripherie de Kinshasa (R. D. Congo)[END_REF]. Long term preservation of Raphia wine for commercialization still faces challenges and is the main limitation of its large scale distribution [START_REF] Levi | The preservation and bottling of palm wine[END_REF]. Sorbic acid appears as the most suitable for preservation, due to its wide bactericidal spectrum and few side effects [START_REF] Okafor | Preliminary Microbiological Studies on the Preservation of Palm Wine[END_REF]. So far, pasteurization at 75 °C for 45 min is the most successful method [START_REF] Obahiagbon | Sugar and macrominerals composition of sap produced by Raphia hookeri palms[END_REF].

The wine is distilled to produce a spirit which is known as kaikai [START_REF] Ikegwu | The value of palm wine tapping in the food production practices of Igbo-Land : A case study of Idemili South Local Government Area, Anambra State[END_REF] in the Anambra State of Nigeria, afawfaw [START_REF] Mphoweh | The Degradation Of Raffia Palms and its Socio-Economic and Ecological Repercursions: The Case Study of Bamunka, Ndop, North West Province Cameroon[END_REF]) in the North West region of Cameroon, odontol or ha in southern Cameroon, ogogoro, kaikai or apeteshi in West Africa [START_REF] Adeleke | Physico-chemical properties of commercial local beverages in Osun State, Nigeria[END_REF][START_REF] Iwuoha | Nigerian indigenous fermented foods: their traditional process operation, inherent problems, improvements and current status[END_REF][START_REF] Obot | The measurement of drinking patterns and alcohol problems in Nigeria[END_REF] and sodabi in southeastern Benin [START_REF] Profizi | Biologie et modes de gestion des marécages à Raphia hookeri Mann and Wendland au Sud-Est du Bénin[END_REF]). The alcohol content of the distilled wine varies between 38% and 60% [START_REF] Ababio | Organic chemistry. New school chemistry[END_REF][START_REF] Adeleke | Physico-chemical properties of commercial local beverages in Osun State, Nigeria[END_REF][START_REF] Oladeinde | African Journal of Biomedical Research[END_REF]. As such, this spirt has been outlawed in certain parts of Cameroon due to cases of death from hazardous consumption and difficultly in controlling the total percentage of alcohol after distillation [START_REF] Kouagheu | Au Cameroun, la difficile lutte contre l'odontol, un alcool artisanal qui fait des ravages[END_REF]. On average 19-20 l of Raphia wine is required to produce 1 l of distillate (280-480 l of fermented sap for 20 l of ethanol) [START_REF] Profizi | Biologie et modes de gestion des marécages à Raphia hookeri Mann and Wendland au Sud-Est du Bénin[END_REF][START_REF] Ohimain | Traditional Fermentation and Distillation of Raffia Palm Sap for the Production of Bioethanol in Bayelsa State, Nigeria[END_REF]. Production takes between five to over 12 h using 105-155 kg of fire wood equivalent to 5.25-7.75 kg of wood/l. Palm spirit is served in bars in rural areas and dispensed at plantations and farms to motivate laborers, to preserve food, applied to soak dried fruit to bake Christmas cake by foreigners residing in the Indemili area [START_REF] Ikegwu | The value of palm wine tapping in the food production practices of Igbo-Land : A case study of Idemili South Local Government Area, Anambra State[END_REF] or in the sterilization of razor blades in barber shops [START_REF] Mphoweh | The Degradation Of Raffia Palms and its Socio-Economic and Ecological Repercursions: The Case Study of Bamunka, Ndop, North West Province Cameroon[END_REF].

Finally, some studies have reported that R. hookeri sap can be used as a good source of bioethanol production although yields were highly variable between distillates [START_REF] Ohimain | Traditional Fermentation and Distillation of Raffia Palm Sap for the Production of Bioethanol in Bayelsa State, Nigeria[END_REF].

Raphia fibers

Fibers are an important Raphia raw material (Kicińska-Jakubowska et al. 2012), extracted either from the leaflets or from the petioles and sheaths (Fig. 3C). Fibers are extracted from the spear leaves, which are the young and unfolded leaves (Tuley 1995). After drying, the stripped fibers are twisted into a thread, used to weave an array of products, such as mats, baskets, hats, bags, ropes, hammocks, toys, clothes, traditional attires among others (Table S1). Today, Raphia fibers are even used in the production of clothes and "haute couture" for example in Douala (Mogue Kamga, pers. obs.). The fibers are often dyed with both natural and chemical dyes (Fig. 3C). In the past, Raphia fiber was mainly obtained from R. farinifera in southern Africa or Madagascar (Beccari 1910b;Tuley 1995). Other species also provide useful fibers (Table 1), e.g. R. gentiliana, R. hookeri, R. sudanica R. textilis and R. vinifera [START_REF] Elenga | Thin-layer drying of Raffia Textilis Fiber[END_REF][START_REF] Ouattara | Ethnobotany and conservation of palms from Ghana[END_REF]. Raphia fibers have been particularly recommended for strengthening composite materials [START_REF] Elenga | Effects of alkali treatment on the microstructure, composition, and properties of the Raffia textilis fiber[END_REF].

Fibers extracted from the disintegrating sheaths of palm leaves are collectively referred to as piassava fiber. They are used as ropes, mattress stuffing, brooms and for furniture (Fig. 3D). In the past considerable amounts of piassava from Raphia species were exported from Southeastern Nigeria, Southern Cameroon, Sierra Leone, and Liberia (Beccari 1910a;[START_REF] Chevalier | Nouvelles recherches sur les Palmiers du genre Raphia (Suite et fin)[END_REF][START_REF] Russell | The Raphia palms of West Africa[END_REF]Tuley 1995;Obahiagbon 2009). However, due to strong competition from synthetic materials, the demand for Raphia fiber has declined significantly. It is documented that fiber extracted from the leaf bases of R. hookeri is suitable for paper production, but largescale commercial production has never been performed (Tuley 1995).

White grubs -beetles' larvae

Raphia stems are an important source for grubs, but this has been little acknowledged in the literature [START_REF] Dounias | L'exploitation méconnue d'une ressource connue: la collecte des larves comestibles de charançons dans les palmiers raphias au sud Cameroun[END_REF]Obahiagbon 2009) despite their important nutritive importance across tropical Africa [START_REF] Muafor | Exploitation, trade and farming of palm weevil grubs in Cameroon[END_REF]. Several species of insects from the families Dryophthoridae (Curculionoidea) and Dynastidae (Scarabaeoidea) use the decomposing stems of Raphia for oviposition. The resulting larvae, commonly called grubs (Fig. 4A), are considered a delicacy in the Congo Basin and West Africa. It is from Raphia that most of the grub collection is undertaken in central Africa, more than from the oil palm Elaeis guineensis [START_REF] Dounias | L'exploitation méconnue d'une ressource connue: la collecte des larves comestibles de charançons dans les palmiers raphias au sud Cameroun[END_REF]. These grubs are known under various common names (Figure 4A): ukolo, edon or ugbean in Nigeria (Obahiagbon 2009); fos [START_REF] Defoliart | Hypothesizing about palm weevil and palm rhinoceros beetle larvae as traditional cuisine, tropical waste recycling, and pest and disease control on coconut and other palms -can they be integrated?[END_REF][START_REF] Dounias | L'exploitation méconnue d'une ressource connue: la collecte des larves comestibles de charançons dans les palmiers raphias au sud Cameroun[END_REF], mbé, poseh, and tumbu in Cameroon [START_REF] Muafor | Exploitation, trade and farming of palm weevil grubs in Cameroon[END_REF]; Kpitran in Benin [START_REF] Riggi | Observations of entomophagy across Benin -practices and potentials[END_REF]. Three different species of Raphia-associated edible larvae are known [START_REF] Muafor | Exploitation, trade and farming of palm weevil grubs in Cameroon[END_REF]: two of the genus Rhynchophorus (Rhynchophorus phoenicis Fabricius and R. quadrangulus Quedenfeldt; Dryophthoridae): Ukolo and Edon collected from living and dead Raphia trunks, respectively and the third is Oryctes rhinoceros L. (Dynastidae), inhabiting dead Raphia trunks.

The traditional collection method for grub's extraction consists of walking through Raphia populations identifying young stems naturally infested with the grubs [START_REF] Dounias | L'exploitation méconnue d'une ressource connue: la collecte des larves comestibles de charançons dans les palmiers raphias au sud Cameroun[END_REF][START_REF] Muafor | Exploitation, trade and farming of palm weevil grubs in Cameroon[END_REF]. This is a difficult and hard task, and can take considerable time [START_REF] Dounias | L'exploitation méconnue d'une ressource connue: la collecte des larves comestibles de charançons dans les palmiers raphias au sud Cameroun[END_REF][START_REF] Muafor | Exploitation, trade and farming of palm weevil grubs in Cameroon[END_REF]. A semi-farming method is also described in Cameroon [START_REF] Muafor | Exploitation, trade and farming of palm weevil grubs in Cameroon[END_REF] and consists of felling the individual, making an incision on the trunk at about 1 m from the base of the crown and covering the incision with fresh Raphia leaves. The felled trunk is left for around 1 month allowing the grubs to grow and mature. The harvesters then split the trunk and collect the grubs. This type of harvest yields more grubs per stem than the traditional method [START_REF] Muafor | Exploitation, trade and farming of palm weevil grubs in Cameroon[END_REF].

The larvae are rich in several essential nutrients, containing proteins, carbohydrates and fats, equivalent in energy values to beef or fish [START_REF] Fogang Mba | Growing conditions and morphotypes of African palm weevil (Rhynchophorus phoenicis) larvae influence their lipophilic nutrient but not their amino acid compositions[END_REF]. The larvae of Rhynchophorus phoenicis are an important protein source similar in content to termites, cow milk, eggs, and even beef [START_REF] Banjo | The nutritional value of fourteen species of edible insects in southwestern Nigeria[END_REF][START_REF] Ekpo | Nutritional Potentials of the Larva of Rhynchophorus phoenicis (F)[END_REF][START_REF] Fogang Mba | Growing conditions and morphotypes of African palm weevil (Rhynchophorus phoenicis) larvae influence their lipophilic nutrient but not their amino acid compositions[END_REF][START_REF] Okunowo | Nutritional Value of Rhynchophorus phoenicis (F.) larvae, an edible insect in Nigeria[END_REF][START_REF] Opara | Studies on the production trend and quality characteristics of palm grubs in the tropical rainforest zone of Nigeria[END_REF]. They contain elements such as Ca, Cu, Fe, Mg, Mn, P, K, and Na [START_REF] Muafor | Exploitation, trade and farming of palm weevil grubs in Cameroon[END_REF] and provide several essential and non-essential amino acids and lipids [START_REF] Fogang Mba | Growing conditions and morphotypes of African palm weevil (Rhynchophorus phoenicis) larvae influence their lipophilic nutrient but not their amino acid compositions[END_REF][START_REF] Ogbuagu | Fatty acid and amino acid profiles of the larva of Raffia Palm Weevil: Rhynchophorus phoenicis[END_REF]. Rhynchophorus phoenicis has been recommended in the diets of children and adults [START_REF] Onyeike | Nutritive value of the larvae of raphia palm beetle (Oryctes rhinoceros) and weevil (Rhyncophorus pheonicis)[END_REF][START_REF] Opara | Studies on the production trend and quality characteristics of palm grubs in the tropical rainforest zone of Nigeria[END_REF].

Raphia leaves

Thatch

Across its distribution, Raphia species constitutes one of the main resources for thatching (Fig. 2B) (Beccari 1910b;[START_REF] Calvo-Gutiérrez | Uso y conservación de humedales de yolillo Raphia taedigera (Arecaceae) en el area de Conservación Tortuguero, Costa Rica[END_REF][START_REF] Chevalier | Nouvelles recherches sur les Palmiers du genre Raphia (Suite et fin)[END_REF]Obahiagbon 2009;Tuley 1995). After harvesting of leaves, the leaflets are typically woven into thatching units (Fig. 4B) while still attached to the midrib or after their removal. The thatching units are used as ceiling material especially in rural areas. Raphia roofs are not as durable as synthetic alternatives, nevertheless, they remain popular because they are cheaper, have better cooling properties and are not noisy during heavy downpours (Fig. 4B). They are reported to last between three to five years [START_REF] Manvell | Thatch Geographies in Liberia: An Introduction[END_REF].

Nowadays, Raphia thatches are used around hotels and touristic sites to give a more authentic rural look. To a lesser extent, thatching units are also used for constructing fences. The leaves of R. monbuttorum and R. zamiana are more desirable for thatching because they remain straight when dried and last longer. Others like R. hookeri, are less resistant and easily damaged by insects. Extracted leaf midribs are often used to produce brooms, for example in Cameroon [START_REF] Couvreur | A l'Echelle du Raphia / On Raphia and Man[END_REF]. Finally, the entire leaves can be used as rough wrapping material to pack bulky farm products and fire wood for transport.

Raphia Leaf Petiole and Rachis ('Bamboo')

The petiole and rachis serve various purposes and are referred to as bamboo because the long slender poles are reminiscent to Asian bamboo (Tuley 1995). They are useful for fences, walls, ceilings, barns, and roof constructions, to manufacture handicrafts (e.g., beds, tables, and chairs, Fig. 2D,E), tools, and art work, or as support structures for the cultivation of crops (Table S1). In the Amazon esturay, R. taedigera (jupatí) is used for house construction and to make several domestic utensils [START_REF] Carney | Raphia taedigera in the Amazon Estuary[END_REF][START_REF] Wallace | Palm trees of the Amazon and their uses[END_REF]. Traditionally, R. vinifera bamboo is used in the construction of palaces in the West and North West regions of Cameroon [START_REF] Geary | Palace and chieftaincy in Weh (North West Province, Cameroon)[END_REF].

Fibers extracted from the inner tissue of the Raphia poles are split into thinner units and used to make ropes, baskets, and for weaving mats and other articles [START_REF] Mfouapon | Risques économicoenvironnementaux liés aux emballages traditionnels pour vivres frais utilisés dans le bassin maraîcher de Foumbot (Cameroun)[END_REF]. The baskets are particularly popular for storing and transporting agricultural produce to the markets. When dry, the inner part of the petiole is robust and is used in the manufacture of artifacts. The hard outer layer of Raphia poles is also suitable for making basketry, mats and fishing baskets or fishing traps (Fig. 4C, [START_REF] Couvreur | A l'Echelle du Raphia / On Raphia and Man[END_REF], for example the "matapí" traps made out of R. taedigera in the Amazon estuary [START_REF] Carney | Raphia taedigera in the Amazon Estuary[END_REF]. Finally, the pith of the Raphia bamboo is an excellent and flexible material to make crafts and toys (Fig. 4D) of all sizes [START_REF] Carney | Raphia taedigera in the Amazon Estuary[END_REF][START_REF] Couvreur | A l'Echelle du Raphia / On Raphia and Man[END_REF].

Combined Use of Raphia Parts in the Construction of Musical Instruments

In Central Africa, the rachis is used for making various musical instruments. In the case of the traditional harp-zither, commonly referred to as mvet (Fang, Bulu), mbet (Beti-Eton) or ngiang (Ngumba) and ngombi (Baka). The name nú-peké, which translates to Raphia bird in English, refers to the way the instrument is constructed [START_REF] Brisson | Trade in palm products in North-Western South America[END_REF]. The entire instrument is built of one Raphia bamboo and the three or four strings are made of narrow peels of the bamboo epidermis. The strings are stretched over a central wooden bridge and tied firmly at the end of the sound board. These strings are not very durable and are now typically replaced by metal strings. Whereas most of the harp-zithers feature calabashes for sound amplification (Fig. 4E), those of the Baka and Bakoya Pygmies in Cameroon and Gabon (Fürniss, 2012a;[START_REF] Le Bomin | The musical heritage of the Gabon Bongo: Heritage under influence[END_REF] use pieces made of the pith of the Raphia petiole (Fig. 4F, see [START_REF] Couvreur | A l'Echelle du Raphia / On Raphia and Man[END_REF] for sound).

The lamellaphone (Fig. 2C), commonly referred to as sanza in Central Africa or mbira in East Africa, is another musical instrument partly made from Raphia extracted materials. The sanza is composed of a sound board mounted by small lamella, and the instrument is played with the thumbs or forefingers. The sound board can be made from three or four petioles joined together [START_REF] Borel | Les Sanza[END_REF]Oloa Biloa 2015), while the epidermis of the bamboo can be used for the lamella. Raphia made sanzas are widely used across Cameroon, and are also found in communities living alongside the upper Oubangui-river in the Central African Republic and the DRC [START_REF] Laurenty | Les Sanza du Congo[END_REF].

The pith of Raphia petioles are also used to make the sound board of the one-string harp or támìntúbà of the Bagyeli in Cameroon (Fürniss, 2012b;[START_REF] Oloa Biloa | La musique des pygmées Bagyéli (Sud-ouest Cameroun). Étude des conséquences du changement de mode de vie sur le patrimoine musical[END_REF]. The rigid epidermis of Raphia bamboo provides the string of the musical bow mbèlà instrument of the Aka and Ngbaka peoples in the Central African Republic [START_REF] Fürniss | Existe-t-il des instruments de musique pygmé ?[END_REF].

Other Leaf Products

Palm heart or palm cabbage, specifically the immature, whitish leaves from the center of the crown shaft, is reported as a good food source (Beccari 1910c). The palm heart is also an important component of the lowland gorilla's (Gorilla gorilla gorilla) diet in central Africa [START_REF] Fay | Gorillas (Gorilla gorilla gorilla) in the Likouala swamp forests of north central Congo: Preliminary data on populations and ecology[END_REF].

Another interesting reported use in Madagascar for R. farinifera is the extraction of a wax from the dry petioles (Beccari 1910b;[START_REF] Haller | Sur la cire du palmier Raphia Ruffia de Madagascar et sur l'alcool arachique[END_REF][START_REF] Jumelle | Le Raphia Ruffia. palmier à cire[END_REF]; Perrier de la Bathie 1931) and leaves. However, Perrier de la Bathie (1931) suggests this use is longer continued.

Raphia fruits

The pericarp of Raphia fruits is composed an outer layer or epicarp, covered by vertical rows of large scales, and a typically thick, bright yellow, and oily mesocarp, enclosing the seed. The endocarp is not differentiated [START_REF] Dransfield | Genera Palmarum: The evolution and classification of palms[END_REF], which makes the fruit a berry. The fleshy mesocarp is edible in certain species like R. hookeri, R. sese, R. sudanica, R. vinifera and R. zamiana [START_REF] Mogue Kamga | Two new species of Raphia (Palmae/Arecaceae) from Cameroon and Gabon[END_REF][START_REF] Ouattara | Lectotypification de Raphia sudanica A. Chev. (Arecaceae, Calamoideae), et commentaires sur la biologie et la conservation de l'espèce[END_REF]. The mesocarp varies among species in thickness, texture, and chemical composition (Tuley 1995). In West and Central Africa and in Madagascar, the fruits are consumed raw, cooked or boiled [START_REF] Wildeman | Sur quelques palmiers congolais[END_REF][START_REF] Perrier De La Bathie | Le Raphia de Madagascar[END_REF][START_REF] Russell | The Raphia palms of West Africa[END_REF]) and generally eaten alongside other starchy products such as cassava [START_REF] Moyene | Valorisation des produits forestiers non ligneux des plateaux de Bateke en peripherie de Kinshasa (R. D. Congo)[END_REF][START_REF] Mphoweh | The Degradation Of Raffia Palms and its Socio-Economic and Ecological Repercursions: The Case Study of Bamunka, Ndop, North West Province Cameroon[END_REF]). Raphia hookeri fruits represent a rich source of vitamin A, B, and E, comparable to those in citrus fruits, or chick peas [START_REF] Ogbuagu | Vitamins, phytochemicals and toxic elements in the pulp and seed of raphia palm fruit (Raphia hookeri)[END_REF]. In addition, young soft endosperm of immature seeds are edible. In Amazon estuary or Central America, here is no real mention of fruits of R. taedigera being eaten [START_REF] Carney | Raphia taedigera in the Amazon Estuary[END_REF]Hiraoka 1997), although Calvo-Gutiérrez et al. (2013) suggest that it could have been in the past.

Besides food, fruits also provide different products such as oil, used for hair care and cooking (Beccari 1910b;[START_REF] Carney | Raphia taedigera in the Amazon Estuary[END_REF][START_REF] Wildeman | Sur quelques palmiers congolais[END_REF][START_REF] Lely | The useful trees of Northern Nigeria. London, The Crown Agents For The Colonies[END_REF]Tuley 1995).

The fatty substances of the mesocarp contain saponin, a powerful ichthyotoxin used to paralyze fish [START_REF] Ogbuagu | Vitamins, phytochemicals and toxic elements in the pulp and seed of raphia palm fruit (Raphia hookeri)[END_REF]. Older seeds area source of vegetable ivory which is crafted into smaller items such as key holders and jewelry (Fig. 2G). In 2014, for example, more than 107 tons of dry Raphia seeds were exported from Cameroon to China [START_REF] Tieguhong | Mapping of Forestry Investments and timber trade involving Chinese companies in Cameroon[END_REF].

Cultural Significance of Raphia

A key element of social life: daily life, festivals and traditional ceremonies

Raphia wine plays a central role in the social fabric of rural villages and urban areas. Sharing wine is a sign of gratitude in business deals and personal relationships [START_REF] Douglas | Raffia cloth distribution in the Lélé Economy[END_REF][START_REF] Mbuagbaw | The palm wine trade: Occupational and health hazards[END_REF]. It especially important during traditional ceremonies, such as birth celebrations, endowments, dowry ceremonies, enthronement of chiefs, marriages and funerals, during which it is consumed and offered to ancestors and chiefs.

Traditional clothing and accessories woven from Raphia fibers hold high cultural value in traditional and customary ceremonies, like weddings, chief enthronements, funerals, and birth celebrations. These uphold existing customs in, the Western Highlands of Cameroon or the Teke in Gabon, Republic of Congo, and DRC. In western Cameroon, special bags woven from Raphia fibers are used to place gifts and worn by women who gave birth to twins during the Si ("Ceremony of the twins") (Cosiaux, pers. com.). In the pre-colonial era along the Loango coast (western part of the Republic of the Congo and Gabon), Raphia cloth was essential in many events: newborns were laid on Raphia cloth; youths wore Raphia cloth skirts during initiation; the suitor of a young girl brought Raphia clothing and wine as gifts to her family, and the dead were dressed in Raphia clothing [START_REF] Martin | Power, Cloth and Currency on the Loango Coast[END_REF].

Raphia is a primary material linked to customary authority, nobility or royalty. Thatch from Raphia leaves associated to it fiber marks the presence of traditional authorities and notables in parts of Cameroon. In addition, hats and garments of kings, chiefs and traditional authorities are woven from Raphia fiber.

The stripped fibers woven into skirts are closely linked to dance in Central Africa. These "skirts" embellish the mendzan-xylophones among the Bulu-Beti-Fang people in Cameroon, Gabon, and Equatorial Guinea [START_REF] Lacombe | Les archives sonores comme terrain d'investigations : l'exemple des xylophones fang[END_REF]. In southeastern Cameroon, long and large Raphia fibers skirts represent spirits among Pygmy populations and their neighbors [START_REF] Bahuchet | Histoire d'une civilisation forestière I[END_REF]Fürniss, 2011[START_REF] Oloa Biloa | La musique des pygmées Bagyéli (Sud-ouest Cameroun). Étude des conséquences du changement de mode de vie sur le patrimoine musical[END_REF]. Among the Baka, a special song and dance a accompanies the presentation of raw Raphia fibers to the community, before their transformation into the habit of the forest spirit Jengi [START_REF] Arom | Une perspective interdisciplinaire[END_REF].

Raphia as pre-colonial currency

Raphia clothing in different socio-cultural groups, served as barter or as a currency, especially during pre-colonial times [START_REF] Clist | Une archéologie des provinces septentrionales du royaume Kongo[END_REF][START_REF] Douglas | Raffia cloth distribution in the Lélé Economy[END_REF][START_REF] Edoumba | Aperçu sur les monnaies d'Afrique[END_REF][START_REF] Martin | Power, Cloth and Currency on the Loango Coast[END_REF][START_REF] Rivallain | Monnaies d'Afrique: visions africaines et visions européennes[END_REF]. Raphia cloth was used to buy food, household items, tools [START_REF] Martin | Power, Cloth and Currency on the Loango Coast[END_REF], pay legal fees, such as for adultery, liquidate debts, pay tributes and taxes, or purchase slaves [START_REF] Douglas | Raffia cloth distribution in the Lélé Economy[END_REF]. Cloths made of Raphia were bartered for a wide range of goods (e.g., knives, iron items, pottery, fish) from the Kasai region (Njembe, Dinga, Nkutu). The Lele, Teke, Telela, Basonge and Kongo kingdom in DRC used Raphia cloths as a currency [START_REF] Clist | Une archéologie des provinces septentrionales du royaume Kongo[END_REF][START_REF] Edoumba | Aperçu sur les monnaies d'Afrique[END_REF]. The woven square of 40 × 40 cm, called lubongo or libongo (pl. mbongo) in the Kongo, tsulu by the Teke or madiba by the Basonge [START_REF] Douglas | Raffia cloth distribution in the Lélé Economy[END_REF][START_REF] Edoumba | Aperçu sur les monnaies d'Afrique[END_REF]. These pieces of Raphia were gathered together for making larger cloths: one makuta composed of 10 mbongo, and one nta of 30 tsulu. Nowadays, the term mbongo is still used in the Kikongo and Lingala languages to designate money and "wealth" [START_REF] Tem | Lingala/Falansé Français/Lingala[END_REF].

Raphia cloth and "bamboo" are essential for marriage exchanges (bride price) [START_REF] Gautier | Le raphia face à a crise: une production complémentaire qui pourrait être (re)valorisée[END_REF][START_REF] Rivallain | Monnaies d'Afrique: visions africaines et visions européennes[END_REF]). Among the Lélé, a man can "acquire" a wife for 90 Raphia cloths. [START_REF] Douglas | Raffia cloth distribution in the Lélé Economy[END_REF] concludes that "In a sense, raffia [Raphia] keeps its high value, not because of its use as a textile, but because it gives command over women, and, in a polygamous society, women are always scarce". Although the colonial and present day currencies have been adopted and included in the dowry, they cannot replace the legal tender of Raphia products, which underlines their socio-cultural value.

Raphia products used in magico-religious practices

Raphia products are used in various ritual practices [START_REF] Quiroz | The cultural importance of plants in Western African religions[END_REF]. In northwestern Cameroon, Raphia wine is used to control the human embodiment of mischievous or evil spirits called "juju", and is given to chiefs and notables during village festivals, funeral ceremonies and ritual libations (Mogue Kamga pers. obs.). The wine is given to ancestors during ritual offerings [START_REF] Salpeteur | Du palais à l'autopsie: les doublures animales dans une chefferie bamiléké (Cameroun)[END_REF]. Gifts are placed in Raphia fiber bags used to pay tribute to ancestors. Raphia leaves play an important role in spiritual protection e.g., in Liberia and certain parts of Cameroon, leaves of R. vinifera are woven into a curtain, representing a barrier against witchcraft, poison or evil [START_REF] Gruca | Ritual uses of palms in traditional medicine in sub-Saharan Africa : a review[END_REF]. The Baka in Cameroon use this curtain to separate the initiated from the public (Fig. 4G). In south-eastern Cameroon, Raphia fibers chase evil spirits during the circumcision ceremonies [START_REF] Fürniss | Bèkà. Rituel de circoncision chez les Baka occidentaux du Cameroun[END_REF][START_REF] Fürniss | The adoption of the circumcision ritual bèkà by the Baka-Pygmies in Southeast Cameroon[END_REF]. Leaves of R. farinifera in Madagascar are burnt as incense in churches and used to make crosses [START_REF] Gruca | Ritual uses of palms in traditional medicine in sub-Saharan Africa : a review[END_REF]. Dried roots of R. vinifera are sometimes burnt inside dwellings to expel evil spirits [START_REF] Mphoweh | The Degradation Of Raffia Palms and its Socio-Economic and Ecological Repercursions: The Case Study of Bamunka, Ndop, North West Province Cameroon[END_REF]). In the circumcision rituals of the people in southeastern Cameroon, the inner fibrous parts of leaf petioles and rachis are crafted into imitations of weapons and kepis that identify the members of the ritual association as warriors. Finally, the newly circumcised can be recognized by her very long Raphia skirt [START_REF] Fürniss | Bèkà. Rituel de circoncision chez les Baka occidentaux du Cameroun[END_REF]Fürniss, 2011). The Yoruba deity Sopona, the god of illness (smallpox) is usually covered in a mask and a body suit of Raphia palm [START_REF] Voeks | Sacred leaves of Candomble: African magic, medicine and religion in Brazil[END_REF]. This tradition spread to Cuba and Brazil where Raphia fiber is still imported for important ceremonies.

Medicinal Importance of Raphia

Raphia species are used for medicinal purposes including inflammatory, digestive system, muscular-skeletal system, circulatory system, blood system, mental pain, subcutaneous tissue, pregnancy and birth disorders bruises, and poisoning [START_REF] Gruca | African palm ethno-medicine[END_REF]. Medicinal herbs are often added to palm wine to cure a wide range illnesses such as malaria, measles and jaundice. They also are used to prevent diarrhea and headaches [START_REF] Bassir | Observations on the fermentation of palm wine[END_REF][START_REF] Donfack | Importance of Forest products from raffia forests (Rafia farinifera) and impacts of some harvesting methods on natural regeneration in West regions of Cameroon[END_REF][START_REF] Guedje | Garcinia lucida Vesque (Clusiaceae): from traditional uses to pharmacopeic monograph for an emerging local plant-based drug development[END_REF][START_REF] Mphoweh | The Degradation Of Raffia Palms and its Socio-Economic and Ecological Repercursions: The Case Study of Bamunka, Ndop, North West Province Cameroon[END_REF]). Raphia sap stimulates lactation in nursing mothers [START_REF] Donfack | Importance of Forest products from raffia forests (Rafia farinifera) and impacts of some harvesting methods on natural regeneration in West regions of Cameroon[END_REF][START_REF] Okon | Economic Analysis of Raphia Palm (Raphia spp .) Wine Production in Akwa Ibom State, Nigeria[END_REF]. The seeds of R. hookeri are specifically used to stop pulsation of the fontanels on babies' head [START_REF] Gruca | Ritual uses of palms in traditional medicine in sub-Saharan Africa : a review[END_REF]. Recently, unfermented and fermented Raphia hookeri wine was suggested to have antidiabetic and antioxidative potentials [START_REF] Erukainure | Raffia palm (Raphia hookeri) wine inhibits glucose diffusion; improves antioxidative activities; and modulates dysregulated pathways and metabolites in oxidative pancreatic injury[END_REF]. In Gabon, the fruit mesocarp of R. zamiana is used to treat diabetes and high blood pressure (Table 1). The seeds of R. sese are used to regulate and stabilize the sugar level in patients suffering from diabetes [START_REF] Moyenne | Etude de l'exploitation et du marché des produits forestiers non ligneux à Kinshasa[END_REF]. A liquid extracted from the spathes of R. farinifera is used as laxative and the roots for treating toothaches [START_REF] Pernet | Pharmacopée de Madagascar[END_REF].

Conclusion and Perspectives

Due to their multiple uses and cultural value, Raphia species are among the most important species providing NTFPs in Sub-Saharan Africa [START_REF] Ingram | Win-wins in forest product value chains? How governance impacts the sustainability of livelihoods based on non-timber forest products from Cameroon[END_REF]. [START_REF] Barfod | SE Asian palms for agroforestry and home gardens[END_REF] referred to palms as being versatile, based on a number of desirable properties as seen from an agricultural perspective. According to this definition species of Raphia score high in versatility, especially R. hookeri, R. farinifera and R. vinifera, which produce a wide range of products (Table 1). Sap is the main product extracted from species of Raphia. It is used in daily life and plays a key role in traditional ceremonies. The exploitation of wine provides an important and quick access to cash for rural people. The extraction of edible grubs from Raphia stems represent other important product being a major source of protein. Entomophagy has been suggested as a sustainable and viable alternative to meat consumption and can contribute to global food security [START_REF] Machovina | Biodiversity conservation: The key is reducing meat consumption[END_REF][START_REF] Sunderland | Food security: why is biodiversity important?[END_REF][START_REF] Van Huis | Insects to feed the world[END_REF]. Thus, the consumption of Raphia grubs will play a fundamental role in this sense, especially because sustainable and better harvesting methods are now being developed increasing yield and safety [START_REF] Muafor | Improving rural livelihoods through participative research on the domestication and breeding of the palm weevil larvae (Rhynchophorus phoenicis Fabr.) : the African palm weevil project : Final project report 22[END_REF]. Other products such as Raphia fibers might gain importance with an increased interest for plastic alternatives [START_REF] Kocak | The use of palm leaf fibres as reinforcements in composites[END_REF]. Finally, Raphia taxonomy remain challenging, but recent taxonomic progress has improved species identification [START_REF] Couvreur | Flore du Gabon-Palmae[END_REF][START_REF] Mogue Kamga | Two new species of Raphia (Palmae/Arecaceae) from Cameroon and Gabon[END_REF]), which will be useful for future anthropological and socio-economic studies. Although substantial information on the different uses of Raphia has been published, more data is still needed on aspects of commercialization such as market structure, yields, trade volumes, geographic extent of trade, and value chains.

The distribution of continental African palm species will be seriously affected by climate change [START_REF] Blach-Overgaard | Globalchange vulnerability of a key plant resource, the African palms[END_REF]. In particular, Raphia stands are under significant pressure, generally being highly transformed by man (Profizi 1988) or destroyed for urbanization and agricultural expansion [START_REF] Calvo-Gutiérrez | Uso y conservación de humedales de yolillo Raphia taedigera (Arecaceae) en el area de Conservación Tortuguero, Costa Rica[END_REF][START_REF] Mphoweh | The Degradation Of Raffia Palms and its Socio-Economic and Ecological Repercursions: The Case Study of Bamunka, Ndop, North West Province Cameroon[END_REF]). In addition to the versatility of its species and use by man, Raphia swamps in South America, Africa and Madagascar, represent important wetland biodiversity reservoirs harboring numerous plants species and wildlife [START_REF] Calvo-Gutiérrez | Uso y conservación de humedales de yolillo Raphia taedigera (Arecaceae) en el area de Conservación Tortuguero, Costa Rica[END_REF], including iconic megafauna species [START_REF] Rainey | Survey of Raphia swamp forest, Republic of Congo, indicates high densities of Critically Endangered western lowland gorillas Gorilla gorilla gorilla[END_REF]. For example, the endangered western lowland gorilla is recorded in high densities throughout the year in Raphia dominated swamps in the Republic of Congo [START_REF] Rainey | Survey of Raphia swamp forest, Republic of Congo, indicates high densities of Critically Endangered western lowland gorillas Gorilla gorilla gorilla[END_REF]. Efficient management of Raphia products and environments where it occurs will contribute to alleviate poverty, fight against hunger and conserve tropical biodiversity in the decades to come. Thus, we need to convey governments, decision makers and stakeholders with a deeper understanding of the ecosystem services offered by Raphia species.

Fig. 1 .

 1 Fig. 1. A sample of Raphia species. A: R. zamiana from northern Gabon. B. R. regalis from Gabon. C. R. vinifera from northwestern Cameroon, growing in maize and banana fields; D: Close up of a Raphia gabonica fruit. Photos Thomas Couvreur.

Fig. 3 .Fig. 4 .

 34 Fig. 3. The multitude of uses of Raphia species -1. A. Raphia hookeri wine tapping, drone view, Lomié, Cameroon. B. Raphia vinifera wine shared between notables during a ceremony at Oku, Cameoon. C. Recently dyed drying Raphia fibers near Douala, Cameroon. D. Raphia fiber uses for furniture construction near Douala, Cameroon. Photo credits: A: Foulques Couvreur; B: Thomas Couvreur; C-D: Suzanne Mogue Kamga

Table 1 . Raphia species uses reported from the literature with their matching use category (

 1 This species is what Otedoh called R. vinifera. However, the name applies to a different species, and for now we do not know what species this refers to. See Mogue Kamga et al. (2019) for more details

see text for definition).

*

Acknowledgements

This study was supported by Agropolis Fondation (RAPHIA project) under the reference ID 1403-026 through the « Investissements d'avenir » program (Labex Agro:ANR-10-LABX-0001-01), under the frame of I-SITE MUSE (ANR-16-IDEX-0006) and the Agence Nationale de la Recherche (grant number ANR-15-CE02-0002-01), both to TLPC.

Use and Cultural Significance of Raphia Palms

Suzanne MOGUE KAMGA* 1 , Grischa BROKAMP 2 , Ariane COSIAUX 3 , Abdon AWONO 4 , Susanne FÜRNISS 5 , Anders S. BARFOD 6 , Fogoh John MUAFOR 7 , Philippe LE GALL 8 , Bonaventure SONKÉ 1 , and Thomas L. P. COUVREUR 3