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Study area and environmental variables
The data set used in this study included 192 sites in which lepidochronological data and shoot density were acquired between 1994 and 2003. Clearly, the rhizome primary production of P. oceanica was estimated as defined by Pergent-Martini et al.12.
The spatial coverage of the data set was not uniform across the Italian Seas. In fact, the sampling sites were mainly concentrated in five Italian regions, i.e. Liguria, Tuscany, Lazio, Basilicata and Apulia (Fig. 1).
Figure 1

Sampling sites from which field data and indirect measurements have been collected (red circles). Data about several sampling stations are available at each site (N = 6 to 15).

Full size image

The environmental variables were all acquired from maps and other related information sources (Table 1), according to the main aim of the study. A detailed explanation of these variables and of the methodology for their acquisition is given in the supplementary materials.
Table 1 Environmental factors used as predictive variables for developing P. oceanica models.
Full size table

Since these environmental factors were used as predictive variables in the modeling procedure, their selection was based on the ecological nature of the modelled processes, taking into account their influence on the latter. For instance, it is well known that depth plays a crucial role in determining the properties of P. oceanica meadows, such as density and productivity, as it is strictly related to other fundamental environmental factors, e.g. light. Therefore, both depth and gradient were considered as predictive variables, as well as the profile of the isobaths, described as either linear, convex or concave. The presence of sources of disturbance, such as sewage discharge or similar pollution, was also taken into account, as an increase in turbidity following an excessive enrichment from nutrient inputs might entail a reduction of water transparency and light penetration, which in turn can alter the ecological proprieties of a P. oceanica meadow. As for the sea floor typologies, i.e. sand, rock and matte, sources of disturbance have been represented as binary variables because of the intention of using only indirect methods for data acquisition, e.g. maps. Clearly, with such types of data source it was possible to perform, with good confidence, only a qualitative assessment. A quantitative coding of those predictive variables would indeed require expensive and time-consuming efforts for field activities, leading to a major drawback of the proposed approach.
The data set was partitioned into two subsets, i.e. training and test sets, for modeling purposes. Data partitioning represents a critical step in modeling, whose aim is obtaining two subsets that are as much as possible independent from each other, while simultaneously representative of the modelled problem, in order to avoid modeling artifacts and to ensure the applicability of the resulting models18.
Accordingly, the partitioning was not based on random selection of the data, rather the subsets were obtained on the basis of the following approach. The data were stratified according to depth, i.e. they were sorted on the basis of their depth and assigned to one of the following bathymetric classes, i.e.[0,5] m, (5,10] m, (10,15] m, (15,20] m, (20,25] m, (25,35] m. These classes comprised 16.67%, 23.96%, 27.08%, 17.71%, 9.90% and 4.69% of the total number of records, respectively. Subsequently, within each bathymetric class, about 70% of the data, i.e. n = 136, were assigned to the training set, while the remaining ones, i.e. n = 56, to the test set. While the former subset comprising the majority of the data was used for the training procedure of the Machine Learning algorithm, i.e. Random Forest19, the test subset was only used a posteriori to evaluate model performance.
The rationale behind the aforementioned approach is that the depth has a paramount ecological role in regulating both P. oceanica shoot density and rhizome primary production, as previously noted. In fact, a wide range of environmental conditions are related to depth, such as light, water movement and sedimentation flows, which in turn strictly affected the structure, the functioning and the ecological condition of P. oceanica meadows. Therefore, using the abovementioned strategy in the data allocation, the inherent variability of the ecological patterns was properly distributed among the subsets, thus ensuring the possibility of obtaining ecologically sound models.
Random Forest
The Random Forest (RF) is a Machine Learning technique which fits an ensemble of Classification Trees and combines their predictions into a single model19.
RF has proven effective in a wide range of applications as it is able to address, for example, both regression and classification problems20, to perform cluster analysis and missing values imputation21,22.
RF has been used for predicting current and potential future spatial distribution of plant species23, as well as for estimating the marine biodiversity on the basis of the sea floor hardness24. RF has been also applied in ecological applications as a classification tool for the assessment of the vulnerability of P. oceanica meadows over a large spatial scale25, and for land cover classification using remote sensing data26,27.
This method relies upon one of the main features of Machine Learning methods, namely that an ensemble of ‘weak learners’ usually outperforms a single ‘strong learner’19. As a matter of fact, each Classification Tree in the forest represents a weak learner, i.e. a single model, trained on a partly independent data subset, i.e. on a bootstrap sample. Each Classification Tree provides predictions based on the data contained in its bootstrap sample, and many trees are combined into an ensemble model, i.e. into a ‘forest’. The overall output of a RF is obtained by averaging the outcomes of all the trees for regression applications, while it is based on majority voting for classification problems.
The diversity of the trees in the forest is ensured by the use of random subsets of data for the tree-building process, i.e. bootstrap samples, as well as by making a random subset of predictive variables available for the tree splitting procedure. These features allow the RF to reduce the correlation among its Classification Trees, while keeping the variance relatively small, thus leading to a more robust model19.
The selection of a random subset of predictive variables at each split ensures maintaining a certain level of randomness during the tree construction process28, and is necessary for the proper functioning of RF. As a matter of fact, the size of the random subset of predictive variables available for the tree splitting procedure represents a tuning parameter, defined as mtry. The latter together with the minimum number of records to be contained in each leaf, called nodesize, are the main tuning parameters that deeply affect RF performance21,29.
In its original work, Breiman19 suggested to set the mtry value equal to p/3 for regression applications, being p is the total number of predictors, and tuning it from half to twice its original value. On the other hand, nodesize and ntree (the latter parameter is the total number of Classification Trees in the forest) are more related to the generalization ability of the RF, and to the overall complexity of the model. Growing a very large forest, e.g. ntree  > 500, or growing the trees to achieve a high degree of purity at their leaves, e.g. nodesize  More

Main features of mycorrhizal genomes
We compared 62 draft genomes from mycorrhizal fungi, including 29 newly released genomes, and predicted 9344–31,291 protein-coding genes per species (see “Methods”, Supplementary Information and Supplementary Data 1). This set includes new genomes from the early diverging fungal clades in the Russulales, Thelephorales, Phallomycetidae, and Cantharellales (Basidiomycota), and Helotiales and Pezizales (Ascomycota). We combined these mycorrhizal fungal genomes with 73 fungal genomes from wood decayers, soil/litter saprotrophs, and root endophytes (Fig. 1 and Supplementary Data 2). There was little variation in the completeness of the gene repertoires, based on Benchmarking Universal Single-Copy Orthologs (BUSCO) analysis (coefficient of variation, c.v. = 7.98), despite variation in assembly contiguity (Fig. 1). Genome size varied greatly within each phylum, with genomes of mycorrhizal fungi being larger than those of saprotrophic species (Figs. 1 and 2, and Supplementary Data 2; P  More

Chemical properties of submicrometer aerosols
Figure 1a presents the time series of the mass concentrations of OC and water-soluble OC (WSOC) in the submicrometer aerosol particles during the cruise. Overall, the patterns of the temporal OC and WSOC variations were similar to each other (r = 0.632, p  More

Quantifying structural sensitivity in models with uncertain component functions
In general, we consider a system of the form:

$$begin{aligned} dot{{mathbf{x }}}={mathbf{G}} left( h_1left( {mathbf{x}} right) ,h_2left( {mathbf{x}} right) ,ldots ,h_pleft( {mathbf{x}} right) , f_1left( {mathbf{x}} right) ,f_2left( {mathbf{x}} right) ,ldots ,f_{m-p}left( {mathbf{x}} right) right) , end{aligned}$$
(1)

where ({mathbf {x}}in {mathbb {R}}) is the vector of d state variables, (h_i, f_i:{mathbb {R}}^{d_i}rightarrow {mathbb {R}}) are the m different component functions describing the inflows and outflows of biomass, energy or individuals due to certain biological processes, with ({mathbf{G}} :{mathbb {R}}^mrightarrow {mathbb {R}}^d) being a composition function describing the general topology of the system. We consider that the precise mathematical formulation of the functions (f_i) are known (or at least postulated) with the only related uncertainty being the precise choice of their parameters. The functions (h_1,ldots h_p) are considered to have unspecified functional form. Instead, they are represented by bounds on their derivatives matching the qualitative properties we would expect from such a function. For example, the per-capita reproduction rate of a population is generally decreasing, at least at large population numbers, while a feeding term described by a Holling type II functional response of a predator should be increasing and decelerating. The (h_i) may also have quantitative bounds on their values:

$$begin{aligned} h_i^{text {low}}left( {mathbf{x}} right)1)), an important question remains: which of the unknown functions contribute the most to the degree of structural sensitivity in the system? The degree of structural sensitivity does not distinguish between the various sources of uncertainty and therefore cannot quantify the relative contributions of the unknown functions to the uncertainty in the model dynamics.
To determine the contribution of each unknown function (h_i), one can allow the error terms (left( varepsilon _1,varepsilon _2,ldots ,varepsilon _pright)) to vary with the goal of investigating how the degree of sensitivity varies with them. For the purpose of this section, let us denote the initial error terms by (varepsilon _i^0). We might be tempted to use the dependence on the (varepsilon _i) to perform global optimisation under certain constraints to find the best possible reduction of (left( varepsilon _1,varepsilon _2,ldots ,varepsilon _pright)). However, one should bear in mind that this analysis would depend on the base functions (hat{h}_i) considered. While these functions are ideally fitted to experimental data, they are only accurate within the error terms (varepsilon _i^0). Excessively reducing the (varepsilon _i) will force all admissible functions to conform strongly in their shape to these base functions far beyond their demonstrated accuracy of fit.
The dependence of the degree of sensitivity on (varepsilon _i) should therefore only be evaluated locally by calculating the gradient (left( frac{partial Delta }{partial varepsilon _1},ldots ,frac{partial Delta }{partial varepsilon _p}right) |_{left( varepsilon _1^0,ldots varepsilon _p^0right) }) giving the direction for the best local reduction of the errors. To adjust for the fact that the error terms may be of different orders of magnitude, when handling the vectors of error terms we should use the norm

$$begin{aligned} left| varvec{varepsilon }right| = sqrt{sum _{i=1}^{p} left( frac{varepsilon _i }{varepsilon _i^0}right) ^2}. end{aligned}$$
(9)

Working in this norm, the gradient needs to be weighted by the initial error terms to provide the direction for the best local reduction of the error terms, this is described by the following structural sensitivity gradient.
Definition 2
The structural sensitivity gradient in a model with p unknown functions each having an error of magnitude (varepsilon ^0_i) is defined as

$$begin{aligned} left( -varepsilon _1^0cdot frac{partial Delta }{partial varepsilon _1},ldots ,-varepsilon _p^0cdot frac{partial Delta }{partial varepsilon _p}right) |_{left( varepsilon _1=varepsilon _1^0,ldots ,varepsilon _p=varepsilon _p^0right) }, end{aligned}$$
(10)

where (Delta left( varepsilon _1,ldots ,varepsilon _pright)) is the degree of structural sensitivity of the system considered as a function of the error terms (varepsilon _i).
One possible problem with the structural sensitivity gradient is that the degree of structural sensitivity in the system may not be an increasing function of the magnitude of the errors. Consider the case that the exact system is structurally unstable, e.g. at a bifurcation point. Then no matter how small the error terms are, there may still be very high levels of structural sensitivity, while larger error terms may cause the level of uncertainty to decrease5. In this case, the structural sensitivity gradient will indicate that one or more of the functions has a negative contribution to the uncertainty of the system, and cannot be taken as a basis for sensitivity analysis.
An alternative approach to quantifying the individual impact of unknown functions which avoids this issue is the computation of partial degrees of sensitivity with respect to each (h_k). To do this, we fix every unknown function except (h_k), a set which we denote ({mathbf{H}} _{sim k}), by fixing the (x_j^*), (h_ileft( {mathbf{x}} ^*right)), and (frac{partial h_i}{partial x_j} left( {mathbf{x}} ^* right)) that are consequently determined by the isocline equations. Denoting by (V_k) the cross-sections of V where only (h_k) varies, and the cross-sections for ({mathbf{H}} _{sim k}) by (V_{sim k}), the local partial degree of structural sensitivity can be defined as follows.
Definition 3
The local partial degree of structural sensitivity with respect to (h_k), is the degree of structural sensitivity in the model when (h_k) is unspecified and all other functions (h_i in {mathbf{H}} _{sim k}) are fixed:

$$begin{aligned} Delta _k({mathbf{H}} _{sim k}):= 4 cdot int _{V_{k_{text {stable}}}} rho _{mathbf{H} _kvert {mathbf{H}} _{sim k}} , d{mathbf{H}} _k cdot left( 1 – int _{V_{k_{text {stable}}}} rho _{mathbf{H} _kvert {mathbf{H}} _{sim k}} , d{mathbf{H}} _k right) , end{aligned}$$
(11)

where (rho _{mathbf{H} _kvert {mathbf{H}} _{sim k}}) is the conditional probability density function on ({mathbf{H}} _{sim k}):

$$begin{aligned} rho _{mathbf{H} _kvert {mathbf{H}} _{sim k}} = frac{rho }{int _V rho d{mathbf{H}} _{sim k }}, end{aligned}$$

with (rho) the (joint) probability distribution over V.
The local partial sensitivity (Delta _k({mathbf{H}} _{sim k})) is a function of ({mathbf{H}} _{sim k}) in that it depends upon the particular values at which the elements of (V_{sim k}) are fixed. As with the degree of structural sensitivity, it can be interpreted as either the probability that the stability of the given equilibrium will be different for two independent choices of the function (h_k) when the (h_{sim k}) are fixed at the given values, or in terms of variance as (Delta _k({mathbf{H}} _{sim k})=4cdot {text {Var}}_k(Yvert {mathbf{H}} _{sim k})) (Y is the Bernoulli variable for stability). If the joint probability distribution (rho) is uniform in V, then (Delta _k({mathbf{H}} _{sim k})) can be expressed purely in terms of the fraction of the volume of (V_k) which gives a stable equilibrium:

$$begin{aligned} Delta _k({mathbf{H}} _{sim k}) = 4 cdot frac{int _{V_{k_{text {stable}}}} d{mathbf{H}} _k}{int _{V_k} d{mathbf{H}} _k} cdot left( 1 – frac{int _{V_{k_{text {stable}}}} d{mathbf{H}} _k}{int _{V_k} d{mathbf{H}} _k} right) . end{aligned}$$
(12)

To obtain a global measure for the sensitivity of the model to (h_k), we can take the average of (Delta _k) over (V_{sim k}).
Definition 4
The partial degree of structural sensitivity with respect to (h_k) is given by

$$begin{aligned} bar{Delta }_k := int _{V_{sim k}} rho _{mathbf{H} _{sim k}} cdot Delta _k({mathbf{H}} _{sim k}) , d{mathbf{H}} _{sim k} end{aligned}$$
(13)

where (rho _{mathbf{H} _{sim k}}) is the marginal probability density function of ({mathbf{H}} _{sim k}).
Recalling the variance-based interpretation of the degree of sensitivity, we obtain (bar{Delta }_k = 4cdot E_{sim k}({text {Var}}_kleft( Yvert {mathbf{H}} _{sim k}right) )). In other words, (bar{Delta }_k) gives the scaled average variance when all functions except (h_k) are fixed. We can also relate the partial degree of structural sensitivity to indices used in conventional variance-based sensitivity analysis. Dividing (bar{Delta }_k) by the overall degree of structural sensitivity in the model gives us (frac{bar{Delta }_k}{Delta }=frac{E_{sim k}({text {Var}}_kleft( Yvert {mathbf{H}} _{sim k}right) )}{{text {Var}}(Y)}=S_{T_k}), the total effect index23 of (h_k) on the stability of the equilibrium. This is a measure of the total contribution of (h_k) to the sensitivity—both alone and in conjunction with the other functions ({mathbf{H}} _{sim k}). However, since the space of valid functions V is in general not a hypercube, the functions (h_i) are not independent factors, and a total decomposition of variance is not possible. Indeed, even if the joint probability distribution (rho) is uniform, the marginal probability distribution (rho _{mathbf{H} _{sim k}}) will generally not be: instead it will equal the volume of the corresponding cross-section (V_k) for ({mathbf{H}} _{sim k}), divided by the volume of V. An alternative to using the partial degrees of sensitivity would be to consider the first-order sensitivity indices (S_k=frac{{text {Var}}_kleft( E_{mathbf{H} _{sim k}}left( Yvert h_k right) right) }{{text {Var}}(Y)}). However, these do not take into account possible joint effects of the (h_i) on the structural sensitivity of the system, so a small (S_k) does not indicate that (h_k) is not a source of sensitivity, whereas (bar{Delta }_k=0) means that (h_k) does not contribute to the structural sensitivity in the system at all.
Similarly to the gradient of the total degree of sensitivity (Delta) as a function of the respective error tolerances, the vector (left( -bar{Delta }_{h_1},ldots , -bar{Delta }_{h_p}right)) needs to be scaled by the elements of (varepsilon ^0) to give us the optimal direction of decrease in (Delta) if the error terms (varepsilon _i) are subject to a proportional reduction. This is described by (left( -varepsilon _1^0 bar{Delta }_{h_1},ldots , -varepsilon _p^0bar{Delta }_{h_p}right)).
Outline of an iterative framework of experiments for reducing structural sensitivity
When dealing with partially specified models, an important practical task is the reduction of the overall uncertainty in the system by decreasing the uncertainty in the system processes (i.e. the unknown model functions). Here we propose an iterative process of such a reduction based on improving our empirical knowledge of the uncertain functions (h_k).
As a starting point, we assume that experiments have produced data on the unknown functions (h_1,ldots ,h_p), to which we can fit some base functions (hat{h}_1,ldots ,hat{h}_p) with initial errors (varepsilon _1^0,ldots ,varepsilon _p^0). We assume that it is possible to perform additional experiments on all uncertain processes in order to obtain more data such that the (varepsilon _i) can be decreased, but with the natural constraint that the total error can only be reduced by a magnitude of (0 More