Gilles Pujol
email: pujol@emse.fr

Simplex-based screening designs for estimating metamodels

Keywords: computer experiments, sensitivity analysis, factor screening, elementary effect, simplex design

The screening method proposed by Morris in 1991 allows to identify the important factors of a model, including those involved in interactions. This method, known as the elementary effects method, relies on a "one-factor-at-a-time" (OAT) design of experiments, i.e. two successive points differ only by one factor. In this article, we introduce a non-OAT simplex-based design for the elementary effects method. Its main advantage, compared to Morris's OAT design, is that the sample size doesn't collapse when the design is projected on sub-spaces spanned by groups of factors. The use of this design to estimate a metamodel depending only on the (screened) important factors is discussed.

Introduction

The framework of this article is experimentation with deterministic computer codes (simulation models), as presented for example by Santner et al. [START_REF] Santner | The design and analysis of computer experiments[END_REF]. Computational models are used when the direct investigation of some real phenomena is expensive, dangerous, or even impossible. However, there are three main obstacles to the study of a computational model: the computation time, the number of inputs, and the size of the input space. Increasingly, the purpose of computer experiments is to study the model over a large range of inputs, rather than around some specific values; see for example Jones et al. [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF].

In such situations, reducing the input dimensionality is a necessity. For this purpose, screening methods aim at identifying the non-important input parameters at a low computational cost. The screening design proposed by Morris [START_REF] Morris | Factorial sampling plans for preliminary computational experiments[END_REF] may be effective because it doesn't rely on a strong prior assumption about the model. In a recent work, Alam et al. [START_REF] Alam | Using Morris' randomized OAT design as a factor screening method for developing simulation metamodels[END_REF] use this design for developing a metamodel, i.e. an approximation of the model. The procedure has two steps: a first design (Morris's OAT) for the screening, and a second design (latin hypercube design in the subspace spanned by the important inputs, while the non-important inputs are fixed) for estimating the metamodel.

In this article, we propose a screening design that can be reused for the estimation of the metamodel, thereby improving the economy of the second set of simulations. Indeed, Morris's design is not well-suited for metamodel estimation: there is an important loss of points when this design is projected into the subspace formed by ignoring some inputs (here, the non-important inputs), so-called collapsing of the design.

Section 2 recalls Morris's screening method with the last improvements by Campolongo et al. [START_REF] Campolongo | An effective screening design for sensitivity analysis of large models[END_REF]. Using this method in the framework of metamodel estimation is discussed in section 3. Section 4 introduces the new design, and its efficiency is illustrated through an example. The starting point of Morris [START_REF] Morris | Factorial sampling plans for preliminary computational experiments[END_REF] is that traditional screening methods, based on the theory of designs of experiments (Fisher [6]), rely on strong assumptions, such as monotonicity of outputs with respect to inputs, or adequacy of a loworder polynomial approximation. In contrast, the method he proposed doesn't rely on such assumptions. His method is referred to as the elementary effects method, and is more and more popular thanks to its moderate computational cost and its graphical aspects (as we shall see).

Let the function

f : x ∈ Ω → y ∈ R denote the computational model. The input space Ω is a subset of R p ; in most cases it is a hypercube: Ω = p i=1 [a i , b i ].
The p input parameters x = (x 1 , . . . , x p ) are called the factors, and the scalar output y is called the response. In practice, however, simulation models have multiple outputs. Assuming that the response is scalar isn't restrictive in this article since we only consider initial designs, and not adaptive ones; therefore, the screening can be done separately for each response based on the same design.

The objective of screening is to split the factors into two subsets x = (u, v), where u (resp. v) is the vector of the important (resp. non-important) factors. "Non-important" means that these factors can be fixed, while having little impact on the model response (see for example Alam et al. [START_REF] Alam | Using Morris' randomized OAT design as a factor screening method for developing simulation metamodels[END_REF]). In section 3, we present an alternative approach to factor fixing, based on regression.

The design of experiments

Morris's design has the following characteristics; also see figure 1(a):

(1) the design belongs to the family of fractional factorial designs, i.e. the points are sampled from a p-dimensional regular grid; (2) the design is structured in groups of points, called trajectories; these trajectories are random, but follow a specific scheme: (a) the trajectories are one-factor-at-a-time (OAT), i.e., two successive points differ by one factor only; (b) this occurs exactly once for each factor and for each trajectory.

To illustrate the construction of a trajectory, a base point x 0 is randomly chosen on the grid, and each coordinate is increased or decreased in turn:

x i = x i-1 + ∆ i e i for i = 1 . . . p, where ∆ i is a multiple of the grid spacing in the ith direction, e ii = 1, and e ij = 0 if i = j. In practice, the trajectories aren't generated with this scheme, but in one step with a matrix approach (see Morris [START_REF] Morris | Factorial sampling plans for preliminary computational experiments[END_REF]).

Composed of R trajectories of p+1 points each, the design has R(p+1) points; for example in figure 1(a), R = 10 and p = 3. Then, the number of points of the design is linear with respect to the number of factors. The number of trajectories R should be large enough to compute statistics such as means and standard deviations (see section 2.2). In the following, x (r) i denotes the ith point of the rth trajectory (i = 0, . . . , p, r = 1, . . . , R).

A randomly generated design can have a poor coverage of the space, especially if the number of points is low with respect to the input space dimension. Space-filling-designs (SFD) were introduced to assure a better spread of the points over the input space (see for example Santner et al. [START_REF] Santner | The design and analysis of computer experiments[END_REF]). Following these principles, Campolongo et al. [START_REF] Campolongo | An effective screening design for sensitivity analysis of large models[END_REF] improved Morris's design by maximizing the distances between the trajectories.

The screening method

The structure of the design allows to calculate, for each trajectory r = 1, . . . , R, one elementary effect per factor, i.e. the increase or the decrease of the response when the considered factor is disturbed, while the other factors are fixed:

d (r) i = f (x (r) i ) -f (x (r) i-1 ) ∆ i , i = 1 . . . p (1)
(recall that x (r) i = x (r) i-1 + ∆ i e i ).
The elementary effects are then post-processed into statistics expressing the sensitivities of the factors. The first statistic is the mean μi ,

μi = 1 R R r=1 d (r) i , i = 1 . . . p . (2)
μi is a measure of the ith factor's importance. Noting that elementary effects with opposite signs cancel each other, Campolongo et al. [START_REF] Campolongo | An effective screening design for sensitivity analysis of large models[END_REF] suggest to consider instead the mean of the absolute value:

μ * i = 1 R R r=1 |d (r) i | , i = 1 . . . p . (3)
Empirical studies [START_REF] Campolongo | An effective screening design for sensitivity analysis of large models[END_REF] tend to show that μ * i proxies the so-called "total sensitivity index", denoted S T i (Homma and Saltelli [START_REF] Homma | Importance measures in global sensitivity analysis of model output[END_REF]). The third statistic is the standard deviation σi ,

σi = 1 R -1 R r=1 (d (r) i -μi ) 2 , i = 1 . . . p . (4)
σi is either a measure of the non-linearities with respect to the ith factor, or a measure of the interactions involved with the ith factor, or both. Morris highlights that his method doesn't allow to distinguish between non-linearities and interactions, remarking however that data analysis could give insight into these phenomena.

To screen the factors, the statistics μi , μ * i and σi are simultaneously considered. In practice, a graph representing σi versus μ * i for i = 1, . . . , p is sufficient to distinguish between three groups of factors: 3 Using the Elementary Effects Method with metamodeling Alam et al. [START_REF] Alam | Using Morris' randomized OAT design as a factor screening method for developing simulation metamodels[END_REF] present the elementary effects method as a tool to reduce the dimensionality, when estimating complex metamodels such as neural networks, support vector machines, kriging, etc. Indeed, these metamodels are used when the relationship between the inputs and the output cannot be represented by simple models, such as polynomial approximations; in this case, the screening procedure should not rely on a strong assumption about the form of the simulation model. That is why the elementary effects method is one of the rare screening methods that can be used in that context.

After the screening phase, the common practice is to use a second design for the regression where the non-important factors are fixed at their nominal values. This implies that more simulations must be run. However, even if the computational cost of the elementary effect method can be considered as moderate (linear with respect to the number of factors), it often reaches the maximally allowed number of simulations. Then, there is no other choice than estimating the regression metamodel from the screening design points only.

We recall that x = (u, v) denotes the splitting between important and nonimportant factors, according to the screening results (i.e. the values of the statistics μ * i and σi ). The objective is to build a metamodel f that depends only on important factors:

y = f (u, v) = f (u) + ε(u, v) , ( 5
)
where ε is a residual random variable representing the gap between the simulation model and the metamodel; we assume that f(u) and ε(u, v) are noncorrelated, and that E(ε(u, v)|u) = 0. The variance of ε gives information about the quality of the screening. Indeed,

var(ε) var(y) = 1 - var( f (u)) var(y) (6) = 1 - var(E(y|u)) var(y) (7) = E(var(y|u)) var(y) , (8)
where [START_REF] Iooss | Global sensitivity analysis of computer models with functional inputs[END_REF] follows from (7) by the total variance law. Hence, var(ε)/var(y) is the so-called "total sensitivity index" with respect to the non-important factors, denoted S Tv (Homma et Saltelli [START_REF] Homma | Importance measures in global sensitivity analysis of model output[END_REF]). Here, the non-important factors v are considered as noise in the model, and the index S Tv is a measure of this noise over the response y (as in Iooss and Ribatet [START_REF] Iooss | Global sensitivity analysis of computer models with functional inputs[END_REF]). A low value confirms that the dropped factors are really non-important. In this sense, the regression model provides a post-validation of the screening results. It must be noted that, when fixing the non-important factors, this post-validation cannot be provided: the practitioner has to trust the screening results.

To estimate a metamodel depending only on the important factors (u), the orthogonal projections of the points of the design onto the subspace formed by the u-coordinates are considered. In Morris's OAT design, the aligned points collapse through the projections, leading to a loss of points1 . For example, figure 1(a) represents such a design in p = 3 dimensions with R = 10 trajectories (altogether 40 points), and figure 1(b) shows the projection of this design onto the (x 1 , x 2 )-plane (27 remaining points), and onto the x 1 -axis (7 remaining points). The loss of points can be explained by the OAT structure (one point lost per eliminated dimension and per trajectory), but also by the grid structure of the design. To illustrate, if α% of the factors are important, the loss of points is greater than (1α)% for the regression. This can be dramatic in high dimensions, because α is expected to be lower than 20%: the loss of points is then greater than 80%. Therefore, we develop simplex-based designs.

4 Simplex designs for computing the elementary effects [Fig. 2

about here.]

To avoid the loss of points, the idea is to allow a better flexibility in the way the trajectories are done. In our new design, the trajectories are assumed to be simplexes. A simplex is the p-dimensional analogue of a triangle in two dimensions. Specifically, a simplex is the convex hull of a set of p + 1 linearly independent points. In this article, the term "simplex" is referring only to the nodes, i.e. a sequence of p + 1 points x i = (x i1 , . . . , x ip ), i = 0 . . . p. To illustrate, figure 2(a) represents such a design in three dimensions.

The design is composed of R different random simplexes in the domain. The simplexes are successively generated, and the space filling improvement referred to in section 2.1 (Campolongo et al. [START_REF] Campolongo | An effective screening design for sensitivity analysis of large models[END_REF]) can also be applied. The simplexes can be of any shape. Technical details about simplex generation are discussed in appendix A. The simplex-based designs enable the computation of the elementary effects. Fitting a first-order polynomial to each simplex,

y (r) i = c (r) 0 + p j=1 c (r) j x (r) ij , r = 1 . . . R , i = 0 . . . p , (9)
implies the assumption that the model is without interactions. The coefficients c

(r) i are then proxies of the elementary effects d (r)

i . Hence, the statistics μi , μ * i and σi can be computed by ( 2)-( 4).

It is important to notice that the size of the simplexes must be chosen properly. Small simplexes imply that the model has to be linear (in the factors x) without interactions at a local scale, whereas with simplexes that spread over the input space each, the model is assumed to be globally linear without interactions.

Example

[Fig. 3

about here.]

We illustrate the elementary effects method with a simplex-based design through the function introduced by Morris [START_REF] Morris | Factorial sampling plans for preliminary computational experiments[END_REF]:

y = f (x 1 , . . . , x 20 ) = β 0 + 20 i=1 β i w i + 20 i<j β ij w i w j + 20 i<j<l β ijl w i w j w l + 20 i<j<l<s β ijls w i w j w l w s , (10)
where Ω = [0, 1] 20 , w i = 2(x i -1/2) except for i = 3, 5, 7, where w i = 2(1.1x i /(x i + 0.1) -1/2). Coefficients of relatively large value are assigned as: β i = 20, i = 1, . . . , 10; β ij = -15, i, j = 1, . . . , 6; β ijl = -10, i, j, l = 1, . . . , 5; β ijls = 5, i, j, l, s = 1, . . . , 4. The remaining first and second order coefficients are generated independently from a standard normal distribution N (0, 1), and the remaining third and fourth order coefficients are set to zero.

The non-important factors are x 11 , . . . , x 20 . Concerning the important factors, their effects are linear, except for x 3 , x 5 and x 7 which are non-linear; factors involved in interactions are x 1 , . . . , x 4 (second, third and fourth-order interactions), x 5 (second and third-order interactions) and x 6 (second-order interactions). Hence, the expected result for the elementary effects method is

(1) non-important factors: x 11 , . . . , x 20 ;

(2) factors with linear effects without interactions: x 8 , x 9 and x 10 ;

(3) factors with non-linear effects and/or interactions: x 1 , . . . , x 7 .

We applied the elementary effects method with a simplex-based design and compare it with Morris's OAT design. In both cases, the number of OAT trajectories/simplexes is R = 10; for the OAT design, the grid spacing is 1/4 (5 levels for each factor); for the simplex design, the simplexes are orthonormal, with an homothety parameter of 1/4 (see appendix A); hence, the simplexes are random rotations of OAT trajectories. The OAT trajectories/simplexes were generated at random (no space filling improvement). Both methods were repeated 100 times. The results are presented in figure B.3. In both cases, the results are as expected. Moreover, the 100 replicates allow to see that:

• the estimation of the statistics based on the elementary effects has a large variability, due to the small size of the samples (R = 10); • with the simplex-based design, the effects of the factors of the third group (interactions and/or non-linear effects) are underestimated; indeed, for these factors, the model isn't linear at the scale of a simplex (here 1/4); however, although underestimated, these effects are properly identified as interactions or non-linearities.

Conclusion

In this article, we have proposed simplex-based designs as an alternative to Morris's OAT designs for the elementary effects screening method. Contrary to Morris's designs where no explicit assumption is made, the simulation model is now supposed to be linear without interactions at the scale of a simplex; however, if the simplexes are small enough, this assumption isn't too restrictive. The advantage of simplex-based designs over Morris's designs is that they keep most of their points (non-collapsing) after the projections on the subspaces formed by groups of factors (typically, important factors), so they are well-suited for metamodel estimation.

The methodology presented in this article -screening followed by regression -may be compared with the forward (stepwise) variable selection procedures where the screening is incorporated into the metamodel estimation procedure. Although these methods are efficient when the assumptions underlying the metamodel fit the problem, it may be difficult to choose, a priori, a kind of metamodel. For example, Welch et al. [START_REF] Welch | Screening, predicting, and computer experiments[END_REF] assume that the model can be approximated through a kriging metamodel, and their screening procedure relies on this assumption. With the methodology presented in this article, the metamodel can be chosen after the screening. Then, several metamodels (regression, kriging, etc.) can be estimated, in order to choose the best one.

The two following issues may be investigated in future research.

The first issue concerns the quality of the design through the different projections. With the space filling algorithm proposed by Campolongo et al. [START_REF] Campolongo | An effective screening design for sensitivity analysis of large models[END_REF], the design is optimized to cover the space over the p dimensions, but it doesn't imply that the subspaces of dimensions lower than p would be correctly filled (i.e. have a nearly uniform distribution). Moreover, it seems impossible to control the quality of the design through all the projections. One possibility could be to optimize the design only in the subspaces formed by single factors, in order to obtain something like a latin hypercube design (following for example Griensven et al. [START_REF] Van Griensven | A global sensitivity analysis tool for the parameters of multivariable catchment models[END_REF]). Another possibility could be to use a priori knowledge of the experimenter, who says that a specific group of factors must be important, and to optimize the design in the subspace formed by these factors.

The second issue concerns the extraction of simplex-based designs from any given design. In particular, this would permit to apply the elementary effects method on existing databases of simulations that don't have the required structure. Two methods have been tested. The first was to sketch an ideal simplex-based design, and then to take the simplexes defined by the nearest points of the real design. The second was to build the Delaunay tessellation of the design (see for example Watson [START_REF] Watson | Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes[END_REF]). However, these two approaches seem to be computationally expensive in high dimensions.

A Generating a random simplex in the input space A random simplex is generated by applying two random transformations to a reference simplex: a rotation to orient it, and a translation to place it in the input space. In the following, we detail how to construct the reference simplex, and how to rotate it.

The reference simplex is represented by a (p + 1) × p matrix

S ref =        x 01 . . . x 0p . . . . . . . . . x p1 . . . x pp        . (A.1)
The reference simplex can be of any shape. The two common shapes are:

(1) the orthonormal simplex, given by: x 0 = (0, . . . , 0), x i = e i , i = 1 . . . p.

(2) the isometric simplex, given by: x 0 = (0, . . . , 0), x 1 = e 1 , and for i ≥ 2,

x ij = 1 n k<i x kj if j < i, x ii = 1 -k<i x ik 2 and x ij = 0 if j > i.
A homothety must be applied to the reference simplex, in order to scale it according to the assumed linearity of the model (see section 4).

The rotation is represented by a p × p matrix belonging to the special orthogonal group SO(p), i.e. having determinant = +1. Let R ij (θ ij ) denote the element of SO(p) which rotates the basis vector e i through an angle θ ij towards the basis vector e j inside the (e i , e j )-plane, leaving the complementary coordinates fixed. R ij (θ ij ) is a p × p diagonal matrix having 1 on its diagonal, where the 2 × 2 sub-matrix indexed by (i, j) × (i, j) is the plane rotation of angle θ ij . The full rotation matrix is the product of all the matrices R ij (θ ij ), and the angles θ ij are referred to as "Euler angles"; there are (p 2p)/2 Euler angles (the number of coordinate-planes in R p ); see for example Arnold [START_REF] Arnold | Mathematical methods of classical mechanics[END_REF].

B Software

The programs used in this article were written in R [START_REF] Development | R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing[END_REF]. The elementary effects method, either with Morris's OAT and simplex-based designs, is implemented in the R package sensitivity, available at http://cran.R-project.org .

List of Figures

B.1 In the background, the superimposition of 100 independent studies. In the foreground, the means over the 100 replicates. 15 Projection of the points of the design on the (x 1 , x 2 )-plane, and, in dotted lines, on the x 1 -axis (see section 3).

2

  Morris's Elementary Effects Method [Fig. 1 about here.]

( 1 )

 1 negligible factors (low μ * i ); (2) factors with linear effects without interactions (high μ * i and low σi ); (3) factors with non-linear effects and/or interactions (high μ * i and σi ). See, for example, figure B.3 (discussed later in the paper).

Figure 2 (

 2 Figure 2(b) shows the projections of the design of figure 2(a). As expected, most of the points don't collapse through the successive projections.

  (a) Morris's OAT design in p = 3 dimensions with R = 10 trajectories. (b) Projection of the points of the design on the (x 1 , x 2 )-plane, and, in dotted lines, on the x 1 -axis (see section 3). 13 B.2 (a) Simplex-based design in p = 3 dimensions with R = 10 simplexes. (b) Projection of the points of the design on the (x 1 , x 2 )-plane, and, in dotted lines, on the x 1 -axis. 14 B.3 Elementary effects sensitivity test with (a) a simplex-based design and (b) a Morris's OAT design, on Morris's function at the computation cost of N = 210 (p = 20, R = 10). Plots of σ versus μ * .

Fig. B. 1 .

 1 Fig. B.1. (a) Morris's OAT design in p = 3 dimensions with R = 10 trajectories. (b)Projection of the points of the design on the (x 1 , x 2 )-plane, and, in dotted lines, on the x 1 -axis (see section 3).

Fig. B. 2 . 7 x 8 ( 1 , x 2 , x 4 x 3 , x 5 x 6 x 7 x 8

 278143678 Fig. B.2. (a) Simplex-based design in p = 3 dimensions with R = 10 simplexes. (b) Projection of the points of the design on the (x 1 , x 2 )-plane, and, in dotted lines, on the x 1 -axis.

Although this is commonly referred to as a "loss of points" (see for example Morris[START_REF] Morris | Factorial sampling plans for preliminary computational experiments[END_REF]), this is in fact a loss of information: all the points will be used for the regression, but one projected point could match several points of the original design, and so, several values of the response.

Acknowledgements

This work was conducted within the frame of the DICE (Deep Inside Computer Experiments) Consortium between ARMINES, Renault, EDF, IRSN, ONERA and TOTAL S.A.


Brent Strickland
email: stricklandbrent@gmail.com

Valentina Aristodemo

Jeremy Kuhn

Carlo Geraci

The categorical role of structurally iconic signs

Goldin-Meadow and Brentari argue that for sign language users gesture, in contrast to linguistic sign, is iconic, highly variable, and similar to spoken language cospeech gesture. We discuss two examples (telicity and absolute gradable adjectives) that challenge the use of these criteria for distinguishing sign from gesture.

Running head: The categorical role of iconic signs

In their target article, Goldin-Meadow and Brentari present a subtle view of the distinction between gesture and "categorical" properties of sign language. In both spoken and sign language, illustrative gesture and linguistic material interact within a single unified system that combines meanings in real time. For example, if one were to say the English sentence "a drunk man is walking around" while gesturing with one's hand in a swervy motion, a listener would need to integrate the meaning of the sentence with that of the gesture to correctly interpret the utterance. On the other hand, language and gesture generate meaning in distinct ways. Language is said to use categorical elements that can combine into higher-order structures (e.g. using grammatical rules), while gesture is said to express meaning holistically.

Goldin-Meadow and Brentari argue that the same two expressive categories exist in sign language as in spoken language. However, since many gestures are implemented in the manual modality, it can be difficult to distinguish gesture from the "categorical" elements of sign language. Accordingly, Goldin-Meadow and Brentari provide three heuristics for distinguishing between these. First, gesture tends to be highly variable within a linguistic community while linguistic signs tend to be uniform.

Second, gesture tends to be iconic and imagistic whereas sign does not. Finally, gesture in signers tends to be accessible to non-signers, and is often expressed via co-speech gesture in spoken language.

While these criteria can be useful for some examples, here we want to focus on a set of "problem" cases that do not neatly fit within the heuristics laid out by Goldin-Meadow and Brentari. In each example, a representation used by signers is highly Running head: The categorical role of iconic signs uniform within signing communities while nevertheless being iconic/imagistic in systematic ways. We argue that careful consideration of these cases requires altering the Goldin-Meadow and Brentari theoretical model and also suggests that cleaner criteria should be adopted for distinguishing between gesture and sign.

First, we consider the case of telicity. [START_REF] Wilbur | Representation of telicity in ASL[END_REF][START_REF] Wilbur | Complex predicates involving events, time, and aspect: Is this why sign languages look so similar?[END_REF] argues that the phonetic form of verbs in several sign languages systematically reflects the telicity of the predicate: telic verbs (like decide, whose meaning has an intrinsic culmination point) are marked with rapid deceleration to an abrupt stop; atelic verbs (like ponder, with no intrinsic culmination) are not. [START_REF] Strickland | Event representations constrain the structure of language: Sign language as a window into universally accessible linguistic biases[END_REF] confirm this generalization across several sign languages (including Italian Sign Language, LIS), and show that even hearing people with no exposure to sign language are able to infer the telicity of a predicate based on the phonetic form of a verb. Following Goldin-Meadow and Brentari's criteria, the accessibility of this correspondence to non-signers likely qualifies it as a gestural component of sign languages.

On the other hand, the phonetic marking of telicity interacts with fully grammaticalized signs in LIS, in a manner invariant from speaker to speaker within a linguistic community. Specifically, in LIS, the phonetic marking of telicity is in complementary distribution with the perfective aspectual marker DONE. Signs like LEAVE, DECIDE or SELL may express perfectivity either by the presence of a clear boundary or by using the lexical sign DONE. In the latter case, telic predicates do not end in an abrupt stop.

Next, we consider absolute gradable adjectives like "full." What defines absolute adjectives is that the adjectival scale includes a maximum degree: when something is completely full, it cannot be any more full. This contrasts with relative adjectives like "rich" which have no maximum degree.

In LIS, many absolute adjectives include an iconic component that indicates that the maximum degree of the scale has been reached. Unlike their spoken language counterparts (and other, non-iconic adjectives in LIS), these signs do not allow for imprecise readings [START_REF] Aristodemo | Comparative constructions and visible degrees in LIS. Talk given at FEAST[END_REF]. For instance, the LIS equivalent of the English sentence this glass of wine is full but you can still pour a bit more is not grammatical because the sign FULL iconically conveys the meaning that the glass is maximally full. The same effect can be obtained in spoken Italian by accompanying the adjective with a co-speech gesture (an open hand waving on the horizontal plane). This fact shows that an iconic component that is an obligatory part of a sign and an independent co-speech gesture in spoken language contribute similar core meanings to utterances.

In the examples discussed above (i.e. telicity and absolute gradable adjectives), there is an iconic/imagistic representation that is comprehensible to non-signers (as in the telicity example) and can optionally be employed during co-speech gesture (as in the absolute scale example). On the Goldin-Meadow and Brentari framework, one might therefore conclude that they are a part of the gestural system. Nevertheless these representations are not variable across members within signing communities, and seem to play a key role in determining grammaticality. On these grounds, one might conclude that they are part of the "categorical" signing system. This leaves open two possible theoretical positions, either of which would Running head: The categorical role of iconic signs represent a substantial alteration of the Goldin-Meadow and Brentari framework. The first possibility is that we treat the markers of telicity and absolute scale in LIS as gesture. On this view gesture would be capable of interacting with the grammatical system of sign languages in highly intricate ways. In this case, we must accept that some types of gesture do not have a tendency to be variable across speakers within a language, that gesture is not necessarily holistic, and that gesture can have virtually identical functions to well known grammatical markers in spoken languages (such as event-finality in Slavic languages).

The second possibility is that we treat the markers of telicity and iconic absolute adjectives in LIS as being categorical elements of the signing system. On this view, one would need to accept that some properties of sign are highly iconic, and that overlap with co-speech gestures from spoken language should not be taken as a criterion for distinguishing sign from gesture.

For the moment, both views are plausible. Subsequent research and refinement of analytic techniques would be necessary to distinguish between them.

Acknowledgements

The research was supported by the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.313610, ANR-10-IDEX-0001-02 PSL*, ANR-10-LABX-0087 IEC, and a Fyssen Foundation Postdoctoral grant.

Running head: The categorical role of iconic signs