Simon Bernard
email: simon.bernard@univ-rouen.fr

Laurent Heutte
email: laurent.heutte@univ-rouen.fr

Sébastien Adam
email: sebastien.adam@univ-rouen.fr

A Study of Strength and Correlation in Random Forests

Keywords: Random Forests, Ensemble of Classifiers, Strength, Correlation

In this paper we present a study on the Random Forest (RF) family of classification methods, and more particularly on two important properties called strength and correlation. These two properties have been introduced by Breiman in the calculation of an upper bound of the generalization error. We thus propose to experimentally study the actual relation between these properties and the error rate in order to confirm and extend the Breiman theoretical results. We show that the error rate statistically decreases with the joint maximization of the strength and minimization of the correlation, and this for different sizes of RF.

Introduction

Recently, Leo Breiman has proposed a new family of ensemble methods called Random Forest (RF) [START_REF] Breiman | Random forests[END_REF], that can be defined as a generic principle of classifier combination that uses L tree-structured base classifiers. The particularity of this kind of combination is that each decision tree is built from a random vector of parameters. It can be built for example by randomly sampling a feature subset (as in Random Subspace Method [START_REF] Ho | The random subspace method for constructing decision forests[END_REF]), and/or by randomly sampling a training data subset (as in Bagging [START_REF] Breiman | Bagging predictors[END_REF]).

RF is now known to be one of the most efficient classification methods ( [START_REF] Breiman | Random forests[END_REF][START_REF] Boinee | Ensembling classifiers -an application to image data classification from cherenkov telescope experiment[END_REF][START_REF] Banfield | A comparison of decision tree ensemble creation techniques[END_REF]). In a recent paper ( [START_REF] Bernard | On the selection of decision trees in random forests[END_REF]), we have shown the interest of designing a dynamic RF induction method, that would add trees to the ensemble by making them growing according to the trees already added to the committee. But for that purpose, it is crucial to find a criterion that could guide the tree induction so that it could suit to the rest of the ensemble. The main objective here is to minimize the error rate of the ensemble while adding more and more trees. For that purpose, we have followed Breiman's idea that the properties of strength and correlation could be very helpful for better understanding and controlling the behavior of RF ( [START_REF] Breiman | Random forests[END_REF]). Our idea is to determine the relation between these two properties and the performance of RF. To do so, we have decided to generate a large set of forests that exhibit different error rates on the same test set, and to monitor their strength and correlation. Those forests are actually sub-forests generated thanks to a classifier selection method applied to a large RF: the principle is to grow a large pool of trees and to select different subsets from it, in order to simulate the growing of several RF, with different sizes and different performance. By this means, we show that error rates statistically decrease for a joint increasing strength and decreasing correlation.

The paper is thus organized as follows: we recall in section 2 the Forest-RI principles; in section 3, we explain our approach and describe our experimental protocol, the datasets used, and the results. We finally draw some conclusions and future works in the last section.

The Forest-RI algorithm

One can see Random Forests as a family of methods, made of different decision trees ensemble induction algorithms, such as the Breiman Forest-RI method ( [START_REF] Breiman | Random forests[END_REF])li often cited as the reference algorithm in the literature. In this algorithm the Bagging principle is used with another randomization technique called Random Feature Selection. The training step consists in building an ensemble of decision trees, each one trained from a bootstrap sample of the original training set -i.e. applying the Bagging principle -and with a decision tree induction method called Random Tree. This induction algorithm modifies the "traditional" splitting procedure for each node, in such a way that the selection of the feature used for the splitting criterion is partially randomized. That is to say, for each node, a feature subset is randomly drawn, from which the best splitting criterion is then selected.

In the literature, only few research works have focused on the number of trees that have to be grown in a RF. When introducing RF formalism in [START_REF] Breiman | Random forests[END_REF], Breiman demonstrated that for an increasing number of trees in the forest, the generalization error converges to a maximum. This result indicates that the number of trees in a forest does not have to be as large as possible to produce an accurate RF. However, noting that above a certain number of trees no improvement can be obtained by adding more "arbitrary" trees in the forest does not mean obviously that the optimal performance has been reached. Besides, in [START_REF] Bernard | On the selection of decision trees in random forests[END_REF] we have shown that a subset of individual trees is able to outperform the whole ensemble. This led us to the conclusion that one could benefit from building trees in a more dependent way than it is actually done in traditional RF induction algorithm like Forest-RI for example. In this perspective, we have decided to focus on the two crucial properties of strength and correlation, that Breiman stated in [START_REF] Breiman | Random forests[END_REF] that they could be very helpful in a better understanding of RF behaviors. In the next section, we detail the definitions of these properties, and the result that led him to this conclusion.

Strength and Correlation

A RF is usually noted as an ensemble of individual classifiers {h(x, Θ k ), k = 1, ...L} where {Θ k } is a family of independent identically distributed random vectors, and x is an input data. In [START_REF] Breiman | Random forests[END_REF], Leo Breiman introduces the strength and the correlation properties through an upper bound of the generalization error noted P E * :

P E * ≤ ρ(1 -s 2 ) s 2
In this bound, s stands for the strength and ρ stands for the correlation. The main conclusion of this result is that the lower the ratio ρ s 2 , the better the forest, since it gives better chances to obtain a low error rate.

Through the demonstration of this result, Breiman defines the margin function of a RF by the following equation:

mr(x, y) = PΘ(h(x, Θ) = y) -max j =y PΘ(h(x, Θ) = j)
where x is an input data, y its class, and where the subscripts Θ indicate that the probability is over the {Θ k } family of random vectors. The strength is then defined as the expectation of this margin over the data space:

s = Ex,y[mr(x, y)]
Then, Breiman defines the raw margin function as

rm(Θ, x, y) = I(h(x, Θ) = y) -I(h(x, Θ) = (x, y))
where (x, y) represents the index of the "best" class among the wrong classes, defined by:

(x, y) = arg max j =y PΘ(h(x, Θ) = j)
What is called here correlation is actually the statistical mean correlation between rm(Θ, x, y) and rm(Θ ′ , x, y) over all pairs of (Θ,Θ ′ ).

The demonstration of these results strongly depends on the assumption that RF contains a "large" number of trees. However, we have already mentioned in the previous section that the number of trees does not have to be that large to obtain an accurate RF. As a consequence, we have decided in this work to focus on RF made of reasonably small number of trees, i.e. from 50 to 200 trees, in order to determine the actual relation between strength, correlation and error rates. The goal is to confirm Breiman's theoretical results for RF of different sizes, and thus to give elements of understanding of RF behaviors that could be helpful for guiding the RF induction. We detail our classifier selection approach for that purpose in the next section.

Tree Selection

For studying the relation between the ratio ρ s 2 and the error rate, we generate a pool of forests and we measure for all of them the strength, the correlation and the error rate. To be able to accurately determine this relation it is necessary to have at our disposal an ensemble of RF that exhibit error rates in a large range of values. For that purpose, we have chosen to apply a classifier selection approach to RF, in order to generate a large ensemble of sub-forests with different error rates on the same test set. Among the classifier selection approaches that we can be found in the literature, we are interested in what is called the wrapper approach, since it specifically consists in selecting the subset of classifiers that a posteriori optimizes the combination performance ( [START_REF] Kohavi | Wrappers for feature subset selection[END_REF]). Moreover, since our goal is to generate a large pool of sub-forests so that we could observe significant differences between these sub-forests in terms of error rates, we have decided to use Genetic Algorithms (GA) ( [START_REF] Santos | A dynamic overproduce-and-choose strategy for the selection of classifier ensembles[END_REF]) for decision tree selection in RF. This approach allows to produce sub-forests that could potentially exhibit high and low error rates.

Hence, we have applied this classifier selection strategy on several datasets described in the next subsection.

Datasets used

The 20 datasets used in our experiments are described in Table 1. The first 17 datasets in this table have been selected from the UCI repository [START_REF] Asuncion | UCI machine learning repository[END_REF]. The last 3 datasets are handwritten character recognition databases; the MNIST database ( [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF]) with a 85 multiresolution density feature set (1 + 2 × 2 + 4 × 4 + 8 × 8) built from greyscale mean values; Digits and DigReject both described in [START_REF] Chatelain | A two-stage outlier rejection strategy for numerical field extraction in handwritten documents[END_REF], on which a 330-feature set has been extracted, made from three state-of-the-art kinds of descriptors, as detailed in [START_REF] Bernard | Influence of hyperparameters on random forest accuracy[END_REF].

Experimental protocol

First, each dataset has been randomly split into a training and a test subset, containing respectively two thirds and one third of the original dataset. We denote by T r the training set and by T s the test set. Then, a RF has been grown from T r , with a number L of trees fixed to 500. The value of the hyperparameter K, which denotes the number of features randomly selected at each node of the trees, has been fixed to √ M , M being the dimension of the feature space, which is a default value commonly used in the literature ( [START_REF] Bernard | Influence of hyperparameters on random forest accuracy[END_REF]). A classifier selection process using a GA has been then applied to this forest. The size of sub-forests generated during this process is fixed, so that all of them could be fairly compared with each other, as the number of trees could affect the calculation of strength and correlation. The selection procedure through GA is conducted for the following sizes of sub-forests: 50, 100, 150 and 200. Concerning the GA parameters, they have been fixed to the following classical values: number of generations fixed to 300, population size to 50, mutation probability to 0.01, crossover probability to 0.60 and selection proportion to 0.80. At each new generation, several new sub-forests, are generated using the crossover and mutation operators. Therefore, for each size of subforest (50, 100, 150, 200) and for each dataset (among the 20 datasets used), a total of 15000 sub-forests are studied, i.e. 50 sub-forests for each generation step multiplied by 300 generations. For each of these 15000 RF, the error rate, the strength and the correlation have been measured on T s . All these measures are analyzed and discussed in the next subsection.

Results and Discussion

The results obtained with the experimental protocol detailed in the previous subsection can be illustrated with a 2D-point cloud, each point being a sub-forest represented by its error rate and its value of ρ s 2 . Figure 4.3 gives the results obtained with sub-forests made of 50 trees. Note that the tendancies observed on this figure and the conclusion drawn from them can be extended to all the sub-forest sizes tested in these experiments (please see [START_REF] Bernard | Forêts aléatoires : De l'analyse des mécanismes de fonctionnement à la construction dynamique[END_REF] for other sizes). To have a better idea of the shape of the clouds, a regression line has been drawn on each diagram. These lines illustrate the relation between ρ s 2 and error rates. One can clearly observe a decrease of ρ s 2 for decreasing error rates. This observation is consistent with Breiman's theoretical result ( [START_REF] Breiman | Random forests[END_REF]) since it indicates that the smaller the ratio, the better chances for the error rate to be low. Note however that this relation is thus still verified even for small forests, which was not demonstrated by Breiman since his result lied on the assumption of a large number of trees grown in the RF. In the perspective of designing a dynamic RF induction algorithm, this result is interesting because it indicates that minimizing the ratio ρ s 2 should allow to minimize a priori the error rate. Thus, using strength and correlation as criteria for guiding the tree induction in a RF growing process should be a promising idea. Hence, a direct perspective of this work is to study a mean to guide the tree induction process to make it fit to the rest of the forest, by maximizing the current ensemble strength and jointly minimizing its correlation. This could be done for example by weighting the training data so that each tree would focus its induction on previous errors, which is actually part of a work in progress that aims at designing a complete dynamic RF induction algorithm.

Conclusion

In this paper we have studied the strength and correlation properties of RF in a perspective of designing a dynamic RF induction process. The interest of inducting a RF in a more dependent way than it is traditionaly done in a RF induction algorithm has already been demonstrated in a recent paper ( [START_REF] Bernard | On the selection of decision trees in random forests[END_REF]), but it requires the use of a criterion for the guidance of such a sequential procedure. We have thus proposed in this paper a study on the feasability of using the strength and correlation, two crucial properties of RF, for that purpose. We have firstly shown that the relation between these two properties and the performance, theoretically established by Breiman for RF made of a large number of trees, is still verified with smaller RF, i.e. made of 50 to 200 trees. We have thus shown that such a dynamic algorithm can be designed according to the joint objectives of maximizing the strength and minimizing the correlation.

Fig. 1 .

 1 Fig. 1. Error rates (y-axis) according to ρ s 2 values (x-axis) for all the sub-forests of 50 trees, obtained during the selection process. The red line is the regression line of the cloud.

Table 1 .

 1 Datasets description

	Dataset	# Samples # Features # Classes Dataset	# Samples # Features # Classes
	Diabetes	768	8	2	OptDigits	5620	64	10
	Gamma	19020	10	2	Page-blocks 5473	10	5
	Isolet	7797	616	26	Pendigits	10992	16	10
	Letter	20000	16	26	Segment	2310	19	7
	Madelon	2600	500	2	Spambase	4610	57	2
	Mfeat-factors	2000	216	10	Vehicle	946	18	4
	Mfeat-fourier	2000	76	10	Waveform	5000	40	3
	Mfeat-karhunen	2000	64	10	Digits	38142	330	10
	Mfeat-zernike	2000	47	10	DigReject	14733	330	2
	Musk	6597	166	2	Mnist	60000	85	10


Clarisse Didelon

S De Ruffray

Mathias Boquet

Nicolas Lambert

A world of interstices: A fuzzy logic approach to the analysis of interpretative maps

Keywords: Mental map, World, interpretative map, fuzzy logic

This paper proposes a methodology based on fuzzy logic to analyze specific mental maps at world scale. Mental maps at all scales and especially at world scale raise specific issues related to the imprecision of the drawing and uncertainty linked to the object drawn. Here, the sample studied was asked to divide the world in regions. The interpretation dimension when building regions reinforce both the uncertainty and imprecision. The fuzzy logic is used here to focus on the world regions limits. It allows providing cartography of the vagueness of regions borders.

Introduction

In order to understand how the world is perceived, spatial psychologists and geographers have traditionally sought to make surveys with mental maps at global scale. The objective was to capture a cognitive perception of space rather than produce an interpretation. Moving beyond this approach our aim is to analyse how places are clustered when interpretative maps are made. In the framework of a major research programme 5 we implemented a survey on the ways in which the world is perceived. While this approach is consistent with the heritage of mental map making it nevertheless raises a number of specific methodological problems which suggest resort to the relevant mathematical solutions proposed by fuzzy logic theory. The aim here is to identify and to characterise interstitial areas.

I. World scale "interpretative" mental maps.

In order to problematise the relation between territory, identity and space practice while moving beyond the more often studied intra-urban [START_REF] Lynch | The image of the city[END_REF][START_REF] Cauvin | Pour une approche de la cognition spatiale intra-urbaine[END_REF] and regional [START_REF] Gould | Mental maps[END_REF][START_REF] Frémont | La région, espace vécu[END_REF] scales, our approach is based on the hypothesis that the world level is a subjective space. This hypothesis may, however, initially seem paradoxical. Indeed one's knowledge of a space is related to the cognitive effort to make. The later is defined as more important when the "distance" increases [START_REF] Moles | collection "synthèses contemporaines[END_REF]. The world would be then the most difficult level to know. This assertion is however increasingly questionable as the world can be, at least theoreticaly, 'known' through the filters of education, television, the Internet and newspapers [START_REF] Paulet | Les représentations mentales en géographie[END_REF]. Indeed a general knowledge of the world is now as easy to gain as a general knowledge of one's own country or region. Therefore, if the knowledge of close at hand spaces relies mainly on experience, knowledge of far away spaces is made easier thanks to the availability of a multitude of new information sources, many of them numerical. Our approach is to be found at the crossroads of methods and concepts in relation to mental maps with the aim of capturing the various interpretations of the world that are revealed.

1.1.

Mapping the subjective space : the polysemous concept of mental maps.

Psychologists were the first to focus on the notion of cognitive space forwarding the hypothesis that space is full of meanings and values. The later are specific by each individual, but with collective studies, it is possible to sum up more or less shared characteristics. The term "mental map" covers various notions taken from the field of "cognitive-behaviourism". This movement emerged in the USA in the 1970s [START_REF] Gold | The meaning of neighbour[END_REF], mixing the fields of perception and imagination, it sought to analyse the link between representation and action [START_REF] Lynch | The image of the city[END_REF][START_REF] Gould | Mental maps[END_REF]. In the literature four rather different concepts and methods are generally included under the catch-all term "mental map".

The notion of "mental map" can refer to a cognitive map, i.e. an individual's un-mapped spatial knowledge, its subjective space [START_REF] Kuipers | Modeling spatial knowledge[END_REF]. Such a map is created from a selection of elements in the space perception process, primarily physical factors that determine one's sensorial capabilities but also individual (gender, character) and sociocultural (scale of value) factors. It is then the result of the "reshaping" of space from its spatial perception and would be a kind of guideline to move and act through the space. This is why geographers are interested in "subjective space": the space is one factor in the shaping of this subjective space while similarly, the later is a factor in the shaping of space. Doubts however remain over the potential for a real cartographical pattern to exist in the mind. [START_REF] Friedman | Reasoning about geography[END_REF] stressed that the "mental map methaphor is a misleading analogy for the representation of geographical knowledge". This then raises the problems of how to capture subjective space and how analyse it.

Mental map can also have the meaning of a "sketch map". Such a map is realised in the framework of a survey where people are asked to draw a specific space on a blank page. The aim here is, primarily, to study people's itineraries or how they position themselves in the space. Researchers often then try to identify and to explain the knowledge level or judgement on the space by highlighting the various perception filters in play. The analysis here can also include the construction of a statistical explanative model such as, for example, gravity models (Pinhiero, 1998). These models are based on the proxemics hypothesis according to which the closer one is from a space, the more information one obtains on it [START_REF] Moles | collection "synthèses contemporaines[END_REF]. They also introduce some information describing surveyed people. One of the major difficulties with this approach however concerns the actual interpretation of the maps [START_REF] Cauvin | Pour une approche de la cognition spatiale intra-urbaine[END_REF]. Indeed there is difference between the respondent's level of knowledge on space and their ability to translate it in visual form. In addition, the implementation of such analysis is often complicated further because the landmarks and scales of the maps produced are rarely comparable. To produce a workable synthesis of invididual representations then is a difficult task.

The third type of mental map, is characterised by the instructions provided during a survey on a particular space: one has to provide its appreciation on a space or to delimit a phenomenon. We suggest naming such maps, "interpretative maps". The reasercher does not try to know how one draws a space nor to determine a respondent's knowledge level but does desire to understand how the respondent defines a particular phenomenon in a space on the base map provided. The analysis then is easier to undertake than in the previous case because the researcher can quickly and easily compare the "real" and the "interpreted" space. This kind of approach is often used when we want analyse risks representations [START_REF] Bonnet | Risques industriels : les territories vulnérables de l'estuaire de la Seine[END_REF][START_REF] Matei | Fear and Misperception of Los Angeles Urban Space, A Spatial-Statistical Study of Communication-Shaped Mental Maps[END_REF]. The level of knowledge and its effects certainly have an influence on the realisation of the maps, but the drawing of a space in map form allows us to substantially eliminate them and to employ a deeper level of analysis.

By extention then the cartographic synthesis of individual results obtained from a survey on the space [START_REF] Paulet | Les représentations mentales en géographie[END_REF] is often also termed 'a mental map'. This synthesis or reconstruction can be made from a classical survey questionnaire where people are asked about their knowledge or judgement on space. The mental map is then the synthesis of all answers and is a cartographic representation of the indicators of knowledge and judgement.

Mental maps at world scale.

Research initiated by T.F. Saarinen on world-scale sketch maps was often undertaken with a view to demonstrating the variety of points of view on the world or to test the geographical knowledge level on some sample of the world's population [START_REF] Saarinen | Centering of mental maps of the world[END_REF][START_REF] Saarinen | World patterns of geographic literacy based on sketch map quality[END_REF]. The material gathered during these surveys is analysed in two main approaches. The first focuses on the 'centering' of students according to their place of residence [START_REF] Saarinen | Centering of mental maps of the world[END_REF]. The second, and more often utilised approach analyses the quality of mental maps i.e. the number or countries mapped, their relative proportions, their location etc. As such, in the literature we can source many authors and numerous places or surveys following this approach; see, for example, Saarinen & MacCabe in Finland (1989), in Germany (1990), in the USA (1995), Pinheiro in Brazil (1998), Boyowa in Africa (2003) etc. Beyond the descriptive approach some authors seek to explain the frequency of place representations via statistical approaches based on the gravity model. Even if they have hypothesis on the factors that could explain the differences in perception and knowledge, the link with the scholar or media knowledge produced (on the world) is rarely deepened and explanatory factors are often simply considered individually without linkage. Moreover, while such research may have been very fruitful from a theoretical perspective it is often disappointing from a methodological perspective : the rich vein of material accumulated is actually often underused. Beyond the setting up of statistical models, authors have never produced a world map synthesing their results allowing them to provide a global picture of the world. A further field of analysis emerged at the beginning of this century with researchers focusing on distortion in cognitive maps while exploring the explanatory factors for this (Friedman & Montello, 2006[START_REF] Battersby | The effect of global scale map projection knowledge on perceived land area[END_REF][START_REF] Battersby | Area estimation of World regions and the Projection of the global scale cognitive map[END_REF][START_REF] Friedman | The role of categories and spatial cuing in global-scale location estimates[END_REF].

1.3.

Challenges raised by an "interpretative" survey at the global scale.

Our survey can be included in the interpretative mental maps category. The question (to divide the world into different regions and give each a name) led respondents to think about the world in order to construct their regions. As stated by [START_REF] Montello | Regions in geography: Process & contents[END_REF] the creation of geographical regions is a necessary approach to effectively organising knowledge of the world. Indeed, when grouping places that they found similar and when separating others they found different respondent give their interpretation of the world and allow us to understand how they organise their knowledge of the world. The maps produced in this manner however raise specific challenges both from a theoretical and a methodological point of view.

Unlike the production of conventional mental maps, the starting point here is not the individual knowledge or experience but the space that has to be concieved, globaly, when doing the exercice. In addition, whatever the scale chosen, the combination of interpretation and incomplete knowledge of the world generally leads to the creation of imprecise and uncertain mental maps. Moreover, as the global scale is likely to be the that which is furthest from individual experience, a survey at this scale inevitably raises the level of level in the mapping of regions.

A second challenge here concerns the notion of regionalisation. Respondents are unlikely to have a precise idea of the limits between regions: they are unknown or fuzzy [START_REF] Paulet | Les représentations mentales en géographie[END_REF]. Regions are merely defined in terms of their visual structure, of specific activity, of ethnic composition and of other characteristics that are not strictly related to spatial cognition [START_REF] Kuipers | Modeling spatial knowledge[END_REF] and are built by using and mixing various sources of information [START_REF] Friedman | The role of categories and spatial cuing in global-scale location estimates[END_REF][START_REF] Battersby | Area estimation of World regions and the Projection of the global scale cognitive map[END_REF] They are not necessarily precisely localisable particularly where multiple criteria are used. The regionalisation of the world then induces imprecision and uncertainty.

Last but not least, the transition from individual maps to a collective synthetic map is based on the assumption that, beyond the multiplicity of personal representations, common mechanisms are expressed in terms of knowledge and perception. Indeed, while individual respondents are seen as unique individuals they are nevertheless also socialised. This implies the existence of a relative level of consistency in terms of the mental pattern of the sample. The transition from individual to collective level is then made easier by the choice to use interpretative maps: clear instructions and a basemap were provided. If, however, individual representations are precise (limits are clearly draw) the transition to the collective level will inevitably introduce a level of imprecision.

II. Fuzzy logic: a methodological answer to the problem of analysing interpretative mental maps

Imprecision and uncertainty in interpretative mental mapping

The main feature of 'interpretative' mental maps is that the spatial objects respondents are asked to draw are deliberately undefined. Uncertainty in respect of the delimitation of regions here is a factor of the differing capability levels of the observers relating, primarily, to their education, character, culture, history etc. Moreover, respondents can have hesitations or doubts when faced with the difficult exercice of mental mapping. Imprecision emerges in respect of how world regions are composed when the interpretative mental maps are drawn. It is difficult, perhaps even impossible, to define a world region as the sum of spatial elements belonging to it without any ambiguity. In addition, the localisation of world region does not conform to an objective appreciation of geographical distance but rather to proximity and neighbourhood. Thus, the synthetic interpretative mental map of a group of people (figure 1) can be characterised as a fuzzy geographical space [START_REF] Rolland-May | Périphériques, bordures, marges territoriales : sous les mots, les concepts[END_REF], 2003).

Figure 1: interpretative mental map

Given the objectives of the analysis, fuzzy sets and possibility theory [START_REF] Zadeh | Fuzzy sets[END_REF] suits this survey. Indeed, it formalises the subjectivity, the uncertainty and the imprecision according to human perception, imagination and thinking from models and representations [START_REF] Bouchon-Meunier | Logique floue et applications[END_REF][START_REF] Ruffray (de) | L'imprécision et l'incertitude en géographie. L'apport de la logique floue aux problématiques de régionalisation[END_REF]. Moreover, the fuzzy set concept has been introduced in order to deal with poorly defined limits, intermediate situations, the gradual transition from one property to another, approximate values and categories, thus avoiding the use of arbitrarily rigid boundaries.

The regionalisation of belonging: formalisation with fuzzy logic

Each region drawn by a group of people is a fuzzy geographical object that implies the definition of a membership function included in the [0,100] interval [START_REF] Zadeh | Fuzzy sets[END_REF] for each spatial component of the geographical space, i.e. the elementary spatial unit that is here a grid cell (figure 2). When the membership function is "zero", the cell of the grid is not included in an interpretative region. This means it belongs to the non-perceived, or at least to areas not represented by respondents.

If the membership function is "100", the spatial unit is always included in an interpretative region of the world. However the characteristics of the region can vary.

In most cases the membership function of the spatial unit is included in the ]0,100[ interval. This reflects the gradation of the belonging of the cell to interpretative regions.

This approach allows us to take three important characteristics into account in the analysis: the differentiation of the region, the imprecision of limits, and the fact that regions overlap. The internal differentiation of geographical space is valuated thanks to the membership value of a world region. Indeed, a space is differentiated when the cells present different degrees of inclusion in a region, as in the "Europe" region (figure 4). The "core" is a homogeneous area of full membership. Differentiation of space introduces the idea of spatial gradient. The imprecision of its limits with its own environment is the second characteristic of the fuzzy geographical space. From a theoretical point of view, the limit of a fuzzy area is not defined. The lower value of the interval is the limit that can be discussed according to different thresholds that can be fixed both empirically and statistically. Thresholds are set up in order to differentiate the strongest and weakest level of membership.

Mapping all the values in the ]0,100[ interval allows to visualise gradations and discontinuities as in the example presented below.

Focus on overlaps: fuzzy intersection operators

If non fuzzy spaces are disjointed, the principle of the excluded third is negated in a fuzzy space because a place can belong in different degrees to many geographical spaces. The space is therefore overlapped. In order to highlight the intersection between two regions fuzzy set theory provides a set of operators called T-norms that allow the mapping of the places that belong to two or more regions simultaneously. The T-norm functions are adapted operators to be used as intersection operators. Two main intersection operators will be used in this paper. First the Zadeh T-norm [T = min (x,y)] which provides the minimum common membership value between two intersected regions and second, the probabilistic T-norm [T = x*y] which multiplies membership values. The highest values of the T-Norms are located where the memberships are most heavily shared and imprecision and uncertainty reaches the highest level (figure 2). This allows mapping of the interstice areas that characterise places of multi-membership. By definition, an area can belong to many fuzzy regions at the same time. The possibility of membership of two or more different spaces can be illustrated by reference, for example, to Turkey and Mexico which are often shared between two different world regions.

III. A world of fuzzy regionalisation

The maps presented in this section are built on a sample extracted from a worldwide survey made in 2009 on students in different fields (geography, medicine, engineering, arts, economics and political sciences) in 43 cities from 18 world countries6 . In the survey students were asked to draw world regions (up to 15) and name them on a polar projection map. It is well known that every map projection induces many distortions in the earth representation and that they have an effect on "long term spatial memory and then on the cognitive map" [START_REF] Battersby | The effect of global scale map projection knowledge on perceived land area[END_REF]. We chose this projection because it was the most "unusual" for all students and because this projection does not lay stress on a particular area. This section shows the advantages of the fuzzy logic approach in the analysis of interpretative mental maps applied to the students of the first completed city of the survey, namely, Paris.

Topology and toponymy approaches

In choosing the regions we will analyse two main approaches are possible. The topology approach selects all the regions drawn around a specific place or a country without taking into account the various names given to the regions. The map obtained (figure 3), illustrates the cognitive indeterminacy related to the names used by respondents. The advantage of this approach is its focus on spatial concentration level of a region. In the example given, all the areas drawn around Paris have been represented. We can observe that the lines are concentrated around the European region but sometimes also include North America and Russia but rarely other spaces.

Figure 3: representation of a world region selected by topology

The toponymy approach selects the region according to the names they received. The heterogeneity character appears in mapped space limits. Figure 4 shows the fuzzy region called "Europe". The most interesting aspect of this approach is its ability to represent the limits of the region and to highlight the discontinuities. The core of the "Europe" region here shows an important discontinuity on its southern edge (the Straits of Gibraltar) while on the Eastern edge membership values are slowly decreasing.

Figure 4: representation of a world region selected by toponymy

In order to map a fuzzy world, we chose the toponymy approach. We analysed the frequency of the names given to the different regions produced retaining the six world regions with the highest frequency. It is striking that those frequently identified regions match the traditional continental vision of the world [START_REF] Grataloup | L'invention des continents[END_REF]: Europe, Africa, North America, South America, Asia and Oceania are most often quoted (figure 5). Despite the toponymy selection, we observe that the limits used vary significantly and that overlaps are numerous. This approach will later allow us to compare the regions obtained with the relative size and position of those in a classical world regions map as suggested by [START_REF] Battersby | Area estimation of World regions and the Projection of the global scale cognitive map[END_REF]. The highest level of overlap is localised to Mexico, which is often shared by the respondents between North and South America. A second are of interest in this context is the Mediterranean Sea area between Europe and Africa. These overlapping areas are not very large, but their cores are quite dense and discontinuities are magnified. This illustrates a clear belonging in the context of some spaces, mainly in the north. The United States never belongs to the southern part of America and Europe never belongs to Africa. In the south large area of gradation can however be observed. They show the hesitation on the membership of Mexico, Central America and the Caribbean to the North American region or of North Africa to the European region. The intersection between Europe and Asia is the largest. It covers all of Eurasia but its core, localised on Russia, is not very intense. The memberships between the two regions are less marked and the indeterminate area is thus larger. The global map (figure 7), illustrates that only three spaces in the world are not characterised by membership indeterminacy: North America, South America and Sub-Saharan Africa. This approach first stresses the existence of three world regions of strong membership values, even if their borders are fuzzy: South and North America and Sub-Saharan Africa. All other places in the world are characterised by the possibility (at least) of double membership. The cartography of overlaps shows a smooth decrease in membership level (Mexico-USA border to the south) and strong discontinuities (Mexico-USA border to the north): for the respondents the membership uncertainty between two regions varies according to place, but also according to direction. The vagueness of such regional limits cannot therefore be described by means of a linear relationship. These results show how much the definition of world regions by the respondents is blurred, even when the same names are used to describe them. This confirms that various knowledge and information sources are mobilised by respondents to draw regions and to delineate the otherness level of two regions. The results of this cartographic experiment should nevertheless be explored further to provide an explanation of this 'dissymmetry of vagueness' on each part of the border. Moreover, this preliminary analysis must be now extended to other places in the world. The results produced here nevertheless represent an important step in the process of realising that the simplistic regionalisation of the world learned at school or proposed by the media is not "fixed" in the public mind as our respondents tended to mobilise various information forms in their choice determinations and thus, perhaps, that rather more fuzzy representations of the world region could be experimented with and proposed to the general public.

Conclusions & perspectives

From a theoretical point of view the survey proves that the world scale is actually perceived as a subjective space: students seem to have a general knowledge of the world when deciding on how to divide it into specific regions. The preliminary results obtained from the use of the fuzzy logic approach in relation to world interpretative mental maps thus appear to be quite promising. It allows the identification of "hard" and "soft" continents [START_REF] Grataloup | L'invention des continents[END_REF] where membership values to one single zone are high while also localising and characterising overlapping and indeterminate areas. This methodological approach is quite interesting because it allows us to build a world level map of regional representations that stresses that regional borders are more or less blurred in the representations. It may also prove fruitful to explore the reasoning behind the construction of world regions and to compare them with the world region patterns that already exist. Perhaps of particular interest in this respect would be a comparison of the results produced here with other rather sharper divisions such as that made famous by Huntington. As noted by [START_REF] Montello | Regions in geography: Process & contents[END_REF] the setting of such regional borders is in itself an exercice that implies vagueness; the approach forwarded here however allows us to identify variations in the vagueness level in space and will be further enriched to help us explore the factors that can explain this vagueness distribution. This method allows quite interesting perspectives to be developed from both a methodological and a theoretical point of view. It is important to continue this work by analysing the large number of world interpretative maps already in existence.

Figure 2 :

 2 Figure 2: representation of fuzzy set.

Figure 5 :

 5 Figure 5: selected areas utilising the toponymy approach

Figure 6 :

 6 Figure 6: main overlapping areas with Zadeh T-Norm.

Figure 7 :

 7 Figure7: a world of interstices(Zadeh T-norm)

Figure 8 :

 8 Figure 8: main overlapping areas with probabilistic T-norm

The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement n°225260.(www.eurobroadmap.eu)

For more information : www.eurobroadmap.eu