M Koscielny

R Cojean

& I Thénevin

Debris flow hazards due to land use change above source areas in torrent catchments. The case study of Les Arcs (Savoie, France)

Keywords: natural hazards, man land use, debris flows, runoff, erosion, hydrological modelling

For a few decades, land use has changed due to developments in mountainous regions and this has resulted in new pressures on the environment. Consequently, slope instabilities could be a physical expression of such imbalance. Since the 1960s, debris flows events have occurred with an increasing frequency on Les Arcs catchment as it never happened before. At the same time, a large skiing area has been expending on the upper slopes. The study of climatic series shows that rainfall events linked to debris flow triggering were not especially intense compared to the more extreme precipitations recorded in the area for the past middle century. This observation suggests that additional factors have taken part in the debris flow triggering for the last fifty years. Following this logic, the space-time study of land use has underlined the role of winter sport resort expansion on processes such as runoff and erosion affecting torrent banks and beds, directly at the origin of debris flow generation. A complementary analysis of effects on hydrology, supported by an hydrological model (PCRaster Software) has been carried out as well. According to modelling results, the land use conversion is responsible for a change of annual water balance resulting in a significant increase of torrent water flow. Particularly, these effects are emphasised by the localization of converted surfaces (roads, buildings, car parks, ski runs, sport facilities…) in the catchment recharge areas above steep slopes of torrent channels, where materials are liable to be mobilized. Indeed at this interface, during rainfall events the water flow and especially the peak flow are more intense than in the past, due to water rerouting and concentrating that may activate erosive processes above debris flow source areas. At the catchment scale, the increasing number of debris flow events seems to be the answer to this change.

Introduction

The expression "natural hazard" takes all its sense considering the human point of view. Debris flow triggering conditions and propagation have received attention from numerous authors [START_REF] Varnes | Slope Movement/Types and Processes[END_REF][START_REF] Hutchinson | General report: Morphological and geotechnical parameters of landslides in relation to geology and hydrogeology[END_REF][START_REF] Meunier | Les progrès de la connaissance et les méthodes d'étude des phénomènes torrentiels[END_REF][START_REF] Cruden | Landslides types and processes[END_REF][START_REF] Hungr | A review of the Classification of Landslides of Flow Type[END_REF] especially because of their consequences on human life and property. They occur under a critical combination [START_REF] Takahashi | Debris flows. IAHR/AIRH Monograph[END_REF][START_REF] Coussot | Rhéologie des boues et laves torrentielles. Etude des dispersions et suspensions concentrées[END_REF][START_REF] Cojean | Mécanismes d'initiation des laves torrentielles dans les Alpes françaises[END_REF] of materials availability and water input, generally heavy rainfalls [START_REF] Caine | Rainfall intensity-duration control of shallow landslides and debris flows[END_REF][START_REF] Govi | Eventi idrologici e frane[END_REF] on steep slopes. In mountain region, where conditions are extreme, environment disturbances (natural or artificial) may activate such processes at short or large time scale. Climatic variability is a natural phenomenon. Its relationship with mass movements, debris flow or floods has widely been studied for the last years [START_REF] Zimmermann | Climatic Change and Debris Flow Activity in High-Mountain Areas. A Case Study in the Swiss Alps. Greenhouse-Impact on Cold-Climate Ecosystems and Landscapes[END_REF][START_REF] Rebetez | Climatic change and debris flows in high mountain regions : the case study of the Ritigraben torrent[END_REF][START_REF] Jomelli | Geomorphic variations of debris flows and recent climatic change in the French Alps[END_REF][START_REF] Koscielny | Impacts des aménagements sur l'évolution géodynamique des versants. Application au site des Arcs[END_REF]. In other part, some of slope failure [START_REF] Sidle | Hillslope Stability and Land Use, Water resources Monogram[END_REF][START_REF] Slosson | Slope Failures in Southern California: Rainfall Threshold, Prediction, and Human Causes[END_REF] can be induced by human activities In this paper we have focused our work on the role of the geographical position of land management on small mountainous catchments in debris flow triggering. In this aim we studied the case of Les Arcs site, an alpine massif which has been occupied by a large ski domain since the 60's, in order to understand the increasing occurrence of debris flow events during the 40 last years.

Les Arcs site

The Massif des Arcs is located on the Bourg-Saint-Maurice area in the Tarentaise Valley in the Northern Alps (fig. 1). It hangs the Isère torrent on the left bank over 2000 m high. The upper part of the catchment (above 1600 m) is transformed into a large ski area with the Arc 1600 and Arc 1800 high capacity resorts.

Review of debris flow events

The steep slopes of the catchment are drained by seven torrents (fig. 1) the alpine hydrological regime of which are influenced by the snow-pack yield at spring. Considering archives from the Middle Age [START_REF] Mougin | Les torrents de la Savoie[END_REF], first two known debris flow events are dated 1938 but damage was insignificant. In the after years, floods occurred in 1966 in the Ravoire and Eglise torrents, afterwards a series of 26 events of debris flow affected five of seven streams between 1970 and 2000. The year 1973 was particularly noteworthy for the Ravoire torrent that triggered height debris flow in the two summer months of August and September. The erosive processes initiated during this period of activity, induced favourable conditions for the triggering and the feeding in materials of the 31/03 -01/04/1981 catastrophic event [START_REF] Koscielny | Impacts des aménagements sur l'évolution géodynamique des versants. Application au site des Arcs[END_REF]. During this event, there was a deep scouring of the bed larger than 20m in some stretches. At Les Arcs catchment scale, this sudden activity has made alterations to the beds and banks and therefore to the morphologic landscape, particularly at elevation less than 1600m. The figure 2 represents the chronology of debris flow events for each torrent in terms of their relative position on the massif from NE (Moulins) to SW (Preissaz). It shows two consecutive phases of events following this direction. In the early years, from 1970 to 1981, the northern torrents (Moulins, Ravoire) were affected by debris flow. Then, from 1988 to 2000, this hazard concerned the southern ones too (St-Pantaléon, Villard, Preissaz). Such distribution suggests that the triggering factors should have evolved in space and in time during these 40 last years.

Debris flow source areas

Debris flow need some particular conditions to be generated: abundant water and materials susceptible to be mobilized on steep slopes constituting source areas.

Water quantity depends on hydrometeorological conditions. Materials presence depends on geology and erosive processes. On Les Arcs catchment, abundant water could be brought by snowmelt in spring or heavy precipitations due to a mountainous climate which is moreover in evolution [START_REF] Koscielny | Impacts des aménagements sur l'évolution géodynamique des versants. Application au site des Arcs[END_REF]. Thus, the geological nature of non-cohesive materials favours erosion. The massif is formed of carboniferous shale bedrock. The actual morphology of the catchment has been formed during the glacial period and the main part of the catchment is covered by morainic deposits and colluviums. This glacial heritage has induced large and local slope movements and slab jointing.

The longitudinal profile of la Ravoire torrent (fig. 3) shows a typical morphology that resulted from glacial period. In the upper part, the drainage basin presents alternations of glacial benches and steep slopes until the main slope rupture at 1600m corresponding to the outset of the torrent thalweg. Under this point, slopes are steeper, and the torrent tends to deeply embank. Debris flow source areas can be localized in this part of the catchment. Such morphology is similar for all the torrents of the catchment. Figure 4 presents cross sections of the Ravoire bed at various elevations. The first one (a) corresponds to the area immediately located under the main glacial bench, where almost events of debris flow have been triggered. The torrent bed enlargement is due to the successive debris flow events of the 70-80's. The present bed level is 15-20m lower than before events. At this elevation, the left bank is very abrupt and some slope movements could be activated. The profile (b) corresponds to the torrent morphology below. The difference with the (a) section is that the right slope is gentlier due to the bank sapping during the 1981 event. In the cross-section (c), the torrent is much embanked. From the (d) section slopes are gentler still the confluence with the Isère torrent. For all torrents, from stretch (a) to (d), destabilised trees, rocks and masses and banks instabilities testify of the geodynamical activation. In the Savoie region, the 1960-70 decade was characterized by a great economical conversion due to the ski practice revolution. Between 1961 and 1968 the northern part of the catchment was exploited with chairlifts (table 1). Then, the catchment was a virgin site only dedicated to pasture activity. Thus, the building of the first winter sport resort, Arc 1600 (fig. 1) has begun in 1968. Since 1973 buildings and amenities have been expanded to the south of the catchment with Arc 1800 resort creation. Still the 90's constructions and earth works have been carried on southwards. We can also observe that works in les Arcs site have been still in progress, with new buildings construction and ski runs arrangement.

Changes in land use

The development of the winter sport resorts and of the ski area induced many changes to the catchment. These changes and their effects have been identified and analysed [START_REF] Koscielny | Impacts des aménagements en montagne sur les processus hydrologiques et l'évolution géodynamique des versants (Les Arcs, Savoie, France)[END_REF]. The main perturbations are summarized table 2.

Impacts on runoff production

This table shows that land use changes in a winter sport resort play a bad role mainly on runoff and erosion processes. Runoff is principally due to impervious areas development as roofs, car parks, roads and ski runs too that are compacted surfaces inducing an important infiltration reduction [START_REF] Koscielny | Impacts des aménagements sur l'évolution géodynamique des versants. Application au site des Arcs[END_REF][START_REF] Ardvisson | Subsoil compaction caused by heavy sugarbeet harvesters in southern Sweden. I. Soil physical properties and crop yield in six field experiments[END_REF]. Deforestation for roads, buildings and ski slopes acts for runoff production in reducing interception by leaves and absorption by roots. Moreover, the modifications of watershed limits induced by grading works emphasized the hydrological imbalance. Finally, artificial snow production in winter and spring could be responsible for a modification of the global water balance of catchments because the notable quantity of water used for the supply of snow is generally derived from another catchment. It is noteworthy that this quantity of water smelts with some delay at spring during the high-water-flow period.

Impacts on erosion processes

First of all, the disappearance of the vegetal cover increases the impact of rainfalls [START_REF] Sternberg | Enchentes e movimentos coletivos do solo no vale do Paraiba, em dezembro de 1948: influêcia da explotaçao destrutiva da terra[END_REF]. Furthermore, processes of erosion are mainly emphasized during the earth works period. Grading activity (cut and fill) is responsible for a massive production of spoil earth materials. Moreover, it is frequent that some materials are pushed to torrent beds after works. Such a practice is at the origin of the triggering of the first Ravoire debris flow event in 1970. However forty years later, similar practices have been observed during recent works. After all, the man-caused runoff increases the superficial waters action on soils because of the phenomena of rerouting and concentrating in some areas, along roads ramps, ski runs, and more generally because of the increase of the torrent water flow.

Impacts on slope stability

Slope instabilities are frequent in a morphological context such as the one of Les Arcs. But they could be directly enhanced from constructions and deforestation because of the decrease of roots density [START_REF] Endo | On the effect of tree's roots upon the shearing strengh of soil[END_REF][START_REF] Prandini | Behaviour of the vegetation in slope stability : a critical review[END_REF]. Road construction on hillslope inevitably decreases the site stability in the following ways: adding weight to the embankment fill, steepening the slope on both cut and fill surfaces, removing support of the cutslope and rerouting and concentrating road drainage water [START_REF] Sidle | Hillslope Stability and Land Use, Water resources Monogram[END_REF]. Slope stability can be affected by residential development because of: removal of support by excavation, mechanical overloading by fill placement, concentration of water on the site or introduction of additional water [START_REF] Sidle | Hillslope Stability and Land Use, Water resources Monogram[END_REF]. And finally erosion processes represent another factor of instabilities, particularly in banks scouring.

Worsening effects of land management above debris flow source areas

On Les Arcs catchment, the localisation of the ski area and resorts plays a major role on hazard intensification. As it is exposed at figures 1 and 3, because of the drainage basin configuration, massive land use changes are concentrated on the upper slopes in the reception basin, above the thalweg of torrents. Depending on considered catchment, reception basin surface could represent 52 % (Ravoire) to 74% (Preissaz) of the total watershed surface. In all cases, impervious areas and ski runs represent more than 20% of reception basin surfaces. Spoil materials from grading activities above are carried to the thalweg by drainage water inducing erosion. During rain events, water is running from ski slopes and from impervious areas, rather than infiltrates, and then it is concentrated down the urbanized area in the torrent with an increased discharge just above debris flow source areas. Consequences can be erosion, slope instabilities and mobilisation of banks materials. A hydrological modelling approach has allowed underlining the role of the land use change on the torrent flows. Simulations have been conducted with a distributed model [START_REF] Koscielny | Impacts des aménagements sur l'évolution géodynamique des versants. Application au site des Arcs[END_REF] created in the PCRaster language [START_REF] Karssenberg | PCRaster Version 2 Manual[END_REF]. Figure 5 represents the response of La Ravoire torrent during a decennial rainfall depending on the land use, before the ski resort construction and presently. The hyetograph is calculated at the outfall of the drainage basin. The variation in intensity between the two discharge curves shows that the catchment response to a rainfall event solicitation is significant considering the land use factor. The mean discharge calculated during the decennial rainfall was 0,9 m 3 .s -1 before the resort construction but has raised to 1,3 m 3 .s -1 in the 2000's.

This represents an increasing of 44%. But the effect of land use change is particularly significant on the peak flow value estimated at +52% at the exit of the resort above debris flow source areas. At the annual scale, the increase of the discharge is notable to. Considering the mean annual regime of precipitation at Bourg-St-Maurice, the water balance model has calculated an increase equivalent to 24 % of the mean annual discharge at the outfall of the catchment, (with a peak of 26 % during the high water flow period) induced by an increase of 7,5% of the runoff.

At the early 70's, the debris flow occurrence on Les Arcs site has revealed an active geodynamic context. At the same time, important modifications were induced on the upper slopes by construction of two winter sport resorts and their ski area. Our analysis supports the fact that the natural susceptibility of the catchment to debris flow hazard due to geological, morphological and climatic predispositions context has been enhanced by human activities and effects of land use changes. The conversion of the traditional land use to a use dedicated to tourist activities has conducted to a water imbalance attributed to the increase of the runoff induced by four phenomena: deforestation, soils properties degradation, impervious surfaces construction and water transfers. Erosion processes and slope instabilities have resulted from this change too: spoils production, runoff and torrent flow increase. These negative effects are damaging considering that perturbations are concentrated in the reception basin above the debris flow source areas. Indeed, erosion on the upper slopes produces materials liable to be brought to the torrent bed and to favour erosion processes. Moreover, the significant increase of the torrent discharge and particularly the peak flow underlined by the hydrological model, advantages the banks erosion and the bed scouring. Thus, materials and local landslides resulting from bank erosion processes are susceptible to feed debris flow events. This analysis confirms the hypothesis of a space-time concordance between land use change in upper slopes and debris flow triggering downstream following a north-south axis.

Figure 1 .

 1 Figure 1. Location map of les Arcs site.

Figure 2 .

 2 Figure 2. Space-time series of floods and debris flows occurred on les Arcs site.

Figure 3 .Figure 4 .

 34 Figure 3. Longitudinal profile of la Ravoire catchment (from DEM) and location of Arc 1600 resort.

Figure 5 .

 5 Figure 5. Ravoire torrent calculated hyetograph at the outfall of the resort above source areas depending on land use.

Table 1 :

 1 Main dates of chronology for buildings at Les Arcs

	Dates	Buildings	Position
	1961-1968	Courbaton chairlifts
	1966 -1968 1968	Access road to Arc 1600 Arc 1600	North
	1968-1973	Access road to Arc 1800
	1973	Arc 1800
	1978	Arc 1800 -Villards village
	1984	Arc 1800 -Charmettoger village	South
	?	Arc 1800 -Chantel village
	2008	Works still in progress

Table 2

 2

		:	Main impacts of land
			management on ski area
	Management types	Runoff	Water rerouting & concentrating	Vegetation decrease	Erosion	Slope instability

Deforestation X X X Impervious areas X X X X Roads X X X X Ski runs X X X X X Irrigation (golf) X X X Artificial snow X X X X X


Inès Jomaa

Catherine Da Cunha

Emilie Poirson

Jean-François Petiot

PREFERENCE BASED SYSTEM: AN ASSISTANCE FOR CHOOSING A COMIC

Keywords: Recommender Systems, Collaborative Filtering, Emotions, Semantics, Preference

This paper presents a contribution to design an online preference based system. The objective of the system is to assist a customer in the products selection process. The product considered here to describe the application is the comic. Current e-commerce recommendation systems assist customers in this process. Nevertheless, quality of the recommendations produced remains a real challenge. There are products that are not recommended to customers though they would appreciate them and others recommended to them though they do not appreciate them. Quality and relevance of recommendations is addressed in this paper. The customer's choice is not only based on product characteristics but also on his/her perceptive expectations. Therefore, products recommendations are considered relevant since they meet customer's expectations and particularly perceptive ones. The suggested algorithm aims to recommend spontaneously comics to an active customer. It is mainly based on collaborative filtering and neighbourhood formation. Cluster of neighbours is formed. Neighbours share common perceptive preferences with the active customer. Favourite comics of the neighbours may interest the latter. Selection of the appropriate comics is based on the product characteristics expected by the active customer. The algorithm produces relevant comics which meet the active customer's expectations and particularly perceptive ones.

Introduction

Appeared in the mid 1990s, recommender systems are used by electronic commerce websites. These websites offer a huge number of products and services for sale, therefore choosing the right product is not an easy task for a website visitor. Recommender systems have emerged in order to help people to choose products or services that best meet their needs and preferences. These systems are based on a variety of filtering techniques such as collaborative [START_REF] Resnick | GroupLens:An Open Architecture for Collaborative Filtering of Netnews[END_REF], [START_REF] Hill | Recommending and Evaluating Choices in a Virtual Community of Use[END_REF], content-based [START_REF] Lieberman | Letizia: An Agent that Assists Web Browsing[END_REF], [START_REF] Pazzani | Syskill & Webert: Identifying Interesting Web Sites[END_REF], demographic filtering [START_REF] Krulwich | LifeStyle Finder: Intelligent User Profiling Using Large-Scale Demographic Data[END_REF], [START_REF] Pazzani | A Framework for Collaborative, Content-Based and Demographic Filtering[END_REF] and other techniques [START_REF] Burke | Hybrid Recommender Systems: Survey and Experiments[END_REF], [START_REF] Montaner | A Taxonomy of Recommender Agents on the Internet[END_REF].

To recommend products to an active customer, most of recommendation algorithms focus on finding similar customers from similar products they purchased or rated. In fact, collaborative filtering [START_REF] Shardanand | Social information filtering: algorithms for automating "Word of Mouth[END_REF], [START_REF] Konstan | GroupLens: Applying Collaborative Filtering to Usenet News[END_REF] is one of the earliest and most promising recommender technique, it makes recommendations by matching people with the same interest. It predicts products a customer would like, based on the opinions (evaluations or ratings) of other people for these products. In order to identify people who share the same interests, similarity between customers could be calculated by determining the proximity between them. The predicted score attributed by the active customer to unseen products is determined using the evaluation scores of his/her neighborhood.

Collaborative filtering offers to customer the possibility to discover other various domains. In fact, diversity and novelty characterize the recommendations produced. Collaborative filtering also works well for complex objects such as movies and music [START_REF] Burke | Hybrid Recommender Systems: Survey and Experiments[END_REF]. However, collaborative filtering suffers from several limitations such as the "Sparsity problem" [START_REF] Claypool | Combining Content-based and collaborative filters in an online newspaper[END_REF], [START_REF] Montaner | A Taxonomy of Recommender Agents on the Internet[END_REF], the "Gray sheep" [START_REF] Claypool | Combining Content-based and collaborative filters in an online newspaper[END_REF] and the "quality of recommendations for the users" [START_REF] Sarwar | Analysis of Recommendation Algorithms for E-Commerce[END_REF]. Indeed, sparsity problem occurs when there are not enough common products rated by customers. For example, if the number of customers is small relative to the large number of products, the matrices containing the customers' ratings of products could be sparse, thus it is hard to find correlations between customers and consequently to find neighborhoods. "Gray sheep" problem occurs in a small community of customers, when there are customers having no close neighbors and whose ratings don't match any group. Consequently, they get poor recommendations.

Collaborative filtering challenge addressed in this paper is the quality of recommendations. In fact, there are products that are not recommended to one customer though he/she would appreciate them and others recommended to him/her though he/she does not appreciate them [START_REF] Sarwar | Analysis of Recommendation Algorithms for E-Commerce[END_REF]. Thus, recommendations produced are not always relevant. For example, let's imagine that Paul and Edouard like detective stories; but their choice of liking cookery books are different as Paul like them but Edouard don't. Collaborative filtering could recommend cookery books to Edouard, even though these recommendations are not relevant for him.

The problem of relevance of recommendations produced by collaborative filtering algorithm is particularly addressed in this paper. In fact, the aim is to propose one possible extension of collaborative filtering based on emotional, sensory and semantic product characteristics. Our suggested algorithm includes an improvement of understanding perceptive customer needs and can improve the finding products of potential interest. The product chosen within the framework of our project is the comic. In fact, the comic market is growing and attendances various festivals devoted to comics are very important.

The paper is organized as follows. In Section 2, we present the process of determination of the comic's characteristics named comic's attributes. Section 3 describes the system framework. Finally, in Section 4, we present the algorithm that will be implemented.

Determination process of the product attributes

Two categories of product attributes are considered here: objective and perceptive attributes. Characteristics of the product that are independent of the user and that will be used to define the product sheet are called Objective attributes (e.g. number of pages, author, …). Perceptive attributes are divided into two categories: emotional category and sensory and semantic category. We name emotional attributes those related to the affective state of the user, pleasure and displeasure. Sensory and semantic attributes are all subjective attributes that don't belong to the emotional category.

Determination of objective attributes

Objective attributes will be used to define the comic's sheet. They will also be used by the customer in his/her search of comics. In order to determine them, two comics' experts (a bookseller and comic association journalist, a comic bookseller) were questioned about relevant comic objective attributes. A comic website was also referred. A list of objective attributes was obtained. The comic price is considered important in the definition of a comic since it represents generally a criterion for buying a product for customer. Thus, we keep the most relevant objective attributes which are:

Title of series (or Title of album if there is only one album); volume numbertitle of volume; screenwriter(s); cartoonist; colorist; publisher; collection; number of pages; edition release date ; summary (either one for the series or one for the volume, but often, there is one summary per volume), price

Determination of emotional attributes

A summary list of emotions described in the literature was presented to ten comics' fans. They were asked to identify the main representative emotions provoked by reading a comic.

The most frequently cited emotions were the following: joy, surprise, pleasure, curiosity, fascination, relaxation, boredom, satisfaction, awaiting, and calm. This list of terms was retained for future use.

Determination of sensory and semantic attributes

Terms related to sensory and semantic comics' characteristics were collected by two different ways: from readers' comments from two comics' websites, by questioning comics' fans. We also refer to McCloud [START_REF] Mccloud | Making Comics[END_REF].

The most occurring terms were considered. Among the identified terms, the term "emotional" was got. This term was considered encompassing all emotions identified in the previous section; therefore it was omitted from the final list of sensory and semantic attributes. The term "calm" was also omitted since it was considered as an emotional term. Finally, the following list of ten comic's sensory and semantic attributes was kept:

Gripping, nice, classic, clear, imaginary, original, artistic, funny, dynamic, of quality.

Preference based system framework

The purpose of our system is to assist customers (users of the system) in their choice of an appropriate comic; this comic is assumed to meet their expectations. The system uses three data bases (figure 1), namely:

-Customers' data base: stores personal information of each customer. The customer's expectations (that is to say the expected objective attributes of the comic) will also be stored. In order to be registered, each customer has to fill this information in a form. Data in customer's form (figure1) represents the values of his/her personal information (name, age…) and the expected objective attributes (collection, publisher…). An attribute can have multiple values (ex. One customer likes more than one comics' publisher).

-Products' data base: stores the value of the objective attributes for each comic (comics' sheets, figure1). This data base is updated once a new comic appears in the market.

-Preferences data base: stores the ratings of each comic by the customer. To that end, a product evaluation form is provided (figure1). Global rating e ij represents the overall evaluation of a comic i by a customer j. It reflects customer's objective and perceptive preferences regarding the comic that he/she had read and it is valued on 10 point-scale. Perceptive comic's attributes are evaluated on Semantic Differential Scale [START_REF] Osgood | The Measurement of Meaning[END_REF]. Here, we consider a 7 point-scale (Product evaluation form, figure1). The system use case considered is: "The system spontaneously recommends products to registered customer". Recommendations are based on customers' comics' evaluations. The algorithm proposed helps to solve this use case. Figure 2 illustrates by an example the different stages of the algorithm. In this example, we aim to recommend comics to an active customer called Paul.

Figure 2. Stages of the suggested algorithm

In order to recommend comics to Paul, the algorithm searches groups of customers with similar evaluations of comics. This stage represents the first filtering based on the overall comics' ratings. These ratings exist in each product evaluation form of the preferences data base (figure 1). Thus Paul's neighbours are identified. They have similar preferences to him and they form cluster1. This cluster is based on the overall comics' ratings. The second stage, called here selection is a filtering on perceptual attributes evaluations of comics. It provides among cluster1's customers who share Paul's perceptual point of view; they are neighbours who evaluate the comics in the style of Paul and for the same perceptive reasons. They constitute Cluster selection . The highly rated comics by the customers of Cluster selection represent the favourite comics. They are obtained after applying Filering comic (figure2). Favourite comics not yet rated by Paul may interest him. The system checks whether these comics' objective attributes meet the Paul's expectations. These expectations are expressed in Paul's form represented in the customer database (figure 1). Comics obtained are recommended to Paul.

The different stages of the algorithm are described as follows:

Filtering Eval

This stage consists in applying a filtering on all customers of the system on the basis of the ratings (or the overall evaluation) assigned to each comic. The rating of the comic i given by the customer j is assigned by e ij .

Let n be the number of comics in the system and c the number of customers. We consider the matrix that comprises the n comics in rows and the c customers in columns (figure 3). Each entry (e i, j ) of the matrix fluctuates between 1 and 10 (see "Product evaluation form", figure 1).

Figure 3. Illustration of the overall ratings of comics by customers

The objective of this stage is to determine customers who have similar preferences regarding the comics. A measurement of the proximity between two customers is required. This measurement can be achieved by the method of the cosine measure [START_REF] Breese | Empirical Analysis of Predictive Algorithms for Collaborative Filtering[END_REF].

In fact, each customer A of the matrix (figure 3) can be represented as a vector A  .Cosine method calculates the cosine of the angle between two given customers A and B that represents the similarity between them. A similarity matrix (c X c) of all customers is obtained. The next stage is the neighborhood formation; different methods can be used, such as "Center-based". This method consists in forming a neighborhood for a particular customer by selecting his k nearest neighbors; k represents the maximum number of neighbors or threshold that has to be set. Sarwar [START_REF] Sarwar | Analysis of Recommendation Algorithms for E-Commerce[END_REF] also presented an algorithm named "Aggregate Neighborhood" to determine the neighborhood of a customer.

In our case study, comics' recommendations will be addressed to Paul. Considering the approach outlined above, we get Paul's neighborhood called cluster1 (figure 2). This cluster is based on overall comics' ratings. In order to form Paul's neighborhood, the measurement of the proximity between Paul and the customers of the system is required. Customers who are compared with Paul must have assessed at least two comics in common with him. The proximity considered here between Paul and one customer x is the similarity coefficient sim Paul,x , computed from the matrix of the overall ratings of comics (figure 3). It is determined as follows [START_REF] Resnick | GroupLens:An Open Architecture for Collaborative Filtering of Netnews[END_REF]: . . sim x represents the vector of the customer x. This vector components are (e ix ) (figure 3) that represent the evaluation of the comic i (i=1…n) by the customer x. If one comic is not evaluated by the customer x, it is not taken into account in the formula. x represents the euclidian norm of x A threshold that reflects the maximum number of Paul's neighbors has to be set later. Paul's nearest neighbors will be selected. Each one has a similarity coefficient above the threshold. Cluster1 includes ten neighbors who have preferences similar to Paul.

Selection (threshold selection , index similarity )

This stage aims to form cluster Selection (figure 2) by selecting among cluster1 the customers who share common perceptive ratings with Paul. These customers must have similar preferences than Paul regarding comics and they must have similar perceptive ratings. In order to form cluster Selection , let p be the number of comics evaluated by Paul. For each comic evaluated by Paul, a perceptive proximity between Paul and one customer x is determined (customer x belongs to cluster1). This proximity is the similarity between Paul and the customer based on the perceptive evaluations of the considered comic.

Figure 4. Illustration of similarity between Paul and cluster1's customers based on perceptive evaluations of p comics

Sim (Paul, customer x) comic,i, Paul is determined as follows: for each comic evaluated by Paul, a matrix is associated: perceptive attributes are in rows and customers are in columns (Figure 5). The coefficient of this matrix (e attribute, customer ) is the evaluation of a comic perceptive attributes by a given customer. The evaluation of each attribute is rated on 7 point-semantic scale (see "Product evaluation form", figure 1). Each customer of the matrix can be represented as a vector. Vector components of customer x are (e attribute, customerx ). If comic i was evaluated by the customer x, (Paul, customer x) comici, Paul would be determined by calculating the cosine coefficient between Paul and customer x.

Let p' be the of comics evaluated by Paul and customer x at the same time (p'< p). From the matrix presented in figure 4, it is possible to determine the overall similarity coefficient between Paul and Customer x. The overall similarity coefficient is given by the following formula: In order to identify the relevant neighbors, a selection depending on index similarity is realized (figure 2). This selection consists in setting a threshold called threshold Selection that represents the size of Paul's perceptive neighborhood. The choice of neighbors depends on index similarity . Neighbors having the higher index similarity will be selected. This selection lead to Cluster selection ; it includes neighbors who share Paul's perception of emotional, sensory and semantic attributes.

Favorite comics

In this step, we determine the favorite comics of each customer who belongs to cluster selection . Favorite comics that have an average evaluation above 7/10 are chosen.

Comics recommended to Paul

The filtering Filtering comic used here restricts favorite comics to those not evaluated by Paul. Moreover, these comics should meet Paul's relevant objective attributes (i.e. his ideal comic given in the customer's form).The used method is based on a threshold coef obj . This threshold defining the maximum distance between Paul's relevant objective attributes and comic's objective attributes. In order to compute this distance, a dummy coding (binary matrix) is used (figure 6).

Conclusion and perspectives

The purpose of this project is to design a preference based system that helps customers in their purchase decision. Different use cases are considered. In this paper, the use case that consists in making spontaneous recommendations to the customers of the system is chosen. To that end, an algorithm recommends a set of products (in our case comics) is suggested. This algorithm is based on the traditional collaborative filtering, selection techniques and clusters constitution based on the similarity between the customers. Clusters gather the customers who share similar preferences toward products. The algorithm takes into account the products' evaluations, and a refining of perceptive (emotional, sensory and semantic) data; here is our contribution. In fact, refining improves the relevance of recommendations. Recommended products answer customer objective and perceptive expectations. However, "Sparsity" and "Gray sheep" problems which represent limitations in collaborative filtering persist in the suggested algorithm. Moreover, threshold parameters remain to be set.

As perspectives of this work, different types of perceptive data can be separated to distinguish different evaluation methods and to obtain different corresponding customers' clusters. The test and the validation of the proposed algorithm remain to be done. This will validate our work and allows the determination of the various parameters that have to be set. Furthermore, other use cases of the system remain to be released. These use cases represent other possible interactions between the customers and the system, namely: a customer enters his/her personal information, and his/her expectations in the system; a customer searches comics depending on different criteria. Another use case in relation to the storing and the update of the products in the product data base has to be realized.

Figure 1 .

 1 Figure 1. Illustration of the data bases' structure

2

 2

  Sim (Paul, Customer x) comic i, Paul represents the proximity considered between Paul and customer x based on perceptive evaluations of comic i,Paul (figure 4); the index of the comic evaluated by Paul.

  Customerx represents the i th comic evaluated by Paul and Customer x at the same time. index similarity is determined for each Paul's neighbor. It represents the similarity between Paul and Customer x depending on the perceptive evaluation of comics.

Figure 5 .

 5 Figure 5. Illustration of the matrix showing the comic perceptive attributes rated by cluster1's customers (this comic was rated by Paul)

Figure 6 .

 6 Figure 6. Dummy coding of the objective attributes-Matrix showing the absence or presence of Paul's relevant objective attributes in the favorite comicsIf one favorite comic meets one of the Paul's relevant objective attributes (or expectations), "1" is inserted in the corresponding row. Otherwise, "0" is placed. For each comic (column), the number of "1" is counted and divided by the number of objective attributes. The result represents the distance d j between Paul's ideal comic and a favorite comic j (j=1…c):


Mercedes Bonet
email: mercedes.bonet-semenas@inserm.fr

Béatrice Blondel

Babak Khoshnood

Evaluating regional differences in breastfeeding in French maternity units: a multilevel approach

Keywords: Breastfeeding, Regional variations, Social inequalities, Multilevel models

Objectives: To study how individual and regional characteristics might explain regional variations in breastfeeding rates in maternity units and to identify outlier regions with very low or high breastfeeding rates.

Design: Individual characteristics (mother and infant) were collected during hospital stay. All newborns fed entirely or partly on breast milk were considered breastfed. Regional characteristics were extracted from census data. Statistical analysis included multilevel models and estimation of empirical Bayes residuals to identify outlier regions.

Setting: all births in all administrative regions in France in 2003.

Subjects: a national representative sample of 13 186 live births.

Results: Breastfeeding rates in maternity units varied from 43% to 80% across regions. Differences in the distribution of individual characteristics accounted for 55% of these variations. We identified two groups of regions with the lowest and the highest breastfeeding rates, after adjusting for individual-level characteristics. In addition to maternal occupation and nationality, the social characteristics of regions, particularly the population's educational level and the percentage of non-French residents, were significantly associated with breastfeeding rates.

Conclusions: Social characteristics at both the individual and regional levels influence breastfeeding rates in maternity units. Promotion policies should be directed at specific regions, groups within the community, and categories of mothers, to reduce the gaps and increase the overall breastfeeding rate.

Introduction

Evidence on the short-and long-term beneficial effects of breastfeeding continue to increase 1 2 and exclusive breastfeeding is recommended for the first six months of life 3 4 . However, breastfeeding rates in maternity units vary strongly from country to country, and the level in France at the turn of this century was particularly low (63%) [START_REF] Zeitlin | EURO-PERISTAT Project with SCPE EUROCAT EURONEOSTAT[END_REF] . National rates mask important regional differences, as observed in the United Kingdom 6 7 , Italy [START_REF] Giovannini | Monitoring breastfeeding rates in Italy: national surveys 1995 and 1999[END_REF] , the United States 9 10 , Australia [START_REF] Donath | Rates of breastfeeding in Australia by State and socio-economic status: evidence from the 1995 National Health Survey[END_REF] , and France 12 13 . Understanding these geographic variations is essential for several reasons. First, public health policies, including breastfeeding promotion policies, are conducted at the level of regions or states within countries 9 14 . Identification of geographic zones with particularly high or low breastfeeding rates could thus facilitate the orientation of these policies.

Secondly, analysis of regional differences may contribute to better knowledge of the determinants of breastfeeding. Many factors influence breastfeeding practice and interact at various levels.

Besides factors at the individual level, the contextual factors that characterise women's environments also play an important role  factors such as family, social network, and community 15 16 . Nonetheless, we know relatively little about the respective roles of individual and contextual characteristics in breastfeeding and how these characteristics interact at different levels. To our knowledge, few studies have examined geographic variations of breastfeeding rates within countries, after adjusting for individual factors 9 17 . Moreover, studies that have assessed the role of contextual characteristics analysed them at the individual (e.g., for newborns) instead of group level (e.g., geographic areas) 9 18 19 .

Among the entire set of factors that influence breastfeeding practices, social and cultural factors occupy a particularly important place. In high income countries, breastfeeding is more common among women of higher social class, among immigrants 6 15 20 , and metropolitan residents 9 18 . Moreover, the decision to breastfeed depends on the attitude of family and friends and on the general opinion of the population about breastfeeding. Public beliefs about breastfeeding vary according to the general population's economic and culture level 21 22 . It is therefore important to know the extent to which the social characteristics of women and of the general population may explain some regional differences in breastfeeding.

Our objective was to investigate how regional variations in breastfeeding in maternity units might be explained by differences in the distribution of individual maternal characteristics between regions, and whether regional social characteristics were associated with breastfeeding, independently of individual-level factors. We also used empirical Bayes residuals to identify regions with extremely high or low breastfeeding rates, after adjustment for individual-level characteristics. This analysis, which used multilevel models [START_REF] Raudenbush | Hierarchical linear models : Applications and Data Analysis Methods[END_REF] , was conducted with data from a national representative sample of births in France in 2003.

Materials and methods

Data

Individual-level data were obtained from the most recent French National Perinatal Survey conducted in 2003. The survey's design has been described in detail elsewhere [START_REF] Blondel | Enquête nationale périnatale 2003 : situation en 2003 et évolution depuis[END_REF] . It included all births in all administrative regions at or after 22 weeks of gestation or of newborns weighing at least 500 grams, during a one-week period. Two sources of information were used: 1-medical records, to obtain data on delivery and the infant's condition at birth and, 2face-to-face interviews of women after childbirth, to obtain data about social and demographic characteristics and breastfeeding. Around 50% of mothers were interviewed within 48 hours of the birth and 38% on the third or fourth postpartum day. The information regarding infant feeding refers to the feeding method (only breast-fed, breast-fed and bottle-fed, or only bottle-fed) reported by the mother at the interview.

The final study population consisted of 13 186 infants, after exclusion of infants born in French overseas districts (n=636), infants transferred to another ward or hospital (n=975), infants whose mother was hospitalized in an intensive care unit for more than 24 hours (n=26), and those with an unknown feeding status (n=393).

Regional-level data came from census data from 1999 and 2003. We distinguished 24 regions: 21 administrative regions and a further subdivision of Ile-de-France: Paris, Petite Couronne (Paris inner suburbs) and Grande Couronne (Paris outer suburbs).

Outcome and predictor variables

We analysed breastfeeding as a binary variable, considering that newborns were breastfed if they were fed entirely or partly breast milk at the time of interview.

At the individual level, we included variables identified in a previous analysis as related to breastfeeding in our population [START_REF] Bonet | Differential trends in breastfeeding according to maternal and hospital characteristics: results from the French National Perinatal Surveys[END_REF] . Social and demographic variables included maternal age, parity, nationality, maternal occupation (current or last occupation), and partnership status (marital status/living with a partner). Other variables were included in the models as potential confounders: mode of delivery, characteristics of the infants (gestational age, birth weight, and multiple birth), status of maternity units (university, other public, private hospital) and size (number of births per year).

Social context at the regional level was characterized by four indicators: the percentage of urban population (percentage of population in communes that include an area of at least 2000 inhabitants with no building further than 200 m away from its nearest neighbour), the percentage of residents with a university educational level (percentage of residents aged 15 years old or older with at least a three-year university degree), the average annual salary per employee (in euros), and the percentage of non-French residents.

Statistical analysis

We estimated breastfeeding rates by region with corresponding 95% binomial exact confidence intervals. We used a two-level hierarchical logistic regression model [START_REF] Raudenbush | Hierarchical linear models : Applications and Data Analysis Methods[END_REF] with infants (level-1) nested within regions (level-2). First, we estimated a random-intercept model, without any predictor variables (model 1, "empty model") to obtain the baseline regional-level variance ( 00

 (1)
). In a second model (model 2), we included variables characterizing mothers, infants and maternity units.

Model 2 allowed us to estimate the residual regional variation after adjustment for individual-level variables ( 00

 (2)
). We used the proportional change in the variance (PCV), defined as PCV= ( 00

 (1) - 00  (2) / 00  (1)
) x 100, to assess the extent to which regional differences may be explained by the compositional factors (i.e., possible differences in the distribution of individual-level characteristics) of the regions.

Next, we investigated whether regional variables were associated with breastfeeding independently of individual-level factors. We included regional-level variables in four separate models (model 3a to 3d), after adjustment for individual-level variables: percentage of urban population (model 3a), percentage of residents with a university educational level (model 3b), average annual salary (model 3c), and percentage of non-French residents (model 3d). Additional analysis allowed us to investigate the effect of the regional characteristics most strongly associated with breastfeeding when put together in the same model (model 4). Cut-off points for regional variables were established at the 50 th percentile and the reference category for each variable was equal or inferior to the 50 th percentile of the distribution of each variable. Analyses using quartiles showed comparable results.

We also examined whether the effects of certain individual-level social characteristics differed across regions. We did so by estimating random-coefficient (random-intercept and random-slope) models to assess whether associations between breastfeeding and maternal nationality and occupation varied from one region to another. In addition, we tested whether the association between breastfeeding and maternal occupation varied according to the educational level of the population in each regions, and whether the association between breastfeeding and maternal nationality varied according to the percentage of non-French population in each region, by examining cross-level interactions.

Finally, we analyzed regional differences in breastfeeding using empirical Bayes residuals, in order to identify outlier regions (those with unusually high or low breastfeeding rates). Empirical Bayes residuals are defined by the deviation of the empirical Bayes estimates of a randomly varying level-1 (individual level) coefficient from its predicted value based on the level-2 (regional level) model [START_REF] Bonet | Differential trends in breastfeeding according to maternal and hospital characteristics: results from the French National Perinatal Surveys[END_REF] . We computed empirical Bayes residuals based on a random-intercept model that included only individual-level variables. Hence, these residuals reflect differences across regions after adjustment for individual-level characteristics. Computation of the residuals for each region took into account the number of infants in the region. As a result, the fewer the number of infants in a region, the more the value of the regional residual shrinks towards the average breastfeeding rate across regions. This is done so that small regions do not appear as outliers due purely to chance. We compared ranking of regions for all breast-fed infants (only breast-fed and breast and bottle-fed only breast-fed and for) and also for infants only breast-fed.

Descriptive analysis was performed using STATA 9 software (StataCorp LP, College Station, TX, USA). Multilevel analysis was performed using Hierarchical Linear and Nonlinear Modeling (HLM version 6) software (Scientific Software International, Inc., Lincolnwood, IL, USA).

Results

Figure 1 shows breastfeeding rates across regions in France. They were higher in Ile de France, Rhône-Alpes, Provence-Alpes-Côte d'Azur, and Alsace (from 67% to 80%) and lower in Auvergne, Pays-de-la-Loire, and Picardie (from 42% to 51%).

Breastfeeding rates also varied according to regional characteristics (Table 1). They were higher in regions with a high percentage of urban population, of residents with a university educational level, and of non-French residents, and in regions with a high average salary.

Variations in breastfeeding rates across regions were statistically significant (model 1), with a baseline regional variance of 00

 (1)
= 0.147 (p<.0001) (Table 2). The inclusion of individual-level variables (model 2) decreased the variance in breastfeeding rates across regions but residual differences remained statistically significant ( 00

 (2)
= 0.066; p<.0001). The PCV was 55% (PCV= (0.147-0.066/0.147)x100= 55%). Hence, about half of the regional variations in breastfeeding could be explained by differences in the distributions of individual-level variables across regions. High breastfeeding rates were found mainly among women who were primiparous, non-French, and from higher status occupational groups. The measured characteristics of the infants and the maternity units in our study also had little effect on breastfeeding practice and on regional variations (data available on request)

Next, we introduced one regional variable at a time into four different models (model 3a to 3d) (Table 2). After taking into account individual-level variables, including maternal education and nationality, regions with a high percentage of urban population, of people with university education, or of non-French residents still had higher breastfeeding rates. The association between breastfeeding and average salary was not significant.

Residual regional variance for model 2 (which included individual-level variables only) slightly reduced with the introduction of the percentage of urban population ( 00  (3a) = 0.050). Variance for model 2 was further reduced by 50% with the addition of regional educational level ( 00

 (3b) = 0.031)
or the percentage of non-French population ( 00  (3d) = 0.034) (i.e., PCV= (0.066-0.031/0.066)x100= 53%). Hence, individual and regional variables (educational level or percentage of non-French population) together accounted for 79% of the regional variations in breastfeeding (i.e., PCV= (0.147-0.031/0.147)x100= 79%).

We used random-coefficient models to examine whether the effects of certain individual-level social variables differed across regions. Results from these models did not show significant regional differences in the effects associated with maternal occupation or nationality. In addition, we did not find significant interactions between the effects of factors at the regional and individual levels: maternal occupation and regional educational level (p≥0.2 for almost all occupational groups) and maternal nationality and regional non-French population (p= 0.08).

Next, we included in the same model (model 4) the two regional variables most strongly associated with breastfeeding  percentage of residents with a university educational level and percentage of non-French population. Both variables remained significantly associated with breastfeeding.

Moreover, results from model 4 showed that together individual and regional variables accounted for 90% of the regional variations in breastfeeding (i.e., PCV= (0.147-0.015/0.147)x100= 90%).

Finally, empirical Bayes residuals were used to rank regions according to their breastfeeding rates, after taking into account individual-level characteristics (Figure 2). Formally, the empirical Bayes residuals represent regional differences in the adjusted log-odds of breastfeeding in maternity units after taking into account individual-level characteristics in different regions. Therefore these residuals reflect indirectly adjusted regional differences in breastfeeding rates. We identified a group of regions with the lowest (Picardie, Pays-de-la-Loire, Auvergne and Nord-Pas-de-Calais) and another with the highest breastfeeding rates (Provence-Alpes-Côte d'Azur, Paris, Petite Couronne and Rhône-Alpes). In general, confidence intervals for the empirical Bayes residuals were relatively wide. However, those for regions with the lowest breastfeeding rates did not overlap with those with the highest breastfeeding rates. In a model comparing infants only breast-fed and infants only bottle-fed, ranking of regions for only breast-fed infants did not differ from ranking of regions for all breast-fed infants.

Discussion

Breastfeeding rates varied widely between regions and about half of the regional variations could be explained by differences in the distribution of maternal characteristics across regions. Estimates of empirical Bayes residuals in multilevel models suggested that there were regions with high breastfeeding rates and regions with low breastfeeding rates, independent of individual-level characteristics. In addition, at the regional level, a high percentage of urban population, of people with university education or of non-French population had a positive effect on breastfeeding.

We chose to analyze the geographic differences in breastfeeding at the regional level for several reasons. Health policies in France are beginning to be implemented at the regional level, as stated in the French Public Health Code [START_REF] Jorf | Loi n°2004-806 du 9 août 2004 relative à la politique de santé publique[END_REF] . Following recommendations in the national nutrition program [START_REF] Hercberg | Pour une grande mobilisation nationale de tous les acteurs pour la promotion de la nutrition en France[END_REF] , since 2006, health professional networks and regional committees in charge of perinatal health are including breastfeeding promotion in their objectives. Regional breastfeeding workshops for health professionals are also organized. Analysis at the regional level is also important because regional social and demographic characteristics vary substantially across French regions [START_REF]La France et ses régions, édition 2002-2003[END_REF] . Finally, national statistics, such as census data, are available at the regional level.

However, our analysis at regional level had some limitations. The small number of regions (n=24) in our sample limited the number of regional variables that could be introduced in the same model [START_REF] Av | Multilevel analysis in public health research[END_REF] . Moreover, we were not able to assess the impact of regional breastfeeding promotion policies because data about these policies are not available systematically. Other multilevel studies have shown that policies or legislation in favour of breastfeeding explained a part of the differences between states in the US [START_REF] Singh | Nativity/immigrant status, race/ethnicity, and socioeconomic determinants of breastfeeding initiation and duration in the United States, 2003[END_REF] or between municipalities in Brazil [START_REF] Venancio | Individual and contextual determinants of exclusive breast-feeding in Sao Paulo, Brazil: a multilevel analysis[END_REF] . Nonetheless in France, the effect of these policies in 2003 was probably slight, because programs promoting breastfeeding were introduced only in the early 2000's.

French National Perinatal surveys provide information about a limited number of indicators and are not designed to study specifically questions related to breastfeeding in detail. We were therefore unable to use the complete definitions of breastfeeding using the WHO criteria [START_REF] Who | Indicators for assessing infant and young child feeding practices[END_REF] . Furthermore, no information was collected about practices in maternity wards or breastfeeding duration. The effects of maternity unit practices within a given region are difficult to assess. However, only two out of 618 maternity units had received the Baby Friendly Hospital designation in France in 2003 [START_REF] Cattaneo | Protection, promotion and support of breastfeeding in Europe: current situation[END_REF] .

We identified regions with very high and very low breastfeeding rates using empirical Bayes residuals [START_REF] Raudenbush | Hierarchical linear models : Applications and Data Analysis Methods[END_REF] . Identification of these regions can facilitate targeting policies to promote breastfeeding, particularly in regions with very low breastfeeding rates. They may also be helpful in identifying factors or programs that may favour breastfeeding in regions with high breastfeeding rates. The multilevel approach used in our analysis and in particular the use of empirical Bayes residuals is potentially applicable to a wide spectrum of evaluation studies, aimed at estimating the effects associated with groups (e.g., regions, neighbourhoods, hospitals, or wards). These residuals have distinct advantages because they take into account the hierarchical structure of data (group membership) and produce relatively stable estimates even when the sample sizes per group are modest [START_REF] Raudenbush | Hierarchical linear models : Applications and Data Analysis Methods[END_REF] .

Despite their advantages, empirical Bayes residuals have limitations as group-level indicators [START_REF] Raudenbush | Hierarchical linear models : Applications and Data Analysis Methods[END_REF] .

There is possible bias due to unmeasured individual-level confounders and/or model misspecifications. This is a general limitation of all multivariable regression models, including multilevel models and empirical Bayes residual estimations. Another important consideration relates to the potential problem of a statistical self-fulfilling prophecy. This can come about as the result of shrinkage of the estimates for empirical Bayes residuals towards the average value in the population for small groups (small regions). Hence, to the extent that data are unreliable for small groups, the group effects are made to conform more to expectations. Consequently, it becomes more difficult to identify small regions that represent outliers. This could be the case for Corse, the smallest region in our sample, which had the second lowest breastfeeding rate in our sample but was not identified as a low outlier region by the empirical Bayes residuals.

Our results showed a strong association between breastfeeding and maternal occupation and nationality. These associations were comparable to those identified in a previous analysis that did not take regional variations into account [START_REF] Bonet | Differential trends in breastfeeding according to maternal and hospital characteristics: results from the French National Perinatal Surveys[END_REF] . In addition, using random coefficients from multilevel models, we showed that associations between breastfeeding and maternal characteristics did not differ across regions or according to regional social context. These results suggest that maternal characteristics play an important and stable role in breastfeeding, independently of the context where the mothers live.

The high proportion of women with maternal characteristics most favourable to breastfeeding in the regions with high breastfeeding rates [START_REF] Blondel | Enquête nationale périnatale 2003 : situation en 2003 et évolution depuis[END_REF] explains nearly half the regional variations in breastfeeding in France. This is the case in Paris and in its immediate suburbs, where the many women with highstatus jobs (e.g., managers, professionals, or technicians) or born in foreign countries appear to contribute to the very high breastfeeding rates in these regions compared to other French regions. In the United States, 25-30% of the variation in maternal breastfeeding between states also appears to be explained by maternal characteristics [START_REF] Singh | Nativity/immigrant status, race/ethnicity, and socioeconomic determinants of breastfeeding initiation and duration in the United States, 2003[END_REF] .

We observed that at the regional level both a highly educated population and a substantial foreign population have a positive influence on breastfeeding. Our results are consistent with those of a recent study in the US [START_REF] Forste | Are US Mothers Meeting the Healthy People 2010 Breastfeeding Targets for Initiation, Duration, and Exclusivity? The 2003 and 2004 National Immunization Surveys[END_REF] that showed that women living in an area that is a high-risk environment for newborns (based on the indicators of the Right Start for America's Newborns program) were less likely to breastfeed. On the other hand, we found no relation between breastfeeding and mean income. In some studies that used only individual-level data, breastfeeding was found to increase with maternal education 10 17 or poverty level 10 18 . However, when the effects of education and poverty level were assessed simultaneously, breastfeeding remained associated with maternal education but not with poverty level [START_REF] Dubois | Social inequalities in infant feeding during the first year of life. The Longitudinal Study of Child Development in Quebec (LSCDQ 1998-2002)[END_REF] .

Our results at the regional level suggest that education and culture play a more important role than standard of living. The influence of the social and culture background on breastfeeding may be related to public knowledge of breastfeeding benefits, beliefs and attitudes, and breastfeeding practices in the general population 15 19 21 22 31 32 . For example, populations of foreign origin are very favourable to breastfeeding for cultural reasons [START_REF] Griffiths | The contribution of parental and community ethnicity to breastfeeding practices: evidence from the Millennium Cohort Study[END_REF] . Similarly, more highly educated people are more receptive to health messages and might be more supportive of health-related behaviour, including breastfeeding 21 22 31 32 .

In this way, a high proportion of foreign residents may produce through different mechanisms an environment that is culturally supportive of breastfeeding, independent of the mother's nationality.

Regions with a high proportion of foreigners today have long been regions with high immigration rates. The role of foreign cultures may remain strong in these regions, including for mothers born in France. That is, the preference for breastfeeding seems to continue from immigrant mothers to first and second-generation mothers [START_REF] Hawkins | Influence of moving to the UK on maternal health behaviours: prospective cohort study[END_REF] . In regions with a high foreign population, there may be many French women of the first or second generation  in families, among healthcare professionals, and in childbirth preparation or breastfeeding support groups  very favourable to breastfeeding. For example, in areas with high immigrant rate, the partners of native-born French women may more often be either foreign or from an immigrant family, and they may incite their partners to breastfeed more frequently [START_REF] Griffiths | The contribution of parental and community ethnicity to breastfeeding practices: evidence from the Millennium Cohort Study[END_REF] .

The socio-cultural context may also have an important impact on health professional practices in maternity units 34 and explain regional disparities. It has been shown that health professionals' knowledge, experiences and beliefs influence attitudes and behaviours on breastfeeding support and promotion 35 36 . However, we do not know how health professionals' support in maternity units varied between regions at the time of the survey. In any case, maternity units are part of, and are influenced by the general socio-cultural context, which is an important determinant of breastfeeding promotion policies. For example, breastfeeding promotion practices could be more easily adopted in maternity units within regions with a highly educated population.

Regional variations in breastfeeding may also stem from the breastfeeding practices of the preceding generation. The regional differences in 2003 are very similar to those observed in 1972 [START_REF] Rumeau-Rouquette | Evolution of breast-feeding in France from 1972 to 1976 (author's transl)[END_REF] , with breastfeeding rates higher in the east and Ile-de-France (Paris and its suburbs) than in the west.

The literature shows that women who were themselves breastfed breastfeed more often [START_REF] Yngve | Breastfeeding determinants and a suggested framework for action in Europe[END_REF] .

Grandmothers transmit not only their own feeding practices and beliefs, but also their confidence that breastfeeding is the normal way to feed an infant if they had breastfed their own children [START_REF] Grassley | Grandmother breastfeeding support: what do mothers need and want?[END_REF] .

Women who give birth in regions where there was a high breastfeeding rate in the past may therefore have received greater support for breastfeeding from their parents, family and friends.

Conclusion

Our study shows that a multilevel analysis including estimations of empirical Bayes residuals can identify regions with particularly high or low breastfeeding rates. This can in turn be helpful in targeting regional policies to promote breastfeeding, especially in regions with low breastfeeding rates. Our results suggest that strategies to be developed must also include, in all the regions, differentiated activities adapted to particular social groups, to improve the attitude of the general population towards breastfeeding, to help mothers in their feeding choices for their newborns and to support health professionals in and outside maternity units in implementing breastfeeding promotion activities. Breastfeeding promotion policies at these different levels might contribute both to decreasing individual and regional differences and to increasing national breastfeeding rates.

FIGURE 1 

 1 FIGURE 1  Breastfeeding rates in maternity units in France in 2003 (n) Alsace (399); Aquitaine (571); Auvergne (239); Basse Normandie (309); Bourgogne (278); Bretagne (624); Centre (488); Champagne-Ardenne (279); Corse (54); Franche-Comté (222); Haute Normandie (403); Ile-de-France : Petite Couronne(1145), Paris (781), Grand Couronne (1094); Languedoc Roussillon (478); Limousin (150); Lorraine (425); Midi-Pyrénées (539); Nord-Pas-de-Calais (995); Pays-de-la-Loire (802); Picardie (354); Poitou-Charentes (284); Provence-Alpes-Côte d'Azur (965); Rhône-Alpes (1308).

FIGURE 2 

 2 FIGURE 2  Regional variations in breastfeeding in maternity units *: Empirical Bayes residuals

TABLE 1  Breastfeeding rates in maternity units according to regional characteristics *Quartiles refer

 1 to the distribution of regional variables; † p<.0001 for all variables.

	Regional characteristics (quartiles)*	% breastfeeding	†
	Percentage of urban population
	<58.6	54.0
	58.6-66.2	56.5
	66.3-75.1	63.0
	≥75.2	67.6
	Percentage of residents with a university
	educational level
	<13.6	53.1
	13.6-14.6	53.1
	14.6-17.9	61.4
	≥18	71.2
	Average annual salary in euros
	<14 926	57.2
	14 926-15 405	59.8
	15 406-15 995	55.6
	≥15 996	70.9
	Percentage of non-French residents
	<3.1	54.6
	3.1-3.9	53.7
	4.0-6.4	62.2
	≥6.5	72.7

TABLE 2  Breastfeeding in maternity units in 2003 according to maternal and regional characteristics: results of the multilevel analysis *Models 2 to 4 were adjusted for all individual-level variables in table and mode of delivery, gestational age, birth weight, multiple birth size and status of the maternity unit; † included regional variables in 4 different models (models 3a to 3d); ‡ included regional variables at the same time in the model; § regional variables were cutoff at 50 th percentile. Reference group  50 th percentile; ║ p<.0001; ¶ PCV= (Τ00 [START_REF] Ip | Breastfeeding and Maternal and Infant Health Outcomes in Developed Countries[END_REF] -Τ00 [START_REF] Who | Evidence on the long-term effects of breastfeeding : systematic review and meta-analyses[END_REF] / Τ00 [START_REF] Ip | Breastfeeding and Maternal and Infant Health Outcomes in Developed Countries[END_REF] ) x 100.


La Méthode De Newton

Tan Lei

Professeur à l'Université d'Angers. (page web) ANS la vie courante, des problèmes mathématiques se modélisent souvent sous forme d'équations. Résoudre ces problèmes revient alors à trouver des solutions à ces équations.

Les dimensions d'une feuille de papier

Voici un exemple : prenons une feuille de papier A4 standard. Sa largeur est de 21 cm (nous l'avons un peu arrondie pour simplifier le calcul [1]). Sa longueur mesure entre 29 cm et 30 cm. Comment expliquer ce rapport « étrange » entre longueur et largeur ? En effet, pour des raisons économiques et esthétiques, ce rapport a été choisi pour qu'après avoir été plié en deux, le papier reprenne la même forme, c'est-à-dire le même rapport longueur/largeur. Ainsi, en notant x la longueur de notre feuille A4, le rapport longueur/largeur avant le pliage est de x 21. Après le pliage, la nouvelle longueur est 21cm, et la nouvelle largeur x 2, avec donc comme rapport 21 (x 2). La longueur x de la feuille A4 est donc solution de l'équation Newton fournit toujours des solutions approchées des équations, et ce avec autant de précision que nécessaire (voir l'article original de Newton, ajouté à la fin de cet article).

Qu'est-ce qu'un « algorithme itératif » ? C'est tout simplement l'opération consistant à répéter un grand nombre de fois la même opération mathématique. Tapez par exemple un nombre au hasard, disons 5, sur une calculatrice. Puis appuyez, disons une trentaine de fois, sur la touche (en espérant qu'elle fonctionne, cette fameuse touche !).

Vous allez voir défiler sur l'écran une succession de nombres décimaux, et ces nombres s'approchent de plus en plus de 1. Essayez cette fois-ci 0 001 à la place de 5, qu'observez vous ? Ce que vous êtes en train de faire, c'est d'itérer l'opération « racine carrée ». Et cette itération vous fournit au fur et à mesure des approximations de plus en plus précises du nombre 1 (ce qui n'est pas très intéressant en soi).

Vous pouvez bien sûr itérer d'autres opérations comme x , sin ou cos si votre calculatrice le permet, tout en variant le choix du nombre de départ. Vous allez constater qu'itérer x en commençant par 1 05 ou 0 95 donne des résultats très différents, tandis qu'itérer sin (ou cos) donne toujours le même résultat final.

Personnellement j'ai été très étonnée lors de la découverte de ces curieux phénomènes. Pas vous ?

Cependant, cette valeur avec toutes ses décimales n'est qu'une approximation de , puisque ce nombre réel n'est pas un nombre décimal [START_REF]C'est-à-dire : il faut, pour l'écrire, une infinité de chiffres après la virgule[END_REF], de même que 1 3 33333333 , , ne sont pas des nombres décimaux. Une des questions est alors de trouver une bonne approximation de ces nombres par des nombres décimaux, en fonction de la façon dont ils sont définis. Cette question n'a pas grand intérêt pour les nombres qui sont des fractions (comme 1 3), car le développement décimal d'un tel nombre est toujours périodique (comme celui de 1 3 où le chiffre 3 se répète indéfiniment dans le développement).

La touche « racine carrée » est en panne

Imaginons qu'un jour la touche ne fonctionne pas. Ou qu'on ait besoin d'une valeur approchée avec beaucoup plus de précision. Ou encore que l'on tombe sur un autre problème dont aucune touche de la calculatrice ne nous fournisse de réponse.

Heureusement, après avoir transformé ces problèmes en équations, nous pouvons utiliser un algorithme itératif inventé par le génial mathématicien et physicien Isaac Newton (1642-1727). Cette méthode de

Essayons avec x

. On obtient, successivement :

2 x x 2 2 -8 = 0 = x 2 -8 - P (x) P (x) - 2x x 82 2 -8 2 1 x + x 882 -P -P = 1
1 i i + 882 i i 2 = - 2 1 -8 i = - i i i 882 + i 882 -882

Avec un polynôme de degré 3

Essayons maintenant la méthode de Newton z (z) P (z) pour un polynôme un peu plus compliqué :

ici z désigne un nombre complexe inconnu et a est le nombre complexe -0 00508 33136 . (En fait on aurait pu prendre n'importe quel nombre complexe pour a, mais celui-ci a été choisi spécifiquement pour illustrer par la suite l'apparition du fameux lapin de Douady. Voir le paragraphe « Choix de a » ci-dessous. )

Les solutions de l'équation P (z) sont 1,et -

. On pourrait croire naïvement qu'une sorte de graphe séparateur jouerait le rôle de la médiatrice (comme dans le cas précédent) : ce graphe séparerait le plan en trois régions, et une valeur initiale (qui est un nombre complexe) prise dans une région donnée nous donnerait des approximations de la racine du polynôme située dans cette région. Par exemple :

-882 882

-P P (z) z ) = z -a + 2 1 z + a + 2 1 ( -1 + 0 i = 0 2 1 + a 2 1 -a
Arthur Cayley (1821-1895) avait tenté en vain de justifier cette intuition (voir son article, ajouté à la fin de cet article). Nous allons tout de suite comprendre pourquoi. Voici ce que notre ordinateur dessine :

Une expérience numérique

x 1 y 1 4 = x + i -P 2 1 + a 2 1 -a
Ainsi, plutôt qu'un simple graphe séparateur, on voit apparaître un surprenant « fractal » qui partage le carré en trois grands lacs, un autour de chaque solution, mais également en beaucoup de petits lacs rouges, verts ou bleus, qui s'entremêlent de manière très compliquée...

Ah ! Si Cayley avait vu cela !

Observons ce fractal de plus près. Voici un agrandissement dans la fenêtre -0 22 22.

La couleur noire indique le lieu des « mauvaises » valeurs de test initiales : si l'on choisit au départ un

x y 0 nombre complexe au sein de cette partie, la méthode de Newton n'approchera jamais d'une solution de P .

Même si ce lieu est relativement petit, on voit tout de même des taches piégées dans son intérieur, des sortes de lacs noirs (en termes mathématiques, l'intérieur de cet ensemble n'est pas vide). La présence de ces lacs met en doute l'efficacité globale de la méthode de Newton.

En plus, ces lacs noirs ne sont pas dans une partie isolée, bien délimitée, du plan. Au contraire ils s'entremêlent partout avec les lacs rouges, verts ou bleus.

Ceci illustre le phénomène « chaotique » de cette méthode itérative : en partant d'un point de la frontière d'un de ces lacs, la moindre erreur numérique nous fait basculer soit d'une solution vers une autre, soit vers un piège de couleur noire.

Il faut dire que la présence de ces taches noires est due à notre choix spécifique du nombre complexe a.

En choisissant un autre nombre pour a, le fractal engendré peut très bien ne plus avoir de taches noires.

On peut même prédire à peu de choses près tous les choix de a pour lesquels cela se produit.

Une question se pose alors : dans ces cas-là, peut-on affirmer que la méthode de Newton est enfin globalement efficace ? Eh bien, ce n'est pas si simple que ça.

Il existe aujourd'hui des logiciels qui produisent des valeurs « aléatoires », c'est-à-dire imitant le hasard. On aimerait dire que la chance qu'un tel logiciel produise un nombre complexe noir est minime, lorsqu'il n'y a pas de tache noire à l'intérieur [START_REF]En termes mathématiques, ce serait dire que le lieu noir est de mesure de Lebesgue nulle[END_REF]. Or ce n'est pas toujours le cas : pour certains choix de a, il n'y a pas de taches noires (l'intérieur de l'ensemble noir est vide), pourtant la probabilité pour qu'un point choisi au hasard soit noir n'est pas nulle [START_REF] Buff | Ensembles de Julia quadratiques de mesure de Lebesgue strictement positive[END_REF].

Lapins

Essayons maintenant de comprendre un peu mieux la structure du fractal de Newton présenté ci-dessus, donc calculé avec le nombre a que nous avons choisi ci-dessus.

La première observation est qu'il y a des taches noires un peu partout à la frontière des lacs colorés, et que ces taches ont des tailles très variées.

La deuxième est que ces taches ont toutes une drôle de forme : elles ressemblent un peu à un lapin à deux oreilles.

Adrien Douady (1935Douady ( -2006) ) Une ressemblance frappante avec le lieu noir que nous avons appelé « lapin », n'est ce pas ? D'ailleurs ce fractal s'appelle le lapin de Douady.

Choix de

Alors pour quelle raison le fractal produit par l'itération du polynôme quadratique ci-dessus apparaît-il dans notre fractal de Newton ?

C'est dû à un phénomène de renormalisation, démontré par Douady et Hubbard : pour des bons choix de a, et pour la méthode de Newton associée au polynôme (qui dépend de a), itérer un nombre pair de fois, par exemple 2n fois, cette méthode dans une région proche de 0, revient à itérer n fois un polynôme quadratique de la forme z (où c dépend de a). Donc si l'on veut voir apparaître le fractal engendré par un c spécifique, par exemple le lapin de Douady, il ne reste plus qu'à ajuster le nombre a.

a P (z) z ) = z -a + 2 1 z + a + 2 1 ( -1 2 + c 2 + c = 0 2 + c = c ; c 2 + c ; ( 2 + c 2 + c ; ( 2 + c 2 + c 2 + c ; solution de l'équation (c ) .
Essayons donc de résoudre cette équation. En divisant par c, elle devient : c(c )

. Cette nouvelle équation a trois solutions, et celle qui nous convient est approximativement -0 1225612 7448618i. Notons la par c . Ainsi pour tout choix de c proche de c , le troixième terme ci-dessus est proche de 0, et itérer z devrait nous donner un lapin. Nous en avons choisi un au hasard, notamment c 0 12 75i, et ça marche ! Revenons à la méthode de Newton. Cette fois-ci il faut choisir un a tel que le sixième itéré de 0 par cette méthode soit proche de 0. Il existe 18 valeurs de a rendant ce terme du sixième itéré nul, et celle qui nous convient est approximativement -0 0051054 3313825i. Notons la par a . Ainsi n'importe quel choix de a proche de a devrait nous donner un fractal de Newton avec des lapins sautillant dedans. Là encore nous en avons choisi un au hasard, notamment a 0 00508 33136i. Et des beaux lapins apparaissent bel et bien dans notre fractal.

(A titre d'information, le sixième itéré de 0 pour ce choix de a est -0 0001958 0005808i.)

Coutures et accouplements

Cette relation avec les lapins n'explique pas encore toute la structure du lieu noir. Il faut encore comprendre comment ces multiples lapins se sont reliés les uns et les autres.

En fait, bien que ce ne soit pas du tout évident à l'oeil nu, ce fractal est obtenu en recousant ensemble deux fractals plus simples, et donc plus facils à comprendre. Mais nous n'allons pas entrer dans les détails ici. Nous nous contentons juste d'illustrer ces deux fractals ainsi que la couture.

Voici le premier fractal, provenant de l'itération de

Q(z) 1 503

)z

Ce fractal est appelé « ensemble de Julia » de Q. Vous remarquerez que de multiples lapins y sont déjà présents.

2 + c 2 + c = 0

+ 1 2 + 1 = 0 + 0 0 0 2 + c = - + 0 + 0 0 0 = - + 0 -0 = z 3 + ( -0 i 2
Et voici le deuxième, provenant de l'itération de z 2 12i)z

Faisons maintenant tourner notre fractal de Newton de 90 pour mieux visualiser la couture : puis sur une sphère [START_REF]La sphère est liée au plan par une projection stéréographique, voir une belle image de projection stéréographique (sur ce site)[END_REF]) :

La théorie derière cette couture a été connue depuis 1997 (voir les références mathématiques ci-dessous). Mais l'illustration en film présente de grandes difficultés techniques, de nature à la fois informatique et mathématique. Arnaud Chéritat a du surmonter toutes ces difficultés afin de réaliser ces remarquables vidéos pour nous. Je lui en suis très reconnaissante.

Au passage ce procédé de couture a été appelé « accouplement » avec humour par Adrien Douady.

Solutions et points fixes

Essayons maintenant de comprendre pourquoi la méthode de Newton se stabilise à une certaine solution r de l'équation P (z) (pour un P quelconque).

Pour cela remplaçons z par une telle solution r dans z (z) P (z). Eh bien, c'est pour approcher le graphe du polynôme P par une droite (une tangente) f (z) z )P (z ) (z ).

Cette fonction f admet la même valeur et la même dérivée en z que le polynôme P . On peut donc imaginer que la solution de f (z) est proche de celle de P (z) . L'équation f (z) est tellement simple qu'on peut la résoudre directement, soit

  aide d'une calculatrice, vous trouverez 29 698485, d'où la longueur en centimètres de notre feuille A4 (et voilà pourquoi A4 est aussi 21 9 7) !

  La « touche » méthode de NewtonAlors quelle est la méthode de Newton pour trouver des solutions approchées de l'équation x 82 ? Notons P (x) 82 pour simplifier la présentation (cela rend également la méthode plus conceptuelle). Newton a réalisé que pour cela, il suffisait d'itérer l'opération x (où P désigne la dérivée de P ), soit x ou encore un peu comme si la calculatrice avait une touche « x (x) P (x) » et qu'on appuie un grand nombre de fois sur cette touche. Nous allons voir un peu plus tard, dans le paragraphe « Solutions et points fixes », la raison mathématique d'utiliser la formule précise x (x) P (x) plutôt qu'une autre. A présent contentons nous de faire quelques expériences numériques.

  c'est le « bassin » d'attraction de la solution ; et à sa droite, celui de . Alors pourquoi partir d'un nombre complexe alors qu'un réel suffisait ? En effet, dans d'autres problèmes plus compliqués, on risque de tomber sur une équation dont la solution recherchée n'est pas réelle. Tester sur des valeurs initiales qui sont des nombres complexes permet donc de repérer également ce genre de solutions. Ce n'est pas tout. Aller des réels vers les complexes nous permet à la fois d'élargir notre champ de vision (comme illustré par de nombreux dessins ci-dessous), et d'avoir accès à de puissants outils mathématiques relevant de l'analyse complexe. Il y a d'ailleurs des problèmes mathématiques purement réels qui n'ont pu être résolus que grâce à l'utilisation des nombres complexes, c'est ce qu'on appelle encore « complexifier » le problème.

Faisons

  une nouvelle expérience numérique, en remplaçant cette fois la calculette par un ordinateur : Plaçons-nous dans une fenêtre carrée du plan, disons -1 4 6, -1 5 5. Découpons ce carré en 400 00 pixels. Pour chaque pixel (donné sous la forme d'un nombre complexe z y), demandons à notre ordinateur d'itérer l'opération z (z) P (z) un grand nombre de fois (par exemple mille fois), et colorions le pixel z en bleu si le résultat est un nombre complexe proche de 1, en vert s'il est proche de -, en rouge, s'il est proche de -... Enfin colorions le pixel en noir si rien de tout cela se produit.

  fait partie des premiers mathématiciens à avoir observé cela. Il a très vite compris que ces lapins proviennent de l'itération d'une autre opération, plus simple cette fois-ci, celle de z -0 12 75i) Faisons l'expérience comme lui, en colorant un pixel dans le carré -1 5 5 en rouge si ses itérés par cette opération s'échappent du carré, et en noir sinon.Voici le résultat :

  Alors comment procéder précisément ? Pour cela, il faut d'abord savoir comment produire le lapin de Douady. Nous allons voir que le choix de c, tout comme celui de a, est relativement flexible. Commençons par itérer z à partir de z . Nous obtenons, successivement, la théorie de Douady et Hubbard, pour obtenir un fractal qui ressemble à un lapin, il faut choisir un c parmi ceux qui rendent le troisième terme ci-dessus relativement petit, donc un c proche d'une

  Voici enfin la couture en vidéo (film réalisé par Arnaud Chéritat), d'abord dans le plan :

  Nous observons que r Cela veut dire que faire opérer la méthode de Newton ne change pas la valeur r : si vous avez tapé r comme valeur initiale, vous pouvez appuyer tant que vous voulez sur la touche « méthode de Newton », vous obtiendrez toujours r. En termes mathématiques, r est un point fixe. Ce que nous avons fait est un exemple d'une méthode assez courante en mathématique : on transforme la résolution d'une équation en la recherche d'un point fixe d'une opération appropriée.Mais on aurait pu choisir une opération plus simple. Par exemple la touche « z (z) », qui remplace z par z (z) : bien sûr si r est une solution de P (z) , appuyer sur cette touche produira toujours r.Alors pourquoi choisir une opération aussi compliquée que celle de Newton, avec la division par la

z 2 P

 2 et voilà d'où vient la division par la dérivée ! La relation entre la solution de P et celle de f se voit facilement sur une figure (ici dans le cas réel) : Poussons-nous un peu plus loin dans notre compréhension : pourquoi l'itération de la méthode de Newton donne-t-elle des valeurs s'approchant de plus en plus d'une solution r (si l'on est parti d'une valeur assez proche), et ce de manière extrêmement rapide ? Pour comprendre cela calculons la dérivée de l'opération z Nous trouvons remplaçons z par la solution r dans cette expression : la dérivée s'annule elle aussi en r.Non seulement la méthode de Newton laisse la valeur r invariante, mais en plus elle y admet une dérivée nulle. On dit que r est un point fixe super-attractif. Cette terminologie illustre le fait que la solution r attire vers elle, de toutes ses forces, les valeurs voisines.Alors pourquoi la vitesse de convergence a-t-elle un quelconque lien avec le fait que la dérivée s'annule en r ? (z)P (z)Nous pouvons essayer de comprendre cela en considérant un exemple plus simple, mais typique de ce genre de situation. Prenons l'opération « élever un nombre au carré », c'est-à-dire « x ». La valeur 0 est fixe (0 !), et la dérivée (2x) est nulle en 0. La valeur 0 est donc bel et bien un point fixe superattractif.Faisons ensuite un test (sur une calculatrice par exemple) de la vitesse de convergence en prenant 0 5 comme valeur initiale. Nous obtenons : En gros le nombre de zéros après la virgule double à chaque itération. Rapide, n'est ce pas ?! Vous pouvez par exemple le comparer avec l'itération de l'opération « diviser un nombre par 100 », c'està-dire « x 100 » (avec donc une dérivée 1 100 qui n'est pas nulle). Qui est-ce qui va l'emporter en fin de compte ? Graphe séparateur Et l'histoire d'un graphe séparant les trois solutions d'un polynôme (quelconque) P (z) de degré 3, dans tout cela ? Une idée si naturelle... vous n'avez peut-être pas envie de l'abandonner complètement. Eh bien, vous avez raison. Pour le voir, rajoutons un facteur h devant le terme P (z) P '(z), notre « touche méthode de Newton » devient z Ce facteur va modifier la vitesse à laquelle la méthode de Newton permet de s'approcher d'une racine de P . Plus h est petit, moindre est la vitesse. Nous fabriquons ainsi toute une famille de « fractals de Newton », un pour chaque valeur de h. Quand h décroît, devenant de plus en plus petit, ces fractals se transforment progressivement en un graphe. Ce graphe sépare le plan en trois régions, contenant chacune une solution de P (z) . Et nous avons enfin trouvé le graphe séparateur que Cayley cherchait ! Voici ce qui se passe avec notre P (z) spécifique, c'est-àPour citer cet article : Tan Lei, La méthode de Newton et son fractal. Images des Mathématiques, . En ligne, URL : http://images.math.cnrs.fr/La-methode-de-Newton-et-son.html

  Remettons ce dernier terme (toujours à la place de x) dans la formule de Newton, et ainsi de suite, cela nous donne -219 25 , puis -251 227 ... On voit que ces valeurs restent imaginaires pures et donc ne se rapprocheront jamais de nos solutions .

	1;	2 1	1 +	1 882	41 5; = 4	2 1	441 5 +	882 441 5	21 749; = 2
	puis
	112 8632;	0 339; 6	7 4782; 3	0 50594; 3	9 70917; 2	9 69849 2	9 69848 2
	9 698485 882 2
										882
	= 1									882
										+ i
				y
					882				882
										+ i
	882
							2 1		x +	x 882
	nous obtenons
					2 1	i +	i 882		=	2
	et, comme i		1, c'est aussi
						(i	82 )	440 5
	Que se passerait-il si nous prenions comme valeur initiale un autre nombre complexe x y ? Il n'est
	pas très difficile de démontrer que, si x est positif, nous allons nous rapprocher de	, et si au

Après seulement quelques coups d'essai, on est déjà très près de la vraie solution . Voilà une méthode bien efficace...

[3] [4]

On pourrait montrer que, en commençant par n'importe quel nombre proche de (à la place de

x ici), les valeurs obtenues en itérant le procédé vont toujours s'approcher de .

Nous allons maintenant tester cette méthode avec des nombres complexes (nous expliquerons pourquoi à la fin de ce paragraphe), c'est-à-dire des nombres sous la forme x y, avec i un nombre (imaginaire) dont le carré est -1, et x deux nombres réels habituels (voir le film Dimensions mentionné ci-dessous pour une explication animée et détaillée des nombres complexes). Le nombre x est appelé la partie réelle, et y la partie imaginaire.

Un tel nombre serait proche de si x est proche de et y est proche de 0. En particulier les nombres imaginaires purs, c'est-à-dire ceux dont la partie réelle est nulle, ceux la forme 0 y, sont très loin de .

Faisons le test avec i comme valeur initiale, cette fois-ci avec un crayon sur une feuille de papier (A4 par exemple !) au lieu d'une calculatrice.

Mettons i (à la place de x) dans la formule de Newton contraire x est négatif, nous allons nous rapprocher de . La droite des nombres complexes imaginaires purs, qui n'est rien d'autre que l'axe des y, est à la fois la médiatrice des deux solutions, et la ligne de séparation de deux comportements différents des suites de valeurs itératives. À sa gauche,