Laure Latruffe

Laurent Piet
email: laurent.piet@rennes.inra.fr

Does land fragmentation affect farm performance? A case study from Brittany, France

Keywords: agricultural land fragmentation, farm performance, cartographic field pattern registry, France JEL classifications: Q12, Q15, D24 morcellement parcellaire agricole, performance des exploitations agricoles, registre parcellaire graphique, France Classifications JEL : Q12, Q15, D24

land fragmentation affect farm performance? A case study from Brittany, France

The SMART-LERECO Working Papers are meant to promote discussion by disseminating the research of the SMART and LERECO members in a preliminary form and before their final publication. They may be papers which have been accepted or already presented in a national or international scientific conference, articles which have been submitted to a peer-reviewed academic journal, or chapters of an academic book. While not peer-reviewed, each of them has been read over by one of the scientists of SMART or LERECO and by one of the two editors of the series. However, the views expressed in the SMART-LERECO Working Papers are solely those of their authors.

Does land fragmentation affect farm performance?

A case study from Brittany, France

Introduction

Fragmentation of agricultural land is widespread around the world and results from various institutional, political, historical and sociological factors, such as inheritance laws, collectivisation and consolidation processes, transaction costs in land markets, urban development policies, and personal valuation of land ownership [START_REF] King | Land fragmentation: notes on a fundamental rural spatial problem[END_REF][START_REF] Blarel | The economics of farm fragmentation: evidence from Ghana and Rwanda[END_REF]. Farm land fragmentation (LF) is a complex concept that encompasses five dimensions covering: i) number of plots farmed; ii) plot size; iii) the shape of plots; iv) distance of the plots from the farm buildings; v) distances between plots (or plot scattering).

From the public economics perspective, LF may generate both positive and negative externalities: it may increase biodiversity and society's economic value of landscape but, in contrast, it may induce additional trips by farmers which may result in extra roadwork, road safety issues, greenhouse gas emissions, etc. However, first and foremost, LF may affect farmers' production decisions and therefore have an impact on the performance of farms. This impact may be negative or positive. There are several reasons why the impact may be negative. Firstly, LF may exacerbate conflicts regarding labour allocation on the farm: it takes time to travel from one plot to another while the labour force could be undertaking more productive tasks. Secondly, production costs may be increased as LF may require additional equipment, secondary farm buildings and/or external service expenses. Thirdly, LF may restrict the choice of production and constrain management practices, especially in terms of herd management. This could be particularly true for regions where dairy production prevails, such as Brittany, a region in the west of France. Fourthly, investments for soil quality improvement, such as drainage, may be reduced on remote plots, potentially reducing yields.

However, the impact of LF on farm performance may be positive. This is the case if LF leads to an increased diversity in land quality so that the allocation of crops across plots may be optimised, resulting in potentially higher overall yields. In addition, LF may give greater opportunities for risk diversification, thereby reducing production risks at the farm level. For example, a fragmented farm would be less affected by a pest outbreak that spreads on contiguous plots only.

Several authors have tested empirically the effects of LF on the performance of farms. For example, [START_REF] Jabarin | Impacts of land fragmentation on the cost of producing wheat in the rain-fed region of northern Jordan[END_REF] investigated the impact of LF on the production cost of wheat in Jordan. In China, [START_REF] Nguyen | Land fragmentation and farm productivity in China in the 1990s[END_REF], [START_REF] Wan | Effects of land fragmentation and returns to scale in the Chinese farming sector[END_REF] and [START_REF] Tan | Impact of land fragmentation on rice producers' technical efficiency in South-East China[END_REF] investigated the effect of LF on the productivity of major crops, crop output of rural households, and technical efficiency of rice producers in the South-East of the country, respectively. Kawasaki (2010) evaluated both the costs and benefits of LF in the case of rice production in Japan, similarly to [START_REF] Rahman | Impact of land fragmentation and resource ownership on productivity and efficiency: The case of rice producers in Bangladesh[END_REF] in Bangladesh. [START_REF] Parikh | Measurement of technical efficiency in the northwest frontier province of Pakistan[END_REF] In most of this research, LF is represented by the number of plots and/or their average size.

These two variables are employed either directly or indirectly by the use of more elaborate measures, such as the Simpson index or the Januszewski index (which are defined further in the text). However, these variables do not account for all dimensions of LF and may not capture all the constraints that LF imposes on production systems. There are a few exceptions to the use of these sole variables. For example, [START_REF] Tan | Impact of land fragmentation on rice producers' technical efficiency in South-East China[END_REF] considered in addition the average distance from the plots to the homestead, while [START_REF] Gonzalez | Evaluation of productive rural land patterns with joint regard to the size, shape and dispersion of plots[END_REF] used more elaborate measures of LF (which accounted for the size, shape and dispersion of plots) to study the productivity gains from land consolidation. However, in this latter case, these measures were not tested on a real sample of farms, but instead were applied to a hypothetical dataset of farms.

The objective of the paper is to analyse the influence of LF on the performance of farms in the case of one French region, the Western region of Brittany, or 'Bretagne'. This is a NUTS2 Commission, 2000), Brittany farms were composed on average of 14 plots, with a mean plot size of 4.35 ha. Twenty-five percent of the farms had 18 plots or more, and 25% of these plots had an average area of 2.42 ha or less. Such figures are quite similar to the national averages for France. In this paper, the relationship between LF and farm performance is investigated for the year 2007 using several performance indicators (production costs, yields, financial results and technical efficiency) calculated from farm-level data, and various LF indicators calculated at the municipality level using data from the RPG. The various fragmentation indicators enable the traditional measures of plot number and mean size of plots, as well as the scattering of plots, to be accounted for.

The paper is structured as follows. Section 2 describes the data and explains the methodology used to calculate the various indicators of performance and of LF. Section 3 presents the methodology used to investigate the effect of LF on performance and the results of the empirical analysis. Section 4 concludes.

Data and methodology

Measuring farm performance

As mentioned above, we used farm-level data for the calculation of farm performance 2010), where the main production is the one which provides at least two-thirds of the farm's gross standard margin. The distribution reflects Brittany's agriculture where dairy, poultry and pig breeding prevail: 29% of the sample specialised in dairy production, and 27% in granivores production. Mixed crop and livestock farming (generally the production of cows' milk and field crops) accounted for 11% of the sample, and the breeding of other grazing livestock (goats and sheep) for 14%. Finally, for 13% of the sample farms the main production was field crops, and for another 6% the main production was crops other than field crops (mainly vegetables). This is higher than the region's average (1.7 AWU) and similar to the region's commercial farms' average (2.1 AWU) (2010 Agricultural Census). The average number of livestock units (calculated using the European standard coefficients applied to each livestock type) on the sample farms was 244.9. This relatively high figure is due to the numerous farms in Brittany specialised in livestock and, in particular, to the poultry and pig head numbers. On average, farms rented in 77% of their utilised area and employed 15% of hired labour force.

Several indicators of farm performance were computed for each farm in the sample. Firstly, various categories of production costs were calculated per farm and per unit of utilised area.

These consisted of costs of fertilisers, seeds, pesticides, fuel, intermediate consumption and hired labour. Secondly, two production yields were calculated: wheat yield in tons of wheat produced per hectare of wheat cultivated; and milk yield in litres of milk produced per cow.

Thirdly, four revenue or profitability results were calculated per farm and per unit of utilised area: the farm gross product, composed of farm sales and insurance compensations; the farm gross margin, obtained from the farm gross product minus variable costs specific to crop and livestock production; the farm operating surplus, obtained from the farm gross margin minus land, labour and insurance costs; and the farm pre-tax profit, given by the farm operating surplus minus depreciation and interest, and before taxes are deducted. Subsidies were not included in the farm gross product, and therefore not included either in the three profitability indicators. Finally, technical efficiency and scale efficiency were calculated for each farm.

Technical efficiency assesses how far farms are located from the maximum production frontier for a given combination of inputs. It is a more complex measure than partial productivity indicators such as yields, since it relates all outputs produced to all inputs used on the farm. Technical efficiency is composed of pure technical efficiency (that is to say, whether farmers operate their farm efficiently) and of scale efficiency (that is to say, whether the farm's production scale is optimal). Technical and scale efficiencies were computed using the non-parametric method Data Envelopment Analysis (DEA) which employs linear programming to construct a frontier that envelops the data used [START_REF] Charnes | Measuring the efficiency of decision making units[END_REF].

Efficiency scores obtained by DEA are between one -for a fully efficient farm (i.e., a farm located on the efficient frontier) -and zero, with smaller scores indicating lower efficiency.

Since the efficient frontier depends on the sample used, the efficiency scores may be overestimated if the most highly performing farms in the population are not included. For this reason, we constructed the efficient frontier for the whole Brittany sample (469 farms). We merged all types of farming into one sample to overcome the limited sizes of the type of farming sub-samples. The DEA model was output-oriented (so that farms were assumed to maximise their output level, given input levels). The model had one single output, namely the farm output produced in Euros, and four inputs: the utilised area in hectares; the labour used in AWU; the intermediate consumption in Euros; and the capital value in Euros. Under the assumption that farms operated under constant returns to scale, the total technical efficiency score for each farm was obtained. Total technical efficiency was then decomposed into pure technical efficiency and scale efficiency. The calculation of the pure technical efficiency was made under the assumption that farms operated under variable returns to scale, and indicated the efficiency of farmers' practices irrespective of farm size. By contrast, scale efficiency, which was calculated for each farm as the ratio between its total technical efficiency and its pure technical efficiency, revealed whether the farm operated at the optimal scale of production. , 2000). This is a Geographic

Information System (GIS) database which is maintained by the 'Agence de Service et de Paiement' (ASP), a public body which gathers the field patterns declared by farmers who apply for support under the framework of the Common Agricultural Policy (CAP)2 and which delivers subsidies to farmers based on these declarations. In fact, farmers are not requested to delineate each of their individual fields but rather each of their 'plots', which we define for this paper as follows: a plot is a set of contiguous fields (which may or may not all bear the same crop) which is both delimited by easily identifiable landmarks (such as agricultural byways, roads, rivers, another plot, etc.) and is stable from year to year.

We used the 2007 registry ('RPG anonyme ASP 2007') which identifies 450,787 plots used by 31,921 farms for the four NUTS3 regions of Brittany. Each farm could be categorised as one of the following: i) a farm that was registered in one of the four Brittany NUTS3 regions and whose plots were all located inside this one region; ii) a farm that was registered in one of the four Brittany NUTS3 regions, but whose plots were partly located outside that region; and iii) a farm that was registered outside Brittany but whose plots were located totally or partly inside one of the four Brittany NUTS3 regions. We retained all farms and plots corresponding to case i). As regards case ii), we only retained those farms whose plots were located in one of the four NUTS3 regions directly neighbouring Brittany (namely 'Loire-Atlantique', 'Maineet-Loire', 'Manche' and 'Mayenne', see Figure 1) and we considered both their plots located in Brittany and their plots located in these four directly neighbouring regions. Similarly, as regards case iii), we retained those farms registered in one of the four above-mentioned NUTS3 regions directly neighbouring Brittany and we considered both their plots located in Brittany and their plots located in one of these four directly neighbouring regions. Finally, in order to ensure that we included 'entire' farms only, we excluded those farms whose total area declared by the farmer in the RPG was 0.02 ha or more different from the area obtained from summing the areas of each individual plot of the farm. In the end, the database used consisted of 29,433 farms and 418,480 plots.

For each farm among these 29,433 farms, ten fragmentation descriptors were computed, which relate to one of the five dimensions of LF as described in the introduction (the formal definitions of the descriptors are given in the Appendix).

1. LF descriptors relating to the number of plots. One descriptor was used, namely the number of plots on the farm ( ).

2. LF descriptors relating to the shape of plots. Two descriptors were used: the weighted average of the shape index of the plots ( ) [START_REF] Akkaya Aslan | Some metric indices for the assessment of land consolidation projects[END_REF]; and the average of the areal form factor ( ) [START_REF] Gonzalez | Evaluation of land distributions with joint regard to plot size and shape[END_REF].

3. LF descriptors relating to the size of plots. Three descriptors were used: the average plot size ( ); and two more elaborate indexes, namely the Simpson index ( ) [START_REF] Blarel | The economics of farm fragmentation: evidence from Ghana and Rwanda[END_REF][START_REF] Hung | The economics of land fragmentation in the north of Vietnam[END_REF][START_REF] Kawasaki | The costs and benefits of land fragmentation of rice farms in Japan[END_REF] and the Januszewski index ( ) [START_REF] King | Land fragmentation: notes on a fundamental rural spatial problem[END_REF].

4. LF descriptors relating to the distance of plots from the farm. Three descriptors were used: the average distance of a hectare from the farm ( ); and two more elaborate indexes, namely the grouping index ( ) [START_REF] Marie | Des pratiques des agriculteurs à la production de paysage de bocage[END_REF] and the structural index ( ) [START_REF] Marie | Des pratiques des agriculteurs à la production de paysage de bocage[END_REF].

5. LF descriptors relating to the scattering of plots (i.e., to the distance between plots).

One descriptor was used, namely the normalised average nearest neighbour distance ( ).

Since there was no information in the registry concerning the location of the farmsteads, we first computed the centroid of each plot (that is, its geometric centre) and inferred from this the barycentre of each farm (that is, its 'centre of mass', with the 'mass' associated with each plot of the farm being the plot's area); we then replaced the distance from the farmstead by the distance from the barycentre of the farm in those LF indicators which use distance in their definition (namely , , and ).

It should be stressed that the relationship between a descriptor and LF may be positive (i.e., a higher value of the descriptor indicates higher fragmentation) or negative (i.e., a higher value of the descriptor indicates lower fragmentation). As can be seen in Table 3, descriptors positively related to LF are the number of plots, the weighted average shape index, the Simpson index, descriptors relating to the distance from the barycentre of the farm and the normalised average nearest neighbour distance, while descriptors negatively related to LF are the average areal form factor, the Januszewski index and average plots size. 4 reports descriptive statistics for the 29,433 farms in our database. On average, the farms registered outside Brittany were the largest (their average area was 75.51 ha) and the farms registered within Brittany were relatively similar across Brittany's NUTS3 regions in terms of average area (around 50 ha with a standard deviation of about 40 ha). Among the four Brittany NUTS3 regions, 'Côtes-d'Armor' appears to be the most fragmented one for most LF descriptors, followed by 'Finistère', 'Ille-et-Vilaine' and finally 'Morbihan'. The fragmentation of farms registered outside Brittany showed greater variation: they were relatively fragmented when considering most descriptors but, in contrast, they presented a lower fragmentation level in terms of mean size of plots, which was higher than that of farms registered inside Brittany. There are two explanations for this contrasting picture. Firstly, the sample of farms registered outside Brittany was smaller. Secondly, when considered together they constituted a heterogeneous category (the structure and main production of farms in the northern neighbour 'Manche' were quite different from those of the southern neighbour 'Loire-Atlantique'). 1) and whose plots are at least partly located in one of Brittany's NUTS3 regions ('Côtes-d'Armor', 'Finistère','Ille-et-Vilaine' and 'Morbihan').

Source: authors' calculations based on the field pattern registry 'RPG anonyme ASP 2007' database As explained in Section 2.1, we analysed the influence of average LF in the municipality where a farm was located on the farm's performance. To do this, we calculated the aggregated fragmentation descriptors at the level of each municipality of the 29,433 farms in the field pattern database. We computed the weighted average of each descriptor considering all farms with at least one plot in , each of these farms being weighted by its share in the total operated area of , or, formally:

∑ (1)
where represents one of the ten fragmentation descriptors, represents farm 's operated area located within municipality and ∑ is the total operated area in municipality . Note that, because the RPG only includes farms which apply for CAP payments and because we excluded almost 8% of the farms (2,488 out of 31,921) from the initial database during the sample selection process (see above), the descriptors calculated at the municipality level should be viewed only as proxies for the true farmland fragmentation of municipalities. We are confident, therefore, that our sample can be regarded as representative of Brittany.

3 The role of LF on farm performance

Methodology

The influence of LF on farm performance was investigated using Ordinary Least Squares (OLS) regressions, where the dependent variables were, in turn, each of the 15 per-farm performance indicators described above. All LF indicators were introduced in turn in the regressions as explanatory variables. Therefore, there were 15 10 150 regressions, which differed according to the dependent variable (each performance indicator) and the LF indicator used as the explanatory variable.

Various explanatory variables, available in the FADN data, were used in all 150 regressions in addition to LF descriptors: farmer's age; farm size in terms of utilised area in hectares; a farm size dummy based on classes of economic size (the dummy is equal to one if the farm is greater than 100 Economic Size Units (ESU), with 1 ESU equivalent to 2,200 Euros of standard gross margin, and zero if it is less than 100 ESU); a farm legal status dummy (equal to one for an individual farm, and zero for a partnership or company); the share of rented land in the farm utilised area; the share of hired labour in total labour used on the farm; the farm capital to labour ratio; the operational subsidies received by the farm, related to hectares of utilised area; a farm location dummy (equal to one if the farm is located in an area subject to nitrate pollution zoning restrictions, and zero if not); and farm production specialisation dummies (based on the categories in Table 1 with 'other crops' being the reference).

For each regression, we computed the confidence interval of the estimated parameters from the White or 'sandwich' estimator of the variance-covariance matrix, which is robust to misspecification problems such as heteroskedasticity and small sample size.

Results

Table 6 summarises the accuracy with which the 150 models fit the data, as measured by the R-squared statistics. This accuracy ranges from an average of 0.167 for the regressions with the milk yield as the dependent variable, to an average of 0.659 for the regressions with the hired labour cost as the dependent variable; 94 out of the 150 regressions exhibited an Rsquared statistic above 0.30, which is fairly satisfactory for such cross-sectional micro data models based on a limited sample. It is also worth noting that the standard deviations of the R-squared statistics are low, indicating that, for a given farm performance indicator, the fit of the model is quite similar whatever the LF descriptor used as a regressor. Source: authors' calculations

Source: authors' calculations

Our results show that, from a methodological point of view, each LF descriptor relates to one or more performance indicators but not to all of them and that, reciprocally, each performance

indicator is explained by one or more LF descriptors but not by all of them. This gives confidence in our strategy of using a wide set of variables for both dimensions. LF descriptors which are most related to farm performance appear to be, first, the grouping index ( ) and the average areal form factor ( ), and then the structural index ( ) and the number of plots ( ). The number of plots proves to be the only LF descriptor which is significantly related to at least one indicator in the four categories of farm performance which we considered, but the significance levels are somewhat limited (never less than 5%). By contrast, the average size of plots, although a traditionally used LF descriptor, seems to have a limited impact on the various dimensions of farm performance, be it directly considered ( ) or indirectly through the more elaborate indexes ( and ).

Most results regarding the detailed links between LF descriptors and performance indicators conform to agronomic and economic intuition. Firstly, production costs are positively related to the number of plots and to their distance from the farm, but decrease with plot size. The results regarding the shape of plots are more surprising since they first suggest that seed cost, pesticide cost and hired labour cost should decrease when plots are more irregularly shaped.

The result regarding seed cost is difficult to interpret but it is significant at the 10% level only.

However, the other two results, which are more significant, may be explained as follows: on the one hand, irregularly shaped plots may impede the spread of pest attacks and hence reduce the use and therefore cost of pesticides; on the other hand, irregularly shaped plots may be more difficult to entrust to the care of hired, often less qualified, people so that the operator will farm them himself or herself, hence reducing the cost of hired labour. Secondly, LF appears to have a negative impact on yields, especially that of wheat, mostly through size and distance. Thirdly, revenue and profitability are found to decrease with the number of plots, the irregularity of their shape and their distance from the farm, but the average size of plots does not seem to have a significant impact. Then again, counter-intuitive results (the positive impact of on the operating surplus and of on the gross margin) are significant at the 10% level only. Finally, total technical efficiency proves to be significantly related to none of the considered LF descriptors. By contrast, conforming to intuition, the number of plots seems to play a role in reducing pure technical efficiency, while the scattering of plots affects scale efficiency. However, the positive and significant impact of the scattering of plots on pure technical efficiency is more difficult to interpret.

In order to present the regression results in a more practical and vivid way, we simulated the impact of a reduction in LF at the municipality level on two key performance indicators;

wheat yield, as a main physical component of farm performance, and pre-tax profit, as a main financial component of farm performance. This reduction in LF could hypothetically be reached by, for example, a consolidation programme. To this end, we computed for each LF descriptor what improvements in pre-tax profit and wheat yield could be obtained by the average farm when moving, at the municipality level, from one LF quartile to the next in the direction of reducing fragmentation. With this, fragmentation improvements are immediately readable in terms of Euros per farm for the pre-tax profit and tons per hectare for the wheat yield. Therefore, this can illustrate the relative importance of LF descriptors whose estimated regression coefficients are not directly comparable with each other.

Table 8 illustrates that the highest benefits in terms of pre-tax profit would be reached by reducing LF in terms of distance of plots from the barycentre of the farm: on average, decreasing the average distance of a hectare ( ) at the municipality level by around 500 m would raise the pre-tax profit by 5,862 Euros per farm -a 43% increase. By comparison, reducing the average number of plots ( ) per farm at the municipality level from 22.5 to 14.5 would lead to a pre-tax profit increase of 4,987 Euros per farm (or 37%).

Concerning the yield of wheat, the highest benefits (almost 0.5 ton per hectare, or a 9% increase) would be obtained from a reduction in the grouping index ( ), i.e., by reducing the maximum distance of plots from their barycentre, rather than their average distance. In the case of the wheat yield, the second best option would consist of improving the size of plots at the municipality level as measured by the Januszewski ( ) and the Simpson ( ) indexes, rather than the number of plots by farms, with expected gains estimated at approximately 0.4 ton per hectare (or an 8% increase).

Such figures may look quite substantial for both performance indicators. However, they are mainly intended to illustrate our results and especially to compare the marginal benefit (or, reciprocally, the relative burden) of each LF dimension on the various aspects of performance.

They should not be viewed as accurate predictions, for at least three reasons. Firstly, the simulated LF improvements may actually be very substantial themselves, hence very costly to implement in real life. Thus these implementation costs should be compared, in addition to comparing the benefits from improving one LF descriptor with respect to the others.

Secondly, it is hardly plausible that a particular consolidation programme would enhance one LF descriptor only, leaving the others unchanged. In general, a consolidation programme would seek to improve several LF dimensions at the same time, e.g., by reducing the number and distance of plots, improving their shapes and increasing their average size. However, these dimensions may be competing among themselves to some extent, so that a compromise would have to be reached, leading to a limited improvement in each dimension -if not to a deterioration for some descriptors in some cases. It is our view that the way in which these multi-dimensional benefits and costs aggregate together remains an empirical question, which may be addressed only thanks to hypothetical simulations such as that of [START_REF] Gonzalez | Evaluation of productive rural land patterns with joint regard to the size, shape and dispersion of plots[END_REF] or for specific case studies. Thirdly, the above average pre-tax profit and wheat yield improvements may also reveal that such heavy consolidation programmes are likely to induce additional changes in farming practices and in farm production, so that they should not be simply compared to the average pre-consolidation figures as if they were ceteris paribus.

Conclusion

We have investigated the relationship between agricultural land fragmentation (LF) and farm performance in 2007 in the French NUTS2 region of Brittany. Various farm performance indicators (in terms of costs, yields, revenue, profitability, technical and scale efficiency) calculated for a sub-sample of FADN farms were regressed on several explanatory variables, including average LF descriptors computed for the municipalities where those farms were located. Among the LF descriptors used, we considered not only the number of plots and the mean size of plots which are traditionally used in the economic literature investigating the impact of LF on farm performance, but also more complex indexes, in order to account for: the shape of plots; (a proxy of) the distance between plots and farmsteads; and the distance between plots themselves (or scattering of plots).

In our view, our analysis highlights that, from a methodological perspective, the measures of LF traditionally used in the literature, namely the number of plots and the average plot size, may not reveal the full set of significant relationships with farm performance because they do not capture all the dimensions of land fragmentation. In particular, they exclude distance considerations. In this respect, the grouping index used here seems to be powerful. However, circumventing the absence of information regarding the location of the farmsteads by computing distances relative to the farm barycentre, as we have done in this paper, may introduce some bias that would be worth investigating.

Considering only the significant relationships, the analysis of farm performance and LF gives three main findings. Firstly, whatever the LF descriptor considered, in general similar conclusions are reached regarding the impact of LF on the various components of farm performance. There are three main conclusions: i) LF tends to increase production costs; ii)

LF has a negative impact on crop yields; iii) LF tends to reduce the revenue and profitability of the farm. Such findings that LF is overall harmful to farm performance are consistent with those found in the previous literature on the subject. Secondly, these very general conclusions should not hide the fact that in some cases, even if very few, the impact of LF on farm performance was the opposite to that expected, and that it was not always possible to find an economic rationale for such results. This is another argument in favour of using several LF indicators to investigate the link between fragmentation and performance. Thirdly, we have shown that the benefits from reducing fragmentation may differ with respect to the improved LF dimension and the performance indicator considered. The overall impact of a real-life consolidation programme, which may modify several LF dimensions at the same time,

remains an empirical open question which should be investigated carefully in each specific case.

We should also stress that, while the general finding of our analysis is a negative impact of LF on farm performance, we do not advocate a 'blind' reduction of farm fragmentation through large scale and systematic consolidation programmes. One reason is the necessity of balancing the private and societal gains of LF reduction. In the Brittany case, fragmented agricultural land is usually associated with hedges and natural corridors which have been shown to be beneficial to, e.g., biodiversity, water fluxes and the environment in general [START_REF] Thenail | Variation of farm spatial land use pattern according to the structure of the hedgerow network (bocage) landscape: a case study in northeast Brittany[END_REF][START_REF] Thenail | The contribution of croprotation organization in farms to crop-mosaic patterning at local landscape scales[END_REF]. This indicates that the private costs of LF at the farm level should be carefully balanced with potential public benefits at society's level. In this respect, programmes which aim at enhancing the structure of field patterns under the constraint of preserving and/or replanting hedges, such as the 'amicable plot exchange' programme put in place in Brittany by the agricultural extension services and the local authorities, may represent an efficient compromise (CA Bretagne, 2011).

Even though these results sound reasonable and generally conform to intuition, our analysis suffers two major limitations which should be considered with great care if such research is to be aimed at proposing a consolidation programme. Firstly, endogeneity issues would have to be investigated carefully: although we can be relatively confident that the relationship between variables is mainly in one direction from a static point of view, namely that municipalities' LF influences the performance of specific farms, it might be that, in a dynamic perspective, efficient farms are more likely to be in a position to decrease their fragmentation at the expense of neighbouring farms. Secondly, drawing any causal conclusions would mean assuming a direct link between the LF of the municipality where the considered farm is located, and the LF within the farm itself: though the approach adopted here -due to data limitations -indeed relies on the hypothesis that the higher the LF of the municipality, the higher the probability for the farm to be fragmented, it may happen that farms which are not very fragmented may be located in a highly fragmented municipality, and vice versa. Finding a way to gain access to a measure of fragmentation at the individual level for the farms in our sample constitutes a major challenge for future work. Although our analysis has shed some light on the relationship between the performance of a farm and the LF in the municipality where it is located, further investigation is therefore needed, especially before any policy recommendations can be made. The LF descriptors are defined as follows:

Appendix. Formal definitions of the LF descriptors used

1. LF descriptors relating to the number of plots -number of plots: ∑ 2. LF descriptors relating to the shape of plots -weighted average plot shape index:

-average plot areal form factor: ∑

Les

  Working Papers SMART-LERECO ont pour vocation de diffuser les recherches conduites au sein des unités SMART et LERECO dans une forme préliminaire permettant la discussion et avant publication définitive. Selon les cas, il s'agit de travaux qui ont été acceptés ou ont déjà fait l'objet d'une présentation lors d'une conférence scientifique nationale ou internationale, qui ont été soumis pour publication dans une revue académique à comité de lecture, ou encore qui constituent un chapitre d'ouvrage académique. Bien que non revus par les pairs, chaque working paper a fait l'objet d'une relecture interne par un des scientifiques de SMART ou du LERECO et par l'un des deux éditeurs de la série. Les Working Papers SMART-LERECO n'engagent cependant que leurs auteurs.

  investigated the influence of LF on the technical efficiency of farms in the North-West Frontier Province of Pakistan, while Manjunatha et al. (2013) carried out a similar investigation in India. In Europe, Di Falco et al. (2010) analysed how LF affects farm profitability in Bulgaria and Del Corral et al. (2011) analysed how LF affects the profits of Spanish dairy farms.

  indicators, and municipality-level averages for the calculation of LF. More precisely, we investigated the relationship between farm performance for a sample of farms, and LF of the municipality where the sample farms are located. The underlining assumption is that a farm's LF is positively correlated with the LF in the municipality where the farmstead is located. The studied farms were extracted from the French Farm Accountancy Data Network (FADN) 2007 database. The FADN database, managed by the French Ministry of Agriculture, contains structural and bookkeeping information for a five-year rotating panel of professional farms. In 2007, 480 farms of the FADN sample were located in Brittany. Among those 480 farms we excluded ten farms which used no land and one farm with inconsistent capital data. The final sample thus consisted of 469 farms. Figure1shows the location of the municipalities of the 469 FADN Brittany farms.

Figure 1 :

 1 Figure 1: Brittany NUTS3 regions and studied municipalities

Figure 2 :

 2 Figure 2: Main productions in Brittany's municipalities

  descriptors relating to the distance of plots from the farm -average distance of a hectare: descriptors relating to the s tter ots ca ing of pl -normalised average nearest neighbour distance: ∑ ⁄

Table 1 : Main characteristics of the farms in the FADN sample used (469 farms) Share of farms in the sample (%)

 1

	According to their main production
	Field crops	13
	Dairy	29
	Other grazing livestock	14
	Granivores	27
	Mixed (crops and livestock)	11
	Other crops	6
	In areas

with nitrate pollution zoning restrictions 4 Mean Std. deviation Minimum Maximum Utilised agricultural area (ha

	)	62.36	44.93	0.12	398.96
	Number of full time labour equivalents	2.44	2.57	1.00	23.96
	Number of livestock units	244.85	359.62	0.00	2,522.11
	Share of land rented in (%)	77	33	0	100
	Share of hired labour (%)	15	25	0	100
	Source: authors' calculations based on the French FADN 2007 database
	Table 1 describes the sample of the 469 farms used to analyse the relationship between farm
	performance and LF. It shows the distribution of these farms according to their main type of
	production, according to the European definition of type of farming (European Commission,

Table 1

 1

	). In 2007, the studied farms utilised on average 62.4 ha,

Table 2 : Performance of the farms in the FADN sub-sample used

 2 tons per hectare, and 269 farms (57% of the sample) produced milk with an average yield of 7,043 litres per cow. The 469 farms generated on average almost 1,800 Euros per hectare of pre-tax profit without subsidies. Their total technical efficiency score was 0.595 on average, indicating that they could increase their output by 40.5% without increasing their

	Farm performance indicator	Average value per farm per hectare	Number of observations
	Production costs (Euros)
	Fertiliser cost	7,083.73	341.87	469
	Seed cost	7,724.19	1,143.79	469
	Pesticide cost	6,010.70	225.95	469
	Fuel cost	4,768.39	156.43	469
	Intermediate consumption cost	194,364.50	13,542.45	469
	Hired labour cost	13,622.62	3,360.09	469
	Yields
	Wheat yield (tons / hectare)	5.3		342
	Milk yield (litres / cow)	7,043		269
	Revenue and profitability without farm subsidies (Euros)
	Gross product	297,385.60	23,901.72	469
	Gross margin	103,021.10	10,359.27	469
	Operating surplus	63,214.24	5,258.55	469
	Pre-tax profit	13,697.48	1,756.50	469
	Efficiency scores
	Total technical efficiency	0.595		469
	Pure technical efficiency	0.680		469
	Scale efficiency	0.883		469

Source: authors' calculations based on the French FADN 2007 database Table 2 presents the descriptive statistics of the performance indicators for the 469 sample farms. Among these, 342 farms (73% of the sample) produced wheat with an average yield of 5.3

Table 3 : Relationship between the studied descriptors and LF

 3

	Descr	y r iptors positivel elated	to LF	Descriptors negatively related to LF
	Number of plots	(	)		Average plot areal f orm fa	ctor (	)
	Weighted averag	e plot s hape in	dex (	)	Average plots size (	)
	Simpson index (				)		Januszewski index (	)
	Average distance		ctare ( e of a h	)
	Grouping index (		)
	Structural index (		)
	Normalised average nearest neighbour distance (	)
	Table

Table 4 : Descriptive statistics of the fragmentation descriptors at the farm level a

 4

	Land fragmentation descriptor	NUTS3 'Côtes-d'Armor'	NUTS3 'Finistère'	NUTS3 'Ille-et-Vilaine'	NUTS3 'Morbihan'	Neighbouring NUTS3 regions b	All
	Number of farms		7,942	6,149	8,653	6,298	391	29,433
	Average farm area (ha)		49.13	54.85	47.49	52.92	75.51	51.00
							(34.98)	(41.64)	(38.72)	(39.60)	(40.96)	(38.82)
	Number of plots (	)	15.11	14.55	12.24	12.32	14.93	13.55
							(11.10)	(11.10)	(10.09)	(9.62)	(9.24)	(10.56)
	Weigh	age plot shape ted aver	1.34	1.32	1.31	1.33	1.37	1.33
	index (	)				(0.19)	(0.17)	(0.18)	(0.19)	(0.18)	(0.19)
	Averag	e plot ar	eal form	0.044	0.044	0.044	0.044	0.042	0.044
	factor (	)			(0.006)	(0.006)	(0.006)	(0.006)	(0.005)	(0.006)
	Average plots' s	ize (	)	3.67	4.41	4.53	4.90	5.74	4.37
							(2.45)	(3.53)	(14.58)	(3.65)	(3.32)	(8.36)
	Simpson index (		)	0.77	0.77	0.72	0.73	0.80	0.75
							(0.22)	(0.21)	(0.25)	(0.24)	(0.15)	(0.23)
	Januszewski index (	)	0.37	0.37	0.42	0.41	0.34	0.39
							(0.19)	(0.18)	(0.21)	(0.20)	(0.13)	(0.20)
	Average distance of an		1,221	1,373	1,246	1,084	3,115	1,256
	hectare (		)		(1,823)	(1,917)	(1,844)	(1,392)	(3,452)	(1,814)
	Grouping index (	)	8.93	8.92	8.74	6.84	18.01	8.55
							(12.68)	(11.86)	(13.35)	(10.33)	(17.45)	(12.41)
	Structural index (	)	3.93	3.42	3.15	2.19	4.13	3.23
							(11.52)	(7.67)	(7.83)	(5.21)	(5.56)	(8.53)
	Normalised average nearest	1.47	1.32	1.66	1.40	2.18	1.49
	neighbour distance (		(3.90)	(3.76)	(4.89)	(3.53)	(5.23)	(4.14)

a Except for the number of farms, averages are presented and standard deviations are shown in brackets and italic font. b Farms registered in NUTS3 regions directly neighbouring Brittany

('Loire-Atlantique', 'Maine-et-Loire', 'Manche' and 'Mayenne', see Figure

Table 5 : Descriptive statistics of the fragmentation descriptors at the municipality level

 5

	Land fragmentation descriptor	Mean	Std. deviation	Min	Max
	Studied municipalities (349 observations)
	Number of farms						60.56	29.73	3	200
	Farmed area (ha)						3,588.31	1,844.17	53.32	11,811.04
	Number of plots (			)		19.29	7.04	8.84	62.09
	Weighted average pl ot shap dex ( e in	)	1.344	0.065	1.172	1.542
	Average plot are al form	fac	tor (	)	0.043	0.002	0.038	0.049
	Average plots' s	ize	(			)	4.77	2.10	0.31	30.24
	Simpson index (			)		0.841	0.043	0.727	0.954
	Januszewski inde	x (		)	0.302	0.047	0.158	0.422
	Average distance	of a h	e ctare (	)	1,675	443	897	4,339
	Grouping index (				)		9.555	2.843	4.332	26.063
	Structural	index (			)		3.179	3.751	0.780	47.152
	Normalised average nearest neighbour	0.986	0.281	0.415	2.444
	distance (

All municipalities in Brittany (1,255 observations)

  In total, 349 municipalities were related to the 469 farms of the FADN, out of the 1,255Brittany municipalities for which we had data in the RPG. Table5reports descriptive statistics for the 349 municipalities, as well as for all the 1,255 Brittany municipalities. It appears from this table and from a further examination of the distributions for all LF descriptors, that our sample of 349 municipalities is skewed towards higher values of LF compared to the full sample of 1,255 municipalities, but that the discrepancy is very slight.

	Number of farms						45.67	28.72	1	200
	Farmed area (ha)						2,781.25	1,704.15	9.01	11,811.04
	Number of plots (			)		20.97	8.34	3.00	85.18
	Weighted average pl ot shap dex ( e in	)	1.347	0.075	1.084	1.848
	Average plot are al form	fac	tor (	)	0.043	0.002	0.026	0.056
	Average plots' s	ize	(			)	4.87	15.23	0.31	540.57
	Simpson index (			)		0.850	0.049	0.404	0.973
	Januszewski inde	x (		)	0.290	0.052	0.124	0.668
	Average distance	of a h	e ctare (	)	1,670	562	217	6,854
	Grouping index (				)		9.358	3.207	1.976	43.073
	Structural	index (			)		3.075	2.620	0.582	47.152
	Normalised average nearest neighbour	0.937	0.350	0.289	5.344
	distance (
			Source: authors' calculations based on the field pattern registry
								'RPG anonyme ASP 2007' database

Table 6 : R-squared statistics for the 150 OLS regressions

 6

	Farm performance indicator (dependent variable)	Obs.	Mean	Std. deviation	Min	Max
	Production costs
	Fertiliser cost per farm	469	0.358	0.001	0.357	0.360
	Seed cost per farm	469	0.439	0.001	0.438	0.441
	Pesticide cost per farm	469	0.581	0.001	0.581	0.583
	Fuel cost per farm	469	0.459	0.003	0.457	0.463
	Intermediate consumption cost per farm	469	0.543	0.000	0.542	0.543
	Hired labour cost per farm	469	0.659	0.001	0.657	0.661
	Yields
	Wheat yield	342	0.260	0.014	0.246	0.288
	Milk yield	269	0.167	0.006	0.161	0.180
	Revenue and profitability without farm subsidies
	Gross product per farm	469	0.577	0.001	0.577	0.579
	Gross margin per farm	469	0.460	0.003	0.458	0.467
	Operating surplus per farm	469	0.251	0.003	0.248	0.259
	Pre-tax profit per farm	469	0.184	0.003	0.181	0.189
	Efficiency scores
	Total technical efficiency	469	0.386	0.001	0.386	0.388
	Pure technical efficiency	469	0.301	0.003	0.299	0.308
	Scale efficiency	469	0.202	0.002	0.201	0.207
	Source: authors' calculations
	Due to space constraints, we do not present the detailed results for each of the 150
	regressions. Instead, we report in Table 7 the signs and significance levels of the regression
	coefficients obtained for each LF descriptor.

Table 7 : Fragmentation and FADN farms' performance: sign and significance of regression coefficients for LF indicators a Farm performance indicator

 7

	Indicators of
	the number	Indicators of plot shape	Indicators of plot size
	of plots
	number of plots ( )	weighted average plot shape index ( )	average plot areal form factor ( )	average plot size (ha) ( )	Simpson index ( )	Januszewski index ( )

a The fragmentation descriptors (columns) are calculated at the municipality level and the descriptors which name is in italic are negatively related to land fragmentation (see text). ***, **, *, °: significance at the 0.1%, 1%, 5% and 10% level respectively; ns: not significant at the 10% level.

Table 7 (continued): Fragmentation descriptors and FADN farms' performance: sign and significance of regression coefficients for LF indicators a Farm performance indicator Indicators of plots' distance from the farm Indicators of plots' scattering

 7

	average distance of a hectare ( )	grouping index ( )	struc tural (	index )	Normalised av. nearest neighbour distance ( )

a The fragmentation descriptors (columns) are calculated at the municipality level and the descriptors which name is in italic are negatively related to land fragmentation (see text). ***, **, *, °: significance at the 0.1%, 1%, 5% and 10% level respectively; ns: not significant at the 10% level.

The Nomenclature of Territorial Units for Statistics (NUTS) provides a single uniform breakdown of territorial units for the production of regional statistics for the European Union (EU). (Source: http://epp.eurostat.ec.europa.eu/portal/page/portal/nuts_nomenclature/introduction)

For more information on the RPG, see the dedicated pages on the website of the ASP.(http://www.asp-public.fr/?q=node/856).

Acknowledgments

Financial support from FP7 project Factor Markets (Comparative Analysis of Factor Markets for Agriculture across the Member States) grant agreement n°245123-FP7-KBBE-2009-3 is acknowledged. The authors thank the Service de la Statistique et de la Prospective from the French ministry of Agriculture for providing the FADN 2007 database. The authors also thank Hugo Juhel and Yann Desjeux for their help and Pierre Dupraz for his useful suggestions.


Eric Marcon
email: eric.marcon@ecofog.gf

Zhiyi Zhang

Bruno Hérault

The Decomposition of Similarity-Based Diversity and its Bias Correction

Keywords: Entropy, Biodiversity, Phylogenetic diversity, Functional diversity

 encompasses classical measures of diversity, including the number of species, Shannon and Simpson diversity, and also phylogenetic and functional diversity as long as distances between species are ultrametric. We derive two estimators to allow applying it to real, often under-sampled data, and its decomposition into alpha, beta and gamma diversity when an assemblage of communities is considered.

Introduction

It is still surprising that a so widely used ecological concept, such as the concept of diversity, is still so debated. Ecology is rich of such recurring debates on concepts (e.g. the "ecological resilience" concept [START_REF] Gallopín | Linkages between vulnerability, resilience, and adaptive capacity[END_REF], or the "functional trait" concept [START_REF] Violle | Let the concept of trait be functional![END_REF]) that could nevertheless be seen as part of a fundamental theoretical corpus. This may be why one may consider that ecology is still a "young" science [START_REF] Weiner | On the practice of ecology[END_REF]. Practically, the notion of diversity is more or less consensual among field ecologists. Very roughly, the diversity of an ecological system is even higher than the individuals in the system are "different". Two ecological systems are even more different than their individuals are, too. Problems arise when we move from words to mathematical quantification. Entropy, viewed as the average surprise provided by the data [START_REF] Shannon | A Mathematical Theory of Communication[END_REF][START_REF] Macarthur | Fluctuations of Animal Populations and a Measure of Community Stability[END_REF], paved the way of a coherent theoretical framework [START_REF] Hurlbert | The Nonconcept of Species Diversity: A Critique and Alternative Parameters[END_REF] that respects a set of meaningful axioms [START_REF] Rényi | On Measures of Entropy and Information[END_REF][START_REF] Patil | Diversity as a concept and its measurement[END_REF] including discarding species identities, continuity relatively to the probabilities of occurrence of species and the Pigou-Dalton [START_REF] Dalton | The measurement of the inequality of incomes[END_REF] property (replacing an individual of a more abundant species by an individual of a less abundant species increases diversity). Generalized entropy, namely HCDT entropy [START_REF] Havrda | Quantification method of classification processes. Concept of structural a-entropy[END_REF][START_REF] Daróczy | Generalized information functions[END_REF][START_REF] Tsallis | Possible generalization of Boltzmann-Gibbs statistics[END_REF], allows moving the cursor, the order of diversity q, from rare to abundant species. Conversion of entropy into Hill numbers [START_REF] Macarthur | Patterns of species diversity[END_REF][START_REF] Adelman | Comment on the "H" Concentration Measure as a Numbers-Equivalent[END_REF][START_REF] Hill | Diversity and Evenness: A Unifying Notation and Its Consequences[END_REF] provides the effective numbers, i.e. the number of equally-frequent entities (e.g. species or communities) which would yield the same diversity value as the data. The product of the effective number of communities [START_REF] Jost | Partitioning diversity into independent alpha and beta components[END_REF], namely β diversity, by α diversity (the average diversity of communities) is equal to the γ diversity of a mixture of communities (i.e. a meta-community). HCDT entropy and Hill numbers have been applied to phylogenetic and functional diversity, and, though always perfectible, solutions have been proposed to most issues: the correspondence between HCDT entropy and Hill numbers [START_REF] Jost | Entropy and diversity[END_REF][START_REF] Marcon | Decomposing phylodiversity[END_REF], the decomposition of diversity according to Jost [START_REF] Jost | Partitioning diversity into independent alpha and beta components[END_REF], Chiu et al. [START_REF] Chiu | Phylogenetic beta diversity, similarity, and differentiation measures based on Hill numbers[END_REF] or Routledge [START_REF] Marcon | Decomposing phylodiversity[END_REF][START_REF] Marcon | Generalization of the partitioning of Shannon diversity[END_REF], providing different definitions of α diversity, and robust estimators [START_REF] Grassberger | Finite sample corrections to entropy and dimension estimates[END_REF][START_REF] Chao | Nonparametric estimation of Shannon's index of diversity when there are unseen species in sample[END_REF][START_REF] Haegeman | Robust estimation of microbial diversity in theory and in practice[END_REF][START_REF] Zhang | Entropic Representation and Estimation of Diversity Indices[END_REF]. choosing the importance of ordinary species. The great improvement of this measure is to build on a species-to-species distance matrix directly. This is a very desirable property that allows taking into account not only evolutionary or functional distances but also any other kind of paired relationships [START_REF] Leinster | Measuring diversity: the importance of species similarity[END_REF]. Let's focus on functional diversity. While evolutionary relationships are naturally represented by a phylogenetic tree, functional diversity is more generally calculated from a matrix of multivariate distances, which is often highly distorted later when transformed into a functional dendrogram [START_REF] Podani | On dendrogram-based measures of functional diversity[END_REF]. In addition to being problematic for the calculation of the functional diversity, this over-simplification of the raw trait matrix is extremely frustrating for field ecologists knowing the many difficulties encountered to measure these batteries of functional traits, especially in hyper-diverse ecosystems [START_REF] Baraloto | Functional trait variation and sampling strategies in species rich plant communities[END_REF]. Knowing that single trait values or single trait variation axis are often inadequate predictors of the species fundamental niche [START_REF] Mcgill | Rebuilding community ecology from functional traits[END_REF], losing the multivariate information when estimating diversity prevents taking into account unique combinations of traits that influence ecosystems that are responsible for the ecosystem effect of a given species [START_REF] Eviner | Functional Matrix: A Conceptual Framework for Predicting Multiple Plant Effects on Ecosystem Processes[END_REF]. Up today, this similarity-based diversity q D Z still lacks both (i) a robust estimator and (ii) a decomposition framework. First, it has long been recognized that the observed diversity depends on the sample size [START_REF] Wolda | Similarity indices, sample size and diversity[END_REF], so that estimation-bias corrected estimators are required [START_REF] Beck | Undersampling and the measurement of beta diversity[END_REF][START_REF] Butturi-Gomes | Computer intensive methods for controlling bias in a generalized species diversity index[END_REF]. Second, Leinster and Cobbold have said very little about q D Z decomposition into α, β and γ components. If the path is relatively well marked for diversity indices based on dendrograms, it remains to extend this analysis to the similarity-sensitive ones.

In this paper, we first recall the definitions and duality between similarity-based diversity and Ricotta and Szeidl's [START_REF] Ricotta | Towards a unifying approach to diversity measures: Bridging the gap between the shannon entropy and rao's quadratic index[END_REF] entropy q H Z . We discuss the definition of the similarity matrix in depth. We explicit its parametric construction argued by Leinter and Cobbold and the differences with an alternative approach that lead Chiu and Chao [START_REF] Chiu | Distance-based functional diversity measures and their decomposition: a framework based on hill numbers[END_REF] to propose a different measure of functional diversity. We propose two estimationbias corrected estimators: the first one is an implementation of the Horvitz-Thompson [START_REF] Horvitz | A generalization of sampling without replacement from a finite universe[END_REF] estimator, the other is built on the estimation of the ordinariness of unobserved species. Then we derive the decomposition of both q D Z and q H Z and provide a definition of β diversity and entropy.

Methods

Notations

Consider a random sample taken from a meta-community made of several local communities. Abundances of species in each local community is denoted n s,i (s = 1, 2, . . . , S is the index of species, i the index of communities). n s is the number of individuals of species s in the meta-community, n i the number of individuals sampled in local community i and n the total number. The same notations are used for probabilities of occurrence p s,i which are unknown but estimated with ps,i = n s,i /n i . Community weights are w i : they may be equal to n i /n but any positive values summing to 1 are al-lowed. We assume that p s = ∑ i w i p s,i for all species. This may be understood as the definition of our meta-community: the assemblage of communities whose species probabilities are the weighted average of those of communities. Diversity of the meta-community is γ diversity. Diversity of local communities is α diversity.

Species similarity is introduced as a square matrix Z of dimension S ×S whose elements z s,t are the similarity between species s and t. Similarity is between 0 and 1 by definition, and z s,s = 1: any species is completely similar to itself. A detailed presentation of the possible matrices can be found in [START_REF] Leinster | Measuring diversity: the importance of species similarity[END_REF] and is not repeated here. A matrix of particular interest is Z = I S , the identity matrix where each species is completely different from the others, used to measure neutral diversity. The definition of q D Z can be extended to relatedness matrices, which only require that all terms z s,t are positive, and the diagonal terms strictly positive (see appendix A5 of [START_REF] Leinster | Measuring diversity: the importance of species similarity[END_REF]). Our results remain valid for relatedness matrices except for our new estimator which explicitly supposes that a species similarity with itself is 1.

2.2 Definition of q D Z and q H Z A species ordinariness is defined as its average similarity in the community:

(Zp) s = ∑ t p t z s,t (1)
q D Z is the inverse of the generalized mean of order q -1 [START_REF] Hardy | Inequalities[END_REF] of the community's species ordinarinesses:

q D Z = ∑ s p s (Zp) q-1 s 1 1-q ; q = 1 (2) 1 D Z = 1 ∏ s (Zp) p s s (3)
In other words, 1/ q D Z is the average ordinariness of species of the community. The generalized mean allows giving more or less importance to ordinary species. Rarity is not concerned by the parameter: p s is at power 1. When speciesneutral diversity is addressed, ordinariness reduces to the species probability so both notions converge, but this not the general case. Chiu and Chao [START_REF] Chiu | Distance-based functional diversity measures and their decomposition: a framework based on hill numbers[END_REF] have another approach: they define functional diversity as the weighted average distance between pairs of species, where the weight of the frequency of pairs is parameterized by its power q. Then, q allows giving more or less importance to rare pairs of species (hence to rare species). Both approaches are valid, but quite different. In similarity-based diversity, low values of q focus on original (understood as "not ordinary") species, high values on ordinary species. q D Z is the deformed exponential of order q of the entropy of Ricotta and Szeidl [START_REF] Ricotta | Towards a unifying approach to diversity measures: Bridging the gap between the shannon entropy and rao's quadratic index[END_REF]:

q H Z = 1 -∑ s p s (Zp) q-1 s q -1 = ∑ s p s ln q 1 (Zp) s ; q = 1 (4) 1 H Z = -∑ s p s ln (Zp) s (5)
Diversity is a monotonic, increasing function of entropy, which is more easily written with the formalism of deformed logarithms [START_REF] Tsallis | What are the numbers that experiments provide?[END_REF]. For q = 1:

ln q x = x 1-q -1 1 -q (6)
e x q = [1 + (1 -q) x] 1 1-q (7)
The relation between entropy and diversity is the same as that of HCDT entropy and Hill numbers:

ln q q D Z = q H Z (8)
q D Z = e q H Z q (9)

Similarity matrix

A euclidean distance matrix ∆ between species, whose elements δ s,t are the distance between species s and t, can be obtained by a generalization of Gower's distance [START_REF] Pavoine | On the challenge of treating various types of variables: application for improving the measurement of functional diversity[END_REF] from the values of traits measured on each species. An example is given below. The simplest way to obtain a similarity matrix from a distance matrix is to normalize it and take its complement to 1: z s,t = 1δ s,t /max(∆). We denote d s,t the normalized distances.

We follow Leinster and Cobbold to suggest z s,t = e -ud s,t where u is a positive constant that can be interpreted as a scaling factor of the distance matrix. The negative exponential transformation means that the similarity between species decreases exponentially with respect to distance rather than linearly. From a theoretical point of view, this definition is supported by the theory of categories [START_REF] Leinster | The Magnitude of Metric Spaces[END_REF] which provides several useful results. One of them is that the maximum value of q D Z (given Z) is obtained for a unique distribution of probabilities whatever q [START_REF] Leinster | A maximum entropy theorem with applications to the measurement of biodiversity[END_REF], and this value is the magnitude of the subset of euclidean space defined by the distance matrix. Magnitude is an invariant of metric spaces, analogous to the cardinality of sets. Although this theory is not necessary to define what diversity is, it is particularly appealing because of the evident analogy with the effective number of species. Empirically, u controls the convexity of the transformation of distances into similarities. When u = 0, similarity is 1 whatever the distance. u = 1 is close to a linear transformation. When u increases beyond 1, similarities are more and more packed close to 0. When u → +∞, the similarity matrix tends to the identity matrix and q D Z tends to species-neutral diversity. So u is a parameter that controls for the distribution of similarity values. This may be seen as an issue, since it adds one degree of freedom to the measure of diversity. On the other hand, building a distance matrix relies on arbitrary choices of methods. The ordination of distances clearly has an ecological meaning (more distant species are less similar) but the distribution of distances has less support. The parameter u allows to chose how similar to each other species are.

Estimation bias correction

As many other measures of diversity, q D Z suffers estimation bias [START_REF] Dauby | Sampled-based estimation of diversity sensu stricto by transforming Hurlbert diversities into effective number of species[END_REF]. The plug-in estimator, built by simply plugging ps into the formula of q D Z , is biased for two reasons: unobserved species [START_REF] Chao | Nonparametric estimation of Shannon's index of diversity when there are unseen species in sample[END_REF] and the nonlinearity of diversity with respect to probabilities [START_REF] Bonachela | Entropy estimates of small data sets[END_REF].

Bias-corrected estimators exist for HCDT entropy [START_REF] Marcon | Generalization of the partitioning of Shannon diversity[END_REF], including the Horvitz and Thompson [START_REF] Horvitz | A generalization of sampling without replacement from a finite universe[END_REF] estimator first adapted by Chao and Shen [START_REF] Chao | Nonparametric estimation of Shannon's index of diversity when there are unseen species in sample[END_REF] for Shannon entropy. We use it here to estimate q H Z . We then propose a new estimator of q D Z .

Chao-Shen estimator

q H Z is the sum over species of the measure h s = p s ln q 1 (Zp) s . Unobserved species are responsible for unobserved terms of the sum. An unbiased estimator of such a sum has been derived by Horvitz and Thompson [START_REF] Horvitz | A generalization of sampling without replacement from a finite universe[END_REF]: each observed term is divided by the probability for the species to be sampled: the lower it is, the higher h s is weighted in the sum.

Chao and Shen [START_REF] Chao | Nonparametric estimation of Shannon's index of diversity when there are unseen species in sample[END_REF] proposed to combine it with the estimator of the sample coverage. The total probability of occurrence of unobserved species is by definition 1 minus the sample coverage [START_REF] Good | On the Population Frequency of Species and the Estimation of Population Parameters[END_REF], which can be estimated from the data following Zhang and Huang [START_REF] Zhang | Turing's formula revisited[END_REF]:

Ĉ = 1 - n ∑ r=1 (-1) r+1 n r -1 f r (10)
n is the sample size and f r the number of species observed r times in the sample. This is an improvement of the wellknown Turing's formula Ĉ = 1f 1 /n. The observed species probabilities can be better estimated by Ĉ p1 , . . . , Ĉ pT where T is the number of observed species. 1 -Ĉ is the estimated total probability of unobserved species.

We also have to estimate (Zp) s . The observed Z matrix lacks the S -T columns of unobserved species, and the vector of probabilities lacks the corresponding probabilities { pT+1 , . . . , pS }. Nothing is known about the similarity of unobserved species with observed ones. A reasonable assumption is that their average similarity with any other species is identical to the average similarity between observed species, z = ∑ s,t =s z s,t /[T (T -1)]. Since the sum of missing probabilities of species is known to be 1 -Ĉ, an estimator of (Zp) s is:

Zp s = ∑ t Ĉ pt z s,t + 1 -Ĉ z (11)
The first term is the sum over observed species of their similarity with species s and the second term is the expected sum over unobserved species. Plugging the estimator of p s and that of (Zp) s into the Horvitz-Thompson estimator, an estimation-bias corrected estimator of q H Z is:

q HZ = ∑ s Ĉ ps ln q 1 ( Zp) s 1 -1 -Ĉ ps n (12)
The estimator of diversity is q DZ = e q HZ q .

Alternative estimator

Instead of correcting the estimator for unobserved species, we now want to explicitly estimate their contribution to the value of diversity.

The observed Z matrix lacks S -T lines and columns. The actual Z matrix is (only bold elements are known):

Z =                  1 • • • z 1,T z 1,T +1 = z • • • z 1,S = z . . . . . . . . . . . . . . . z T,1 • • • 1 z z z T +1,1 = z • • • z 1 z . . . . . . . . . z S,1 = z • • • z z • • • 1                  (13)
If q = 1, diversity can split into two terms:

q D Z = ∑ s≤T p s (Zp) q-1 s + ∑ t>T p t (Zp) q-1 t 1 1-q = (K +U) 1 1-q (14)
The first term K is easy to estimate, as in the Chao-Shen estimator:

K = ∑ s≤T Ĉ ps ∑ t≤T Ĉ pt z s,t + 1 -Ĉ z q-1 (15)
The second one is more problematic:

U = ∑ t>T p t [z (1 -p t ) + p t ] q-1 (16)
The number of terms is unknown, as is each p t , but U can be estimated following Zhang and Grabchak [START_REF] Zhang | Entropic Representation and Estimation of Diversity Indices[END_REF]. We define:

V = ∑ s p s [z (1 -p s ) + p s ] q-1 (17)
V is a linear diversity index, using the terminology of Zhang and Grabchak. It can be estimated (derivation in Appendix S1) by:

V = 1+ ∑ s≤T n s n n-n s ∑ v=1 (1 -z) v v ∏ i=1 i -q i v ∏ j=1 1 - n s -1 n -j (18)
U is the sum of the terms concerning unobserved species, so observed terms must be subtracted from V . The improved estimator of q D Z is thus:

q DZ = K + V -∑ s≤T Ĉ ps z 1 -Ĉ ps + Ĉ ps q-1 1 1-q (19)
The estimator of q H Z is:

q HZ = K + V -∑ s≤T Ĉ ps z 1 -Ĉ ps + Ĉ ps q-1 -1 1 -q (20)
If q = 1, estimating entropy is easier. It can be split into two terms:

1 H Z = -∑ s≤T p s ln (Zp) s -∑ t>T p t ln (Zp) t = L + X (21)
As above, the first term is estimated from the data:

L = -∑ s≤T Ĉ ps ln ∑ t≤T Ĉ p j z s,t + 1 -Ĉ z ( 22
)
The second term addresses unobserved species:

X = -∑ tgtT p t ln [z (1 -p t ) + p t ] (23)
We define the linear diversity index:

W = -∑ s p s ln [z (1 -p s ) + p s ] (24)
W can be estimated by (Appendix S1):

Ŵ = ∑ s≤T n s n n-n s ∑ v=1 (1 -z) v v v ∏ j=1 1 - n s -1 n -j (25)
Finally:

1 HZ = L + Ŵ + ∑ s≤T Ĉ ps ln z 1 -Ĉ ps + Ĉ ps ( 26
) 1 DZ = e 1 HZ (27)

Decomposition of entropy

We decompose entropy and diversity following Marcon et al. [START_REF] Marcon | Decomposing phylodiversity[END_REF]. Entropy and diversity can be measured in each community. When communities are pooled, a meta-community (whose species probabilities are the weighted average of those of communities) is defined. Entropy and diversity of the meta-community are called γ. The α entropy of the metacommunity is the weighted average entropy of communities, i.e. we follow Routledge's [START_REF] Routledge | Diversity indices: Which ones are admissible?[END_REF] definition of α entropy. Its deformed exponential is α diversity (which is thus not the average diversity of communities). Entropy is decomposed additively, according to Patil and Taillie's [START_REF] Patil | Diversity as a concept and its measurement[END_REF] concept of diversity of a mixture. Diversity is decomposed multiplicatively, and both decompositions are equivalent. Our purpose here is to characterize β entropy, beyond defining it only as the difference between γ and α entropy. The α entropy of community i is:

q i H Z α = 1 -∑ s p si (Zp) q-1 si q -1 (28)
The α entropy of the meta-community is the weighted average entropy of communities:

q H Z α = ∑ i w i q i H Z α = 1 -∑ i w i ∑ s p si (Zp) q-1 si q -1 (29)
Algebra detailed in Appendix S2 gives the expression of β entropy:

q H Z β = ∑ i w i ∑ s p si ln q 1 (Zp) s -ln q 1 (Zp) si ( 30
)
β entropy is the generalized Jensen-Shannon divergence [START_REF] Marcon | Generalization of the partitioning of Shannon diversity[END_REF] between the distribution of Zp in each community and in the meta-community.

The α diversity of the meta-community is the deformed exponential of its α entropy:

q D Z α = e q H Z α q = ∑ i w i ∑ s p si (Zp) q-1 si 1 1-q (31)
β diversity can be obtained by taking the deformed exponential of the decomposition of entropy:

q D Z β = e q H Z β 1+(1-q) q H Z α q (32)

Decomposition of diversity

It is also interesting to directly decompose diversity to characterize β diversity. It is usually only defined as the ratio between γ and α diversity [START_REF] Tuomisto | A diversity of beta diversities: straightening up a concept gone awry. Part 1. Defining beta diversity as a function of alpha and gamma diversity[END_REF] or as a transformation of β entropy [START_REF] Marcon | Decomposing phylodiversity[END_REF][START_REF] Marcon | Generalization of the partitioning of Shannon diversity[END_REF][START_REF] Marcon | Characterizing the relative spatial structure of point patterns[END_REF].

The inverse of q D Z γ is the generalized mean of order q -1 of (Zp) s :

1 q D Z γ = ∑ i p s (Zp) s q-1 1 q-1 (33)
From q D γ = q D α q D β and Routledge weighting of alpha diversity, simple algebra (Appendix S3) yields:

q D Z β =   ∑ i w i 1/ q i D Z α 1/ q D Z γ q-1   1 q-1 (34)
As 1/ q D Z γ is the average ordinariness of species, q D Z β is the average normalized ordinariness of communities where the normalized ordinariness of community i is defined as

1/ q i D Z α 1/ q D Z γ ,
that is to say its average species ordinariness divided by that of the meta-community.

We may consider each community as an assemblage of monospecific populations. Then, each population's diversity is q s D Z α = 1, and weights are p s . Introducing these values into eq. ( 34) yields q D Z γ = q D Z β : as shown by Rao and Nayak [START_REF] Rao | Cross entropy, dissimilarity measures, and characterizations of quadratic entropy. Information Theory[END_REF] for entropy, the diversity of an assemblage of species is the β diversity between its monospecific assemblages.

Results

Test of bias correction

We used the Barro-Colorado Island (BCI) 50-ha plot forest inventory [START_REF] Condit | Tropical Forest Census Plots[END_REF][START_REF] Hubbell | Light-Gap Disturbances, Recruitment Limitation, and Tree Diversity in a Neotropical Forest[END_REF][START_REF] Hubbell | Barro Colorado Forest Census Plot Data[END_REF]. Year 2005 census contains 20852 individuals from 229 species, among which 24 have been observed only once. The sample coverage is close to 99.9%, allowing to consider that the inventory is almost exhaustive and to use it as a reference to test the efficiency of estimators applied to subsamples. We set species similarity equal to 2/3 inside a genus, 1/3 inside a family, and 0 outside family. The average similarity between pairs of distinct species is z ≈ 0.01. The similarity matrix is somehow rough but no better data was available for such a number of species.

We simulated inventories between 100 and 2000 individuals in a multinomial distribution respecting same species The plug-in estimator (plain line) is the least effective. The Chao-Shen estimator (dashed line, grey-shaded envelope) performs better than our new estimator (dotted line, dashed envelope) for small values of q but our new estimator is very effective for high values of q. Envelopes are the 2.5% and 97.5% quantiles of simulated values. The horizontal line is the diversity of the whole data set. probabilities. Simulations were repeated 1000 times, estimated entropy averaged and finally transformed into diversity for plotting.

Simulated estimations are plotted in Figure 1. 95% confidence envelopes are shown: simulated samples are realizations of the assumed multinomial distribution of the community; stochasticity is not due to the estimators but to sampling. The best estimator depends on the value of q. Their correction is almost identical at q = 1.2 (not shown). At q = 1.5, diversity can be estimated very accurately by the new estimator with a sample whose size is less than the number of species of the community. Following Marcon et al. [START_REF] Marcon | Generalization of the partitioning of Shannon diversity[END_REF]; we choose a pragmatic estimation-bias correction using the maximum value of the two estimators.

Partitioning functional diversity

We used the same tropical forest dataset as Marcon and Hérault [START_REF] Marcon | Decomposing phylodiversity[END_REF], made of two 1-ha fully inventoried plots in the Paracou field station in French Guiana. 1124 individual trees (diameter at breast height over 10 cm) have been sampled among 229 species. Four key functional traits were addressed: seed mass and tree maximum height [START_REF] Hérault | Functional traits shape ontogenetic growth trajectories of rain forest tree species[END_REF], and specific leaf area and wood specific gravity [START_REF] Baraloto | Functional trait variation and sampling strategies in species rich plant communities[END_REF]. A dissimilarity matrix was first built using the Gower metric by the daisy function of the cluster package [START_REF] Maechler | cluster: Cluster Analysis Basics and Extensions[END_REF] for R [START_REF]R: A Language and Environment for Statistical Computing[END_REF]. Our distance matrix is the square root of the output of the daisy function: it is euclidean [START_REF] Pavoine | On the challenge of treating various types of variables: application for improving the measurement of functional diversity[END_REF]. The similarity matrix Z was defined as z s,t = e -ud s,t . Figure 2 shows the diversity profile of the whole data set (γ diversity) depending on both parameters q and u. The effective number of species varies little whatever the order of diversity when u is small. When u increases, species are considered more Parameter q controls the importance of original species, parameter u the similarity between species. High values of u make functional diversity converge to species-neutral diversity. and more dissimilar and the diversity profile with respect to q converges to that of species-neutral diversity.

Among all possible values of u, we think that one is of particular interest: the one which mazimizes the variance of similarity. We will discuss it in the next section. Considering our distance matrix, it is u = 2.48. We calculated diversity of order 1. Neutral γ diversity is 134 effective species. Similaritybased functional γ diversity is 3.68 effective species. It can be interpreted as: the average species ordinariness is 1/3.68 = 0.27. It is also a Hill number: 3.68 completely different species with equal frequencies would have the same diversity as the observed meta-community made of 229 species. This is a very small value; due to the low functional distances between species in the similarity matrix: z ≈ 0.26. Top left: α diversity of the meta-community; top right: α diversity of each community (P006: plain line, P018: dotted line); bottom left: β diversity; bottom right: γ diversity. Diversity is the number of effective species (effective communities for diversity) against the order q. It is estimated with bias correction.

Neutral β diversity is 1.46 effective communities: the species distributions are very different between the communities. Yet, functional β diversity is 1.02: almost no functional β diversity is detected between the communities. Figure 3 shows the functional diversity profile of the meta-community: its values are almost constant whatever q.

Discussion and Conclusion

In this paper, we derived the decomposition of similaritybased diversity q D Z and proposed two estimators to reduce estimation bias from the level of the plug-in estimator. The Chao-Shen estimator is the best for small orders of diversity but our alternative estimator outperforms it for higher orders.

Similarity-based diversity is preferred to estimate functional diversity from a distance matrix because it does not require building a dendrogram and so it preserves the topology of species in the space of functional traits. Even if we focused mainly on the functional diversity, the method works equally well with any similarity measures. As highlighted by Leinster and Cobbold, similarity can be measured in any meaningful way: a genetic similarity will lead to a decomposition of genetic diversity; a molecular similarity will lead to a decomposition of molecular diversity, and so on. Results are effective number of basic entities and entity assemblages which have the same desirable properties as classical diversity measures.

A critical issue is the definition of the scale parameter u. Chiu and Chao [START_REF] Chiu | Distance-based functional diversity measures and their decomposition: a framework based on hill numbers[END_REF] question the legitimacy of q D Z to measure diversity in terms of effective number of species since many examples show that q D Z almost does not vary with q. The reason may be quite simple: all these examples define similarity as 1 minus the normalized distances, resulting in diversity profiles similar to that of Figure 2 for u = 1. Numerically, when u is small, similarities are higher, so are the values of species ordinariness (Zp) s . (Zp) q-1 s can be approached by its Taylor expansion of order 1 when it is close to 1. Simple algebra shows that the consequence of this linear approximation is that the generalized mean of order q -1 reduces to the arithmetic mean, allowing further simplifications in the calculation of q D Z which can be approximated by the inverse of the average similarity between species pairs: 1/(∑ s ∑ t p s p t z s,t ). Numerical simulations show that the approximation remains quite good for ordinariness values quite far from 1. In other words, if a significant proportion (in terms of sum of probabilities) of species have an ordinariness over say 0.4, this rough approximation appears to work very well on real examples. Another reason may be that diversity actually varies little because species ordinarinesses are very similar [START_REF] Messier | How do traits vary across ecological scales? a case for trait-based ecology[END_REF]. Then there are few original species and they do not contribute much to the value of diversity, whether they are given more importance by q or not.

This problem is partially solved when the appropriate definition of similarity is applied. Yet, another one emerges since no decisive criterion exists to chose a value of u, so a profile is required. The comparison of the diversity of two communities must be done along the bivariate profile allowing positive values of both q and u, as recommended by Leinster and Cobbold. The effective number of species given by q D Z for a specific couple of values q and u is perfectly defined but difficult to interpret since it varies substantially (from 1 to the species-neutral diversity) according to u. It is clearly useful to be able to choose a particular value of u to be able to synthesize the diversity of a community in a few effective numbers. We propose two solutions.

First, a classical approach in ordination methods consists of finding the best point of view to observe the highest possible variability of the data. We follow it here: considering the distance matrix and the constraints of its transformation into a similarity matrix, there exists a value of u that maximizes the variance of Z. It appears to be a reasonable choice because lower values make species excessively similar to each other, and greater values make them more different (up to the complete dissimilarity defining species-neutral diversity).

Species Distribution

Probabiliy density

Figure 4. Representation of species overlap. Species take place in the multidimensional space of traits. Intraspecific variability causes some uncertainty. Individuals are located around the expected position of each species, following a Gaussian distribution. Variance is assumed to be the same in all dimensions (whatever the trait) and for all species, for simplicity of the conceptual model. Two species are represented with their density of probability projected onto the axis that joins them. The standard deviation of the density of probability on the figure is half the distance between the two species.

The other possible solution consists of considering u as a proxy for species overlap. [START_REF] Lepš | Quantifying and interpreting functional diversity of natural communities: practical considerations matter[END_REF] represent the niche of species in a one-dimension space. It can be generalized to the multidimensional space of traits assuming variability is the same for all species and all traits (Figure 4). Similarity between species may be defined as the area of overlap of the distribution curves. In other words, a species' own niche can be defined as the region of the space of traits where its probability of occurrence is the greatest. Species overlap is the probability for a species to be in the other's niche. Mathematically, supposing both species have a normal distribution with the same variance, their overlap is:

O s,t = 2 +∞ d s,t /2 σ √ 2π e -1 2 d s,t σ 2 ( 35
)
The overlap does not change as long as d s,t /σ remains constant. As distances have been normalized, the variable of interest is σ which represents the intra-specific variability of traits. Let's consider the two most distant species, with d s,t = 1. If σ = 1/6, their similarity is very small (around 1%). Higher values of σ mean that no species can be considered completely dissimilar, so z s,t > 0 whatever the species. Smaller intra-specific variability allows z s,t 0 for closer couples of species.

A quite good approximation of O s,t is e

-d σ √ 2 . Taking u = 1 σ √ 2
, the similarity z s,t = e -ud s,t represents species overlap. The value of u corresponding to σ = 1/6 is about 4. Higher scales correspond to less intra-specific variability.

We established the link between the mathematical properties of the scale parameter u and its biological meaning, even though the assumptions of the model, i.e. equal variability for all traits and all species, is not realistic. Yet, it is better than the usual hypothesis of no variability at all. The diversity of eigenvalues [START_REF] Pavoine | New biodiversity measure that includes consistent interspecific and intraspecific components[END_REF] allows taking into account intra-specific variability rigorously but does not allow estimation-bias correction.

The alternative approach by Chiu and Chao appears to be more straightforward since it directly relies on the distance matrix. Yet, it is influenced by the way the distance matrix is built. If it is based on a generalized Gower's metric, the distribution of distances can be modified by transforming the numerical trait variables (e.g. take their logarithm) or redefining the (arbitrary) numerical values of ordered categorical ones. The parameter u noes nothing else, but it does it explicitly. The fundamental difference between the two definitions of functional diversity is not on the role of u but on that of q, which allows to focus on ordinary or rare species, depending of the chosen measure of diversity.

We estimated functional β diversity between two tropical forest plots previously investigated by Marcon and Hérault [START_REF] Marcon | Decomposing phylodiversity[END_REF] with different methods. The optimal value of u (implying the similarity of the most different species is about 8 %) results in quite flat profiles of diversity (Figure 3). This is not a numerical artifact (the average species ordinariness is 0.26) but due to low variability (its standard deviation is 0.03) despite the efforts made to maximize the variability of similarity. The functional β diversity between these two quite similar forest communities is negligible (very close to 1). The main explanation is probably the high functional redundancy between these two forest plots. They are spaced by only a few hundred meters and are located on similar topography and soils. It is always strange to observe how tropical forests can locally be extremely rich in species but exhibit a very low turnover in space [START_REF] Condit | Beta-diversity in tropical forest trees[END_REF][START_REF] Novotny | Low beta diversity of herbivorous insects in tropical forests[END_REF]. This finding, well-documented for neutral diversity, seems here exacerbated from the perspective of the functional diversity. This supports the view that, in tropical forests, niche differentiation, if any, occur at very fine spatial scale through vertical distribution shaped by light access and micro-habitat complementarity [START_REF] Kraft | Assembly in an Amazonian Forest Tree Community Functional Traits and Niche-Based[END_REF]. Above this micro-habitat scale, functional assemblages are very similar to each other and governed by a few leading and continuous functional axes [START_REF] Baraloto | Functional trait variation and sampling strategies in species rich plant communities[END_REF]. This is supported by the flat shape of diversity profiles: giving more importance to original species (lowering q) has no effect because less ordinary species are not frequent. In other words, according to our limited data and number of traits, functional diversity is brought by frequent species and quickly saturated: adding more species or more communities does not increase it. These conclusion clearly have to be completed by the alternative point of view of Chiu and Chao's functional diversity which can evaluate the importance of rare species rather than original ones.

The entropart package [START_REF] Marcon | entropart, an R package to partition diversity[END_REF] for R allows diversity estimation presented in the paper.

𝑣 𝑖 =1 � (1 -𝑧̅ ) 𝑣 (-1) 𝑣 (𝑝 𝑠 -1) 𝑣 ∞ 𝑣=1 � 𝑠≥1 = � 𝑝 𝑠 �1 + � 1 𝑣! ��(𝑖 -𝑞) 𝑣 𝑖 =1 � (1 -𝑧̅ ) 𝑣 (1 -𝑝 𝑠 ) 𝑣 ∞ 𝑣=1 � 𝑠≥1 = 1 + � 𝑝 𝑠 �1 + � (1 -𝑧̅ ) 𝑣 𝑣! ��(𝑖 -𝑞) 𝑣 𝑖=1 � (1 -𝑝 𝑠 ) 𝑣 ∞ 𝑣=1 � 𝑠≥1 = 1 + � (1 -𝑧̅ ) 𝑣 𝑣! ��(𝑖 -𝑞) 𝑣 𝑖 =1 � ∞ 𝑣=1 � 𝑝 𝑠 (1 -𝑝 𝑠 ) 𝑣 𝑠≥1 Denote 𝜁 1,𝑣 = ∑ 𝑝 𝑠 (1 -𝑝 𝑠 ) 𝑣 𝑠≥1 : 𝑉 = 𝜁 1,0 + � (1 -𝑧̅ ) 𝑣 𝑣! ��(𝑖 -𝑞) 𝑣 𝑖 =1 � 𝜁 1,𝑣 ∞ 𝑣=1 = 𝜁 1,0 + �(1 -𝑧̅ ) 𝑣 �� � 𝑖 -𝑞 𝑖 � 𝑣 𝑖 =1 � 𝜁 1,𝑣 ∞ 𝑣=1
𝑉 is thus a linear diversity index. It can be estimated according to Zhang and Grabchak (2014) by:

𝑉 � = 1 + � 𝑛 𝑠 𝑛 � (1 -𝑧̅ ) 𝑣 �� 𝑖 -𝑞 𝑖 𝑣 𝑖 =1 � �� �1 - 𝑛 𝑠 -1 𝑛 -𝑗 � 𝑣 𝑗=1 � 𝑛-𝑛 𝑠 𝑣=1 𝑠≤𝑇
𝑊 is calculated the same way.

Figure 1 .

 1 Figure 1. Estimation of BCI diversity 0.5 D Z (a) and 1.5 D Z (b) depending on sample size.The plug-in estimator (plain line) is the least effective. The Chao-Shen estimator (dashed line, grey-shaded envelope) performs better than our new estimator (dotted line, dashed envelope) for small values of q but our new estimator is very effective for high values of q. Envelopes are the 2.5% and 97.5% quantiles of simulated values. The horizontal line is the diversity of the whole data set.

Figure 2 .

 2 Figure 2. Bivariate profile of the Paracou meta-community.Parameter q controls the importance of original species, parameter u the similarity between species. High values of u make functional diversity converge to species-neutral diversity.

Figure 3 .

 3 Figure 3. Functional diversity profile of the Paracou forest communities.Top left: α diversity of the meta-community; top right: α diversity of each community (P006: plain line, P018: dotted line); bottom left: β diversity; bottom right: γ diversity. Diversity is the number of effective species (effective communities for diversity) against the order q. It is estimated with bias correction.

2 Γ 3 …Γ 1

 231 Let Γ = -ln[𝑧̅ (1 -𝑝 𝑠 ) + 𝑝 𝑠 ]. We have:Γ = -ln[𝑧̅ + (1 -𝑧̅ )𝑝 𝑠 ] Γ ′ = -1 -𝑧̅ 𝑧̅ + (1 -𝑧̅ )𝑝 𝑠 Γ′′ = (1 -𝑧̅ ) 2 [𝑧̅ + (1 -𝑧̅ )𝑝 𝑠 ] (3) = -2(1 -𝑧̅ ) 3 [𝑧̅ + (1 -𝑧̅ )𝑝 𝑠 ] (𝑣) = (𝑣 -1)! (1 -𝑧̅ ) 𝑣 (-1) 𝑣 [𝑧̅ + (1 -𝑧̅ )𝑝 𝑠 ] 𝑣At 𝑝 𝑠 = 1:Γ (𝑣) = (𝑣 -1)! (1 -𝑧̅ ) 𝑣 (-1) 𝑣We introduce the Taylor expansion of Γ at 𝑝 = 1 in 𝑊:𝑊 = -� 𝑝 𝑠 ln[𝑧̅ 𝑠≥1 + (1 -𝑧̅ )𝑝 𝑠 ] = � 𝑝 𝑠 �� 1 𝑣! (𝑣 -1)! (1 -𝑧̅ ) 𝑣 (-1) 𝑣 (𝑝 𝑠 -1) (𝑞 -1) 1 -∑ 𝑤 𝑖 ∑ 𝑝 𝑠𝑖 (𝒁𝒑)We replace 𝑝 𝑠 by ∑ 𝑤 𝑖 𝑖 𝑝 𝑠𝑖 and factorize: decomposition of diversity Recall the definitions of 𝛾 and 𝛼 diversity:

Acknowledgments

This work has benefited from an "Investissement d'Avenir" grant managed by Agence Nationale de la Recherche (CEBA, ref. ANR-10-LABX-25-01). Funding came from the project Climfor (Fondation pour la Recherche sur la Biodiversité).

Appendix S1: Estimation of 𝑉 and 𝑊

We first estimate 𝑉.

Let Δ = [𝑧̅ (1 -𝑝 𝑠 ) + 𝑝 𝑠 ] 𝑞-1 and Δ′, Δ′′, …, Δ (𝑣) be the 𝑣 th partial derivative of Δ with respect to 𝑝 𝑠 . We have:

At 𝑝 𝑠 = 1, we evaluate these derivatives to be:

< ∞ and 𝑧̅ is a small positive number, we can introduce the Taylor expansion of Δ at 𝑝 = 1 in 𝑉. We have:


Henri Der Sarkissian

Jeanpierre Guedon

Pierre Terve

Nicolas Normand

Imants Svalbe

EVALUATION OF DISCRETE ANGLES ROTATION DEGRADATIONS FOR MYOCARDIAL PERFUSION IMAGING

INTRODUCTION & PURPOSE

For myocardial perfusion imaging, the tomographic transaxial cardiac volume must be reoriented in the standard views for inter-patient comparison and diagnosis accuracy, as advised by the EANM guidelines. The volume is then resampled and displayed in the standard HLA, VLA and SA axis.

Rotation of digital images defined on discrete grid introduce image degradation and artifact arising from the lack of perfect match between continuously rotated grid and the original one. As a result, every rotation has to be considered cautiously and be held in the manner to minimize the quantification error.

The goal of this poster is : 1) to compare existing rotation algorithms for nuclear medicine images (with their specific spectral content), 2) to use a novel method (Svalbe) both based on discrete angle rotation and image sampling to ensure an one-to-one pixel mapping, 3) to measure rotation operator defaults using the same discrete angles for all tested methods. 2 kind of experiments are used : 1) kN rotations of 2π/N angle. 2π/N is chosen to be a discrete angle. This scheme is only used for continuous rotations (c.f. Table 1), 2) Rotation of angle (p 1 ,q 1 ), followed by (p 2 ,q 2 ). The resulting image is then rotated one last time by the global reverse angle (p 1 p 2 -q 1 q 2 ,q 1 p 2 +p 1 q 2 ). This scheme is tested for continuous and discrete rotations (c.f. Table 2). Each row m represents a discrete direction of projection. By re-arranging cleverly the rows, we can perform rotations in the projection space.

Left to right: Original PET image, rotation of angle θ using nearestneighbor interpolation, rotation of angle θ using cubic spline interpolation θ = arctan(1 2)

Discrete angle directions given by Haros-Farey series of order 5

For con(nuous rota(ons, cubic spline seems to give the best results. Even higher order spline give worse results. We discourage the use of bilinear interpola(on since it blurs drama(cally the image. We also discourage the use of even degree splines, as for example SP2 which gives rise to strong aliasing ar(facts.

The discrete FRT--based rota(on does not give the best MSE values, but it does not introduce any blur. Moreover, the original pixel intensi(es are preserved thus the rota(on is totally reversible and lossless, which is very appreciable for quan(ta(ve analysis based on pixel intensity. Some drawbacks, like the magnifica(on of image size increases memory usage and the periodic sampling of FRT needs a specific mapping to achieve discrete rota(on from real acquisi(on.

RESULTS

Experiments

CONCLUSION

The key points of our method are: -The method is fully discrete so no assumption is made about an underlying continuous model, making it fast, exact and reversible -It is performed into a discrete projection space, close to the data acquisition space. Thus no tomographic reconstruction has to be made prior to reorientation, minimizing the smoothing effects of interpolation. Future work: Effects of both reconstruction and reorientation have to be assessed simultaneously. For this purpose, we need estimators of left ventricle orientation directly into the projection space.

The authors thank Pr. Leguludec and Pr. Rouzet from Bichat Hospital, Paris for the data they provided and the QuantiCardi project from Pays de la Loire Region for supporting this work. Special thanks to Benoit Recur for his help.

Corresponding error images

R∑

  (t, m) = I (< my + t > p ,FRT is an exact, discrete and periodic form of the Radon transform. It maps a p size square image I(x,y) to a p×(p+1) projection image R(t,m), where each pixel of R(t,m) contains the projection value of the original image onto a discrete-angle ray.

  Error measurement using 2 metrics Mean Square Error (MSE) -the l 2 norm of error image : Peak Signal to Noise Ratio (PSNR) :

80 Rotations of 45°

		MSE	PSNR	MSE	PSNR	MSE	PSNR	MSE	PSNR
	Nearest	24.90	34.17	24.90	34.17	145.41	26.50	145.41	26.50
	Bilinear	25.29	34.10	98.55	28.19	155.29	26.22	326.36	22.99
	CubicB-spline	20.64	34,98	20.92	34.93	115.17	27.52	139.64	26.68
	High order (Spline 7)	21.93	34,72	71.57	29.58	116.87	27.45	267.27	23.86

Table 1 : kN rota.ons of 2π/N

 1

	Method

Table 2 : Rota.on of angle (1,1) followed by (2,1) and reverse rota.on by (1,--3)

 2