Safaa Wasof

Jonathan Roger Michel

Per Henri Lenoir

Arild Aarrestad

Inger Greve Alsos

W Scott Armbruster

Gunnar Austrheim

Vegar Bakkestuen

H John

B Birks

Mari Bråthen

Olivier Broennimann

Jonathan Lenoir
email: jonathan.lenoir@u-picardie.fr

Per Arild Aarrestad

W Scott Armbruster

Kari Anne Bråthen

Jörg Brunet

Hans Henrik Bruun

Carl Johan Dahlberg

Martin Dieckmann

Stefan Dullinger

Rasmus Ejrnaes

Jean-Claude Gégout

Bente Jessen Graae

John-Arvid Grytnes

Antoine Guisan

Kristoffer Hylander

Ingibjörg S Jónsdóttir

Jutta Kapfer

Kari Klanderud

Miska Luoto

Ann Milbau

Mari Moora

Bettina Nygaard

Arvid Odland

Harald Pauli

Virve Ravolainen

Stefanie Reinhardt

Marlen Sylvi

Sandvik

Fride Høistad Schei

James Speed

Jens-Christian Svenning

Wilfried Thuiller

Unn Liv

Tveraabak

Vigdis Vandvik

Liv Guri Velle

Risto Virtanen

Pascal Vittoz

Wolfgang Willner

Thomas Wohlgemuth

Niklaus E Zimmermann

Martin Zobel

Guillaume Decocq

Disjunct populations of European vascular plant species keep the same climatic niches

Keywords: Alpine plants, climatic niche, disjunct populations, distant populations, native plants, niche conservatism, niche evolution, niche optimum, niche overlap, niche width, species distribution modelling

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Results

Only one species (3%) with strictly disjunct populations and 58 species (16%) with at least distant populations showed a region effect on their climatic niche. Niche overlap was higher for species with strictly disjunct populations (0.50±0.1) than for species with distant populations (0.42±0.1), and highest for Arctic-alpine species (0.55±0.06). Estimated climatic niches were, on average, larger and located towards warmer and wetter conditions in the Alps.

Main conclusion

Climatic niches seem to be generally conserved between populations that are separated between the Alps and Scandinavia and likely have been so for 10,-15,000 years, despite the fact that populations in the Alps and Fennoscandia became adapted over time to their respective sets of regional climatic conditions. Therefore, the basic assumption of species distribution models that species' climatic niche is constant in space and time -at least one time scales 10 4 years or less -seems to be largely valid for arctic-alpine plants.

INTRODUCTION

A long-standing question in ecology is whether species populations that occupy distant regions have retained similar ecological requirements, i.e. whether their environmental niche remained constant over space and time [START_REF] Lavergne | Biodiversity and climate change: Integrating evolutionary and ecological responses of species and communities[END_REF]. This question departs from the principle of phylogenetic niche conservatism, the tendency of closely related species to be more similar than expected, and which involves evolutionary time scales [START_REF] Losos | Phylogenetic niche conservatism, phylogenetic signal and the relationship between phylogenetic relatedness and ecological similarity among species[END_REF][START_REF] Lavergne | Biodiversity and climate change: Integrating evolutionary and ecological responses of species and communities[END_REF][START_REF] Wiens | Niche conservatism as an emerging principle in ecology and conservation biology[END_REF]. Determining whether or not geographically distant populations have evolved to occupy distinct niches, and to what extent, and in which conditions, is of paramount importance in predicting future biodiversity under climate change.

Indeed, most correlative models used to predict species range changes take the conservative assumption that species' environmental niches are relatively constant in space and time [START_REF] Guisan | Unifying niche shift studies: insights from biological invasions[END_REF].

One difficulty when trying to address this question is that species' fundamental environmental niche reflecting their physiological tolerances cannot be estimated from empirical field data because other factors, such as biotic interactions and dispersal limitation, often restrict the range of conditions species may encounter. The latter is often called the realised ecological niche [START_REF] Hutchinson | Concluding Remarks[END_REF][START_REF] Austin | Measurement of the Realized Qualitative Niche: Environmental Niches of Five Eucalyptus Species[END_REF]. For example, changes in biotic interactions or dispersal limitation across regions may create two distinct realised niches within the same global fundamental niche. Reversely, lack of change of the realised niche usually indicates that the underlying fundamental niche has been largely conserved and no strong regional adaptations have occurred.

Invasive species colonising newly separated regions have been particularly studied in this context (see [START_REF] Guisan | Unifying niche shift studies: insights from biological invasions[END_REF] for a review). Range-filling processes in the invaded range are indeed strongly limited or at an early stage and the potential for divergence of niche requirements are largely unexploited [START_REF] Williamson | The distribution of range sizes of native and alien plants in four European countries and the effects of residence time[END_REF]. Instead, assessments of Wasof et al. climatic niche differences between distant populations originating from distinct postglaciation colonisation events are more likely to inform us about species' abilities to modify their ecological requirements under future climate change. However, such comparisons have rarely been undertaken (but see [START_REF] Pellissier | Thermal niches are more conserved at cold than warm limits in arctic-alpine plant species[END_REF].

Here, we extend such niche comparison between distant populations of native species by using a representative sample of the pool of vascular plants (n = 888) that have successfully recolonised both the European Alps and Fennoscandia. These two geographically distant mountainous regions have the advantage of having different recolonisation histories during the last glacial-interglacial cycle and sharing analogous macroclimatic gradients.

During the Last Glacial Maximum (LGM, ca 23,000-18,000 yr BP), Fennoscandia was largely covered by the Eurasian ice-sheet [START_REF] Svendsen | Late Quaternary ice sheet history of northern Eurasia[END_REF], and largely recolonized from …. In contrast, numerous ice-free refugial areas were located relatively close to each other at the eastern and southern periphery of the much smaller ice caps that covered central parts of the Alps [START_REF] Schönswetter | Molecular evidence for glacial refugia of mountain plants in the European Alps[END_REF]. Thus, many high-elevation plants probably survived the LGM at the margins of both the Alps [START_REF] Schönswetter | Molecular evidence for glacial refugia of mountain plants in the European Alps[END_REF] and the Eurasian ice-sheet [START_REF] Birks | Plant macrofossils and the Nunatak Theory of per-glacial survival[END_REF], and on the nunataks or other ice-free pockets in the Alps [START_REF] Stehlik | Nunatak survival of the high Alpine plant Eritrichium nanum (L.) Gaudin in the central Alps during the ice ages[END_REF]. The realised climatic niche of a given Arctic-alpine plant species should thus be wider in the Alps than in Fennoscandia since the process of post-glacial recolonisation of all climatically suitable sites (i.e. range-filling) has probably been less constrained by dispersal limitations [START_REF] Dullinger | Post-glacial migration lag restricts range filling of plants in the European Alps[END_REF]. In addition, proximity of the Alps to the main southern refugia of the temperate European flora (Iberian, Italian, and Balkan Peninsulas) [START_REF] Birks | Alpines, trees, and refugia in Europe[END_REF], with larger population sizes and genetic diversity [START_REF] Hewitt | The genetic legacy of the Quaternary ice ages[END_REF], may have enhanced population adaptability, and also genetic variation in the Alps. This may have fostered a species' capability to exploit available post-glacial ice-free areas and thus widen its fundamental climatic niche as well as shift its fundamental climatic niche towards warmer conditions. In contrast, the long-distance recolonisation of Fennoscandia is likely to have involved repeated founder events, successively reducing the genetic variation and most likely also the fundamental climatic niche (e.g. [START_REF] Giesecke | Moving front or population expansion: How did Picea abies (L.) Karst. become frequent in central Sweden?[END_REF]. Additionally, such long-distance recolonisation of Fennoscandia not only originated from the main southern refugia located in southern Europe, but also from eastern refugia located in Russia [START_REF] Eidesen | Genetic roadmap of the Arctic: plant dispersal highways, traffic barriers and capitals of diversity[END_REF], which may have contributed to shift the fundamental climatic niche towards somewhat colder conditions due to founder events from cold-adapted populations.

Conversely, total vascular plant species richness in the Alps is much higher than in Fennoscandia [START_REF] Lenoir | Cross-scale analysis of the region effect on vascular plant species diversity in southern and northern European mountain ranges[END_REF], generating different competition regimes, which may result in different realised climatic niches between populations in the Alps and in Fennoscandia [START_REF] Pellissier | Thermal niches are more conserved at cold than warm limits in arctic-alpine plant species[END_REF]. The greater the number of species in a region, the narrower the species' realised niche width should theoretically be (regional diversity-niche width hypothesis; [START_REF] Macarthur | Geographical Ecology: Patterns in the Distribution of Species[END_REF], due to contraction at the least stressful margin of the environmental gradient. This would lead to the expectation of narrower realised niches through contraction of the warmer part of the range, resulting in the species being observed towards colder conditions in the Alps.

Finally, potential differences in the realized climatic niche between distant populations of the same species are likely to be related to its life-history traits. For instance, we can assume that a generalist or opportunistic species is able to occupy wider fundamental niches and thus can more easily adjust its realised niche, depending on the region, than a specialist or late-successional species whose fundamental niche is too narrow to allow large shifts of the realised niche [START_REF] Pearman | Niche dynamics in space and time[END_REF]. But we could also assume that disjunct populations of a dispersal-limited species are likely to be isolated from gene flow thus favouring local adaptation and increasing the chances for strong differences between their respective realized climatic niches compared with distant populations of a widespread species not limited by its Wasof et al. dispersal abilities. Linking the extent to which two distant populations of the same species have similar realized climatic niche (cf. niche overlap) with traits is therefore informative. Knowing all this, we aim to answer five questions: (i) Do the realised climatic niches of species native to both the Alps and Fennoscandia differ between the two regions? (ii) To what extent does the realised climatic niche of a species overlap between the two regions? (iii) Is there a trend towards wider (supporting the genetic diversity hypothesis) or narrower (supporting the diversity-niche width hypothesis) realised climatic niches in the Alps? (iv) Are the realised climatic niches of species shifting towards warmer (supporting the geneticdiversity hypothesis) or colder (supporting the diversity-niche width hypothesis) conditions in the Alps? (v) How much of the variation in species' niche overlap can be explained by species' life-history traits?

MATERIALS AND METHODS

Study area and species pool

The study area covers two geographically distant regions: the European Alps as delineated by the Alpine Convention Boundary (http://www.alpconv.org/) and Fennoscandia as delineated by the administrative boundaries of Norway, Sweden, Finland and Denmark (excluding islands such as the Svalbard archipelago) (Fig. 1(a)). To select the list of terrestrial vascular plant species occurring in both study regions, we assembled more than 70,000 vegetation plots from the most recent versions of two different European vegetation databases: (i) the Alps vegetation database (AVD, n = 31,524, http://www.givd.info/ID/EU-00-014) [START_REF] Lenoir | The Alps Vegetation Database -a geo-referenced community-level archive of all terrestrial plants occurring in the Alps[END_REF] and (ii) the Nordic vegetation database (NVD, n = 41,785, http://www.givd.info/ID/EU-00-018) [START_REF] Lenoir | Local temperatures inferred from plant communities suggest strong spatial buffering of climate warming across Northern Europe[END_REF]. A total of 888 species occurred in both databases. This corresponds to the common species pool obtained by matching species lists from AVD and NVD (see Appendix S1 in Supporting Information).

Because most of the 888 species common to the Alps and Fennoscandia also occur outside this study area, it is likely that the realised climatic niche of at least one of the two studied populations will be truncated thus potentially affecting our results. To account for such effects, we screened each of the 888 species' range maps (Comparative Chorology of the Central European Flora; [START_REF] Meusel | Vergleichende Chorologie der zentraleuropäischen Flora[END_REF][START_REF] Meusel | Vergleichende Chorologie der zentraleuropäischen Flora[END_REF][START_REF] Karten | Vergleichende Chorologie der zentraleuropäischen Flora[END_REF] to subdivide this pool into two subsets: (i) species having disjunct distribution restricted to mountainous areas or Arcticalpine tundra with at least one population strictly confined to the Alps and another strictly confined to Fennoscandia, hereafter referred as the "restricted" group for which the entire realised climatic niche is likely to be captured (n = 105); and (ii) species having either a contiguous distribution between the Alps and Fennoscandia, with these simply constituting distant populations, or a patchy distribution not restricted to mountainous areas or Arcticalpine tundra with populations extending outside the Alps or Fennoscandia, hereafter referred as the "enlarged" group for which the realised climatic niche will be truncated in at least one of the two studied regions (n = 783). We analysed and reported all our results separately for the restricted and enlarged groups.

Climatic data

We used nine bioclimatic variables expected to have direct ecophysiological impacts on plant distributions [START_REF] Prentice | Special Paper: A Global Biome Model Based on Plant Physiology and Dominance, Soil Properties and Climate[END_REF]. Three related to temperature: growing-degree-days above 0°C (GDD); absolute minimum temperature (AMT); and continentality index (CI). The remaining six were water-related variables: annual aridity index (AI: note that higher AI values represent more humid conditions); annual potential evapotranspiration (PET); annual actual evapotranspiration (AET); Priestley-Taylor alpha coefficient (ALPHA = AET/PET); water balance over the year (sum of monthly precipitation minus monthly potential evapotranspiration) (WBAL); and soil water content of the most stressful month (SWC). All variables were derived from globally available datasets at a spatial resolution of 30 arc seconds which is less than one kilometre at the equator: WorldClim [START_REF] Hijmans | Very high resolution interpolated climate surfaces for global land areas[END_REF] http://www.worldclim.org/) and CGIAR (CGIAR-CSI: http://www.cgiar-csi.org/) (Appendix

S2).

Presence and absence data

For each of the 888 species, we followed a set of data-handling procedures (Appendix S3).

We first compiled geo-referenced presence records by querying AVD, NVD, and the Global Biodiversity Information Facility (GBIF: http://www.gbif.org/) and geo-referenced absence records from AVD and NVD. Presence (AVD, NVD, GBIF) and absence (AVD, NVD) data were then aggregated across the study area at a 1-km resolution (cf. climatic grids) to compute a probability of presence (pr) of the focal species per grid cell by dividing the number of presence records found within a given grid cell by the total number of presence and absence records of that species in it. Note that we could not compute a pr value for grid cells without information on presence and absence data and thus grid cells lacking presence and absence data were not used in the subsequent analyses. Finally, we subsampled all available grid cells for which a pr value had been computed to limit spatial autocorrelation issues and to balance sampling effort between regions (Appendix S3).

Trait data

To explain the observed differences in the realised climatic niche of species between the Alps and Fennoscandia, we assembled data on species' traits according to their biogeography, ecology, and biology. For biogeographic data, we extracted information on chorology (i.e. biogeographic origins) from [START_REF] Landolt | Flora indicativa: ecological indicator values and biological attributes of the flora of Switzerland and the Alps[END_REF]. For ecological data, we used Ellenberg indicator values (Ellenberg et al., 1991) for light (L), soil nutrients (N), soil pH (R), soil moisture (F), temperature (T), and continentality (K). For biological data, we collected data on plant height, specific leaf area (SLA), diaspore mass and Raunkiaer's life-forms from several sources (Appendix S1). Diaspore mass and plant height values were log-transformed prior to analyses.

Data analyses

To compare the realised climatic niches between two distant populations of the same species, we used two complementary approaches. First, we tested for any overall regional differences (region effect) in the realised climatic niche of each species using an environmental niche modelling (ENM) approach based on generalised linear models (GLMs). Second, we used a bivariate analysis based on the two most statistically significant climatic layers found in the GLMs, including one energy-related variable (GDD) and one water-related variable (AI) to Wasof et al. quantify the degree of niche overlap and niche shifts between regions for each species. Once differences in the realised climatic niches between distant populations in the Alps and Fennoscandia were assessed, we linked species' niche overlap values to species' life-history traits.

Region effect

For each of the 888 species, all 1-km 2 grid cells retained after the selection process (Appendix S3) were split into two subsets: one subset for model development (training dataset) including 2/3 of the selected grid cells and one subset for model validation (test dataset) consisting of the remaining 1/3 of the grid cells. Because of this splitting, 550 species, of which 65 belong to the restricted group, were sufficiently frequent (i.e. having a pr value exceeding 0 in at least 50 grid cells within each region of the training dataset) to be retained for model development.

For each of the 550 selected species, we ran a series of GLMs for proportion data with pr as the dependent variable to select the most influential bioclimatic variables (GDD, AMT, CI, PET, AET, WBAL, ALPHA, AI, SWC). We also evaluated the effect of the region variable (Alps vs. Fennoscandia) and the potential interaction with any of the selected climatic variables in the models (Appendix S4). Finally, we updated the model by incorporating the total number of presence/absence records per grid cell as a covariate to correct for the effect of sampling effort across the study area and checked for overdispersion. In those cases where overdispersion was found a quasi-binomial error function was used.

We then evaluated the quality of the GLM using the test dataset for each species and the area under the receiver-operating characteristic curve (AUC) ranging from 0.5 (poor models) to 1 (perfect models) [START_REF] Swets | Measuring the accuracy of diagnostic systems[END_REF]. We decided to retain only the best models by focusing on species that had AUC values greater than 0.8 for further comparisons of the realised climatic niche between regions. A total of 389 species, of which 31 belong to the restricted group, out of 550 fulfilled this requirement (Fig. S5 in Appendix S5). To transform the pr into presence/absence data (Fig. S3 in Appendix S3), we computed the sensitivity-specificity sum maximiser criterion (MST) based on the test dataset [START_REF] Jiménez-Valverde | Threshold criteria for conversion of probability of species presence to either-or presence-absence[END_REF].

Finally, for each of the 389 species, we merged the training and test datasets to recalibrate the final model based on all the available data and extracted several model statistics (coefficient estimates, p-values, AIC, null deviance, residual deviance). To test for differences in the realised climatic niche of a given species, we focused on the significance of the region effect and its interaction with the climatic variables (Fig. 2 and Fig. S6 in Appendix S6).

Observed niche overlap

For each of the 389 species, we estimated niche overlap between the two regions within the bi-dimensional climatic space defined by GDD and AI (Fig. 1(b)). These two variables were selected since they were important for many of our species. We divided the bi-dimensional climatic space into a grid of 500 × 500 cells bounded by the minimum and maximum values of GDD (first axis) and AI (second axis) (Fig. 1(b), Appendix S7). For each region separately, we mapped the occurrence data (pr > 0) in the bi-dimensional space defined by the subset of grid cells in which the focal species was recorded as present (pr > 0) or absent (pr = 0) to map its realized climatic niche within the available climatic space. We applied a kernel smoother to standardise species densities across the bi-dimensional climatic space independently of the sampling effort and the resolution in the climatic space (Fig. S7 in Appendix S7, Broennimann et al., 2012).

We used Schoener's D similarity index [START_REF] Schoener | Nonsynchronous spatial overlap of lizards in patchy habitats[END_REF]Broennimann et al., 2012) (see formula in Appendix S7) to compute the observed niche overlap (D obs ). The D index was calculated on the above-mentioned smoothed density of occurrences, (i.e. observed niche overlap, D obs ), and varies from 0 (no overlap) to 1 (complete overlap). We assessed D obs not only across the overall climatic space including both analogue and non-analogue climates, but also within the analogue climatic space alone (Fig. 1). Assessing D obs across the whole climatic space allows to account for non-analogue climatic conditions that may become analogue with future climate change, whereas focusing on the analogue climatic space solely enables D obs assessment to be independent of simple changes in climatic conditions thus precluding to mix up apples and pears. In theory D obs should be higher when focusing on the analogue climatic space alone with difference in D obs values between the analogue (cf. baseline) and the overall climatic space indicating to what extent non-analogue climates matter for D obs . To compute D obs across the analogue climatic space alone, we quantified for each species the part of the bi-dimensional climatic space that was shared by the two regions (i.e., analogue climates) (Fig. S7 in Appendix S7) and then D obs was computed once again as above using Schoener's D similarity index [START_REF] Schoener | Nonsynchronous spatial overlap of lizards in patchy habitats[END_REF]Broennimann et al., 2012).

Niche overlap test

To assess to what extent the realized climatic niche of a given species is conserved between the Alps and Fennoscandia, we compared its empirical D obs value to its simulated niche overlap (D sim ) value obtained under a baseline scenario of climatic-niche identity. By baseline scenario, we mean setting and using exactly the same climatic niche model (cf. climatic-niche identity) for each of the two studied populations of the focal species to successively: (i) predict each population distribution in the Alps and Fennoscandia based on climatic conditions solely; (ii) project these spatial predictions into the GDD-AI climatic space; and

(iii) compute D sim using the same approach as for D obs (see Appendix S7 for more details).

Comparing a species' D obs value in light of its D sim value help to appreciate how much the realized climatic niche of this focal species is conserved. The lower the D obs value compared with D sim , the greater the niche differentiation between the Alps and Fennoscandia.

Note that D obs can be greater than D sim , i.e. the realized climatic niche is more conserved than expected under the climatic-niche identity assumption. On the one hand, D sim can reach low values simply because our simulations account for the potential effects of other climatic variables than the one used to compute D sim (cf. GDD and AI) (Appendix S7). On the other hand D sim is based on model predictions that incorporate climatic predictors only whereas D obs relies on empirical data that reflect not only climatic dimensions of the realised niche but also other abiotic and biotic dimensions not considered in our models and which could involve habitat-compensation effects. For these two reasons, it is not that odd to find greater values for D obs than D sim . We used Student's paired t-test to assess the significance of the average difference between D obs and D sim across all species.

Niche width and niche optimum

We used two other parameters to assess differences in the realised climatic niche of each of the 389 studied plant species between the two regions: niche width (range of climatic conditions suitable for a species) and niche optimum (maximum probability of presence of a species within its realised climatic niche). Species niche-width and niche-optimum values were computed for both the Alps and Fennoscandia from the observed density of occurrences of a given species within both the overall and analogue climatic spaces.

We followed the method of [START_REF] Theodoridis | Divergent and narrower climatic niches characterize polyploid species of European primroses in Primula sect. Aleuritia[END_REF] to compute these parameters and did so for the Alps and Fennoscandia, separately. Species' niche-width and niche-optimum values were assessed by first extracting the scores along GDD and AI of 100 pixels sampled randomly from its bi-dimensional climate space, selecting pixels with a probability according to species' density of occurrences. Once extracted, the inter-decile range (80%) of these 100 values along GDD and AI (i.e. niche width) as well as GDD and AI coordinates of the pixel where the species reached its maximum occupancy (i.e. niche optimum) were computed. This randomisation procedure was repeated 100 times and the mean values of the 100 obtained values for niche width and niche optimum for both the Alps and Fennoscandia were retained.

We used Student's paired t-test to assess the significance of the mean difference in species' niche width and niche optimum between regions.

Insights from plant traits

Finally, we used three independent ordinary least square (OLS) regressions to assess how much of the variation in D obs could be explained by plant biogeography, ecology, and biology.

First, we fitted D obs against a qualitative variable representing species' biogeographic origins (Arctic-alpine, Asia, Eurasia, Europe, Holarctic, Mediterranean, North America) followed by Tukey's HSD test for post-hoc pairwise comparisons of group means. Second, we fitted D obs against all six semi-quantitative Ellenberg indicator values (L, N, R, F, T, K). Third, we fitted D obs against plant growth, photosynthetic performances, and dispersal abilities (plant height, SLA, diaspore mass) as well as Raunkiaer's life-forms. We ran all three models for both D obs computed across the overall climatic space including both analogue and non-analogue climates, but also for D obs computed within the analogue climatic space only.

RESULTS

Regional differences

Among the 31 species belonging to the restricted group and that were retained after the modelling procedure to test for a region effect, only one species (Epilobium alsinifolium) showed regional differences in its response curves, namely along the aridity gradient (AI) (Fig. S8 in Appendix S8). Among the remaining 358 species belonging to the enlarged group, only 58 species showed regional differences in their response curves along at least one of the nine studied climatic variables (Table S8 in Appendix S8). Of these 58 species, 29 showed regional differences in response to growing degree days (GDD) (Fig. 2(b)), 26 in response to soil water content during the most stressful month (SWC), 8 in response to the aridity index (AI), and 5 in response to the continentality index (CI) (Fig. S8 in Appendix S8).

Niche overlap

Based on the overall climatic space defined by GDD and AI, the observed niche overlap (D obs ) for the 31 species belonging to the restricted group was 0.50 on average, ranging from 0.16 to 0.59 (Table S9 in Appendix S9). These values were higher, on average, than the D obs values calculated for the 358 remaining species from the enlarged group (mean = 0.42; min = 0.02; max = 0.66) (two-sample Student's t-tests, p-value = 0.002) (Table S9 in Appendix S9). The simulated niche overlap (D sim ) under the baseline scenario of climatic-niche identity was much lower than D obs for both the restricted (mean = 0.40; min = 0.25; max = 0.52) (paired two-sample Student's t-tests, n = 27, p-value << 0.001) and the enlarged (mean = 0.28; min = 0.005; max = 0.7) (paired two-sample Student's t-tests, n = 195, p-value << 0.001) species groups. Among the 58 species that showed regional differences in their response curve along at least one climatic variable (Table S8 in Appendix S8), only 6 species had distant realised climatic niches that overlapped less than under the baseline scenario (D obs < D sim ): Alyssum alyssoides, Avenula pratensis, Carex ericetorum, Ranunculus bulbosus, Rubus idaeus, and Vincetoxicum hirundinaria. We found similar patterns for the results based on the analogue climatic space solely, except that all niche overlap values were higher (Table S10 in Appendix S10).

Niche width and niche optimum

Irrespective of the species group considered (i.e. restricted vs. enlarged), niche width across the overall climate space was larger in the Alps than in Fennoscandia for GDD (Figs. 3(a S9 in Appendix S9). The same patterns were found for the 31 species of the restricted group when assessing niche width based on analogue climates alone, except that the trend was not significant for AI (p-values = 0.06). However, when assessing niche width based on analogue climates alone for the remaining 358 species of the enlarged group, species' realised climatic niches were larger in the Alps than in Fennoscandia for both variables (Table S10 in Appendix S10).

Niche optima results across the overall climatic space for the 31 species of the restricted group showed no difference, on average, along GDD but a shift towards higher values of AI (more humid conditions) in the Alps compared with Fennoscandia (Figs. 4(a) and4(c)) S9 in Appendix S9). For the remaining 358 species of the enlarged group, niche optima were located, on average, towards higher values of both GDD and AI (warmer and more humid conditions) in the Alps than in Fennoscandia, (Figs. 4(b) and4(d)) (Table S9 in Appendix S9). We found the same patterns when assessing niche optima based on analogue climates alone (Table S10 in Appendix S10).

(Table

DISCUSSION

The realised climatic niche is conserved

Our findings suggest predominant niche conservatism, with only 15% of the 389 studied species displaying regional differences in realised climatic niches. Noteworthy, only one (3%) of the 31 species for which populations are clearly disjunct between the Alps and Fennoscandia showed regional differences in its climatic niche. Additionally, the fact that the observed niche overlap (50%) is higher, on average, than the simulated niche overlap (40%) obtained under a baseline scenario of climatic-niche identity suggests strong niche conservatism. Using a different approach and with a smaller set of plant species (n = 8), [START_REF] Martínez-Meyer | Conservatism of ecological niche characteristics in North American plant species over the Pleistocene-to-Recent transition[END_REF] showed that the current species' distribution in North America predicts well their distribution at the Last Glacial Maximum (LGM) and reciprocally, suggesting an ecological niche conservatism despite changes in climate and environmental conditions over that time period. Our results confirm the general idea that niche conservatism is more likely to be the rule over short to moderate time events (thousands of year or shorter) and support results from previous studies focusing on a limited set of species [START_REF] Martínez-Meyer | Conservatism of ecological niche characteristics in North American plant species over the Pleistocene-to-Recent transition[END_REF][START_REF] Pearman | Niche dynamics in space and time[END_REF][START_REF] Peterson | Ecological niche conservatism: a time-structured review of evidence[END_REF]. Over longer time periods of million to billion years, there is no study testing species niche conservatism, certainly because very few relict species have survived since the Palaeogene/Neogene without involving speciation events. However, many studies have tested phylogenetic niche conservatism between closely related species and found evidence for niche differentiation (see Table 1 in [START_REF] Peterson | Ecological niche conservatism: a time-structured review of evidence[END_REF].

The observed niche overlap is greater for species with strictly disjunct populations

Niche overlap was higher for species having disjunct distribution with at least one population confined to the Alps and one population confined to Fennoscandia (cf. the restricted group: 0.50±0.1) than for species having contiguous or patchy distribution across Europe with populations extending outside the Alps or Fennoscandia (cf. the enlarged group: 0.42±0.1). This difference may stem from the fact that the entire realized climatic niche is likely to be captured in the restricted group whereas in the enlarged group the realized climatic niche will be truncated in at least one of the two studied regions and thus we may underestimate niche overlap for the enlarged group.

Considering species' biogeographic affinities, niche overlap was highest for arcticalpine species (Fig. 5), supporting former findings of [START_REF] Pellissier | Thermal niches are more conserved at cold than warm limits in arctic-alpine plant species[END_REF], who found a strong conservatism of the low-temperature limit of 26 arctic-alpine plant species.

Consistently with their findings that thermal niches are more conserved at cold than at warm limits in aArctic-alpine plant species, we found that niche overlap for arctic-alpine plant species increased with increasing preference for cold temperatures and moist soils. On the other hand, species displaying dissimilar realised climatic niche were geographically and/or edaphically widespread species (e.g., Alyssum alyssoides, Calluna vulgaris, Carum carvi, Fragaria vesca, Galeopsis tetrahit, Juniperus communis, Ranunculus bulbosus, Rubus idaeus, Vaccinium myrtillus, Viburnum opulus) (Table S8 in Appendix S8). This supports the idea that generalist species are tolerant of a wider range of environmental conditions given their often great diversity of phenotypes and thus are more able than a specialist species to respond to regional differences by exploiting a greater diversity of habitats [START_REF] Colles | Are specialists at risk under environmental change? Neoecological, paleoecological and phylogenetic approaches[END_REF].

Generalist or opportunistic species are able to occupy wider fundamental niches and thus can easily shift their realised niche depending on the region, leading to a lower niche overlap than for a specialist or late-successional species whose fundamental niche is too narrow to allow large shifts of the realised niche [START_REF] Pearman | Niche dynamics in space and time[END_REF]. This contradicts recent findings from [START_REF] Early | Climatic niche shifts between species' native and naturalized ranges raise concern for ecological forecasts during invasions and climate change[END_REF], who compared native vs. naturalised ranges of the same species and found greater disequilibrium (cf. lower niche overlap) for species with native ranges that are small and occupy a narrow range of climatic conditions. However and as suggested by [START_REF] Early | Climatic niche shifts between species' native and naturalized ranges raise concern for ecological forecasts during invasions and climate change[END_REF], the stronger climatic disequilibrium for range-restricted species is likely due to non-climatic factors such as dispersal and biotic factors constraining the native range whereas in the naturalised range especially the latter biotic constraints may be alleviated (cf. enemy-release hypothesis), thus leading to strong climatic disequilibrium. Here, rangerestricted species are arctic-alpine species whose distribution is chiefly constrained by climatic factors and less so by biotic factors, which could explain such discrepancy.

Niche overlap also increased as plant species prefer acid grounds. Interestingly, siliceous bedrock predominates across Fennoscandia and calcareous bedrock is rare and local. Thus, acidic soils developed rapidly across Fennoscandia soon after deglaciation (e.g., [START_REF] Boyle | Measuring ecological niche overlap from occurrence and spatial environmental data[END_REF], leading today to a larger pool of acidophilous species in Fennoscandia than in the Alps [START_REF] Lenoir | Cross-scale analysis of the region effect on vascular plant species diversity in southern and northern European mountain ranges[END_REF]. Therefore, in comparison to calciphilous species, postglacial recolonisation of acidophilous species in Fennoscandia might have been less constrained by dispersal limitation, leading to greater niche overlap with their distant populations in the Alps.

In brief, the regional imprint on of the Quaternary climate changes on …. is weak and the realised climatic niches of terrestrial vascular plants seem largely conserved. This is particularly true for specialist species like Arctic-alpine plants.

The realised climatic niche is wider in the Alps

Despite the realised climatic niche being globally conserved, we find regional differences trended towards wider niches in the Alps than in Fennoscandia (Fig. 3), especially when constraining our analyses to the analogue climatic space (Table S9 in Appendix S9). This suggests that species have filled a greater proportion of their fundamental climatic niche in the Alps and runs counter to MacArthur's (1972) hypothesis for island biogeography, which proposes that species' realised-niche width is constrained by the size of the regional species pool so that more intense competition in species-rich islands should lead to narrower realised niches. The same reasoning for mainland biogeography does not seem to hold. Our findings, however, are consistent with a recent study on the realised-niche width of plants which concludes that competition from ecologically similar species plays a relatively minor role in species' establishment in different habitat types [START_REF] Manthey | Niche expansion after competitor extinction? A comparative assessment of habitat generalists and specialists in the tree floras of south-eastern North America and south-eastern Europe[END_REF].

The wider realised climatic niches in the Alps support our hypothesis that greater genetic diversity due to different refugia close to the Alps, corresponding to greater habitat heterogeneity, increases the likelihood for a species to fill its fundamental climatic niche.

Many refugia provided ice-free areas with various habitat conditions where high elevation plants could survive the glacial stages in the Alps [START_REF] Schönswetter | Molecular evidence for glacial refugia of mountain plants in the European Alps[END_REF]. This has likely played an important role in the intraspecific diversification of many alpine plants in the Alps [START_REF] Alvarez | History or ecology? Substrate type as a major driver of spatial genetic structure in Alpine plants[END_REF], and is also reflected by the higher levels of genetic diversity within species in the Alps than Fennoscandia [START_REF] Eidesen | Genetic roadmap of the Arctic: plant dispersal highways, traffic barriers and capitals of diversity[END_REF]. In contrast, cryptic glacial refugia in the north [START_REF] Stewart | Refugia revisited: individualistic responses of species in space and time[END_REF] were probably confined to areas outside the margin of the massive Fennoscandia ice-sheet [START_REF] Birks | Plant macrofossils and the Nunatak Theory of per-glacial survival[END_REF]) and fewer, smaller, genetically depauperate populations survived, with these populations contributing little to total gene pool after post-glacial population recolonisation [START_REF] Edwards | Constraints on post-glacial boreal tree expansion out of far-northern refugia[END_REF], thus resulting in lower genetic diversity and narrower realised climatic niches in modern populations. However, such differences in realised niches width may also result from time-lagged range expansion following post-glacial warming and disequilibrium with contemporary climate for populations in Fennoscandia [START_REF] Svenning | Limited filling of the potential range in European tree species[END_REF].

The realised climatic niche is located towards warmer and wetter conditions in the Alps

The large and diverse set of terrestrial vascular plants occurring in two distant regions in the present study show that species generally have their climatic optima in warmer and more humid conditions in the Alps than in Fennoscandia, irrespective of the climatic space considered (whole or analogue). At least two different processes could explain these interregional differences.

First, shifts in the position of the fundamental niche itself may be involved through local adaptation [START_REF] Davis | Evolutionary responses to changing climate[END_REF][START_REF] Leimu | A Meta-Analysis of Local Adaptation in Plants[END_REF]. Results from common garden experiments, where species' populations from different geographic areas are together in several climates, reveal local adaptation of most tree and herbaceous plant species to their local home environments compared to the foreign environments [START_REF] Davis | Evolutionary responses to changing climate[END_REF][START_REF] De Frenne | Temperature effects on forest herbs assessed by warming and transplant experiments along a latitudinal gradient[END_REF][START_REF] Alberto | Potential for evolutionary responses to climate change -evidence from tree populations[END_REF].

Second, species may change their optimum position along one climatic axis (e.g., growing degree days or aridity index) of the n-dimensional niche hypervolume (sensu [START_REF] Hutchinson | Concluding Remarks[END_REF] to compensate for differences in local conditions other than climatic factors. It has been shown that vascular plant species can spatially shift their realized-niche optima for soil nutrients and pH [START_REF] Diekmann | Shifts in ecological behaviour of herbaceous forest species along a transect from northern central to North Europe[END_REF][START_REF] Wasof | Ecological niche shifts of understorey plants along a latitudinal gradient of temperate forests in north-western Europe[END_REF]. We did not account for this process in our analyses (e.g. soil pH or land-use), thus it could be a confounding…?

We conclude that the realised climatic niche of terrestrial vascular plants that successfully recolonised northern Europe in the last glacial-interglacial cycle has largely been conserved and that this is especially true for specialist species like arctic-alpine plants and species less constrained by dispersal limitation. This has important consequences for predicting future biodiversity under climate change. Notably, the basic assumption that species' realised climatic niche is constant in space and time (thousands of years) seems to be valid to a large extent. Indeed, if the aim is to track the geographic distribution of species with low dispersal limitations this might well work. However, we also show that regional differences in niche width and optimum are rather common. Hence, if the aim is to track precisely the geographic position of some key species distribution parameters (range limit or center), then the basic assumption of species distribution models is likely to be flawed, simply because it does not account for regional subtleties in niche width and optimum values. (SRTM) at 1-km resolution and projected into the 'ETRS89 / ETRS-LAE; code EPSG: 3035' projection system. Dryas octopetala is a typical Arctic-alpine plant with a strictly disjunct distribution across Europe and which populations in the Alps and Fennoscandia are confined to mountainous areas or Arctic-alpine tundra (cf. the "restricted" group). Fragaria vesca is a widespread plant which distribution is patchy across Europe but not necessarily confined to mountainous areas or Arctic-alpine tundra (cf. the "enlarged" group). according to species' biogeographic groups when considering the 31 species belonging to the restricted group (a) and the 358 species belonging to the enlarged group (b). Chorology is a qualitative variable representing species' biogeographic origins: Arctic-alpine (Ar.-al.), Asia, Eurasia (EU), Europe (EUR), Holarctic (Hol.), Mediterranean (Med.), North America (NA) [START_REF] Landolt | Flora indicativa: ecological indicator values and biological attributes of the flora of Switzerland and the Alps[END_REF]. Numbers in brackets correspond to the number of species in each biogeographic group. Box plots show median, 25 and 75% percentiles, whiskers extend to 1.5

BIOSKETCHES

x interquartile range, and points are values lying outside this range.

  ) and 3(b)) and larger in Fennoscandia than in the Alps for AI (Figs. 3(c) and 3(d)) (Table

Safaa

  Wasof is a PhD student at Jules Verne University of Picardie in France and is interested in plant community ecology and in how plant communities are affected by contemporary environmental changes. SW prepared the data, ran the analyses, interpreted the results, and wrote the first draft of the manuscript; JL designed the study, provided part of the data, analysed the data and interpreted the results; PAA, GA, VB, KAB, HJBB, HHB, CJD, MD, SD, MD, RE, JCG, BJG, JAG, AG, KK, ML, AM, BN, AO, HP, VR, SR, SMS, FHS, JS, WT, LUT, VV, LGV, RV, PV, WW, TW, NEZ, and JK, provided the data; GD, WT, AG, HJBB, IGA, MD, KH, SD, RV, JAG, WSA, VV, RE, JCS, TW, WW, MM, MZ, and JCG contributed to the design of the study and to the interpretation of the results. All authors contributed substantially to elaborate the final version of the manuscript.

FIGURE LEGENDS Figure 1

 LEGENDS1 FIGURE LEGENDS

Figure 1

 1 Figure 1 Delineation of the two study regions (a, b), i.e. the Alps in black and Fennoscandia

Figure 2

 2 Figure 2 Response curves of Dryas octopetala (a) and Fragaria vesca (b) along the growing

Figure 3

 3 Figure 3Histograms showing the differences between species' realised-niche width in Alps

Figure 4

 4 Figure 4Histograms showing the differences between species' realised-niche optimum in

Figure 5

 5 Figure 5 Box plots showing the distribution of observed niche overlap values across the

Figure 6

 6 Figure 6 Scatter plots showing the distribution of the observed niche overlap across the

  Fig. 1 895

  Fig. 2 899

  Fig. 6 907

ACKNOWLEDGMENTS

We are grateful to the many people who collected plot data and to those who managed and provided the databases for use. We wish to thank two anonymous referees and Linda Beaumont for insightful comments as well as Emilie Gallet-Moron for help in preparing geographical maps. We greatly acknowledge the Syrian Ministry of Higher Education for funding SW's PhD thesis. JCS and WT were supported by the European Research Council (ERC-2012-StG-310886-HISTFUNC and ERC-2011-StG-281422-TEEMBIO, respectively). MD acknowledges financial support from the Swedish Research Council Formas and AM from the Swedish Research Council VR. MM and MZ were supported by Centre of Excellence FIBIR and IUT 20-28. PAA, GA, VB, KAB, HHB, BJG, JAG, JK, KK, AO, VR, SR, SMS, FHS, JS, LUT, VV, and LGV were supported by the Norwegian Research Council. JD and KH were supported by the EkoKlim Program at Stockholm University.

Insights from plant traits

Whatever the species group (i.e. restricted vs. enlarged) and the climatic space (whole vs.

analogue) considered, we found similar patterns (Appendices S11 and S12). For clarity, we only report results based on the full climate space (Appendix S11). Species' biogeographic origins of the restricted and enlarged species groups explained 50% vs. 7%, respectively, of the variation in D obs , with Arctic-alpine species showing the highest D obs values and standing out of the other biogeographic groups (Fig. 5(a) and 5(b)). Species' ecological indicator values of the restricted and enlarged species groups explained 53% vs. 21%, respectively, of the variation in D obs , with contrasting patterns depending on the species group. For the restricted group, D obs increased with increasing preference for cold temperatures and moist soils (Fig. 6(a) and6(c)). In contrast, for the enlarged group, D obs increased with increasing preference for acid soils and for fertile soils (Fig. 6(a) and 6(c)). There was no clear relationship between D obs and species' biological traits (Appendix S11).

SUPPORTING INFORMATION

Additional Supporting Information may be found in the online version of this article: Appendix S1 Pool of terrestrial vascular plant species common to the Alps and Fennoscandia.

Appendix S2 Details about the nine climatic variables considered in this study.

Appendix S3

List of data-handling manipulations on presence and absence data of each of the 888 terrestrial vascular plant species belonging to the common species pool. Appendix S11 Full results on niche overlap and plant traits.


Roméo Molina
email: romeo.molina@ijclab.in2p3.fr

Vincent Lafage
email: vincent.lafage@ijclab.in2p3.fr

David Chamont
email: david.chamont@ijclab.in2p3.fr

Fabienne Jézéquel
email: fabienne.jezequel@lip6.fr

Investigating mixed-precision for AGATA pulse-shape analysis

The AGATA project aims at building a 4π gamma-ray spectrometer consisting of 180 germanium crystals, each crystal being divided into 36 segments. Each gamma ray produces an electrical signal within several neighbouring segments, which is compared with a data base of reference signals, enabling to locate the interaction. This step is called Pulse-Shape Analysis (PSA). In the execution chain leading to the PSA, we observe successive data conversions: the original 14-bit integers given by the electronics are finally converted to 32-bit floats. This made us wonder about the real numerical accuracy of the results, and investigate the use of shorter floats, with the hope to speedup the computation, and also reduce a major cache-miss problem. In this article, we first describe the numerical validation of the PSA code, thanks to the CADNA library. After the code being properly instrumented, CADNA performs each computation three times with a random rounding mode. This allows, for each operation, to evaluate the number of exact significant digits using a Student test with 95% confidence threshold. In a second step, we report our successes and challenges while refactoring the code so to mix different numerical formats, using high precision only when necessary, and taking benefit of hardware speedup elsewhere.

Introduction

AGATA 1 (Advanced GAmma-ray Tracking Array) is a European research project with the objective of building a detector with significantly higher resolution than existing ones. It will be used for experiments with intense stable and radioactive ion beams, to study the nucleus structure. In the present work we will deal with two problems: the uncertainty of the results in numerical simulations due to the floating point arithmetic that we will refer to as accuracy and the need to accelerate the processing of data coming from the detector. We will get interested in the way these both aspects are connected by the kind of precision formats that are used and from which depend both the performance and the accuracy of the computations.

Today's computers are usually using floating-point operations to simulate calculations on real numbers. There exist several floating-point formats with varying numbers of bits. Until now, fp64 (called double precision) and fp32 (called single precision) have been the most common formats but we observe a rise of half precision formats in the last years, in particular on GPUs.

Applications requiring high definition and data processing, with numbers of very different magnitudes usually use the fp64 and fp32 formats, whose performance have been optimized in hardware and embedded in development practices. Those formats enable results with 7 or 15 significant digits and allow to deal with large ranges of values. In the counterpart, they are expensive in terms of execution time, energy and memory. Under the influence of low-precision applications, half (fp16, bf16) and even lower-precision formats are developing strongly, with dedicated hardware delivering excellent performance. On NVIDIA A100 GPUs, for example, fp16 or bfloat16 are 16 times faster than fp32. This performance is attracting interest in fields requiring both speed and high-accuracy results, such as particles and nuclear physics. To combine these two needs, mixed-precision approaches are being developed relying on the wise use of different kinds of precision formats in the same code to benefit from their respective advantages.

However, since floating-point arithmetic relies on approximations that accumulate as computations proceed, uncontrolled computations can lead to a completely false result. We therefore need to control the accuracy of our computations, in particular when we intend to mix several precisions. In the present paper we are using a software named CADNA which enables us to track the evolution of the number of exact significant digits in our code and detect numerical instabilities.

In this paper we will present our results on the stability of the code of the AGATA pulseshape analysis by comparing the results obtained with fp64 format and the stochastic types from CADNA. We will also present the results obtained using a lower precision format fp16 on the naive full grid algorithm, and on the coarse-fine algorithm used in practice.

The AGATA experiment

Gamma-ray spectroscopy is a fast, non-destructive analysis technique for identifying radioactive isotopes in a sample. The energy of incident gamma rays is measured by a detector and compared with those known to be emitted by isotopes that allows to identify the isotope at the origin of the gamma ray.

High Purity Germanium (HPGe) detectors are the currently best solution for gamma and X rays spectroscopy. They are much more efficient than silicon ones that cannot be larger than few millimeters while germanium crystals can be few centimeters large and then absorb full gamma rays from a few tens of keV to beyond 10 MeV. Germanium crystals have to be very pure and work at really low temperatures around -196 • C that are achieved with nitrogen cryostats.

The processing of data from high-energy and nuclear physics experiments is historically divided into two steps. The first step, called online, takes place immediately on leaving the detector, and generally involves electronic processing to sort the data and eliminate noise. Today, it is performed using electronic components and FPGAs. The second step, called offline, consists in reconstructing the signal from the data obtained and stored, so that the computations can be performed several times, with the aim of refining them. However, the explosion of data volumes leads to more online computing. This is because physicists are seeking to identify increasingly rare and discreet phenomena hidden in what used to be considered as noise. The trend, therefore, is to introduce more software processing in the online part, to cope with the influx of data. This calls for high-performance algorithmic processing, which is the goal of our work.

AGATA is a 4π array detector that will in time consist of 180 36-fold segmented HPGe detectors with optimized geometry grouped into triplets for a total of 6480 channels. It relies on the gamma-ray tracking. This technique consists in determining every point of interaction of a gamma-ray in the germanium crystal to track its whole path and then measuring the energy and the angle at which it has been emitted.

Each Germanium detector of AGATA is split into 36 segments (6 slices along the axis times 6 sectors around the axis). When a photon arrives into the Germanium crystal, it loses energy scattering electrons off atoms. This happens multiple times, producing a shower of electrons that generates an electric current and then an analog signal.

Analog signals are converted into digital signals using an analog-to-digital converter, sampling at 100 MHz over 14 bits. These data are then serialized and preprocessed by FPGAs, which reduce them by a factor of 100, retaining only those detectors that have recorded a relevant signal. In addition, a moving-window deconvolution algorithm determines the energy deposited in each detector. The raw data, energy information and time stamp are then transmitted to the computer system via optical fiber. From there, the NARVAL data acquisition system enables data to be stored in ADF (AGATA Data Flow) format and processed through a series of actors running as individual processes on a network of computer nodes.

The first actor is the crystal producer, which reads the measurements, casts it in fp32, which generates an increase in storage size, and passes it on to the preprocessing actor. This actor defines thresholds and corrects signal noise. At the output, it produces energy-calibrated and time-aligned data. This is followed by the Pulse Shape Analysis (PSA), in which the signals obtained are compared with a signal database to determine the position of the interactions. This eliminates the need for all the data, and allows only the energy and position information to be transmitted to the next actor.

The post-PSA actor sets thresholds on core energy and segments that can be recalibrated and moved over time. It can also apply a correction for the effects of detector wear. The next step merges data from all AGATA crystals using time stamps. A filter can be used to set minimum event thresholds. Data from auxiliary detectors are then merged by the next actor, again using time stamps. The last actor is the tracking filter. It tracks the various events in the detector to determine how the gamma rays have interacted in the complete detector.

The CADNA tool for numerical validation

The representation of a floating-point number y, described by the IEEE754 standard [1], is y = s • m • β e with s the sign, β the base, e the exponent such that e min ≤ e ≤ e max and m the mantissa which satisfies 1 ≤ m < β for normalized numbers. Let us denote by t the precision, i.e. the number of bits in the mantissa plus an implicit bit set to 1. In the present paper, we are only dealing with floating point numbers in base β = 2. Thus we can define u the machine epsilon as u = 2 -t in round-to-nearest mode. Depending on the format used, more or fewer bits are used to store the mantissa and exponent, as described in Table 1.

Elementary rounding errors accumulate and can lead to invalid results. This problem is perfectly illustrated by a polynomial introduced by S. Rump [START_REF] Rump | Reliability in computing[END_REF]. Let us consider P(x, y) = 333.75y 6 + x 2 (11x 2 y 2 -y 6 -121y 4 -2) + 5.5y 8 + x/(2y) with x = 77617 and y = 33096. The results obtained with different working precisions are: -1.85901e+30 in fp32, -1.18059e+21 in fp64, 1.1726 in fp128, while the correct result is 54767 66192 ≈ -0.827396059946821368141165095479816292. Stochastic arithmetic [START_REF] Vignes | [END_REF] is a probabilistic method that involves performing each arithmetic operation several times with a random rounding mode: each result is rounded up or down with the same probability. Then the number of exact significant digits is estimated using a Student's test with a confidence level of 95 %. It has been shown [START_REF] Vignes | [END_REF] that, in some reasonable sense, the optimal sample size is N = 3. Indeed the estimation with N = 3 is more reliable than with N = 2 and increasing N does not significantly improve the quality of the estimation. CADNA2 [START_REF] Chesneaux | Study of the computing accuracy by using probabilistic approach[END_REF][START_REF] Jézéquel | [END_REF][6][7] is a library that implements stochastic arithmetic to estimate the number of exact significant digits of a floating-point number resulting from numerical computation. CADNA enables one to use new numerical types, called stochastic types on which numerical validation can be performed. CADNA also provides the definition of all arithmetic operations and mathematical functions involving stochastic variables. CADNA is available in C/C++ and Fortran and is a relatively lightweight tool, requiring little code rewriting thanks to operator overloading, with a memory overhead of 4 and an execution time overhead of around 10. With CADNA each arithmetic operation is performed 3 times before the next arithmetic operation is executed, thereby propagating the rounding error differently each time. This synchronous execution enables the detection of numerical instabilities, usually due to results that have no exact significant digits. CADNA can be used for the numerical validation of HPC simulations: either CPU codes that rely on MPI and/or OpenMP or GPU codes written in CUDA. It makes it possible to control the numerical stability of a computation and to experiment with different precisions, including native or emulated half [START_REF] Jézéquel | Numerical validation of half precision simulations[END_REF].

Profiling of AGATA computations 4.1 Performance analysis

The code of AGATA is almost fifteen years old, and suffers from various problems which put a strain on its performance. Furthermore, it is not adapted to take advantage of modern architectures capabilities. We first performed a static analysis that revealed the massive use of loops and sometimes arbitrary use of fp32 and fp64 formats. Using perf3 , a Linuxbased performance analysis tool, we have explored its weaknesses and determined the functions that shall be targeted to improve its performance. This analysis clearly showed that the Chi2InnerLoop function concentrates 69 % of the CPU cycles, 80 % of the cache-references and more annoying, 73 % of the cache-misses. All this makes this function, and the PSA that includes it, a prime target for accelerating the execution of the code of AGATA. The number of memory accesses, due to the fact that we are processing a very large database, has led us to seek ways of reducing the amount of data accessed.

In addition, we have observed that in the execution chain leading to this PSA, there are successive data conversions. Data coming from the detector are originally sampled in 14 bits by the electronics, which correspond to a dozen bits of truthful information. These 14 bits are later extended into 16-bit integers to fit in computer format, and finally cast into 32-bit floats to perform floating-point operations. This made us wonder if lower floating-point precision can be used, while maintaining the accuracy.

Our work therefore has two concomitant objectives: reducing the cache-miss of the PSA by reducing the volume of data accessed and take advantage of the arithmetics of modern architectures, while controlling the numerical quality of our results.

Pulse-Shape Analysis

The PSA aims at comparing the obtained signal with a signal database to determine the position of the interaction. The naive algorithm consists in calculating a chi2 between the signals obtained in the interaction segment and its neighbors with those associated with each interaction point calibrated in a preliminary phase. Considering 9 neighboring segments, with 1200 sampled points each, over 60 time steps and 32 bits of data, we obtain 2.6 Mb, which exceeds the size of the cache.

Our work takes place in a context of numerous discussions with physicists to understand their needs and the logic behind their choices of implementation such as the use of a "chi2" using a power 0.3. These exchanges aim at defining what is an acceptable result from the PSA and therefore what room for maneuver we have in terms of accuracy.

A large amount of work has already been dedicated to this objective, with an algorithmic approach, denoted as coarse-fine algorithm from now, consisting in dividing the search into two phases: a first minimum search on a coarse grid and a second step that searches for the real minimum on a fine grid in the neighborhood of the coarse minimum. This approach, currently used in practice, considerably reduces the amount of data to be processed and accelerates the computation by a factor 5. However, it does not achieve exactly the same results as the naive full grid algorithm, as we explain hereafter. Another approach explored but not used in practice, consists in transforming the data to perform a wavelet analysis ; we do not develop this approach here.

We have noticed the use of a look-up table in the chi2 computation. This approach that reduces the accuracy, in particular for absolutely small numbers, also makes the use of CADNA inefficient, because data read from a table are considered exact, whereas loss of accuracy can be taken into account by CADNA in numbers resulting from numerical computation. Probably introduced to accelerate the code, it actually slows it down due to increased cache usage. We replaced it by a direct computation.

Accuracy control

We have then been particularly interested in checking the accuracy of computations carried out using the CADNA tool and experimenting the use of different precision settings. To do so, we have instrumented the PSA code so that we can easily use different precisions and also stochastic types of CADNA. We needed to experiment using CADNA to control the accuracy because we are performing a minimum search in a 504-dimensional space that carries a risk to accumulate catastrophic cancellations. We have first analyzed the version of the code implementing the full grid search and observed that the code is slightly sensitive to perturbations. Indeed, using stochastic fp32, stochastic fp64, or classic fp64, instead of the original classic fp32, the code gives 0.02 % difference in the points found. This first analysis validates the original results and allows us to envisage a further reduction of the precision used.

Using reduced precision formats

Half-precision computation

Having confirmed the numerical quality of the results obtained by the PSA, we sought to use reduced-precision formats within it. We therefore used a library4 that emulates fp16 half precision. This format, introduced in the 2008 revision of the IEEE754 standard, enables one to perform floating-point operations on 16 bits only, and is becoming increasingly widely available, in particular on recent GPUs. It does not only reduce storage and energy consumption, but also delivers significant speed-up. For example, a computation performed in fp16 instead of fp32 can be accelerated by up to ×16. This format, driven by the dynamism and massive investment in neural network computation that often uses reduced precision, is set to continue developing, and thus appears to be a central element in the performance gains for years to come. We therefore felt it necessary to explore the possibilities of exploiting such a format, in a context requiring a certain level of accuracy, namely nuclear physics, and in this case the execution of AGATA PSA.

Thanks to our initial instrumentation, we have been able to easily perform the full grid algorithm in fp16 which resulted in determining 7.76 % points differently from the full grid fp32 version. We have also been able to perform the coarse-fine algorithm in fp16 that resulted in 14.04 % of the points found differently.

The acceptability of those differences is to be discussed with physicists and is analysed more closely in section 5.3.

Mixed-precision computation

Mixed-precision computation is an emerging paradigm that mixes several precisions in the same code with the goal of getting the performance benefits of low precisions while preserving the accuracy and stability of high precision. It is based on the idea that not all computations are equally important, considering for example that in an addition between two numbers of different magnitudes, most of the significant digits of the smaller one may be eliminated. The main problem for this paradigm is to decide in which precision each operation shall be performed. Our work takes place in the area of precision tuning that aims at using a tool that evaluates the number of correct digits in a result for different precision configurations.

We therefore looked for a way of using low precision while remaining close to the original results. We turned to the algorithm used in production, which performs a coarse-fine search. Such an algorithm naturally lends itself to a mixed-precision approach, in which the coarse search is performed with low precision and the fine search with higher precision. Running the coarse-fine algorithm in fp32 yields 8.22 % different points with respect to the direct algorithm, while with the mixed-precision version we obtain 8.55 %, which seems comparable and then acceptable. Of course, we could also perform both steps in low precision, but under the same conditions, this would result in 14.04 % of differences from the fp32 full grid algorithm as already mentioned in section 5.1. In the case of the coarse-fine algorithm, the coarse part represents around 20 % of the computation time, so this is a limit to the time savings we can expect if we do not change the grid size to relatively increase the coarse part. These preliminary results confirm the interest of mixed precision for applications to improve the performance while maintaining the accuracy and strengthen our determination to continue in this direction.

Evaluation of error acceptability

In this work, we were confronted with different ways of analyzing computation errors. Firstly, we were interested in numerical accuracy in the sense of the number of exact significant digits, an analysis enabled by CADNA and stochastic arithmetic. Then, we looked at the number of interaction points found differently using the possible combinations of methods and precision settings. In this section we try to measure how different the points found are and in which energy class they lie. To this end, we have produced the following figures. In Figure 1 we can observe that most of the points found differently from the full grid fp32 method are less than 3 mm far from the original points and that the number of points found differently from the full grid fp32 points decreases with distance. Moreover, as we can observe on Figure 2, most of the unfaithful points rely, as any point, in the smaller energy class that requires less accuracy. Also, we can observe that the distribution of unfaithful points is the same for the classic coarse-fine algorithm performed in fp32 and the mixed precision one using half precision for the coarse search. All in all, these data confirm that the mixed precision solution can be trusted as well as the uniform precision fp32 coarse-fine algorithm.

Conclusion and perspectives

We have demonstrated the benefits of using low precision in the AGATA PSA code, and specified the error control procedures required to obtain satisfactory results. To this end, we instrumented the AGATA code to allow the use of different combinations of precisions, using the CADNA tool which, with the aid of stochastic arithmetic, estimates the number of exact significant digits in a result. We faced a difficulty, since the result we were seeking was not a floating-point number, but a series of discrete points which we wanted to be as close as possible to the actual points of interaction. Finally, we opted for a mixed-precision computation based on the existing coarse-fine minimum search algorithm used in practice, with fp16 in the coarse step and fp32 in the fine step. This strategy enabled us to obtain very similar results to those provided by the coarse-fine algorithm in uniform precision fp32. We were also able to verify that points found differently from the full grid fp32 algorithm were at a small distance and very similar to those found by the coarse-fine fp32 algorithm. Finally, we checked the energy classes in which the different points were found and observed the same similarities and a distribution essentially in the low energies.

Following on from this work, it would appear necessary to implement our mixedprecision solution on hardware really supporting and check the expected performance benefit. Another interesting lines of research would consist in refining the hierarchy of coarse and fine grids, either by adding intermediate grid sizes using various precisions, or by varying the respective weights of coarse and fine grids. Indeed, our strategy shows the expected benefit on the coarse part, but this remains limited due to the relative weight of the coarse search, which currently represents only 20 % of the total computation.

Figure 1 .

 1 Figure 1. Distances between points found by the full grid fp32 algorithm and alternative methods

Figure 2 .

 2 Figure 2. Energy classes in which the points found by various methods belong to

Table 1 .

 1 Various floating-point formats

		Bits
		Mantissa (t) Exp. Range	u = 2 -t
	fp128	113	15	10 ±4932 1 × 10 -34
	fp64	53	11	10 ±308	1 × 10 -16
	fp32	24	8	10 ±38	6 × 10 -8
	fp16	11	5	10 ±5	5 × 10 -4
	bfloat16	8	8	10 ±38	4 × 10 -3

http://www.lip6.fr/cadna

https://perf.wiki.kernel.org

https://half.sourceforge.net


Sofia Kaisaridi
email: sofia.kaisaridi@icm-institute.org

Hugues Chabriat

Sophie Tezenas Du Montcel

A multivariate bayesian mixed-effect model (Leaspy) to analyze the trajectory of cognitive decline in CADASIL

 . We used a bayesian mixed effect model(Leaspy) to model the multivariate disease progression and explore the spatiotemporal relationships between these measures. We were also able to identify different groups of individual evolution according to the time shift and acceleration rate by taking into account the gender, education level, smoking, hypertension and the mutation's position.

Introduction

According to the population parameters describing the starting acceleration point of the logistic curves we were able to distinguish 3 groups of evolution :

Data & Method

We analysed data obtained from 395 patients recruited at the French National Referral Centre CERVCO across a median number of 4 visits per individual (IQR:3-7) (total visits: 2007).The follow-up duration varied from 6 months to 19 years with a median interval of 6.5 years (IQR:3.5-11). They were 45% men, 72% with a high school diploma or less, 18% current smokers and 21% presenting hypertension with a median inclusion age of 53 years (IQR:45-61). They were 66% presenting the mutation in the EGFr domains considered as high risk 1-6,8,11 and 26 [2] . The outcomes analyzed were 16 clinical scores measuring cognitive efficiency, cognitive speed and mental flexibility, memory performance, life quality, disability and neurological deficits.

We used a logistic multivariate bayesian mixed-effect model(Leaspy) to analyze the progression of these outcomes. As with every mixed effect model we obtained the fixed effects (population parameters representing the shape and position of logistic curves) and the random effects (individual parameters that represent the time shift (𝜏), acceleration rate (𝜉) and inter-marker spacing). The bayesian framework allowed us to minimize the errors between the estimated curves and the individual timepoints by performing iteratively joint estimations of these parameters. Leaspy also offered us the advantage to model the evolution without knowing the exact time of the disease onset for each individual.

Results

Conclusion

We implemented a Gaussian Mixture Model classification for the individual Leaspy parameters describing the temporal variability(𝜏 and 𝜉) and the 5 covariables that showed significant interactions with these parameters in the non-parametric tests (gender, education level, smoking, hypertension and the position of the mutation). As a result, we got a group of patients that show early evolution ( ̅ 𝜏 ! =47, ̅ 𝜉 ! =0.5), a group of intermediate ( ̅ 𝜏 ! =65, ̅ 𝜉 ! =0.3), and a group of late and slow evolution ( ̅ 𝜏 ! =72, ̅ 𝜉 ! =-0.3).

• Cognitive alterations develop heterogeneously during the progression of CADASIL and vary widely according to the stage of the disease. • The disease progresses more aggressively for patients that are male, less educated, smokers, presenting hypertension and having the mutation in the EGFr domains 1-6,8,11 and 26.

  Early age : VADAS-Cog Backward digit span (empan), VADAS-Cog Symbol digit test (coderep), VADAS-Cog Digit cancellation task (barragecor), Trail making test B time (tmtbt) , Grober & Buschke total free recall (gblibre_total) Mid age : Rankin index (indexscrankin), Trail making test A time (tmtat), VAS Quality of life (echelv), VADAS-Cog Total score (vadas_cog) Late age : Mattis DRS Initiation (initiation), Grober & Buschke index of sensitivity to cueing (ind_react), Mattis DRS Total Score (scoretot), Trail making test B errors (tmtbe), Grober & Buschke delayed total recall (rdtotal), NIHSS index (indexnihss), Barthel index (indexscbarthel)