Search

Partitioning a set of elements into a given number of classes to find a globally optimal solution can be challenging due to the combinatorial explosion of the problem size. In the univariate case, where elements can be ordered, the number of partitions is significantly lower than in the multivariate case, and the problem is easier to handle. In this article, we focus on the univariate case and propose using complete enumeration to find a globally optimal solution. Although complete enumeration may also be computationally prohibitive as the number of elements and classes increases, it can be feasible in some situations. For such cases, we propose an algorithm that generates all contiguous partitions for a variable number of classes to be used with any objective function or set of constraints. • We compare exact problem sizes and approximate time complexities for multivariate and univariate partitioning. • We fill a technical gap in the literature by providing a valuable tool for researchers or engineers who need to exactly solve unusual univariate partitioning problems. • We use a convenient data structure for representing partitions of elements into classes and an iterative algorithm that simulates nested loops for any depth level, allowing for efficient generation of all possible contiguous partitions.

International audience; The cost-effectiveness and reduced human effort employed in setting up acoustic monitoring in the field makes bioacoustics an appealing option for wildlife monitoring. This is especially true for secretive vocal species living in remote places. However, acoustic monitoring still raises questions regarding its reliability when compared to other, human-driven methods. In this study we compare different approaches to count rock ptarmigan males, an alpine bird species which lives at high altitudes. The monitoring of rock ptarmigan populations is traditionally conducted using a point-count protocol, with human observers counting singing males from a set of different points. We assessed the (1) feasibility and (2) reliability of an alternative counting method based on acoustic recordings followed by signal analysis and a dedicated statistical approach to estimate the abundance of males. We then (3) compared the results obtained with this bioacoustics monitoring method with those obtained through the point-count protocol approach over three consecutive years. Acoustic analysis demonstrated that rock ptarmigan vocalizations exhibit an individual stereotypy that can be used to estimate the abundance of males. Simulations, using subsets of our recording dataset, demonstrated that the clustering methods used to discriminate between males based on their vocalizations are sensitive to both the number of recorded signals, as well as the number of individuals to be discriminated. Despite these limitations, we highlight the reliability of the bioacoustics approach, showing that it avoids both observer bias and double counting, contrary to the pointcount protocol where this may occur and impair the data reliability. Overall, our study suggests that bioacoustics monitoring should be used in addition to traditional counting methods to obtain a more accurate estimate of rock ptarmigan abundance within Alpine environments.

Graphical abstract, Variance estimation in multistage sampling without replacement usually requires considerable computational effort. One option is to implement explicit formulas on a computer, at least for some specific sampling designs. This approach becomes quite cumbersome to handle beyond two stages, both from the formulation and computer implementation points of view. Another option is to provide a general method to compute variance estimates for any number of stages. Such an approach may involve data structures and estimators which are recursively defined. • The solution we present in this article is intended to be both general and computationally efficient by relying on a full-iterative implementation. • The definition of the estimators remains implicit as in the recursive approach, but is expressed in terms of recurrence relations translated into iterative algorithms. • These algorithms rely only on (dense) array data structures. Moreover, most of the necessary computer memory is only used during preliminary steps and is not required when performing the statistical calculations.

Afinde répondre à la demande croissante d'information et d'analyse de données environnementales pour ses nombreux publics, l'Office français de la biodiversité se dote de réseaux de compétences sur la data et le numérique. La Direction de la Surveillance, de l'Evaluation et des Données et son service Système d'Information, des Outils, des Analyses et des Données portent trois réseaux correspondants : le réseau des géomaticiens, le réseau des ingénieurs et développeurs dataet le réseau Mathématiques Appliquées - Statistique. Ces deux derniers réseaux, plus récents, rassemblent les ingénieurs et techniciens de domaines larges et variés, recouvrant la discipline des sciences des données. La capacité d'articuler le travail des différents experts, pour répondre aux besoins croissants de traitement, de gestion et d'analyse de données, est crucial pour l'OFB. Le réseau des ingénieurs et développeurs datapropose du 9 au 11 février 2021 un hackathon et un mini-séminairesur les nouveaux modes de collaborations des agents à profil scientifique, et à profil plus technique, les uns détenant les clés des méthodes et calculs, les autres celles des architectures permettant de réaliser les opérations informatiques et l'accès aux services (les APIs). Les 9 et 10 février, une équipe mixte participera à un hackathon (espèces confinées: une API indiquant la probabilité de présence d'un individu d'une espèce donnée, dans un rayon donné autour de soi). Le 11 février, plusieurs présentations de projets et une table ronde permettront d'explorer sur une demi-journée la thématique de l'intégration de modèles scientifiques dans de les nouvelles architectures numériques, et de mieux distinguer les besoins et les projets possibles pour les agents de l'OFB et leurs partenaires. Ces journées, conçues avec l'appui du réseau jumeau Mathématiques Appliquées - Statistiqueen cours de préfiguration et de la société experte Oslandia, sont ouvertes à tous les agents publics de l'Environnement - mais s'adressent essentiellement à des ingénieurs et des techniciens de niveau scientifique ou technique avancé, désireux de progresser sur le vaste domaine des sciences des données.

For migratory birds, sustainable harvest management based on quantitative modelling needs cross-border hunting bag statistics. At the European scale, proper modelling requires both reliable and mutually compatible hunting bag data between regions and countries. Owing to the absence of harmonisation among the different hunting bag collecting schemes in Europe and the lack of methodological metadata, adaptive management at the flyway scale is currently extremely challenging for a number of species. For improving the current state of affairs, we expose statistical concepts, terminology and issues inherent to hunting bag data collection schemes; identify the multiplicity of error sources for being able to judge the quality of hunting bag statistics; call for a harmonisation process; discuss the origin of the hurdles in the production of standardised hunting bag statistics at the European scale; and suggest some potential avenues for future actions for overcoming them.

A variable taking positive values from a lognormal distribution and null values with a given probability is distributed according to the so-called delta-lognormal distribution. Two situations arise depending on whether the data are regarded as a random sample from an infinite population (superpopulation) or from a finite population, itself considered as a random sample from a superpopulation. In the case of an infinite population, estimating the mean can be accomplished using a uniformly minimum-variance unbiased estimator (UMVUE). Likewise, the prediction of the mean in the case of a finite population may be based on the UMVUE. In both cases, one expects a gain in precision when taking into account the shape of the distribution by relying on the UMVUE rather than on the sample mean, which is a nonparametric estimator (or predictor).1.For the infinite population case, the relative efficiency results presented in this article are more complete and more accurate than those published so far.2.The article fills a gap regarding the question of relative efficiency in the case of a finite population.3.Calculations were performed using the exact expression for the variance of the UMVUE of the mean, expressed in terms of the confluent hypergeometric limit function.

International audience; This paper presents an efficient algorithm called SolveFuzzySystem, or SFS, for finding real solutions of polynomial systems whose coefficients are symmetrical L-R fuzzy numbers with bounded support for which the spread functions L and R are bijective. The real solutions of such a system are deduced from solutions of some polynomial systems with real coefficients. This algorithm is based on new results that are universal because they are independent from the spread functions. These theoretical results include the management of the fuzzy system's solutions signs. An implementation in the Fuzzy package of the free computer algebra software SageMath and a parallel version of the algorithm are described.

Hunting bag statistics are often the only available data for performing ecological studies about harvested species, and total harvest is sometimes used as a proxy of abundance of the game species under study in a given geographical area and period of time. This practice raises at least two questions, (i) are the total hunting bag estimates good indices of population abundance, and if so, for what uses?, (ii) what is the reliability of given hunting bag statistics and is it possible to evaluate and take into account their uncertainty without relying on uncheckable assumptions? This methodological paper is aimed at answering the second question, from the point of view of the hunters' sampling. Through Monte Carlo simulations, we illustrate the potential selection bias induced by relying on volunteer samples of hunters. We expose the statistical causes and remedies to this issue. We put the emphasis on the paramount importance of random sampling, both for avoiding selection bias and to perform statistical inferences on a sound basis, in a framework free of statistical assumptions. We explain under what circumstances not taking into account unequal inclusion probabilities at the estimation stage could result in biased estimation. The acknowledgement that for a selection bias to occur, it is necessary that both the unequal inclusion probabilities are not accounted for in the estimators and these probabilities are correlated to the individual hunting bags is a statistical result that is neither widely known nor appreciated by most wildlife ecologists — and perhaps also, some wildlife statisticians.

International audience; This paper presents an efficient method for finding the positive solutions of polynomial systems whose coefficients are symmetrical L-R fuzzy numbers with bounded support and the same bijective spread functions. The positive solutions of a given fuzzy system are deduced from the ones of another polynomial system with real coefficients, called the real transform. This method is based on new results that are universal because they are independent from the spread functions. We propose the real transform T (E) of a fuzzy equation (E), which positive solutions are the same as those of (E). Then we compare our approach with the existing method of the crisp form system.

Lithium-doped nickel oxide and undoped nickel oxide thin films have been deposited on FTO/glass substrates by a surfactant-assisted ultrasonic spray pyrolysis. The addition of polyethylene glycol in the sprayed solution has led to improved uniformity and reduced light scattering compared to films made without surfactant. Furthermore, the presence of lithium ions in NiO films has resulted in improved electrochromic performances (coloration contrast and efficiency), but with a slight decrease of the electrochromic switching kinetics.

Tungsten oxide thin films are known for their excellent electrochromic efficiency, nevertheless, most of deposition methods reported in the literature are not suitable to produce high quality films and at low cost. In this paper, tungsten oxide thin films have been deposited on FTO–glass substrates by a surfactant-assisted ultrasonic spray pyrolysis. This deposition method, associated to the presence of surfactant, has led to improved electrochromic performance of tungsten oxide films, without excessive production cost. Tungsten oxide films made by the surfactant-assisted ultrasonic spray pyrolysis show a high specific surface area, which results in one of the highest coloration contrast (83%) ever reported for tungsten oxide thin films. WO3 thin films also show excellent reversibility (94%), as well as good coloration efficiency (28 cm2/C).

Reliable hunting bag statistics are a prerequisite for sustainable harvest management based on quantitative modeling. Estimating the total hunting bag for a given game species is faced with a multiplicity of error sources. Of particular concern is the nonresponse error. We consider that the major cause of nonresponse bias is when the reluctance to respond is related to a null harvest, which leads to a potentially important overestimation. For tackling the nonresponse bias issue, we advocate the repeated subsampling of nonrespondents, with a final phase of personal interview by phone, intended to be without nonresponse. When a 100% response rate is actually reached at the last phase, both total and sampling variance can be estimated without bias, whatever the response rates at the previous phases. The actual case of imperfect response at the last phase is studied using Monte Carlo simulations. For imperfect response at the last phase, we show that the estimators we advocate are biased downwards but that these bias remain very moderate if the response rate at the last phase is high enough, depending on the circumstances. Furthermore, we illustrate that increasing the number of phases improves the nonresponse bias attenuation. In case of a hunting bag collecting scheme prone to a high nonresponse rate, for obtaining a very satisfying nonresponse bias attenuation we advocate relying on the multiphase sampling strategy with two- or three-phases, and a response rate in the last phase of at least 90%.

The Eurasian woodcock Scolopax rusticola is a valued game species in southwestern Europe, which is locally under a high hunting pressure. Disturbance due to hunting can lead to avoidance of wintering areas or have an impact on population dynamics owing to behavioural changes. We studied the impact of disturbance on 54 radio-tagged woodcocks under an experimental framework. The birds were allocated to the following three groups: simulated hunting disturbance (controlled disturbance; CD), hunted (H) and control (C). We studied the following four behavioural elements: space use, activity duration, number of commuting flights and date of spring migration departure. We found no differences in behavioural responses between the groups except for space use. A majority of CD birds used larger ranges during daytime than did H and C birds but did not change their nocturnal sites. To a large extent, this nocturnal site fidelity likely accounted for the weak disturbance effect, in so far as food intake was ensured. Ou...

Many ecologists use two-dimensional systematic sampling to estimate mean density of individuals over the domain sampled. They usually calculate the variance of the mean as if the sample were a simple random sample, using the unbiased estimator under this sampling design, that is, o2/n. This practice leads to a selection bias, i.e., incorrect inclusion probabilities of population units in the sample are used in the estimator of variance of the mean. The magnitude of the bias varies with the underlying spatial autocorrelation structure. Design-based inference and model-based inference are two conceptual frameworks for tackling estimation of variance of the mean in a systematic sample. This paper reports use of the geostatistical estimation variance cr2 with the model-based approach. This variance of the overall spatial mean depends on the spatial autocorrelation structure of the data. We illustrate the method by considering the density of acorns fallen under a sessile oak during one season. Acorns were numbered in square quadrats of 0.25 m2; the data set was ex- haustive. We drew nine one-start systematic samples from the whole population of quadrats. We computed semivariograms for the whole population and each sample and fitted them to exponential models without nugget. Using geostatistical theoretical results, we calculated a variance of the mean density of acorns by Monte Carlo integration. We show that our variance estimate depends on (1) the origin of the systematic sample, the central sample being the most accurate, (2) quadrat size, with a decrease in the variance when quadrat size increases, (3) the semivariogram model, (4) discretization of the domain used in Monte Carlo integration, and (5) the random number generator. Considering all sources of variation, the variance estimate we calculated ranged from -2 to 36 in our example, for an overall mean equal to 95 acorns per quadrat. Geostatistical variance mainly reflects the locations and size of the sampled quadrats, and the spatial autocorrelation function.

International audience; Background: Data collected to inform time variations in natural population size are tainted by sampling error. Ignoring sampling error in population dynamics models induces bias in parameter estimators, e.g., density-dependence. In particular, when sampling errors are independent among populations, the classical estimator of the synchrony strength (zero-lag correlation) is biased downward. However, this bias is rarely taken into account in synchrony studies although it may lead to overemphasizing the role of intrinsic factors (e.g., dispersal) with respect to extrinsic factors (the Moran effect) in generating population synchrony as well as to underestimating the extinction risk of a metapopulation.Methodology/Principal findings: The aim of this paper was first to illustrate the extent of the bias that can be encountered in empirical studies when sampling error is neglected. Second, we presented a space-state modelling approach that explicitly accounts for sampling error when quantifying population synchrony. Third, we exemplify our approach with datasets for which sampling variance (i) has been previously estimated, and (ii) has to be jointly estimated with population synchrony. Finally, we compared our results to those of a standard approach neglecting sampling variance. We showed that ignoring sampling variance can mask a synchrony pattern whatever its true value and that the common practice of averaging few replicates of population size estimates poorly performed at decreasing the bias of the classical estimator of the synchrony strength.Conclusion/Significance: The state-space model used in this study provides a flexible way of accurately quantifying the strength of synchrony patterns from most population size data encountered in field studies, including over-dispersed count data. We provided a user-friendly R-program and a tutorial example to encourage further studies aiming at quantifying the strength of population synchrony to account for uncertainty in population size estimates.

20 pages Version Preliminaire en Francais : hal-00406357 (2009); International audience; This paper presents an algorithm for computing algebraically relative resolvents which enhances an existing algorithm by avoiding the accumulation of superfluous powers in the intermediate computations. The superfluous power generated at each step is predetermined over a certain quotient ring. As a byproduct, an efficient algorithm for extracting an n-th root of an univariate polynomial is obtained.

International audience; In this paper we significantly extend the limits of existing algebraic algorithms for computing Lagrange resolvents. Such algorithms are fundamental in effective Galois theory. However, they usually are of high complexity. For the case of absolute resolvents, N. Rennert has shown the value of a multimodular approach to reduce the complexity in space and time. We improve and generalize his work to the computation of any resolvent. In addition, we introduce a decomposition formula which allows us to split modular resolvents into resolvents of smaller degrees, thus both speeding computations and making it possible to efficiently parallelize them.

International audience; The approach of solving geometric constraints involving inequalities proposed by Hong and others uses triangular decomposition, solution formulas, and quantifier elimination. We show that for generating dynamic diagrams automatically the performance of this approach can be enhanced, in terms of stability of numeric computation and quality of generated diagrams, when the used solution formulas of cubic and quartic equations are replaced by newly introduced real solution formulas with inequality constraints. Several examples are presented to illustrate the enhanced approach and to demonstrate the advantages and effectiveness of the new solution formulas. An implementation of the enhanced approach in Java with interface to Epsilon and QEPCAD for automated generation of dynamic diagrams is outlined and some experimental data are provided.

In the Camargue (southern France), drastic changes in wetlands have occurred (notably extension of agriculture and salt extraction) since the 1960s, which affect the resources available to migratory waterbirds. Winter diets of Mallard (Anas platyrhynchos) and Teal (A. crecca) in 2006-2008 were assessed by analyses of gullet contents. Using PCA-based methods, duck diets were described and the main feeding habitats used by each duck species were then determined with a typology analysis. The same four food items were most important (in terms of occurrence and average dry weight) in the diet of Mallard and Teal: Oryza sativa (rice), Echinochloa sp., Scirpus maritimus and Potamogeton pusillus seeds. However, Teal diet was more diversified, with eleven feeding habitat types, compared to only five in Mallard. Both species were found to be dependent on ricefields and ricefield-like habitats. Compared to previous studies in the same area between 1964 and 1981, permanent freshwater habitats now appear to be used more intensively by Mallard and Teal, while temporary marshes are used to a lesser extent. Since the 1960s, temporary marshes have been partially replaced by permanent freshwater in order to attract more ducks, mostly for hunting. The flexibility of duck diet in response to changing food availability may explain why duck populations have not decreased in the Camargue or in Europe despite changes in land use.

Abundance indices are widely used to study changes in population size in wildlife management. However, a truly appropriate measure of precision is often lacking in such studies. Statistically, the two crucial issues regarding the use of an abundance index are sampling and observability, which lead one to consider two kinds of errors, namely sampling and observation errors. The purpose of this methodological paper is to relate the number of counts to the precision of an abundance index by introducing the Hansen–Hurwitz–Bershad model which takes into account both sampling and observation errors. We illustrate this statistical approach in the case of a European rabbit (Oryctolagus cuniculus) abundance index based on spotlight counts, for two fixed spatial sampling units located in different ecological contexts. We show (i) that the usual sampling variance estimator is a downward-biased estimator of the total variance of the abundance index, (ii) that the bias of the usual variance estimator does not decrease when increasing the sampling size, (iii) that correlated observation errors may have a dramatic impact on the total variance, especially when the sampling size increases. The acknowledgement that the (pure) sampling variance underestimates the total variance because of observation errors is a statistical result that is neither widely known nor appreciated by most wildlife ecologists. The magnitude of this underestimation may be important and, therefore, observation errors cannot be always considered as a priori negligible in assessing the precision of a count-based abundance index.

Since the recent spread of highly pathogenic (HP) H5N1 subtypes, avian influenza virus (AIV) dispersal has become an increasing focus of research. As for any other bird-borne pathogen, dispersal of these viruses is related to local and migratory movements of their hosts. In this study, we investigated potential AIV spread by Common Teal (Anas crecca) from the Camargue area, in the South of France, across Europe. Based on bird-ring recoveries, local duck population sizes and prevalence of infection with these viruses, we built an individual-based spatially explicit model describing bird movements, both locally (between wintering areas) and at the flyway scale. We investigated the effects of viral excretion duration and inactivation rate in water by simulating AIV spread with varying values for these two parameters. The results indicate that an efficient AIV dispersal in space is possible only for excretion durations longer than 7 days. Virus inactivation rate in the environment appears as a key parameter in the model because it allows local persistence of AIV over several months, the interval between two migratory periods. Virus persistence in water thus represents an important component of contamination risk as ducks migrate along their flyway. Based on the present modelling exercise, we also argue that HP H5N1 AIV is unlikely to be efficiently spread by Common Teal dispersal only.

Use of food resources by herbivores depends on intrinsic constraints, essentially body size and morpho-physiological characteristics, which determine the range of foods they tolerate and environmental constraints, such as seasonality and interspecific interactions, which determine the availability of resources. We analysed a collection of rumen contents samples from sympatric populations of red deer Cervus elaphus and roe deer Capreolus capreolus and tested several theoretical predictions relating to the impact of intrinsic and environmental constraints on diet composition, diversity and similarity. Red deer consumed more slowly digestible, fibrous forage than roe deer and had a more diverse diet throughout the year, which supports predictions deriving from specific body size and morpho-physiological characteristics. In conformity with the optimal foraging theory, both species consumed more slowly digestible forage in times of low food availability (i.e. during winter) than during the rest of the year. An increase in diet similarity in winter, along with predictions from the theory on competitive interaction processes, led us to assume that food resources were not limiting and that exploitative competition between red and roe deer was unlikely in our study area. We underline the importance of studies of the use of food resources by sympatric herbivores in answering applied ecological questions at the local scale, and we suggest that the Euclidean geometrical approach we used is particularly well suited for the analysis of resource matrices, a common end-product of long-term field data gathering on the feeding habits of animals.

Research and industrial interests for electrochromic “smart” windows have considerably grown those recent years due to their potential use as an energy-saving material for buildings. Their CO2 emission can be significantly reduced by decreasing their energy consumption up to 30%. The crucial issues of such devices are the coloration efficiency as well as the reversibility upon coloration and bleaching of the electrochromic layers. The combination of WO3 and Li-doped NiO surface films (separately as working and counter electrodes) affords a good coloration contrast between colored and bleached states of the device, since tungsten oxide colors upon electrons and ions insertion while nickel oxide gets colored after their extraction. In order to improve the performances of those electrochromic films, we have studied the effects of a surfactant-assisted deposition process of both individual layers through Ultrasonic Spray Pyrolysis (USP). This technique indeed constitutes a low-cost alternative to industrial vacuum processes for manufacturing high quality thin films. The crystallization and morphology of those films, as well as their optical, electrochromic and electrochemical properties have been investigated. The presence of surfactant in the precursor solution has led to improved wetting of liquid droplets sprayed on the heated FTO/glass substrate. The resulting films testify for a better homogeneity and a higher active surface, which also allows for enhanced reversibility and coloration contrast.

Colloque avec actes et comité de lecture. internationale.; International audience; This paper presents methods for zero and ideal decomposition of partial differential polynomial systems and the application of these methods and their implementations to deal with problems from the local theory of surfaces. We show how to prove known geometric theorems and to derive unknown relations automatically. In particular, an algebraic relation between the first and the second fundamental coefficients in a very compact form has been derived, which is more general and has smaller degree than a relation discovered previously by Z.~Li. Moreover, we provide symmetric expressions for Li's relation and clarify his statement. Some examples of theorem proving and computational difficulties encountered in our experiments are also discussed.

International audience; Four methods for solving polynomial systems by means of triangular sets are presented and implemented in a unified way. These methods are those of Wu (1987), Lazard (1991), Kalkbrener (1991) and Wang (1993b). They are compared on various examples with the emphasis on efficiency, conciseness and legibility of the output.

International audience; Different notions of triangular sets are presented. The relationship between these notions are studied. The main result is that four different existing notions of good triangular sets are equivalent.

Special Issue on Algorithmic Galois Theory. Article dans revue scientifique avec comité de lecture.; International audience; In this paper we show that some ideals which occur in Galois theory are generated by triangular sets of polynomials. This geometric property seems important for the development of symbolic methods in Galois theory. It may and should be exploited in order to obtain more efficient algorithms, and it enables us to present a new algebraic method for computing relative resolvents which works with any polynomial invariant.

not Available.

Thirty-eight patients (32 men and six women, mean age 48.1 years) were operated upon for acute dissection involving the ascending aorta. The surgical procedure included multiple peripheral arterial cannulations, resection of the initial intimal tear if found (35 cases), and obliteration of the false channel by double cuffing with Teflon of the two layers of the dissecting process proximally and distally. When present (29 cases), aortic regurgitation was usually (21 cases) managed by conservative remodeling of the aortic anulus; 34 prosthetic replacements of the ascending aorta and four replacements of the arch were achieved. The operative mortality was 7.9% (3/38) and the overall hospital mortality was 23.7% (9/38). Nonfatal complications occurred in 11 patients (29%). There were three late deaths (10.3%). Mean follow-up was 3.4 years (2 months to 8 years, 8 months). Twenty-three (88.5%) of the 26 patients were asymptomatic. Contrast tomodensitometry was performed in 14 patients: In type II (two patients), the aorta was normal; in type I (12 patients), residual abnormalities were noted: patency of the false channel (10 cases), aneurysmal dilatation (seven cases), and reduction of the true lumen by the false channel (four cases). These results emphasize the need for scrupulous long-term follow-up in surgically treated aortic dissections.

Article dans revue scientifique avec comité de lecture.; International audience; Finding one point on each semi-algebraically connected component of a real algebraic variety, or at least deciding if such a variety is empty or not, is a fundamental problem of computational real algebraic geometry. Although numerous studies have been done on the subject, only a few number of efficient implementations exist. In this paper, we propose a new efficient and practical algorithm for computing such points. By studying the critical points of the restriction to the variety of the distance function to one well chosen point, we show how to provide a set of zero-dimensional systems whose zeros contain at least one point on each semi-algebraically connected component of the studied variety, without any assumption neither on the variety (smoothness or compactness for example) nor on the system of equations which define it. From the output of our algorithm, one can then apply, for each computed zero-dimensional system, any symbolic or numerical algorithm for counting or approximating the real solutions. We report some experiments using a set of pure exact methods. The practical efficiency of our method is due to the fact that we do not apply any infinitesimal deformations, unlike the existing methods based on a similar strategy.

L’autocorrélation spatiale peut être définie comme la ressemblance des valeurs prises par une variable, exprimée en fonction de leur localisation géographique. L’analyse de l’autocorrélation permet de quantifier la régularité spatiale d’un phénomène (une forme de complexité spatiale) et de déterminer la portée de la dépendance spatiale afin, notamment, de définir un dispositif d’échantillonnage garantissant l’indépendance des données, autorisant ainsi l’utilisation des tests statistiques usuels. L’article aborde trois points : i) définition opératoire de l’autocorrélation pour des variables quantitatives ou qualitatives ; ii) utilisation des tests de randomisation pour tester l’hypothèse nulle d’absence d’autocorrélation ; iii) illustration des deux points précédents par des exemples. Trois jeux de données simulés sont présentés pour illustrer les différences entre les statistiques : le premier ne présente aucune structure spatiale, le deuxième est caractérisé par une structure spatiale périodique, le troisième est un gradient linéaire. Deux jeux de données géomorphologiques sont également analysés : i) une série de segments fluviaux élémentaires, distribués longitudinalement, et sur lesquels ont été mesurées la largeur et l’incision d’un lit fluvial ; ii) la cartographie des formes d’érosion d’un bassin versant, traitée comme une image matricielle. Dans le premier cas, la structure emboîtée des tronçons géomorphologiquement homogènes est mise en lumière, et ce à différentes échelles spatiales. Le second exemple montre qu’une analyse omnidirectionnelle peut conduire à sous-estimer la portée de l’autocorrélation lorsque le phénomène étudié présente une orientation géographique privilégiée. Dans ce cas, il peut s’avérer impossible de définir un échantillon de données spatialement indépendantes, répondant aux exigences des tests statistiques classiques., Spatial autocorrelation can be defined as the similarity of values of a given variable, in relation with their spatial location. Autocorrelation functions are used to quantify the spatial regularity of a phenomenon (a form of spatial complexity) and to assess the lag of spatial dependence in order to design a sampling procedure for which the data are independent, which permits the use of traditional statistical tests. Three points have been developed: i) Definition of autocorrelation functions used for categorical and quantitative variables: Geary’s c, Moran’s I, semivariogram, non ergodic covariance and correlation, J statistics; ii) Definition of the randomization tests used to test the null hypothesis of no autocorrelation: iii) examples illustrating the two previous objectives. Three sets of simulated data were used to compare different autocorrelation functions (Geary’s c, non ergodic covariance and correlation): the first one has no spatial structure, the second one has a periodic spatial structure whereas the third one is characterized by a linear gradient. Spatial autocorrelation has also been assessed on measured geomorphological data. Two sets were studied: i) a set of elementary channel segments of 500 m in length on which mean active channel width and degradation have been measured, ii) a set of pixels of an image of two basins representing the different types of hillslope erosion forms. In the first case, the nested structure of homogeneous geomorphological reaches is highlighted at different spatial scales. The second example, which illustrates autocorrelation assessment on a categorical variable, shows that omnidirectional analysis can underestimate the autocorrelation lag when the studied phenomenon is characterized by a preferential geographical orientation. In this particular case, it may not be not possible to define a sample of data which are spatially independent and on which it is possible to use classical statistical tests., El método de correlación espacial puede definirse como la semejanza de valores que adopta una variable expresada en función de su localización geográfica. Dicho método permite cuantificar la regularidad espacial de un fenómeno (un tipo de complejidad espacial) y determinar el alcance de la dependencia espacial a fin de definir una estrategia de muestreo que garantice la independencia de los datos y que permita la utilización de pruebas estadísticas usuales. El presente trabajo aborda los siguientes puntos: i) Explicación del funcionamiento de la autocorrelación aplicada a variables discretas o continuas; ii) utilización de pruebas aleatorias para aceptar o rechazar la hipótesis nula de ausencia de autocorrelación; iii) ilustración de los dos puntos anteriores por medio de ejemplos. Tres grupos de datos ficticios son presentados para ejemplificar las diferencias entre las estadísticas: el primero carece de estructura espacial, el segundo se caracteriza por una estructura espacial periódica y el tercero representa un gradiente lineal. Así mismo se analizan dos grupos de datos geomorfológicos: i) una serie de segmentos fluviales elementales distribuidos longitudinalmente y de los cuales se ha medido la amplitud e incisión del lecho fluvial; ii) la cartografía de los tipos de erosión de la pendiente de la cuenca tratado como imagen matricial. En el primero de los casos, la estructura constituida de elementos geomorfológicos homogéneos y puesta en evidencia a distintas escalas espaciales. El segundo ejemplo muestra que un análisis omnidireccional pude conducir a subestimar el alcance de la autocorrelación cuando el fenómeno estudiado muestra una orientación geográfica privilegiada. En este caso, puede ser imposible definir una muestra de datos espacialmente independientes, respondiendo así a las exigencias estadísticas clásicas.