Comparaison par simulation de Monte-Carlo des propriétés de deux estimateurs du paramètre d'échelle de la loi exponentielle : méthode du maximum de vraisemblance (MV) et méthode des moindres carrés (MC)

La loi exponentielle est très répandue en hydrologie : elle est faiblement paramétrée, de mise en œuvre aisée. Deux méthodes sont fréquemment utilisées pour estimer son paramètre : la méthode du maximum de vraisemblance et la méthode des moments, qui fournissent la même estimation. A côté de ces deux méthodes, il y a celle des moindres carrés qui est très rarement utilisée pour cette loi. Dans cet article, nous comparons le comportement asymptotique de l'estimateur de la méthode des moindres carrés avec celui de la méthode du maximum de vraisemblance en partant d'une loi exponentielle à un seul paramètre a connu, puis en généralisant les résultats obtenus à partir de la dérivation des expressions analytiques. L'échantillon historique disponible en pratique étant unique, et de longueur généralement courte par rapport à l'information que l'on désire en tirer, l'étude des propriétés statistiques des estimateurs ne pourra se faire qu'à partir d'échantillons de variables aléatoires représentant des réalisations virtuelles du phénomène hydrologique concerné obtenus par simulations de Monte Carlo. L'étude par simulation de Monte Carlo montre que pour de faibles échantillons, l'espérance mathématique des deux estimateurs tend vers le paramètre réel, et que la variance de l'estimateur des moindres carrés est supérieure à celle de l'estimateur du maximum de vraisemblance.Exponential distributions are frequently applied in hydrology, for example: frequency analysis of the duration and severity of water flow conditions MATHIEU L. and al. (1991); regional frequency of storm intensities ARNAUD P., LAVABRE J. (1999); partial duration of hydrological droughts KJELDSEN T. R. and al. (1999) ; and daily rainfall modelling CHAPMAN T.G. (1997) ; KABAILI Z. (1983). This method has only one parameter, and it is easy to use. Its parameter is mainly estimated using the maximum likelihood estimator (MLE) or the method of moments estimator (MOME), but the least square estimator (LSE) can also be applied. For the one-parameter exponential distribution, MOME and MLE give the same expression for the parameter:         Ex         Σ xk         k = 1X^0 = ________         ExUsing LSE requires a Ex size sample of exponential variables and involves the following steps:1. Sorting the Ex variables in the sample in ascending order2. Associating to each quantile xk whose rank is k in the sorted sample an empirical frequency F^k = k - 0.5 / Ex3. Plotting Ex against ln(1-F^k) and using LSE to calculate x^0 by :x^0 = - (ExΣk=1Xk ln(1-F^k)) / (ExΣk=1 [ln(1-F^k)]2In this paper we compare the asymptotic behaviour of the statistical properties (mean and variance) of the MLE and the LSE. These comparisons must be made by using a great number of sample parameter estimations. In practice, only one historical sample of variables issued from a known exponential distribution was available, from which only one parameter can be calculated. To overcome this difficulty, samples of variables whose original theoretical exponential distribution is known are generated using the Monte Carlo numerical method. Samples of estimated parameters (using the MLE or the LSE) are then created from the above samples of random variables, and the statistical properties of the two estimators are then calculated. These different successive steps are summarised below:1. Generate sample of finite size Ex for known exponential variables2. Use this sample to estimate one parameter using MLE or LSE3. Do steps 1 to 2 Np times to collect a Np size sample of parameter estimations4. Use this sample to calculate statistical properties (mean and variance) for the two estimators.According to this approach, sizes Ex and Np should influence the statistical properties of the two estimators. We have verified this with a one-parameter exponential law, with a known theoretical parameter X0=1. Samples of estimated parameters of size Np have been generated from virtual samples of size Ex issued from a population following the above statistical distribution. During this operation, one of these sizes, Ex or Np, has been held constant, while the other, Np or Ex, changed with a constant step. Statistical properties of the estimators have then been calculated for each of the two cases.Let Var Ex (x^0(Np)) and EEx (x^0(Np)) be statistical properties (variance and mean) of the two estimators for fixed values of Ex, and VarNp(x^0(Ex)) and ENp(x^0(Ex)), the same statistical properties for fixed values of Np.Plotting VarEx(x^0(Np)) for Ex=10 and Ex=100 shows that for large values of Np (1000 to 5000) variance tends towards a constant value, close to 0.1 for Ex=10 (Fig. 1a) and to 0.01 for Ex=100 (Fig. 1b), both equal to 1/Ex, when the MLE is used. When parameters are estimated with the LSE, variance tends towards a constant value, greater than the preceding ones (Fig. 1a and Fig. 1b). Plotting VarNp(x^0(Ex)) when Np=1000 is constant, the variance decreases as Ex grows whatever the estimator, but for a given value of Ex, the variance is always greater when the LSE is used (Fig. 3). These two calculations show that asymptotic variance depends only on size Ex of samples of known exponential distribution.Plotting EEx(x^0(Np)) when Ex=10, for important values of Np, the mean is close to the true parameter for the MLE, and greater than this true parameter for the LSE (Fig. 2a). When Ex=100, the mean is close to the true parameter for the two estimators (Fig 2b). From these calculations we notice that the asymptotic mean depends only on the size of Ex for known exponential variables and on the used estimator. The MLE seems to present no bias for the mean, while the LSE presents a bias for small values of Ex, but this bias disappears as Ex increases. To quantify this degree of dependence, we have plotted ENp(x^0(Ex)) for Np=1000 (Fig. 4). For the two estimators, the mean presents an initial bias, when Ex is low and the bias disappears when Ex becomes higher. The initial bias is more important with the LSE.In summary, the asymptotic statistical properties of the two estimators (mean and variance) depend only on the size of Ex for known exponential distribution variables.Empirical plots are unstable for low sample sizes, are sensitive to sampling, and are very difficult to explain. Analytical expressions for the asymptotic statistical properties of the two estimators are needed for realistic comparison. According to formulae (1) and (2), statistical properties depend on E∞(Xk) and E∞(X2k) respectively and the asymptotic mean of Xk and X2k.E∞(Xk) and E∞(X2k) have been derived using the density of probability of Xk through statistics of rank. Asymptotic statistical properties of the two estimators have then been evaluated using the expressions of E∞(Xk) and E∞(X2k).We let E∞[x^0(Ex)] be the asymptotic mean of estimator, Var∞[x^0(Ex)] be the asymptotic variance, and x0 be the theoretical parameter of exponential distribution. By plotting E∞[x^0(Ex)] for X0=1, we note that this expression has a constant value, equal to unity when the MLE was applied, and that it decreases quickly to unity when the LSE was applied (Fig. 6). By plotting Var∞[x^0(Ex)] for x0=1, we also note that the theoretical asymptotic variance diminishes as Ex grows, but is greater when the LSE was applied (Fig. 5). By comparing with empirical plots when x0=1, we establish the same trends.Theoretical derivations of asymptotic statistical proprieties have confirmed empirical experiences: · The MLE for a one-parameter exponential presents no bias · The LSE for a one-parameter exponential is a consistent estimator of the simple exponential parameter

Analysis of dendrometric characteristics of Acacia senegal (L.) Willd. (Fabaceae) in the semi-arid Sahelian area: A case study of silvopastoral zone in Senegal

The study gives an analysis of the dendrometric parameters of a middle Sahelian species (Acacia senegal) in semi-arid environment to get better knowledge of its behavior. The research lays the principle that the species behaves differently according to the ecogeographical stations. So, its characteristics change from one environment to another. That is why it is important to do a specific study for every big zone. The study was done in Silvopastoral zone, more particularly in the area where A. senegal is popular in the country. The study aims to deepen the knowledge of A. senegal dendrometric parameters. The following parameters were measured: diameter of the base, breastheight diameter (1.3 m), breast- height circumference, total height of the tree, length of the trunk and width of the houppier. The sample consists of a population of 76 feet locust trees in Senegal. The study confirmed the hypothesis that the characters of the species vary according to the zone where it is found.Keys words: Acacia senegal, dendrometry, Sahel, Senegal

Élaboration de modèles allométriques d’Acacia Sénéga

La présente étude s’est appesantie sur l’élaboration de modèles allométriques d’une espèce soudanosahélienne (Acacia Sénégal ) dans le but d’asseoir une base de calcul des stocks de carbone. La recherche s’est effectuée en zone sylvopastorale, au nord du Sénégal. Au préalable, l’étude a tenté d’approfondir les connaissances sur la dendrométrie de Acacia Sénégal. Sur le terrain, les paramètres suivants ont été mesurés et fait l’objet d’analyses et interprétations : diamètre à la base, hauteur totale de l’arbre, longueur du tronc, diamètre à hauteur de poitrine, circonférence à hauteur de poitrine et largeur du houppier. La pesée des échantillons de chaque partie (tronc, branche, rameau) de l’échantillon d’arbres composés de 38 pieds d’Acacia Sénégal et leur séchage à l’étuve ont permis de connaître la biomasse sèche de chaque sujet. Ensuite, des modèles allométriques spécifiques (entre la biomasse sèche et les paramètres dendrométriques) ont été générés et les coefficients de corrélation déterminés. Les différentes analyses ont permis d’affirmer que le diamètre à hauteur de poitrine (DHP) est la variable qui offre la meilleure corrélation avec la biomasse aérienne. Parmi les modèles, et par ordre décroissant, nous avons obtenu la relation polynomiale, logarithmique et de puissance. Les modèles retenus ont comme entrée le DBH. De ce fait, le modèle polynomiale (y = 0,032DBH3 - 1,016DBH2 + 10,87DBH + 7,429) présente la meilleure corrélation (R² = 0,963), suivi du modèle logarithmique (y = 13,61ln(DBH) + 17,89) avec une corrélation R² = 0,933 et du modèle de puissance (y = 22,23DBH0,357) pour une corrélation R² = 0,909. L’étude a confirmé l’hypothèse selon laquelle, au sein d’une même espèce, l’élaboration de modèle spécifique reflète plus la réalité et permet de faire une quantification des stocks de carbone plus proche de la réalité que le modèle générique.Mots-clés : modèle allométrique, grande muraille verte, Acacia Sénégal, séquestration de carbone

Variabilité climatique et son incidence sur les ressources en eaux de surface : cas des stations de Bocanda et de Dimbokro, Centre-Est de la Côte d'Ivoire en Afrique de l’Ouest

L’étude du changement climatique et de la variabilité climatique est d’une très grande importance pour la gestion et la planification des ressources en eau. Cet article examine l’impact du changement climatique sur l’évolution du N’zi (principal cours d’eau de la zone). Les séries chronologiques de pluies et de débits ont été utilisées à cet effet. L’application des méthodes de Nicholson, de Maillet ainsi que des tests statistiques de détection de ruptures (test de Buishand, test de Hubert, test de Lee Heghinian et test de Pettitt), et de tendances (test de Mann-Kendall, test de Sen’s) ont permis d’une part de mettre en évidence une variabilité climatique caractérisée par une alternance de périodes humides, normales et sèches, et d’autre part de détecter des ruptures en 1968 pour Dimbokro et en 1975 pour M’bahiakro et Bocanda Ces ruptures s’accompagnent d’une tendance de la baisse de la pluviométrie (10 à 18 %) et des volumes mobilisés par les aquifères (30 à 51 %.). Ce travail constitue une base indispensable pour une meilleure connaissance des ressources en eau des bassins étudiés.Mots-clés: variabilité climatique, séries chronologiques, tendances et ruptures, tests statistiques, Côte d'Ivoire.Climate variability and its impact on water resources of surface: case of stations of Bocanda and Dimbokro (east-central Côte d'Ivoire in West Africa) Climate change and climate variability studies are of great importance for water resources and management and planning in this paper, we put our attention on the impact of these two phenomena on the water resources availability of the N’zi river basin (Main River of the area of study). Rainfall and runoff time series have been used for. Nicholson and Maillet methods and statistical tests for homogeneity (Buishand’s, Hubert’s, Lee and Heghinian’s and Pettitt’s tests) and for trend (Mann Kendall’ test, and Sen’s slope estimator). Nicholson’s and Maillet’s method allow to put in evidence the climate variability with a random succession of periods of excess and deficits. Statistical tests for homogeneity have detected downwards shifts (10% to 51%) in the time series of rainfall beginning in 1968 for Dimbokro and in 1975 for M’bahiakro and Bocanda and in the groundwater (30% to 51%). The work we present in this paper is a preliminary for efficient water management of the N’zi river basin.Keywords: climate variability, times series, statistical tests, homogeneity and trend test, Côte d'Ivoir

Monitoring the introduction of pneumococcal conjugate vaccines into West Africa: design and implementation of a population-based surveillance system.

Routine use of pneumococcal conjugate vaccines (PCVs) in developing countries is expected to lead to a significant reduction in childhood deaths. However, PCVs have been associated with replacement disease with non-vaccine serotypes. We established a population-based surveillance system to document the direct and indirect impact of PCVs on the incidence of invasive pneumococcal disease (IPD) and radiological pneumonia in those aged 2 months and older in The Gambia, and to monitor changes in serotype-specific IPD. Here we describe how this surveillance system was set up and is being operated as a partnership between the Medical Research Council Unit and the Gambian Government. This surveillance system is expected to provide crucial information for immunisation policy and serves as a potential model for those introducing routine PCV vaccination in diverse settings

An auto-luminescent fluorescent BCG whole blood assay to enable evaluation of paediatric mycobacterial responses using minimal blood volumes

Introduction: Understanding protective human immunity against mycobacteria is critical to developing and evaluating new vaccines against tuberculosis. Children are the most susceptible population to infection, disease, and death from tuberculosis, but also have the strongest evidence of BCG-inducible protection. Limited amounts of blood can be obtained for research purposes in paediatrics and therefore there is a need for high-yield, low-volume, human immunology assays. Methods: We transformed BCG Danish with plasmids encoding luciferase full operon derived from Photorhabdus luminescens together with Green Fluorescent Protein and antibiotic selection markers. We characterised the luminescent and fluorescent properties of this recombinant BCG strain (BCG-GFP-LuxFO) using a luminometer and flow cytometry and developed a paediatric whole blood in vitro infection model. Results: Luminescence of BCG-GFP-LuxFO correlated with optical density (Spearman Rank Correlation coefficient r = 0.985, p < 0.0001) and colony forming units (CFUs) in liquid culture medium (r = 0.971, p < 0.0001). Fluorescence of BCG-GFP-LuxFO in paediatric whole blood was confirmed by flow cytometry in granulocytes and monocytes 1 h following infection. Luminescence of BCG-GFP-LuxFO in whole blood corresponded with CFUs (r = 0.7123, p < 0.0001). Conclusion: The BCG-GFP-LuxFO assay requires 225 μL whole blood per sample, from which serial luminescence measurements can be obtained, together with biochemical analysis of supernatants and cellular assay applications using its fluorescent properties. This offers the opportunity to study human-mycobacterial interactions using multiple experimental modalities with only minimal blood volumes. It is therefore a valuable method for investigating paediatric immunity to tuberculosis

Monitoring anti-tuberculosis treatment response using analysis of whole blood Mycobacterium tuberculosis specific T cell activation and functional markers

Background: Blood-based biomarkers have been proposed as an alternative to current sputum-based treatment monitoring methods in active tuberculosis (ATB). The aim of this study was to validate previously described phenotypic, activation, and cytokine markers of treatment response in a West African cohort. Methods: Whole blood immune responses to Mycobacterium tuberculosis ESAT-6/CFP-10 (EC) and purified protein derivative (PPD) were measured in twenty adults at baseline and after 2 months of standard TB treatment. Patients were classified as fast or slow responders based on a negative or positive sputum culture result at 2 months, respectively. Cellular expression of activation markers (CD38, HLA-DR), memory markers (CD27), and functional intracellular cytokine and proliferation (IFN-γ, Ki-67, TNF-α) markers were measured using multi-color flow cytometry. Results: There was a significant increase in the proportion of CD4+CD27+ cells expressing CD38 and HLA-DR following EC stimulation at 2 months compared to baseline (p = 0.0328 and p = 0.0400, respectively). Following PPD stimulation, slow treatment responders had a significantly higher proportion of CD8+CD27–IFN-γ+ (p = 0.0105) and CD4+CD27+HLA-DR+CD38+ (p = 0.0077) T cells than fast responders at baseline. Receiver operating curve analysis of these subsets resulted in 80% sensitivity and 70 and 100% specificity, respectively (AUC of 0.82, p = 0.0156 and 0.84, p = 0.0102). Conclusion: Our pilot data show reductions in expression of T cell activation markers were seen with treatment, but this was not associated with fast or slow sputum conversion at 2 months. However, baseline proportions of activated T cell subsets are potentially predictive of the subsequent speed of response to treatment