Bayes quadratic unbiased estimator of spatial covariance parameters

This work presents Bayes invariant quadratic unbiased estimator, for short BAIQUE. Bayesian approach is used here to estimate the covariance functions of the regionalized variables which appear in the spatial covariance structure in mixed linear model. Firstly a brief review of spatial process, variance covariance components structure and Bayesian inference is given, since this project deals with these concepts. Then the linear equations model corresponding to BAIQUE in the general case is formulated. That Bayes estimator of variance components with too many unknown parameters is complicated to be solved analytically. Hence, in order to facilitate the handling with this system, BAIQUE of spatial covariance model with two parameters is considered. Bayesian estimation arises as a solution of a linear equations system which requires the linearity of the covariance functions in the parameters. Here the availability of prior information on the parameters is assumed. This information includes apriori distribution functions which enable to find the first and the second moments matrix. The Bayesian estimation suggested here depends only on the second moment of the prior distribution. The estimation appears as a quadratic form y'Ay , where y is the vector of filtered data observations. This quadratic estimator is used to estimate the linear function of unknown variance components. The matrix A of BAIQUE plays an important role. If such a symmetrical matrix exists, then Bayes risk becomes minimal and the unbiasedness conditions are fulfilled. Therefore, the symmetry of this matrix is elaborated in this work. Through dealing with the infinite series of matrices, a representation of the matrix A is obtained which shows the symmetry of A. In this context, the largest singular value of the decomposed matrix of the infinite series is considered to deal with the convergence condition and also it is connected with Gerschgorin Discs and Poincare theorem. Then the BAIQUE model for some experimental designs is computed and compared. The comparison deals with different aspects, such as the influence of the position of the design points in a fixed interval. The designs that are considered are those with their points distributed in the interval [0, 1]. These experimental structures are compared with respect to the Bayes risk and norms of the matrices corresponding to distances, covariance structures and matrices which have to satisfy the convergence condition. Also different types of the regression functions and distance measurements are handled. The influence of scaling on the design points is studied, moreover, the influence of the covariance structure on the best design is investigated and different covariance structures are considered. Finally, BAIQUE is applied for real data. The corresponding outcomes are compared with the results of other methods for the same data. Thereby, the special BAIQUE, which estimates the general variance of the data, achieves a very close result to the classical empirical variance.

Diversity, local uses and availability of medicinal plants and wild yams in the Mahafaly region of south-western Madagascar

Short summary: This study was undertaken to assess the diversity of plant resources utilized by the local population in south-western Madagascar, the social, ecological and biophysical conditions that drive their uses and availability, and possible alternative strategies for their sustainable use in the region. The study region, ‘Mahafaly region’, located in south-western Madagascar, is one of the country’s most economically, educationally and climatically disadvantaged regions. With an arid steppe climate, the agricultural production is limited by low water availability and a low level of soil nutrients and soil organic carbon. The region comprises the recently extended Tsimanampetsotsa National Park, with numerous sacred and communities forests, which are threatened by slash and burn agriculture and overexploitation of forests resources. The present study analyzed the availability of wild yams and medicinal plants, and their importance for the livelihood of the local population in this region. An ethnobotanical survey was conducted recording the diversity, local knowledge and use of wild yams and medicinal plants utilized by the local communities in five villages in the Mahafaly region. 250 households were randomly selected followed by semi-structured interviews on the socio-economic characteristics of the households. Data allowed us to characterize sociocultural and socioeconomic factors that determine the local use of wild yams and medicinal plants, and to identify their role in the livelihoods of local people. Species-environment relationships and the current spatial distribution of the wild yams were investigated and predicted using ordination methods and a niche based habitat modelling approach. Species response curves along edaphic gradients allowed us to understand the species requirements on habitat conditions. We thus investigated various alternative methods to enhance the wild yam regeneration for their local conservation and their sustainable use in the Mahafaly region. Altogether, six species of wild yams and a total of 214 medicinal plants species from 68 families and 163 genera were identified in the study region. Results of the cluster and discriminant analysis indicated a clear pattern on resource, resulted in two groups of household and characterized by differences in livestock numbers, off-farm activities, agricultural land and harvests. A generalized linear model highlighted that economic factors significantly affect the collection intensity of wild yams, while the use of medicinal plants depends to a higher degree on socio-cultural factors. The gradient analysis on the distribution of the wild yam species revealed a clear pattern for species habitats. Species models based on NPMR (Nonparametric Multiplicative Regression analysis) indicated the importance of vegetation structure, human interventions, and soil characteristics to determine wild yam species distribution. The prediction of the current availability of wild yam resources showed that abundant wild yam resources are scarce and face high harvest intensity. Experiments on yams cultivation revealed that germination of seeds was enhanced by using pre-germination treatments before planting, vegetative regeneration performed better with the upper part of the tubers (corms) rather than the sets of tubers. In-situ regeneration was possible for the upper parts of the wild tubers but the success depended significantly on the type of soil. The use of manure (10-20 t ha¹) increased the yield of the D. alata and D. alatipes by 40%. We thus suggest the promotion of other cultivated varieties of D. alata found regions neighbouring as the Mahafaly Plateau.