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.

Determination of yield and quality parameters of energy crops applying laboratory and field spectroscopy

A real-time analysis of renewable energy sources, such as arable crops, is of great importance with regard to an optimised process management, since aspects of ecology and biodiversity are considered in crop production in order to provide a sustainable energy supply by biomass. This study was undertaken to explore the potential of spectroscopic measurement procedures for the prediction of potassium (K), chloride (Cl), and phosphate (P), of dry matter (DM) yield, metabolisable energy (ME), ash and crude fibre contents (ash, CF), crude lipid (EE), nitrate free extracts (NfE) as well as of crude protein (CP) and nitrogen (N), respectively in pretreated samples and undisturbed crops. Three experiments were conducted, one in a laboratory using near infrared reflectance spectroscopy (NIRS) and two field spectroscopic experiments. Laboratory NIRS measurements were conducted to evaluate to what extent a prediction of quality parameters is possible examining press cakes characterised by a wide heterogeneity of their parent material. 210 samples were analysed subsequent to a mechanical dehydration using a screw press. Press cakes serve as solid fuel for thermal conversion. Field spectroscopic measurements were carried out with regard to further technical development using different field grown crops. A one year lasting experiment over a binary mixture of grass and red clover examined the impact of different degrees of sky cover on prediction accuracies of distinct plant parameters. Furthermore, an artificial light source was used in order to evaluate to what extent such a light source is able to minimise cloud effects on prediction accuracies. A three years lasting experiment with maize was conducted in order to evaluate the potential of off-nadir measurements inside a canopy to predict different quality parameters in total biomass and DM yield using one sensor for a potential on-the-go application. This approach implements a measurement of the plants in 50 cm segments, since a sensor adjusted sideways is not able to record the entire plant height. Calibration results obtained by nadir top-of-canopy reflectance measurements were compared to calibration results obtained by off-nadir measurements. Results of all experiments approve the applicability of spectroscopic measurements for the prediction of distinct biophysical and biochemical parameters in the laboratory and under field conditions, respectively. The estimation of parameters could be conducted to a great extent with high accuracy. An enhanced basis of calibration for the laboratory study and the first field experiment (grass/clover-mixture) yields in improved robustness of calibration models and allows for an extended application of spectroscopic measurement techniques, even under varying conditions. Furthermore, off-nadir measurements inside a canopy yield in higher prediction accuracies, particularly for crops characterised by distinct height increment as observed for maize.

A probabilistic approach to the estimation of regional photovoltaic power generation using meteorological data

In Germany the upscaling algorithm is currently the standard approach for evaluating the PV power produced in a region. This method involves spatially interpolating the normalized power of a set of reference PV plants to estimate the power production by another set of unknown plants. As little information on the performances of this method could be found in the literature, the first goal of this thesis is to conduct an analysis of the uncertainty associated to this method. It was found that this method can lead to large errors when the set of reference plants has different characteristics or weather conditions than the set of unknown plants and when the set of reference plants is small. Based on these preliminary findings, an alternative method is proposed for calculating the aggregate power production of a set of PV plants. A probabilistic approach has been chosen by which a power production is calculated at each PV plant from corresponding weather data. The probabilistic approach consists of evaluating the power for each frequently occurring value of the parameters and estimating the most probable value by averaging these power values weighted by their frequency of occurrence. Most frequent parameter sets (e.g. module azimuth and tilt angle) and their frequency of occurrence have been assessed on the basis of a statistical analysis of parameters of approx. 35 000 PV plants. It has been found that the plant parameters are statistically dependent on the size and location of the PV plants. Accordingly, separate statistical values have been assessed for 14 classes of nominal capacity and 95 regions in Germany (two-digit zip-code areas). The performances of the upscaling and probabilistic approaches have been compared on the basis of 15 min power measurements from 715 PV plants provided by the German distribution system operator LEW Verteilnetz. It was found that the error of the probabilistic method is smaller than that of the upscaling method when the number of reference plants is sufficiently large (>100 reference plants in the case study considered in this chapter). When the number of reference plants is limited (<50 reference plants for the considered case study), it was found that the proposed approach provides a noticeable gain in accuracy with respect to the upscaling method.