In many practical problems, the underlying structure of an estimated covariance matrix is usually blurred due to random noise, particularly when the dimension of the matrix is high. Hence, it is necessary to filter the random noise or regularize the available covariance matrix in certain senses, so that the covariance structure becomes clear. In this paper, we propose a new method for regularizing the covariance structure of a given covariance matrix. By choosing an optimal structure from an available class of covariance structures, the regularization is made in terms of minimizing the discrepancy, defined by Frobenius-norm, between the given covariance matrix and the class of covariance structures. A range of potential candidate structures, including the order-1 moving average structure, compound symmetry structure, order-1 autoregressive structure, order-1 autoregressive moving average structure, are considered. Simulation studies show that the proposed new approach is reliable in regularization of covariance structures. The proposed approach is also applied to real data analysis in signal processing, showing the usefulness of the proposed approach in practice.

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The main aim of the work is to assess physical parameters of forest woodchips and their impact on the prices achieved by the supplier in transactions with a power plant. During fragmentation of logging residue, high content of green matter and contaminants negatively impacts the quality parameters that serve as basis for settlements. The analysis concerns data on the main parameters -water content, fuel value, sulphur and ash content -from 252 days of deliveries of forest chips to a power plant. The deliveries were realised from forested areas on an average about 340 km from the plant. Average water content and the resultant fuel value of forest chips was within 27-47% and 8.7-12.9 GJ×Mg −1 (appropriately), respectively. They depend on the month in which they are delivered to the power plant. The threshold values for the above-mentioned parameters are set by the plant at a real level and the suppliers have no problems with meeting them. The parameter that is most frequently exceeded is ash content (11.5% of cases). The settlement system does not differentiate on the basis of the transport distance but gives possibility to lower the settlement price when the quality parameters are not met but provides no reward for deliveries with parameters better than the average ones. On the basis of results obtained, it was calculated that average annual settlement price is lower than the contract price by about 0.20 PLN×GJ −1 , which in case of the analysed company may translate into an average daily loss of about 700 PLN.

Based on the extrapolation theory and a sixth order compact difference scheme, new extrapolation interpolation operator and extrapolation cascadic multigrid methods for two dimensional Poisson equation are presented. The new extrapolation interpolation operator is used to provide a better initial value on refined grid. The convergence of the new methods are given. Numerical experiments are shown to illustrate that the new methods have higher accuracy and efficiency.