Typically extrema filtration techniques are based on non-parametric properties such as magnitude of prominences and the widths at half prominence, which cannot be used with data that possess a dynamic nature. In this ...Typically extrema filtration techniques are based on non-parametric properties such as magnitude of prominences and the widths at half prominence, which cannot be used with data that possess a dynamic nature. In this work, an extrema identification that is totally independent of derivative-based approaches and independent of quantitative attributes is introduced. For three consecutive positive terms arranged in a line, the ratio (R) of the sum of the maximum and minimum to the sum of the three terms is always 2/n, where n is the number of terms and 2/3 ≤ R ≤ 1 when n = 3. R > 2/3 implies that one term is away from the other two terms. Applying suitable modifications for the above stated hypothesis, the method was developed and the method is capable of identifying peaks and valleys in any signal. Furthermore, three techniques were developed for filtering non-dominating, sharp, gradual, low and high extrema. Especially, all the developed methods are non-parametric and suitable for analyzing processes that have dynamic nature such as biogas data. The methods were evaluated using automatically collected biogas data. Results showed that the extrema identification method was capable of identifying local extrema with 0% error. Furthermore, the non-parametric filtering techniques were able to distinguish dominating, flat, sharp, high, and low extrema in the biogas data with high robustness.展开更多
Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are ...Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.展开更多
文摘Typically extrema filtration techniques are based on non-parametric properties such as magnitude of prominences and the widths at half prominence, which cannot be used with data that possess a dynamic nature. In this work, an extrema identification that is totally independent of derivative-based approaches and independent of quantitative attributes is introduced. For three consecutive positive terms arranged in a line, the ratio (R) of the sum of the maximum and minimum to the sum of the three terms is always 2/n, where n is the number of terms and 2/3 ≤ R ≤ 1 when n = 3. R > 2/3 implies that one term is away from the other two terms. Applying suitable modifications for the above stated hypothesis, the method was developed and the method is capable of identifying peaks and valleys in any signal. Furthermore, three techniques were developed for filtering non-dominating, sharp, gradual, low and high extrema. Especially, all the developed methods are non-parametric and suitable for analyzing processes that have dynamic nature such as biogas data. The methods were evaluated using automatically collected biogas data. Results showed that the extrema identification method was capable of identifying local extrema with 0% error. Furthermore, the non-parametric filtering techniques were able to distinguish dominating, flat, sharp, high, and low extrema in the biogas data with high robustness.
文摘Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.
