A novel quaternion-valued least-mean kurtosis (QLMK) adaptive filtering algorithm is proposed for three-and fourdimensional processes by using the recent generalised Hamilton-real (GHR) calculus. The proposed QLMK algorithm based GHR calculus minimises the negated kurtosis of the error signal as a cost function in the quaternion domain, thus provides an elegant way to solve a trade-off problem between the convergence rate and steady-state error. Moreover, the proposed QLMK algorithm has naturally a robust behaviour for a wide range of noise signals due to its kurtosis-based cost function. Furthermore, the steady-state performance of the proposed QLMK algorithm is analysed to obtain convergence and misadjustment conditions. The comprehensive simulation results on benchmark and real-world problems show that the use of this cost function defined by the quaternion statistics in the proposed QLMK algorithm allows us to process quaternion-valued signals and thus, significantly enhances the performance of the adaptive filter in terms of both the steady-state error and the convergence rate, as compared with the quaternion-valued least-mean-square algorithm based on the recent GHR calculus.
This tutorial is based on modification of the professor nomination lecture presented two years ago in front of the Scientific Council of the Czech Technical University in Prague [16].It is devoted to the techniques for the models developing suitable for processes forecasting in complex systems. Because of the high sensitivity of the processes to the initial conditions and, consequently, due to our limited possibilities to forecast the processes for the long-term horizon, the attention is focused on the techniques leading to practical applications of the short term prediction models. The aim of this tutorial paper is to bring attention to possible difficulties which designers of the predicting models and their users meet and which have to be solved during the prediction model developing, validation, testing, and applications. The presented overview is not complete, it only reflects the author's experience with developing of the prediction models for practical tasks solving in banking, meteorology, air pollution and energy sector. The paper is completed by an example of the global solar radiation prediction which forms an important input for the electrical energy production forecast from renewable sources. The global solar radiation forecasting is based on numerical weather prediction models. The time-lagged ensemble technique for uncertainty quantification is demonstrated on a simple example.
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