The paper presents the basic theory of complementary statistics and its application in the area of applied probabilistic modeling. By introduction of the complementarity's principle between x-representation (random time series, random process) and p-representation or k-representation (rate of change/velocity of random time series and processes) the probability theory is completed for the "structural" parameter which carries information about the changes of studied time series or the random process. At the end, the basic application of probabilistic modeling is introduced and the presented principle is illustrated on the set of numerical examples with different probability density functions.
The goal of the paper is to analyze the behavior of quantum systems which are connected in more complex circuits through serial, parallel or feedback ordering of various quantum subsystems. The Quantum State Transform (QST) is introduced to define a Quantum Transfer Function (QTF) that can be used to characterize behavior of complex circuits like e.g. stability better. It is shown that ordering more general quantum systems into feedback can yield to the definition of hierarchical quantum systems that are very close to well-known scale-free networks. Finally, all identified mathematical instruments are used to define quantum information/knowledge circuits as ordering of 2-port quantum subsystems covering both input/output information flow and content.
The paper presents iterated algorithm for parameter estimation of non-linear regression model. The non-linear model is firstly approximated by a polynomial. Afterwards, parameter estimation based on measured data is taken as the initial value for the proposed iterated algorithm. As the estimation method, the well-known Least Square Estimation (LSE), artificial neural networks (ANN) or Bayesian methodology (BM) can be used. With respect to the knowledge of initial parameters the measured data are transformed to meet best the non-linear regression criteria (orthogonal data projection). The original and transformed data are used in the next step of the designed iterated algorithm to receive better parameter estimation. The iteration is repeated until the algorithm converges into a final result. The proposed methodology can be applied on all non-linear models that could be approximated by a polynomial function. The illustrative examples show the convergence of the designed iterated algorithm.
The paper presents new methodology how to find and estimate the main features of time series to achieve the reduction of their components (dimensionality reduction) and so to provide the compression of information contained in it under keeping the selected features invariant. The presented compression algorithm is based on estimation of truncated time series components in such a way that the spectrum functions of both original and truncated time series are sufficiently close together. In the end, the set of examples is shown to demonstrate the algorithm performance and to indicate the applications of the presented methodology.
This tutorial summarizes the new approach to complex system theory that comes basically from physical information analogies. The information components and gates are defined in a similar way as components in electrical or mechanical engineering. Such approach enables the creation of complex networks through their serial, parallel or feedback ordering. Taking into account wave probabilistic functions in analogy with quantum physics, we can enrich the system theory with features such as entanglement. It is shown that such approach can explain emergencies and self-organization properties of complex systems.