Eigen-coordinates Method Research Articles

Reduced fractal model for quantitative analysis of averaged micromotions in mesoscale: Characterization of blow-like signals

It has been shown that many micromotions in the mesoscale region are averaged in accordance with their self-similar (geometrical/dynamical) structure. This distinctive feature helps to reduce a wide set of different micromotions describing relaxation/exchange processes to an averaged collective motion, expressed mathematically in a rather general form. This reduction opens new perspectives in description of different blow-like signals (BLS) in many complex systems. The main characteristic of these signals is a finite duration also when the generalized reduced function is used for their quantitative fitting. As an example, we describe quantitatively available signals that are generated by bronchial asthmatic people, songs by queen bees, and car engine valves operating in the idling regime.We develop a special treatment procedure based on the eigen-coordinates (ECs) method that allows to justify the generalized reduced fractal model (RFM) for description of BLS that can propagate in different complex systems. The obtained describing function is based on the self-similar properties of the different considered micromotions. This kind of cooperative model is proposed here for the first time. In spite of the fact that the nature of the dynamic processes that take place in fractal structure on a mesoscale level is not well understood, the parameters of the RFM fitting function can be used for construction of calibration curves, affected by various external/random factors. Then, the calculated set of the fitting parameters of these calibration curves can characterize BLS of different complex systems affected by those factors. Though the method to construct and analyze the calibration curves goes beyond the scope of this paper, this result could benefit future studies that will employ the developed reduced models in diagnosis, prevention, and control of unpredicted and undesired phenomena of some engineering applications that possibly exhibit such BLS.

Application of the linear principle for the strongly-correlated variables: Calculations of differences between spectra

In this paper the authors suggest a new method of detection of possible differences between similar near infrared (NIR) spectra based on the self-similar (fractal) property. This property is a general characteristic that belongs to a wide class of the strongly-correlated systems. As an example we take a set of NIR spectra measured for three systems: (1) glassy carbon (GC) electrodes, (2) GC electrodes affected by azobenzene (AB) substance and finally (3) films (AB-FILM). Besides the physical model that should describe the intrinsic properties of these substances we found the fitting function that follow from the linear principle for the strongly-correlated variables. This function expressed in the form of linear combination of 4 power-law functions describes with the high accuracy the integrated curves that were obtained from the averaged values of the initially measured spectra. The nine fitting parameters can be considered as the quantitative “finger prints” for detection of the differences between similar spectra. Besides this result we established the self-similar behavior of the remnant functions. In other words, the difference between the initially integrated function and its fitting function can be expressed in the form of linear combinations of periodical functions having a set of frequencies following to relationship ω( k) = ω 0 ξ k , where the initial frequency ω 0 and scaling factor ξ are determined by the eigen-coordinates method. This behavior in the NIR spectra was discovered in the first time and physical reasons of such behavior merit an additional research.

Predictions based on the cumulative curves: Basic principles and nontrivial example

In this paper the new prediction method based on analysis of the integrated (cumulative) curves is suggested. This method includes the procedure of the optimal linear smoothing (POLS) for the finding of optimal trends, independent “reading” of relative fluctuations in terms of β-distribution function that are formed after subtraction of the calculated trend and the recognition of the proper fitting hypothesis for the integrated optimal trends by the eigen-coordinates method. The combined noninvasive approach was applied to analysis of temperature data obtained from the site http://data.giss.nasa.gov/gistemp/ related to the global warming (GW) phenomenon. These data are considered as nontrivial examples of verification of new forecasting method. The available data were combined into six files covering the mean/anomalous temperature 1546 month’s points covering the period from the January of 1880 up to October of 2008. Besides the global registered points the combined files included in themselves the north/south data points measured independently for both the Earth’s hemispheres. The combined new method (preliminary verified on mimic data) applied to these files predicts the changing of the GW period by the global cooling (GC) period that will happen during the years 2038–2136. Besides this important result a new method helps to discover the influence of a small but stable oscillating process with a set of self-similar periods Ω n = Ω 0 ξ n , n = 0, ±1, ±2, ±3, ±4 with mean period 〈 T〉 = 12.55 year. This fact should present interest for ecologists and meteorologists working in this field.

Application of new treatment methods for “reading” of the complex capacitance: A quantitative description of the aging phenomenon in polymer glasses

The glassy dynamics have been investigated using electric capacitance measurements for thin films of poly(2-chlorostyrene). The observed relaxation behavior of the electric capacitance has been analyzed using a new smoothing procedure for the noisy curves and the eigen-coordinates method for the fitting trends obtained in the results of the smoothing procedure. Such analysis has given us a fascinating relaxation function related to the glassy dynamics. New aging curve will give us a chance to understand a proper physical mechanism (related to a possible formation of dynamical self-similar interacting clusters) that can describe this interesting phenomenon. Physical consequences for the recognized relaxation function are discussed.

Real time dielectric monitoring of glass transition in n-vinyl pyrrolidone polymerization

Real time dielectric spectroscopy was used for monitoring the dynamics during liquid glass transition in radical bulk polymerization of n-vinyl pyrrolidone. Two different relaxation mechanisms were identified. Initially segmental α-relaxation was dominant. Subsequently, contribution from slower motions became more significant and ε′(tr,ω) underwent maxima in all frequency bands up to 10MHz. Peak amplitude increased and the occurring time was delayed with decreasing frequency. The slow region dynamics were similar to the dynamics of Chamberlin’s domain-like model of glass transition. The glassy phase is formed by a segmental relaxation and a relaxation due to glassy regions. The characteristic relaxation times of slow and segmental motions were well described by a modified Vogel–Fulcher–Tammann relation. The high frequency behavior, associated with the segmental motions, is well described by Havriliak–Negami and Kohlrausch–Williams–Watts formulas, which did not provide satisfactory fittings for the dynamics of slow regions. Both real and imaginary parts of the dielectric data were fully described by a complex dielectric relaxation function, the parameters of which are found by the eigen-coordinates (EC) method.

Experimental confirmation of oscillating properties of the complex conductivity: Dielectric study of polymerization/vitrification reaction

Clear evidence of the existence of fractional kinetics containing the complex power-law exponents were obtained by conductivity measurements of polymerization reaction of polyvinylpyrrolidone (PVP) performed inside a dielectric cell. We established the relationship between the Fourier image R( jω) of the complex memory function K( t) and the time-dependent mean square displacement 〈 r 2( t)〉. This relationship helps to understand the origin of the different power-law exponents appearing in the real part of complex conductivity Re[ σ( ω)] and find a physical/geometrical meaning of the power-law exponents that can form the complex-conjugated values. The complex-conjugated values of the power-law exponents leading to oscillating behavior of conductivity follows from the fractional kinetics suggested by one of the authors (R.R.N.). The relationships [ R( jω)⇔Re[ σ( ω)]⇔〈 r 2( t)〉] are becoming very efficient in classification of different types of collective motions belonging to light and heavy carriers involved in the relaxation/transfer process. The conductivity data obtained for Re[ σ( ω)] during the whole polymerization process of the PVP at different temperatures (80, 90, 100 °C) are very well described by the fitting function that follows from the suggested theory. Original fitting procedure based on the application of the eigen-coordinates (ECs) method helps to provide a reliable fitting procedure in two stages and use the well-developed and statistically stable linear least square method (LLSM) for obtaining the correct values of the fitting parameters that describe the behavior of Re[ σ( ω, T r)] in the available frequency range for the current time of the chemical reaction T r measured during the whole process of polymerization. The suggested theory gives a unique possibility to classify the basic types of motions that take place during the whole polymerization process.

New quantitative "reading" of dielectric spectra of complex biological systems

Dielectric spectroscopy data from measurements on four insect species from 200 MHz to 20 GHz over a temperature range from 10 to 70degC are described through reduction of numerous micromotions to a few collective motions, recognition of a low-frequency dispersion through ratio presentation format, application of a self-consistent iteration procedure, a separation procedure for types of collective motions, and application of the eigen-coordinates method for calculation of seven fitting parameters. Use of such fitting parameters to describe the total system relaxation processes quantitatively should enable practical uses of dielectric spectroscopy for measuring desired qualities or characteristics of biological and other complex materials. This general approach based on the reduction procedure and the use of different formats, including the original fitting procedure, allows more precise determination of the corresponding fitting function and description of dielectric spectra of complex systems. The application of this general methodology can be considered as the basic goal and motivation for this paper

05/01123 Investigations on the Eigen-coordinates method for the 2-parameter weibull distribution of wind speed

05/01123 Investigations on the Eigen-coordinates method for the 2-parameter weibull distribution of wind speed

Application of the generalized mean value function to the statistical detection of water in decane by near-infrared spectroscopy

The generalized mean value (GMV) function, defined as G N ( p ) = ( Δ N ( p ) ) 1 / p (where Δ N ( p ) is the absolute value of the moment of the pth order), was used here to differentiate between statistically close random sequences or those sequences containing large numbers of measured points ( N ⪢ 1 ). The approach taken was to find (with the help of the eigen-coordinates method) the approximate analytical function for any G N ( p ) , and in so doing demonstrate that it is inherently possible to express quantitatively the reduced characteristics of any random sequence, in terms of the ‘universal’ set of fitting parameters defined by this function. The introduction of a ‘universal’ set of the reduced parameters in the moment space then provides the instrument for the comparison of different random sequences. Applications for this new method are evident for many branches of the analytical sciences, but especially in cases when visual ‘labels’ (e.g. resonance lines), which serve as an indication of the presence of an additive, are either absent or ‘contaminated’ strongly by noise. Those fitting parameters from the approximate analytical expression, which depend on the concentration of the small additive can then be used for the construction of the quasi-monotonic line, defined as the calibration curve. Real experiments based on the treatment of near-infrared (NIR) spectra obtained for decane (the initial matrix) with water (the additive) confirm the efficiency of this simple approach. In contrast, the more conventional statistical method, based on cluster analysis, failed to establish the desired calibration curve. This simple and universal approach, which is free from model assumptions, can be used for any set of random sequences (e.g. spectrograms) if it is necessary to compare them quantitatively with each other.

The generalized mean value function approach: a new stastistical tool for the detection of weak signals in spectroscopy

A ‘universal’ set of fitting parameters have been recognized (with the help of the eigen-coordinates method) which fit well to any random sequence of points. The methodology is developed within the space of moments and is based upon the definition of the correct fit to the generalized mean value (GMV) function. By fitting it is possible to express quantitatively the reduced characteristics of any random sequence, thereby providing a possible instrument for differentiating between statistically close random sequences. It is suggested that this new approach might find application in certain spectroscopic measurements, in cases where the signal to noise ratio is low, but the stability of the noise and the influence of other external factors can be maintained. Those fitting parameters from the approximate analytical expression, which depend on the concentration of the small additive, can then be used for the construction of the quasi-monotonic line, defined as the calibration curve. In certain well-defined cases, the new approach might allow significant improvements in the sensitivity of analytical instrumentation particularly when the available analysis methodology itself is non-optimal or even considered unsuitable. To test this possibility we examined the application of the GMV method to the near-infrared detection of model micro-particles (in our case yeast cells) in an aqueous suspension, and thereby demonstrated the possibility of increasing the sensitivity of a certain spectroscopy by at least one order of magnitude.

Investigations on the Eigen‐coordinates method for the 2‐parameter weibull distribution of wind speed

The 2-parameter Weibull distribution is the hypothesis that is widely used in the fitting studies of random series of wind speeds. Several procedures are used to find the set of the two fitting parameters k and c. From an experimental study, the fitting parameters were first determined by the regression method. The basic ideas of the Eigen-coordinates method were reported by previous works, in the case of the 4-parameter Stauffer distribution. In the present paper, the new method is applied to identify the 2-parameter Weibull distribution. The differential equation was identified. Then the study disclosed a linear relationship with two Eigen-coordinates. Two complemental errors ɛ j and e j were introduced, as criteria to assess the goodness-of-fit of the distribution. ɛ j was linked to the linear relationship. e j was used to test the goodness-of-fit between the observed and Weibull cumulative distribution functions. Then the fitting parameters were determined using the Eigen-coordinates method. The results showed a better reliability.

Identification of the generalized Weibull distribution in wind speed data by the Eigen-coordinates method : Al-Hasan, M. and Nigmatullin, R. R. Renewable Energy, 2003, 28, (1), 93–110

Identification of the generalized Weibull distribution in wind speed data by the Eigen-coordinates method : Al-Hasan, M. and Nigmatullin, R. R. Renewable Energy, 2003, 28, (1), 93–110

Fluctuation-noise spectroscopy and a “universal” fitting function of amplitudes of random sequences

A “universal” fitting function has been recognized that makes use of the eigen-coordinates method (Physica A 285 (2000) 547) to accurately describe the distribution of ordered amplitudes within random sequences (taken from a diversity of sources). It is shown that sequences with a discrete structure can be described in terms of specific distributions of relative frequencies with respect to the number of quantum levels. An investigation of these quantum distributions leads to an increase in both sensitivity and selectivity, when attempting the statistical detection of various predominant factors of the hidden signals. The physical meaning of this new function is discussed and a proposal is made as the basis of a new fluctuation-noise spectroscopy, in which the recognized function can be used for the detection of small signals and/or the forerunners of strong signals that are hidden within the random sequences analyzed.

Identification of the generalized Weibull distribution in wind speed data by the Eigen-coordinates method

The real distributions of local wind speed obtained from random temporal series have been analyzed in the framework of the new Eigen-coordinates method (ECs), which helps to identify a true hypothesis with high level of authenticity. The season variations of the wind speed have been registered in Fjage (Jordan) and considered as initial data. It has been shown that histograms of the local wind speed were precisely described by the generalized Weibull model and the distribution model suggested by C. Tsallis should be rejected for present case. The possibilities and recommendations for the application of the ECs method to identify the theoretical curves needed for description of experimental data containing significant deviations are discussed. It has been shown that the ECs method could be used in these cases as a general approach and could be applied for analysis of temporal random series of different nature. PACS: 02.60.Ed., 02.60.Pn., 06.20.Dk., 07.05.Kf., 07.05.Rm.

Recognition of a new permittivity function for glycerol by the use of the eigen-coordinates method

Measurements of real and imaginary parts of the relative complex permittivity of glycerol were carried out in the frequency range 1 mHz–1 MHz at different temperatures between 188 and 263 K. The permittivity data have been analyzed thoroughly by a new data curve-fitting approach that involves the so-called eigen-coordinates method in conjunction with a separation procedure and the inverse permittivity formulas. A new single permittivity function, based on the so-called recap element picture for a self-similar (fractal) structure, has been recognized to describe well such data over the entire frequency range studied. The recognized dielectric function enabled us to infer an electrical equivalent-circuit network for the glycerol sample studied that involves a series combination of two recap elements, reflecting the existence of two different dielectric relaxation processes in glycerol. The temperature dependence of the relaxation times τ 1( T) and τ 2( T) entering into the identified permittivity function was found to obey nearly an Arrhenius behaviour with activation energies E 1≈114 kJ/mol and E 2≈94 kJ/mol. The recognized permittivity function can be justified by presuming that the processes represented by the recap elements characterized by the parameters ( ν 1, τ 1, E 1) and ( ν 2, τ 2, E 2) are linked to `donor-like' and `acceptor-like' charges formed from the infinite hydroxyl hydrogen bonds.

Identification of a new function model for the AC-impedance of thermally evaporated undoped selenium films using the Eigen-coordinates method

The experimental AC-impedance data of undoped amorphous and polycrystalline selenium films were taken at T=300K in the frequency range 100 Hz–10 kHz in the dark and under band-gap illumination. The data were analyzed using the Eigen-coordinates method in conjunction with a ‘separation’ procedure without prior use of equivalent circuits. A unified circuit model with generalized impedances was recognized to parameterize well the observed AC-impedance behavior. The features of this electrical model were linked to the fractality of the heterogeneous structural regions in these selenium specimens via one or two recap elements, each represented by a complex impedance of fractional power-law frequency behavior. These recap elements reflect the electric properties of scale-invariant circuits of self-affine normal (ideal) resistor–capacitor (RC) cells of the Cauer/Forster type.

Recognition of nonextensive statistical distributions by the eigencoordinates method

Nonextensive statistical distributions suggested by Tsallis are studied in the framework of the so-called eigencoordinates (ECs) method which allows to recognize them with high level of authenticity. At first, the possibilities of the ECs method have been demonstrated on model files to exhibit some peculiarities of this new method. Then the amplitude distributions of the real noise tracks before and after earthquakes have been analyzed. It has been shown that this type of noise is fractal and strongly correlated; histograms of the noise amplitudes with high level of reliability are really described by a distribution coming from nonextensive statistics. The possibilities and recommendations for the application of the ECs method to identify theoretical curves claimed for the description of experimental data are discussed.