Academic literature on the topic 'Grassberger -Procaccia correlation integral'

Photon emission accompanying the fracture of an epoxy and single crystal MgO is examined for evidence of deterministic chaos by means of the autocorrelation function, the Fourier transform, the correlation integral of Grassberger and Procaccia, and the fractal box dimension. A positive Lyapunov exponent is also obtained from the epoxy phE data. Each of these measures is consistent with a significant degree of deterministic chaos associated with attractors of relatively low dimension. A typical epoxy fracture surface was analyzed for fractal character by means of the slit island technique, yielding a fractal dimension of 1.32 ± 0.03. The fractal dimensions of the fracture surface and the photon emission data (box dimension) of the epoxy are in good agreement. These observations suggest that fluctuations in photon emission intensity during fracture reflect the production of fractal surface features as they are being produced and thus provide important information on the process of dynamic crack growth.

1. The fractal scaling of heart rate variability, gauged by the correlation dimension (CD), is hypothesized to be characterized by a time structure (chronome), which in health shows differences as a function of gender and age. 2. From 24 h Holter records of 44 clinically healthy male subjects in four age groups (5–10, 20–25, 40–45 and 60–65 years; n = 11 in each group), 500 s sections at 4 h intervals for 24 h were analysed for smoothed R-R intervals sampled at 4 Hz. Using an algorithm modified from Grassberger and Procaccia (Physica D 1983; 9: 189–208), the correlation integral was estimated for embedding dimensions from 1 to 20 with a 1.0 s time lag for each section. Nightly (02.00 hours-06.00 hours) ECG records were similarly analysed in 72 additional clinically healthy subjects of both genders, 5–70 years of age. The single cosinor assessed the circadian characteristics; one- and two-way analyses of variance and linear regression were used to examine changes as a function of gender and age. 3. The 24 h average of CD is largest in the 20–25-year-old men and decreases with age thereafter (P < 0.05). These changes apply in particular to the nightly CD values, which are higher in female than in male subjects (P < 0.001). Increasing age is associated with a decrease in the amplitude and an advance in the phase of the circadian rhythm in CD (P < 0.05). 4. A chaotic end-point from fractal scaling, yielding a non-linear index, such as the correlation integral, undergoes a circadian rhythm and changes with gender and age. This assessment in the chronome represents an added diagnostic tool in cardiology, and provides new end-points for the study of coherence among internal variables of autonomic mechanisms and of influences by external environmental variables upon them.

Aim - combined use of frequency and nonlinear analysis methods for obtaining hypnograms by analyzing electroencephalographic (EEG) signals during somnological studies. Methods. Frequency filtering methods were used for preliminary treatment of EEG signals before the following nonlinear analysis. As non-linear methods of analysis we used fractal methods of deterministic chaos, such as Hurst’s method of the normalized amplitude, approximate entropy method, calculation of the correlation integral by Grassberger and Procaccia’s method. For the possibility of applying the last two methods we used quasi phase space recovery method according to the Taken’s theorem. As a result of non-linear analysis we obtained hypnograms reflecting the transition between the stages of sleep in patients undergoing somnological examination. To assess the reliability of the results, they were compared to the hypnograms obtained by the classical method based on the rules of Rehchaffen and Keyls. Also the problems associated with the occurrence of various types of interference were considered and methods for reducing their influence on the final results were suggested. Results. We can conclude that using these methods with appropriate selection of the parameters, employing the necessary normalization of raw data, and averaging the results allow us to obtain hypnogram having a full match of defined phases of sleep for about half of the periods recorded by EEG. To obtain these results it is sufficient to use only one channel of EEG recording.

In this paper, we study the behavior of the correlation dimension estimated using the Grassberger–Procaccia (GP) algorithm [Grassberger & Procaccia, 1983] in the wavelet domain for functions belonging to Hölder space. We prove that, as the wavelet scale level tends to infinity, the GP correlation dimension estimate tends to zero. Applying this result to the trajectories of the fractional Brownian motion process and using basic properties of the wavelet transform, we show that, for the fractional Gaussian noise process (fGn), the correlation dimension estimated by the GP procedure converges to the zero value. As the fractional Gaussian noise is a stochastic process with 1/fα spectrum, -1 < α < 1, our results confirm Osborne and Provenzale's assertion that colored random noise leads to the convergence of the GP-based correlation dimension estimator. However, our result holds for a different range of the spectrum exponent values. Moreover, for the fGn class of random processes, we found no correspondence between the value of the scaling exponent H and the value of the correlation dimension estimated by the GP algorithm as the latter is simply zero.

This paper reports on the application of the correlation dimension in large rotating machinery fault diagnosis. The Grassberger-Procaccia algorithm and its modified version are introduced. Some important influencing factors relating directly to the computational precision of the correlation dimension are discussed. Industrial vibration signals measured from large rotating machinery with different faults are researched using the above-mentioned methods. The results show that the correlation dimension can provide some intrinsic information on an underlying dynamic system and can be used to classify different faults intelligently.

The correlation dimension D(2) as a characteristic measure of the regular or chaotic behaviour of the solar dynamical system has been calculated. The algorithm suggested by Grassberger and Procaccia (1983) has been applied to time series of relative sunspot numbers and of areas of sunspots and faculae. In the first case, a correlation dimension D(2) ≃ 1.5 has been found; in the other two cases, the algorithm was not convergent, the results obtained being not relevant, due to the too short series of data available.

Abstract. Understanding the complexity of natural systems, such as climate systems, is critical for various research and application purposes. A range of techniques have been developed to quantify system complexity, among which the Grassberger–Procaccia (G-P) algorithm has been used the most. However, the use of this method is still not adaptive and the choice of scaling regions relies heavily on subjective criteria. To this end, an improved G-P algorithm was proposed, which integrated the normal-based K-means clustering technique and random sample consensus (RANSAC) algorithm for computing correlation dimensions. To test its effectiveness for computing correlation dimensions, the proposed algorithm was compared with traditional methods using the classical Lorenz and Henon chaotic systems. The results revealed that the new method outperformed traditional algorithms in computing correlation dimensions for both chaotic systems, demonstrating the improvement made by the new method. Based on the new algorithm, the complexity of precipitation, and air temperature in the Hai River basin (HRB) in northeastern China was further evaluated. The results showed that there existed considerable regional differences in the complexity of both climatic variables across the HRB. Specifically, precipitation was shown to become progressively more complex from the mountainous area in the northwest to the plain area in the southeast, whereas the complexity of air temperature exhibited an opposite trend, with less complexity in the plain area. Overall, the spatial patterns of the complexity of precipitation and air temperature reflected the influence of the dominant climate system in the region.

ABSTRACT .The correlation fractal dimension of strange at tractors of southwest monsoon rainfall of all the 35 Indian meteorological sub-divisions and India as a whole and nonh-east monsoon rainfall of 4 meteorological sub-divisions of peninsular India are estimated using the Grassberger and Procaccia algorithm (1983-). The fractal dimensions provide us the primary information on the number of parameters that are required to understand the dynamics underlying the monsoon dynamic system. The fractal dimensions varied between 2.9 and 7.1 and the saturation occurred between 14 and 21 ~ dimensions. In 5 sub-divisions the fractal dimension could not be determined.

This paper is mainly devoted to the investigation of discrete-time fractional systems in three aspects. Firstly, the fractional Bogdanov map with memory effect in Riemann–Liouville sense is obtained. Then, via constructing suitable controllers, the fractional Bogdanov map is shown to undergo a transition from regular state to chaotic one. Meanwhile, the positive largest Lyapunov exponent is calculated by the Jacobian matrix algorithm to distinguish the chaotic areas. Finally, the Grassberger–Procaccia algorithm is employed to evaluate the correlation dimension of the controlled fractional Bogdanov system under different parameters. The main results show that the correlation dimension converges to a fixed value as the embedding dimension increases for the controlled fractional Bogdanov map in chaotic state, which also coincides with the conclusion driven by the largest Lyapunov exponent. Moreover, three-dimensional fractional Stefanski map is considered to further verify the effectiveness and generality of the obtained results.

Research on the relationships among tectonics, micropores, microfractures, and coalbed methane (CBM) content is important for the optimal selection of CBM production areas. In this study, micropore-microfracture structural parameters of coal samples from the Guojiahe coalfield are determined through the use of X-ray photography, an image recognition algorithm, and a liquid nitrogen adsorption method. The relationship between the micropore-microfracture characteristics of the reservoir and the gas content is quantitatively assessed using the Grassberger and Procaccia (GP) algorithm to calculate the correlation dimension of the parameters. Micropore-microfracture development varies in different tectonic zones. Additionally, the CBM content varies according to the characteristic parameters of hysteresis loops and the pore diameter. The correlation dimension is an effective indicator of the nonlinear relationship between reservoir micropore-microfracture characteristics and the gas content.

This PhD thesis is primarily concerned with the evaluation of seismicity in regional areas in order to analyse the correlation between earthquakes and Acoustic Emissions (AE) detected by monitoring historical constructions. Acoustic emissions transmitted in materials and earthquakes in the Earth's crust are similar phenomena. Despite the fact that they take place at different scales, in both cases they involve a release of elastic energy from a source located in a medium: micro cracks in materials and earthquakes' hypocenters in the Earth's crust. Considering the similarity between the two phenomena and also that earthquakes may have precursors, i.e. phenomena of change in the Earth's physical conditions that take place prior to an earthquake, it was observed that acoustic emission can anticipate the occurrence of certain earthquakes. This is due to micro-cracking of the Earth's crust that anticipates the more properly catastrophic phenomenon. The research into these precursors was carried out through the analysis of correlation between acoustic emissions and seismic events. The monitoring sites, where this experimentation has been applied, are two important historical constructions of the Italian cultural heritage: the "Sacred Mountain of Varallo" in Piedmont region and the "Asinelli Tower" in Bologna, Emilia-Romagna region. The proposed statistical method of analysis is based on the Grassberger-Procaccia (GP) correlation integral. It gives the cumulative probability occurrence of couples of events, AE and seismic events, that occur in the same time windows nearby the monitored site. The analyses conducted for the historical constructions allowed us firstly to confirm the precursory behaviour of acoustic emissions and secondly to verify that between 24 hours and 10 minutes before the main shocks the highest density of acoustic emissions appears. In the second part of the PhD thesis the space correlation between only the earthquakes was studied, in particular those starting from the significant earthquake of the 6th April 2009, with its epicentre close to L'Aquila (Abruzzi region) in Italy, and those starting from the catastrophic and recent earthquake of the 24th August 2016, with its epicentre close to Amatrice (Lazio region), also in Italy. By using the b-value of the Gutenberg-Richter (GR) Law and the Gutenberg-Procaccia (GP) algorithm, the correlation between earthquakes in specific areas has been performed. The aim of the analyses was to recognize the seismic events itself as possible statistic precursory of significant earthquakes. Moreover, starting from the (GP) correlation integral the fractal distribution of events in space, Ds, was evaluated, finding the fault lines activated by the analysed seismic sequence. The analyses conducted for the L'Aquila seismic sequence allowed us to recognize precursory events in the months before the earthquake of April 2009; in fact in the months before April there was a continuous decrease of the b-value trend and of the fractal dimension of seismic events in space, Ds. A relation between the fractal dimension in space of the active fault system during seismic activity, Ds, and the b-value can be established considering the concepts of geometrical self-similarity. The applicability of a fractal relation implies that the seismicity is scale invariant over the range of applicability of the relation, therefore the Gutenberg-Richter relation for earthquakes defining the b-value is a fractal relation with Ds=2b. The analyses, b-value analysis and correlation integral, carried out separately allowed us to verify that Ds≅2b in the month of April 2009. In the case of the Amatrice seismic sequence, a sharp variation in the trend of b-value between July 2016 and August 2016 was recognized; the b-value changes quickly over a short time span. Therefore, in this case, a not so clear precursory behaviour can be noticed. Also in this case, the b-value analysis and the correlation integral, carried out separately, allowed us to verify that Ds≅2b in the month of August 2016. In conclusion, this PhD thesis allows us to evaluate whether there are statistical precursors of significant earthquakes by means of the b-value and the correlation analysis between seismic events, or between seismic events and acoustic emissions by monitoring of the historical constructions. This work represents a first step for the realization of a monitoring and analysis network to assess the behaviour of seismic sequences in order to understand if there are possible precursory events of significant earthquakes, as in the case of L'Aquila, and also to evaluate acoustic emission events as a precursory phenomenon useful for earthquake prediction. The improvement of seismic monitoring, together with structural consolidation and reinforcement, could lead to significant benefits for the mitigation of the devastating effects of earthquakes.

Στην παρούσα διπλωματική υλοποιήθηκε ένα προγραμματιστικό περιβάλλον σε Matlab το οποίο θα μας επιτρέπει να αναλύσουμε σήματα (χρονοσειρές) με δύο τεχνικές: (Ι) με την τεχνική της εμβάπτισης του σήματος σε χώρους φάσεων υψηλών διαστάσεων με σκοπό τον υπολογισμό της φράκταλ (μορφοκλασματικής) διάστασης του ελκυστή που παράγεται στο χώρο φάσεων, χρησιμοποιώντας το θεώρημα εμβάπτισης του Takens και τη μέθοδο Grassberger & Procaccia, και (ΙΙ) με μοντελοποίηση του σήματος με τη μέθοδο NARMAX, ενσωματώνοντας Extended Kalman Filters και τη Θεωρία Διαμελισμού του Λαϊνιώτη για την εύρεση της τάξης (βαθμού πολυπλοκότητας) των NARMAX μοντέλων. Σκοπός της διπλωματικής είναι η σύγκριση των αντίστοιχων αποτελεσμάτων για διάφορες κατηγορίες σημάτων, με σκοπό να διαπιστωθεί κατά πόσο η φράκταλ διάσταση του σήματος σχετίζεται με την τάξη των NARMAX μοντέλων του σήματος.
In this thesis, we implemented in Matlab, a programming environment which allows us to analyze signals (time series) with two techniques: (I) with the technique of immersion of the signal in high-dimensional phase space to calculate the fractal dimension of the attractor generated in phase space, using the theorem of Takens and the method of Grassberger & Procaccia, and (II) signal modeling method NARMAX, incorporating Extended Kalman Filters and the Laynioti partition theorem for finding the degree of complexity of NARMAX models. The aim of the thesis is the comparison of the results for various categories of signals, in order to determine if the fractal dimension of the signal is associated with the degree of complexity of NARMAX models of the signal.

Nonlinear coherent coupling between guided modes may become extremely sensitive to small changes in the input conditions.1,2 This effect has potential use in ultrafast all-optical signal processing and switching in two-mode couplers, such as fiber polarization switches1 or integrated waveguide nonlinear directional couplers. The unavoidable presence of mode-coupling imperfections, however, may lead to chaotic mode conversion that dramatically spoils the power-dependent transmission characteristics.2 Analogous stochasticity may arise whenever two intense light beams counterpropagate in an optical fiber or copropagate in two parallel linearly coupled fiber cores. Numerical integration of chaotic solutions suffers by exponential amplification of errors. However, a statistical description of the properties of a chaotic trajectory turns out to be unaffected by error propagation. Such a characterization has been obtained for the above waveguiding structures by means of different statistical estimators: among others, the maximal Lyapunov exponents (yielding the local rate of divergence of nearby trajectories) and the Grassberger-Procaccia correlation exponent (related to the fractal dimension of the chaotic domain in phase-space). We estimated stochasticity thresholds in terms of key physical parameters (e.g., critical input optical power, perturbation strength, and spatial frequency and maximal useful lengths of nonlinear switching devices.