Fractal Clustering Research Articles

Clustering algorithm based on grid and fractal dimension

Combining the approaches based on grid and fractal,a new kind of clustering algorithm called grid and fractal dimension based clustering algorithm(GFDC) was presented.It overcame the shortcoming of low clustering quality in traditional grid-based clustering method and solved the time-consuming problem of fractal clustering method.In the initial stage of the new algorithm,some initial clusters were got according to the grid density.Then in the expansion stage,the unclassified grids were categorized using the idea of fractal.Experimental results confirm that GFDC is able to capture arbitrary shapes and non-neighboring clustering,and can be applied to the massive and high-dimension dataset.

Topological Self-Similar Networks Introduced by Diffusion-Limited Aggregation Mechanism

We propose a model for growing fractal networks based on the mechanisms learned from the diffusion-limited aggregation (DLA) model in fractal geometries in the viewpoint of network. By studying the DLA network, our model introduces multiplicative growth, aging and geographical preferential attachment mechanisms, whereby featuring topological self-similar property and hierarchical modularity. According to the results of theoretical analysis and simulation, the degree distribution of the proposed model shows a mixed degree distribution (i.e., exponential and algebraic degree distribution) and the fractal dimension and clustering coefficient can be tuned by changing the values of parameters.

Anisotropy and beyond: Geologic perspectives on geophysical prospecting for natural fractures

Recent geologic research on natural fractures challenges assumptions frequently made by geophysicists. Open fractures are not necessarily oriented parallel to the maximum horizontal stress, and fractures do not necessarily close when the fluid pressure within them is reduced. Even in the most mechanically favorable environment, precipitated cements can prop fractures open or seal fractures of any orientation. Fracture sets typically show dispersion in strike, and multiple sets of open fractures can coexist. More importantly, fractures comprise populations that commonly range over orders of magnitude in aperture and length and that occur in nonuniform clusters. Instead of isolated, regularly spaced, large, equally compliant fractures, the Earth presents complex fractal clustering of fractures having a wide range of sizes and variable compliance dictated by natural cements in the fractures and the rock mass. Going beyond anisotropy to document these essential fracture attributes in the interwell space is a ...

Signatures of fractal clustering of aerosols advected under gravity

Aerosols under chaotic advection often approach a strange attractor. They move chaotically on this fractal set but, in the presence of gravity, they have a net vertical motion downwards. In practical situations, observational data may be available only at a given level, for example, at the ground level. We uncover two fractal signatures of chaotic advection of aerosols under the action of gravity. Each one enables the computation of the fractal dimension D(0) of the strange attractor governing the advection dynamics from data obtained solely at a given level. We illustrate our theoretical findings with a numerical experiment and discuss their possible relevance to meteorology.

Vent distribution and crustal thickness in stretched continental crust: The case of the Afar Depression (Ethiopia)

The spatial clustering of vents in basaltic volcanic fields within a stretched continental crust is here used as a proxy for crustal thickness. Basaltic monogenetic vents show self-similar clustering with a power-law distribution defined by the correlation exponent (D) computed in a range with lower and upper cutoffs. The upper cutoff for the fractal clustering of vents yields the thickness of the crust. The spatial distribution of vents is analyzed in the Afar Depression (the northern termination of the East African Rift system in Africa), where the continental crust has thinned considerably. More than 1700 vents were identified and mapped in the Afar region through the use of Landsat ETM+ (enhanced thematic mapper) satellite image mosaics. Vents cluster in seven main groups corresponding to the principal structural features of the Afar Depression. The mapped vents are generally younger than 2 Ma, and most are Holocene age. The Afar vents show self-similar clustering (D = 1.39 ± 0.02) in the ∼2–23 km range. The upper cutoff of ∼23 km matches well the thickness of the crust in the Afar region as derived from seismic and gravity data (∼25 km). The distribution of vents in the Afar Depression is compared with that of vents in the northern Main Ethiopian Rift.

The role of small earthquakes in redistributing crustal elastic stress

Small-magnitude earthquakes have been shown to be negligible when estimating regional seismic moment release or seismic slip along major faults. However, they have a strong control on the redistribution of elastic stresses, and, according to statistical models of earthquake occurrences, on the triggering of subsequent earthquakes. This point is generally overlooked in stress-triggering model that only consider a limited set of earthquakes as stress sources. Elaborating on previous analyses, we here develop a probabilistic approach for estimating, at the hypocentre of an earthquake, the total stress caused by a large set of previous earthquakes, as a function of the magnitude range covered by these stress-generating earthquakes. A generic model that mimics the main features of earthquakes populations (most particularly: the Gutenberg–Richter relation and a fractal distribution of hypocentres with fractal dimension D) is constructed. The cumulative stresses generated in this model by sets of earthquakes are shown to follow a Lévy-stable law with stability index D/3. Similar conclusions are obtained when analysing the M3+ seismicity that occurred in southern California between 1981 and 2000. We show that the contribution to the cumulative static stress, caused by the occurrence of small earthquakes, at the site of pending earthquakes is at least as important as the contribution from the largest earthquakes. This is a direct consequence of the fractal clustering properties of earthquake hypocentre distributions. Including the stress redistribution due to small-scale seismicity should, therefore, significantly improve mechanical models of earthquake triggering.

Critical dynamics of fractal fault systems and its role in the generation of pre-seismic electromagnetic emissions

Regional seismicity is known to demonstrate scale-invariant properties in different ways. Some typical examples are fractal spatial distributions of hypocenters, Gutenberg–Richter magnitude statistics, fractal clustering of earthquake onset times, power-law decay of aftershock sequences, as well as scale-invariant geometry of fault systems. In some regions, the observed scale-free effects are likely to be connected to a cooperative behavior of interacting tectonic plates and can be described in terms of the self-organized criticality (SOC) concept. In this work, we investigate a new SOC model incorporating short-term fractal dynamics of seismic instabilities and slowly evolving matrix of cracks (faults) reflecting long-term history of preceding events. The model is based on a non-Abelian directed sandpile algorithm proposed recently by Hughes and Paczuski [Phys. Rev. Lett. 88 (5), 054302-1], and displays a self-organizing fractal network of occupied grid sites similar to the structure of stress fields in seismic active regions. Depending on the geometry of local stress distribution, some places on the model grid have higher probability of major events compared to the others. This dependence makes it possible to consider a time-dependent structure of the background earth crust geometry as a sensitive seismic risk indicator. We also propose a simple framework for modeling ultra-low frequency (ULF) electromagnetic emissions associated with abrupt changes in the large-scale geometry of the stress distribution before characteristic seismic events.

Fractal clustering of inertial particles in random flows

It is shown that preferential concentrations of inertial (finite-size) particle suspensions in turbulent flows follow from the dissipative nature of their dynamics. In phase space, particle trajectories converge toward a dynamical fractal attractor. Below a critical Stokes number (non-dimensional viscous friction time), the projection on position space is a dynamical fractal cluster; above this number, particles are space filling. Numerical simulations and semi-heuristic theory illustrating such effects are presented for a simple model of inertial particle dynamics.

PERCOLATION AND CRITICAL PHENOMENA OF AN ATTRACTIVE MICELLAR SYSTEM

In this work we study an attractive micellar system for which the percolation curve terminates near the critical point. We have studied such an intriguing situation by means of scattering (elastic and dynamical) and viscoelasticity experiments. Obtained data are accounted by considering in a proper way the fractal clustering processes typical of percolating systems and the related scaling concepts. We observe that the main role in the system structure and dynamics it is played by the cluster's partial screening of hydrodynamic interaction. This behaves on approaching the percolation threshold dramatic effects on the system rheological properties and on the density decay relaxations. The measured correlation functions assume a stretched exponential form and the system becomes strongly viscoelastic. The overall behavior of the measured dynamical and structural parameters indicates, that in the present micellar system, the clustering process originates dilute, polydisperse and swelling structures. Finally, this originates an interesting situation observed in the present experiment. As it has been previously, proposed by A. Coniglio et al., percolation clusters can be considered to be "Ising clusters" with the same properties as the Fisher's critical droplets. Therefore at the critical point the percolation connectedness length (ξp) can be assumed as the diverging correlation length (ξp ≡ ξ) and the mean cluster size diverges as the susceptibility.

Using Self-Similarity to Cluster Large Data Sets

Clustering is a widely used knowledge discovery technique. It helps uncovering structures in data that were not previously known. The clustering of large data sets has received a lot of attention in recent years, however, clustering is a still a challenging task since many published algorithms fail to do well in scaling with the size of the data set and the number of dimensions that describe the points, or in finding arbitrary shapes of clusters, or dealing effectively with the presence of noise. In this paper, we present a new clustering algorithm, based in self-similarity properties of the data sets. Self-similarity is the property of being invariant with respect to the scale used to look at the data set. While fractals are self-similar at every scale used to look at them, many data sets exhibit self-similarity over a range of scales. Self-similarity can be measured using the fractal dimension. The new algorithm which we call Fractal Clustering (FC) places points incrementally in the cluster for which the change in the fractal dimension after adding the point is the least. This is a very natural way of clustering points, since points in the same cluster have a great degree of self-similarity among them (and much less self-similarity with respect to points in other clusters). FC requires one scan of the data, is suspendable at will, providing the best answer possible at that point, and is incremental. We show via experiments that FC effectively deals with large data sets, high-dimensionality and noise and is capable of recognizing clusters of arbitrary shape.

Indications for a successively triggered rupture growth underlying the 2000 earthquake swarm in Vogtland/NW Bohemia

The characteristics of earthquake swarms can neither be described by simple laws nor are the underlying mechanisms presently understood. Swarm activity is often assumed to be caused by an intrusion of fluids into the seismogenic zone. We have studied the earthquake catalog of the large earthquake swarm that occurred in the year 2000 in Vogtland, SE‐Germany and NW‐Bohemia, an area well known for its episodic swarm generation. We observe a significant decrease of the Gutenberg–Richter b value during the swarm evolution as well as a fractal temporal clustering of the earthquakes. The spatial spreading of the swarm's activity, which is approximately confined to one plane, cannot simply be explained by a process of fluid diffusion. Instead, we observe a simple relationship between the spatial spreading and the seismic moment release, which is in good agreement with empirical relationships derived for tectonically driven earthquakes and theoretical crack growth models. This observation points to a progressively growing main fracture underlying the swarm activity. In addition, we find that the swarm earthquakes themselves trigger aftershocks near the border of their rupture area. The stick‐slip behavior of the rupture propagation can be explained by stress transfers and induced fluid flows due to earthquakes in a fluid‐permeated critically loaded fault zone. However, during the first phase, the temporal behavior is found to be different, pointing to intrusion of fluids initiating the swarm activity.

Glass and percolation transitions in dense attractive micellar system

In this work, we study a copolymer-micellar system characterized by clustering processes due to a short-range attractive interaction. This originates a percolation process and a new type of kinetic glass transition. We have studied these intriguing dynamical situations by means of an extensive set of light scattering and viscoelasticity experiments. Obtained data, in both the phenomena, are accounted for by considering in a proper way fractal clustering processes and the related scaling concepts. Near the percolation line the main role in the system structure and dynamics is played by the cluster's partial screening of hydrodynamic interaction, that behaves, on approaching the percolation threshold, dramatic effects on the rheological properties and on the density decay relaxations. The ergodic–nonergodic transition line (glass transition) is studied in terms of the intermediate scattering functions (ISF) in the frame of the mode coupling theory. The measured ISF gives evidence of a logarithmic decay on the density fluctuation followed by a power law behavior. This latter phenomenon is the signature of a high-order glass transition of the A3 type (cusp-like singularity).

Analysis of the time-scaling behaviour in the sequence of the aftershocks of the Bovec (Slovenia) April 12, 1998 earthquake

Analysis of the temporal fluctuations, described by a power spectral density S( f) decaying as 1/ f α with exponent α>0, in the sequence of aftershocks of the Bovec (Slovenia) earthquake of April 12, 1998 has been performed, using the Allan Factor (AF) method. The sequence of the occurrence times of the events with threshold magnitude M th≥2.0 is characterised by a scale-invariant behaviour from the time scale τ∼2·10 4 s with a scaling coefficient α∼0.9, evaluated by a least-square method. By gradually increasing the threshold magnitude up to M th=2.9, the AF curves, associated, respectively with the processes of selected events with magnitude M≥ M th, indicate a monotonic decrease of the value of the scaling exponent α, in the range of time scales stretching from 10 2 to 5·10 5 s. For all the thresholds considered, the AF curves increase monotonically with the counting time and are well fitted by an increasing power-law function for counting times greater than approximately 2·10 4 s. This monotonic power-law increase indicates the presence of fluctuations on many time scales and therefore of fractal clustering.

The Seismicity in the Southern Tyrrhenian Area and its Neural Forecasting

—The Southern Tyrrhenian basin is a region characterized by complex tectonics and large-scale geodynamic deformations, and its seismicity has been recently recognized as following fractal laws in its five dimensional set (magnitude, time and space). In particular, the time occurrences of earthquakes are found to follow a significant fractal clustering within the interval ranging from 16 minutes to 24 hours; for longer intervals, the time clustering regime was not significantly different from that of a random process. An artificial neural network is applied here to the same seismicity data set and further evidence of the nonlinear dynamic process, for which the fractal approach failed, is obtained. The monthly values of magnitude predicted by extrapolation are found to be in fair agreement with the historical data.

Clustering of seismicity at Mount Etna

The aim of this study is to understand the seismic characteristics of fault systems, which play key roles in the geodynamics of the Mt. Etna and the ascending magma. Understanding the seismic behaviour of fault systems and their relationship to volcanic and seismic phenomena may provide a useful contribution to a better understanding of dynamic processes at Mt. Etna. The seismicity in two periods (1874–1913 and 1981–1996), which include some important eruptions, have been analysed. Quantitative analysis of seismicity has been performed. Calculating fractal dimension allows us to distinguish between random, periodic and clustered scale-invariant time distributions of volcanic events. Correlations between fractal clustering evolution at long, mid and short term and eruptive processes has been found.

Fractal, multifractal and sliding window correlation dimension analysis of sedimentary time series

Multifractal analysis, the box counting technique and sliding window correlation dimension analysis are methods for the detection and description of nonlinear variations in sedimentary time/depth series measurements such as well-log data. Combinations of these methods allow the detection of sedimentary discontinuities and fractal clustering of events such as bentonite layers or fossils horizons in sedimentary sections. The methods are used to analyze linear, periodic, fractal and random patterns with different sampling intervals and after setting distinctive measurement thresholds. Application of the methods is illustrated by means of computer simulation models and on the gamma-ray log from a Mid-Cretaceous sedimentary time series.

Fractal clustering of induced seismicity in The Geysers geothermal area, California

We analyse earthquakes recorded at The Geysers geothermal field in California, an area where industrial activity induces seismicity. The seismicity is characterized by the seismic b-value and D, the fractal dimension of earthquake hypocentres measured from sliding windows containing 200 events. We study a group of events strongly clustered around an injection well. Over most of the time period examined we find a positive correlation between b and D. However, during the initiation of injection into a new well we find instead a negative correlation. The differences in correlation are statistically significant at the 1σ level but only marginally so at the 2σ level. These results provide evidence for a transient change in the seismic mechanisms operating, and may be explained by a change from conditions of slow stress loading to rapid loading as a result of the build-up of the rate of water injection into the reservoir.

Scaling properties in landscape patterns: New Zealand experience

In this paper we present a case study of spatial structure in landscape patterns for the North and South Islands of New Zealand. The aim was to characterise quantitatively landscape heterogeneity and investigate its possible scaling properties. The study examines spatial heterogeneity, in particular patchiness, at a range of spatial scales, to help build understanding on the effects of landscape heterogeneity on water movement in particular, and landscape ecology in general. We used spatial information on various landscape properties (soils, hydrogeology, vegetation, topography) generated from the New Zealand Land Resource Inventory. To analyse this data set we applied various methods of fractal analyses following the hypothesis that patchiness in selected landscape properties demonstrates fractal scaling behaviour at two structural levels: (1) individual patches; and (2) mosaics (sets) of patches. Individual patches revealed scaling behaviour for both patch shape and boundary. We found self-affinity in patch shape with Hurst exponent H from 0.75 to 0.95. We also showed that patch boundaries in most cases were self-similar and in a few cases of large patches were self-affine. The degree of self-affinity was lower for finer patches. Similarly, when patch scale decreases the orientation of patches tends to be uniformly distributed, though patch orientation on average is clearly correlated with broad scale geological structures. These results reflect a tendency to isotropic behaviour of individual patches from broad to finer scales. Mosaics of patches also revealed fractal scaling in the total patch boundaries, patch centers of mass, and in patch area distribution. All these reflect a special organisation in patchiness represented in fractal patch clustering. General relationships which interconnect fractal scaling exponents were derived and tested. These relationships show how scaling properties of individual patches affect those for mosaics of patches and vice-versa. To explain similarity in scaling behaviour in patchiness of different types we suggest that the Self-Organised Criticality concept should be used. Also, potential applications of our results in landscape ecology are discussed, especially in relation to improved neutral landscape models.

The Time Clustering of Floodings in Venice and the Cantor Dust Method

The catalogue of sea-floodings in Venice, accurately reconstructed for a period of 12 centuries (interval: 872–1996) has been analysed according to the Cantor Dust method. This provides a means of testing whether clustering in time is a scale-invariant process: if the fraction R of the intervals of length t containing flooding events is related to the time interval by: R∼t (1−D) , then the fractal clustering is occurring with fractal dimension D (0<D<1). The main result is the evidence for a gradual increase of the fractal clustering starting from 1914, when the soil subsidence of the lagoon basin determines an increase in its hydrodynamic response to the marine forcing with a gradual increase of flooding occurrences.

The Power Spectrum in a Strongly Inhomogenous Universe

The standard picture of the matter clustering, the cold dark matter model (and its variants), assumes that, on scales smaller than a certain "flattening scale" λf, the power spectrum increases with the scale, while on much larger scales it decreases so as to match the tiny fluctuations observed in the microwave background. However, there is no consensus on whether the turnaround λf has been detected or not, and on the actual importance of the luminosity segregation effect. We show that, due to the finiteness of the sample, the standard analysis applied to fractal distributions yields a turnaround for scales close to the survey scale, and a systematic amplitude shift with the survey scale. We point out that both features, bending and scaling, are in agreement with recent determinations of the power spectrum, in particular with the CfA2 spectrum in redshift space. Therefore, our conclusion is that the CfA2 sample is consistent with a fractal clustering.