Fractals in nature: From characterization to simulation

Voss, Richard F.

doi:10.1007/978-1-4612-3784-6_1

Cited by 487 publications

(329 citation statements)

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“…This primitive is itself constructed from line segments that are recursively replaced with smaller scaled primi tives. No "drawing" actually occurs until the scale reaches a specified lower limit [see Voss (1988) for a full discussion]. The charm of these systems is that the algo rithm that replaces each line segment of the primitive with smaller versions calls itself recursively but only imple ments pattern drawing at the smallest scale.…”

Section: Recursive Pca Analysismentioning

confidence: 99%

Functional Connectivity: The Principal-Component Analysis of Large (PET) Data Sets

Friston

Frith

Liddle

et al. 1993

J Cereb Blood Flow Metab

1,791

1,097

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Summary:The distributed brain systems associated with performance of a verbal fluency task were identified in a nondirected correlational analysis of neurophysiological data obtained with positron tomography. This analysis used a recursive principal-component analysis developed specifically for large data sets. This analysis is interpreted in terms of functional connectivity, defined as the tem poral correlation of a neurophysiological index measured in different brain areas. The results suggest that the vari ance in neurophysiological measurements, introduced exCooperative and connectionist approaches to un derstanding the integration of brain function are well established (Sherrington, 1941;Hebb, 1949;Edelman, 1978;McClelland, 1988). The nature and organizational principles of extrinsic cortical con nections, particularly the long corticocortical affer ents (e. g. , Goldman Rakic, 1988) has provided a basis for mechanistic descriptions of brain function (e. g. , Mesulam, 1990). These descriptions refer to parallel, massively distributed, and interconnected (sub)cortical areas. Anatomical connectivity is a necessary underpinning for these models and has been used to infer functional connectivity (e. g. , Zeki, 1990). This article describes one way of mea suring functional connectivity using positron emis sion tomographic (PET) measurements of neural ac tivity.PET is in a unique position to acquire data for this Received October 28, 1991; final revision rec::: ived May 8, 1992; accepted June 18, 1992. Address correspondence and reprint requests to Dr. K. 1.Friston at The Neurosciences Institute, 1230 York Ave., New York, NY 10021, U.S.A.Abbreviations used: ANCOV A, analysis of covariance; BA, Broadmann's area; DLPFC, dorsolateral prefrontal cortex; J-PSTH, Joint Peri Stimulus Time Histogram; PCA, principal component analysis; PET, positron emission tomography; rCBF, regional CBF. 5 peri mentally , was accounted for by two independent prin cipal components. The first, and considerably larger, highlighted an intentional brain system seen in previous studies of verbal fluency. The second identified a distrib uted brain system including the anterior cingulate and Wernicke's area that reflected monotonic time effects. We propose that this system has an attentional bias. Key Words: PET-Principal-component analysis-Functional connectivity-Effective connectivity-Verbal fluency Neural networks.sort of analysis because it samples the entire brain state in a uniform fashion. This allows all possible functional connections to be assessed using serial measurements of the same subject in different brain states. The shortcomings of PET include its rela tively poor spatiotemporal resolution and the exact nature of the dependency of measured regional CBF (rCBF) on neural discharge rates. However, PET can be used to address large scale functional connectivity as an important supplement to obser vations on the gross aspects of extrinsic anatomical connectivity and fine time scale effective connectiv ity defined by electrophysi...

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Section: Recursive Pca Analysismentioning

confidence: 99%

Functional Connectivity: The Principal-Component Analysis of Large (PET) Data Sets

Friston

Frith

Liddle

et al. 1993

J Cereb Blood Flow Metab

1,791

1,097

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“…This property rules out representations of topography within river basins through common self-similar processes such as fractional Brownian surfaces. Such surfaces have often been used to model topography at regional scales [Mandelbrot, 1983;Voss, 1988]. Multifractality is an extension of self-similarity.…”

Section: As Was Noted By Veneziano and Niemann [This Issue] (1)mentioning

confidence: 99%

Self‐similarity and multifractality of fluvial erosion topography: 2. Scaling properties

Veneziano¹,

Niemann²

2000

Water Resources Research

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Abstract. In a companion paper [Veneziano and Niemann, this issue] the authors have proposed self-similarity and multifractality conditions for fluvial erosion topography within basins and have shown that topographic surfaces with this property can evolve from a broad class of dynamic models. Here we use the same self-similarity and multifractality conditions to derive geomorphological scaling laws of hydrologic interest. We find that several existing relations should be modified, as they were obtained using definitions of the quantities involved or measurement techniques that are inappropriate under selfsimilarity. These relations include Hack's law, the power law decay of the distributions of contributing area and main channel length, the scaling of channel slope with contributing area, and the self-similarity condition for river courses. Most results are further generalized by replacing main stream flow length and drainage area with generic measures of basin size. The relations we obtain among properly measured topographic variables have simple universal exponents. For example, the exponent of Hack's law is 0.5, the exponent of the distribution of contributing area is -0.5, and the exponent of the distribution of main stream length is -1.0. We also suggest a stochastic condition of drainage network self-similarity that incorporates topological as well as geometric and hydrologic features and a reformulation of Horton's laws using drained area rather than stream order.

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“…Spectral analysis seeks to describe the frequency content and scaling behavior of a signal. The relationship between spectral density and frequency can be used to define the scaling characteristic of any random "noisy" signal [ Voss, 1988]. Since we are dealing with random functions (such as reflection amplitude and well log response) that vary with space coordinates, frequency is replaced by wavenumber.…”

Section: Spectral Analysismentioning

confidence: 99%

Quantifying lateral heterogeneities in fluvio‐deltaic sediments using three‐dimensional reflection seismic data: Offshore Gulf of Mexico

Deshpande¹,

Flemings²,

Huang³

1997

J. Geophys. Res.

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Fractals in nature: From characterization to simulation

Cited by 487 publications

References 0 publications

Functional Connectivity: The Principal-Component Analysis of Large (PET) Data Sets

Functional Connectivity: The Principal-Component Analysis of Large (PET) Data Sets

Self‐similarity and multifractality of fluvial erosion topography: 2. Scaling properties

Quantifying lateral heterogeneities in fluvio‐deltaic sediments using three‐dimensional reflection seismic data: Offshore Gulf of Mexico

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