### Abstract

In the study of the involved geometry of singular distributions, the use of fractal and multifractal analysis has shown results of outstanding significance. So far, the investigation has focussed on structures produced by one single mechanism which were analyzed with respect to the ordinary metric or volume. Most prominent examples include self-similar measures and attractors of dynamical systems. In certain cases, the multifractal spectrum is known explicitly, providing a characterization in terms of the geometrical properties of the singularities of a distribution. Unfortunately, strikingly different measures may possess identical spectra. To overcome this drawback we propose two novel methods, the conditional and the relative multifractal spectrum, which allow for a direct comparison of two distributions. These notions measure the extent to which the singularities of two distributions 'correlate'. Being based on multifractal concepts, however, they go beyond calculating correlations. As a particularly useful tool, we develop the multifractal formalism and establish some basic properties of the new notions. With the simple example of Binomial multifractals, we demonstrate how in the novel approach a distribution mimics a metric different from the usual one. Finally, the applications to real data show how to interpret the spectra in terms of mutual influence of dense and sparse parts of the distributions.

Original language | English |
---|---|

Pages (from-to) | 153-168 |

Number of pages | 16 |

Journal | Fractals |

Volume | 5 |

Issue number | 1 |

Publication status | Published - Mar 1997 |

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### ASJC Scopus subject areas

- General
- Geometry and Topology

### Cite this

*Fractals*,

*5*(1), 153-168.

**Conditional and relative multifractal spectra.** / Riedi, Rudolf H.; Scheuring, I.

Research output: Contribution to journal › Article

*Fractals*, vol. 5, no. 1, pp. 153-168.

}

TY - JOUR

T1 - Conditional and relative multifractal spectra

AU - Riedi, Rudolf H.

AU - Scheuring, I.

PY - 1997/3

Y1 - 1997/3

N2 - In the study of the involved geometry of singular distributions, the use of fractal and multifractal analysis has shown results of outstanding significance. So far, the investigation has focussed on structures produced by one single mechanism which were analyzed with respect to the ordinary metric or volume. Most prominent examples include self-similar measures and attractors of dynamical systems. In certain cases, the multifractal spectrum is known explicitly, providing a characterization in terms of the geometrical properties of the singularities of a distribution. Unfortunately, strikingly different measures may possess identical spectra. To overcome this drawback we propose two novel methods, the conditional and the relative multifractal spectrum, which allow for a direct comparison of two distributions. These notions measure the extent to which the singularities of two distributions 'correlate'. Being based on multifractal concepts, however, they go beyond calculating correlations. As a particularly useful tool, we develop the multifractal formalism and establish some basic properties of the new notions. With the simple example of Binomial multifractals, we demonstrate how in the novel approach a distribution mimics a metric different from the usual one. Finally, the applications to real data show how to interpret the spectra in terms of mutual influence of dense and sparse parts of the distributions.

AB - In the study of the involved geometry of singular distributions, the use of fractal and multifractal analysis has shown results of outstanding significance. So far, the investigation has focussed on structures produced by one single mechanism which were analyzed with respect to the ordinary metric or volume. Most prominent examples include self-similar measures and attractors of dynamical systems. In certain cases, the multifractal spectrum is known explicitly, providing a characterization in terms of the geometrical properties of the singularities of a distribution. Unfortunately, strikingly different measures may possess identical spectra. To overcome this drawback we propose two novel methods, the conditional and the relative multifractal spectrum, which allow for a direct comparison of two distributions. These notions measure the extent to which the singularities of two distributions 'correlate'. Being based on multifractal concepts, however, they go beyond calculating correlations. As a particularly useful tool, we develop the multifractal formalism and establish some basic properties of the new notions. With the simple example of Binomial multifractals, we demonstrate how in the novel approach a distribution mimics a metric different from the usual one. Finally, the applications to real data show how to interpret the spectra in terms of mutual influence of dense and sparse parts of the distributions.

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M3 - Article

VL - 5

SP - 153

EP - 168

JO - Fractals

JF - Fractals

SN - 0218-348X

IS - 1

ER -