### Abstract

The extreme statistics of time signals is studied when the maximum is measured from the initial value. In the case of independent, identically distributed (iid) variables, we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn. Then we turn to correlated periodic Gaussian signals with a 1 fα power spectrum and study the distribution of the maximum relative height with respect to the initial height (MRHI). The exact MRHI distribution is derived for α=0 (iid variables), α=2 (random walk), α=4 (random acceleration), and α= (single sinusoidal mode). For other, intermediate values of α, the distribution is determined from simulations. We find that the MRHI distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all α. The two main distinguishing features of the MRHI distribution are the much larger weight for small relative heights and the divergence at zero height for α>3. We also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some nonperiodic boundary conditions. Finally, we show that, for signals arising from time-translationally invariant distributions, the density of near extreme states is the same as the MRHI distribution. This is used in developing a scaling theory for the threshold singularities of the two distributions.

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

Article number | 041119 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 76 |

Issue number | 4 |

DOIs | |

Publication status | Published - Oct 12 2007 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*76*(4), [041119]. https://doi.org/10.1103/PhysRevE.76.041119

**Extreme statistics for time series : Distribution of the maximum relative to the initial value.** / Burkhardt, T. W.; Györgyi, G.; Moloney, N. R.; Rácz, Z.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 76, no. 4, 041119. https://doi.org/10.1103/PhysRevE.76.041119

}

TY - JOUR

T1 - Extreme statistics for time series

T2 - Distribution of the maximum relative to the initial value

AU - Burkhardt, T. W.

AU - Györgyi, G.

AU - Moloney, N. R.

AU - Rácz, Z.

PY - 2007/10/12

Y1 - 2007/10/12

N2 - The extreme statistics of time signals is studied when the maximum is measured from the initial value. In the case of independent, identically distributed (iid) variables, we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn. Then we turn to correlated periodic Gaussian signals with a 1 fα power spectrum and study the distribution of the maximum relative height with respect to the initial height (MRHI). The exact MRHI distribution is derived for α=0 (iid variables), α=2 (random walk), α=4 (random acceleration), and α= (single sinusoidal mode). For other, intermediate values of α, the distribution is determined from simulations. We find that the MRHI distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all α. The two main distinguishing features of the MRHI distribution are the much larger weight for small relative heights and the divergence at zero height for α>3. We also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some nonperiodic boundary conditions. Finally, we show that, for signals arising from time-translationally invariant distributions, the density of near extreme states is the same as the MRHI distribution. This is used in developing a scaling theory for the threshold singularities of the two distributions.

AB - The extreme statistics of time signals is studied when the maximum is measured from the initial value. In the case of independent, identically distributed (iid) variables, we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn. Then we turn to correlated periodic Gaussian signals with a 1 fα power spectrum and study the distribution of the maximum relative height with respect to the initial height (MRHI). The exact MRHI distribution is derived for α=0 (iid variables), α=2 (random walk), α=4 (random acceleration), and α= (single sinusoidal mode). For other, intermediate values of α, the distribution is determined from simulations. We find that the MRHI distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all α. The two main distinguishing features of the MRHI distribution are the much larger weight for small relative heights and the divergence at zero height for α>3. We also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some nonperiodic boundary conditions. Finally, we show that, for signals arising from time-translationally invariant distributions, the density of near extreme states is the same as the MRHI distribution. This is used in developing a scaling theory for the threshold singularities of the two distributions.

UR - http://www.scopus.com/inward/record.url?scp=35248813014&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35248813014&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.76.041119

DO - 10.1103/PhysRevE.76.041119

M3 - Article

AN - SCOPUS:35248813014

VL - 76

JO - Physical review. E

JF - Physical review. E

SN - 2470-0045

IS - 4

M1 - 041119

ER -