Statistical estimation for multiplicative cascades

Ossiander, Mina; Waymire, Edward C.

doi:10.1214/aos/1015957469

Cited by 80 publications

(50 citation statements)

References 25 publications

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“…Figure 4 shows log-log plots of the empirical moments E (Q max ) q (points) of the standardized discharge maxima inside each group, as a function of the average areaĀ of the grouped catchments, for different moment orders q = 0.5, 1, 1.5, 2, 2.5, 3. The reason why we limit our analysis to moment orders q ≤ 3, is because for highly variable random fields (as is the case of discharge maxima), higher moment orders are underestimated with high probability see e.g., [32,[47][48][49][50][51][52][53].…”

Section: Analysis and Resultsmentioning

confidence: 99%

“…Figure 4 shows log-log plots of the empirical moments [( ′ ) ] (points) of the standardized discharge maxima inside each group, as a function of the average area Ā of the grouped catchments, for different moment orders = 0.5, 1, 1.5, 2, 2.5, 3. The reason why we limit our analysis to moment orders  3, is because for highly variable random fields (as is the case of discharge maxima), higher moment orders are underestimated with high probability see e.g., [32,[47][48][49][50][51][52][53]. Similar to the findings of previous studies see e.g., [4,[8][9][10][11]14,19,28,40], one sees that for all moment orders considered, the initial moments of the standardized discharge maxima ′ vary log-linearly with the drainage area A (see least-squares (LS) fitted lines), with negative slopes corresponding to the empirical moment scaling function ( ) in Equation (4).…”

Section: Analysis and Resultsmentioning

confidence: 99%

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Revisiting the Statistical Scaling of Annual Discharge Maxima at Daily Resolution with Respect to the Basin Size in the Light of Rainfall Climatology

Perdios

Langousis

2020

Water

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Over the years, several studies have been carried out to investigate how the statistics of annual discharge maxima vary with the size of basins, with diverse findings regarding the observed type of scaling (i.e., simple scaling vs. multiscaling), especially in cases where the data originated from regions with significantly different hydroclimatic characteristics. In this context, an important question arises on how one can effectively conclude on an approximate type of statistical scaling of annual discharge maxima with respect to the basin size. The present study aims at addressing this question, using daily discharges from 805 catchments located in different parts of the United Kingdom, with at least 30 years of recordings. To do so, we isolate the effects of the catchment area and the local rainfall climatology, and examine how the statistics of the standardized discharge maxima vary with the basin scale. The obtained results show that: (a) the local rainfall climatology is an important contributor to the observed statistics of peak annual discharges, and (b) when the effects of the local rainfall climatology are properly isolated, the scaling of the standardized annual discharge maxima with the area of the catchment closely follows that commonly met in actual rainfields, deviating significantly from the simple scaling rule. The aforementioned findings explain to a large extent the diverse results obtained by previous studies in the absence of rainfall information, shedding light on the approximate type of scaling of annual discharge maxima with the basin size.

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Section: Analysis and Resultsmentioning

confidence: 99%

Section: Analysis and Resultsmentioning

confidence: 99%

Revisiting the Statistical Scaling of Annual Discharge Maxima at Daily Resolution with Respect to the Basin Size in the Light of Rainfall Climatology

Perdios

Langousis

2020

Water

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“…This paper warns practitioners against the blind use in geophysical time series analyses of classical statistical tools, which neglect dependence and heavy tails in distributions. Ossiander and Waymire (2000) already caution against using high moments in multifractal estimation, but their particular focus is on discrete multiplicative cascade models. Indeed, they demonstrate that the estimators of multiscaling exponents converge almost surely to the structure function of the cascade generators as the sample becomes large for all moment orders within a certain critical interval, whose upper bound is consistent with our results.…”

Section: Discussionmentioning

confidence: 99%

Just two moments! A cautionary note against use of high-order moments in multifractal models in hydrology

Lombardo¹,

Volpi²,

Koutsoyiannis³

et al. 2013

Preprint

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The need of understanding and modelling the space-time variability of natural processes in hydrological sciences produced a large body of literature over the last thirty years. In this context, multifractal framework provides parsimonious models which can be applied to a wide scale range of hydrological processes, and are based on the empirical detection of some patterns in observational data, i.e. a scale invariant mechanism repeating scale after scale. Hence, multifractal analyses heavily rely on available data series and their statistical processing. In such analyses, high order moments are often estimated and used in model identification and fitting as if they were reliable. This paper warns practitioners for blind use in geophysical time series analyses of classical statistics, which is based upon independent samples typically following distributions of exponential type. Indeed, the study of natural processes reveals scaling behaviours in state (departure from exponential distribution tails) and in time (departure from independence), thus implying dramatic increase of bias and uncertainty in statistical estimation. Surprisingly, all these differences are commonly unaccounted for in most multifractal analyses of hydrological processes, which may result in inappropriate modelling, wrong inferences and false claims about the properties of the processes studied. Using theoretical reasoning and Monte Carlo simulations we find that the reliability of multifractal methods that use high order moments (> 3) is questionable. In particular, we suggest to use the first two moments in all problems as they suffice to define the most important characteristics of the distribution

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“…Cascades have been the paradigm for multifractal objects [38][39][40][41]. In this study, we consider the lattice multiplicative random cascade x(t) given by the product of a set of iid processes varying over a range of different time scales:…”

Section: Vt Of Lmrcmentioning

confidence: 99%

Persistent scale free fluctuation in market recovery and recession

Lin

2010

Eur. Phys. J. B

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A counting procedure is introduced to track the persistent trending feature in the market fluctuation based on the second order statistics of the fluctuation. The method is applied to the daily number of the market index in different economic times of the recession and recovery. Similar characteristics with the discrete lattice cascade were found. It suggests the market index fluctuation over a wide range of scales can be modeled as a state of persistent "coordination". Specific time scales of the persistent trending feature are extracted and differences in the recovery and recession are compared.

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Statistical estimation for multiplicative cascades

Cited by 80 publications

References 25 publications

Revisiting the Statistical Scaling of Annual Discharge Maxima at Daily Resolution with Respect to the Basin Size in the Light of Rainfall Climatology

Revisiting the Statistical Scaling of Annual Discharge Maxima at Daily Resolution with Respect to the Basin Size in the Light of Rainfall Climatology

Just two moments! A cautionary note against use of high-order moments in multifractal models in hydrology

Persistent scale free fluctuation in market recovery and recession

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