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| Titel |
New Scaling Model for Variables and Increments with Heavy-Tailed Distributions |
| VerfasserIn |
Monica Riva, Alberto Guadagnini, Shlomo P. Neuman |
| Konferenz |
EGU General Assembly 2015
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 17 (2015) |
| Datensatznummer |
250105752
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| Publikation (Nr.) |
 EGU/EGU2015-5308.pdf |
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| Zusammenfassung |
| Many earth, environmental, ecological, biological, physical, social, financial and other
variables, Y , exhibit (i) asymmetry in sample frequency distributions, as well as (ii)
symmetry in distributions of their spatial and/or temporal increments, δY , at diverse
separation distances (or lags), with sharp peaks and heavy tails which appear to decay
asymptotically (often toward exponential tails of the Gaussian distribution) as lag increases.
No model known to us captures all of these behaviors in a unique and consistent manner.
We propose a new model that does so upon treating Y (x) as a random function
of a coordinate x in the Euclidean (spatial) domain or time, forming a stationary
random field (or process) with constant ensemble mean (expectation) /¨Y /©. We express
the zero-mean random fluctuation Y ’(x) = Y (x) -/¨Y /© in sub-Gaussian form.
In the classical sub-Gaussian form, Y ’(x) = UG(x), where G(x) is a zero-mean
stationary Gaussian random field (or process) and the subordinator U is an independent
non-negative random variable (Samorodnitsky and Taqqu, 2014). We generalize this by
writing Y ’(x) = U(x)G(x) in which U(x) is iid. This enables us to analyze, and
synthetically generate, heavy-tailed non-Gaussian distributions of both Y and δY in
both probability space (across an infinite ensemble of random realizations) and real
space (in a single realization). We derive analytical expressions for probability
distribution functions (pdf s) of Y and δY as well as their lead statistical moments.
We show that when U is lognormal Y follows the well-known normal-lognormal
distribution (NLN, of which the Gaussian distribution is a particular case) with constant
parameters. The NLN pdf has been successfully used to interpret financial (Clark,
1973) and environmental (Guadagnini et al., 2014) data. However, δY is not NLN
and forcing the latter on the former gives a false and widely accepted impression
that (a) parameters of the δY pdf vary with lag and (b) the pdfs of Y and δY
are mutually inconsistent. Our new theory shows that both "observations" stem
from a misconception. We eliminate these apparent inconsistencies by deriving
analytically the pdf of δY from that of Y , given that the latter is NLN. In our model the
tail of the δY pdf scales with the correlation coefficient of G and thereby with
lag; all other parameters of the pdf remaining unaltered with lag. We illustrate key
features of our new model by way of synthetic examples. We then propose and test a
methodology to estimate, accurately and efficiently, all model parameters using
sample moments of both Y and δY . We end by exemplifying the use of our new
model and method of inference by applying them to neutron porosity data from deep
boreholes.
References
Clark, P.K. (1973), A subordinated stochastic process model with finite variance for
speculative prices, Econometrica, 41, 1.
Samorodnitsky, G., and M.S. Taqqu (1994), Stable Non-Gaussian Random Processes,
Chapman and Hall, New York.
Guadagnini, A., S.P. Neuman, T. Nan, M. Riva, and C.L. Winter (2015), Scalable
statistics of correlated random variables and extremes applied to deep borehole porosities,
Hydrology and Earth System Sciences, in press. |
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