A commonly used tool in data analysis is to compute a sample mean. Assuming a
uni-modal distribution, its mean provides valuable information about which value
is typically found in an observation. Also, it is one of the simplest and therefore
very robust statistics to compute and suffers much less from sampling effects of
tails of the distribution than estimates of higher moments.
In the context of a time series, the sample mean is a time average. Due to correla-
tions among successive data points, the information stored in a time series might
be much less than the information stored in a sample of independently drawn data
points of equal size, since correlation always implies redundancy. Hence, the issue
of how close the sample estimate of a time average is to the true mean value of the
process depends on correlations in data. In this thesis, we will study the proba-
bility that a single time average deviates by more than some threshold value from
the true process mean. This will be called the Large Deviation Probability (LDP),
and it will be a function of the time interval over which the average is taken: The
longer the time interval, the smaller will this probability be. However, it is the
precise functional form of this decay which will be in the focus of this thesis. The
LDP is proven to decay exponentially for identically independently distributed
data. On the other hand we will see in this thesis that this result does not apply
to long-range correlated data. The LDP is found to decay slower than exponential
for such data. It will be shown that for intermittent series this exponential decay
breaks down severely and the LDP is a power law. These findings are outlined in
the methodological explanations in chapter 3, after an overview of the theoretical
background in chapter 2.
In chapter 4, the theoretical and numerical results for the studied models in chapter
3 are compared to two types of empirical data sets which are both known to be long-
range correlated in the literature. The earth surface temperature of two stations
of two climatic zones are modelled and the error bars for the finite time averages
are estimated. Knowing that the data is long-range correlated by estimating the
scaling exponent of the so called fluctuation function, the LDP estimation leads
to noticeably enlarged error bars of time averages, based on the results in chapter
3.
The same analysis is applied on heart inter-beat data in chapter 5. The contra-
diction to the classical large deviation principle is even more severe in this case,
induced by the long-range correlations and additional inherent non-stationarity.
It will be shown that the inter-beat intervals can be well modeled by bounded
fractional Brownian motion. The theoretical and numerical LDP, both for the
model and the data, surprisingly indicates no clear decay of LDP for the time
scales under study.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:33763 |
| Date | 17 April 2019 |
| Creators | Massah, Mozhdeh |
| Contributors | Kantz, Holger, Strässner, Arno, Mahecha, Miguel, Technische Universität Dresden, Max Planck Institute for Physics of Complex Systems |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/acceptedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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