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{{refimprove|date=May 2013}}
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'''Surrogate data testing'''<ref name=Theiler1>
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{{cite journal
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|author=J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, J. Doyne Farmer
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|journal=Physica D
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|volume=58
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|page=77
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|year=1992
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|doi=10.1016/0167-2789(92)90102-S
}}</ref> (or the ''method of surrogate data'') is a statistical [[proof by contradiction]] technique and similar to [[Bootstrapping (statistics)|parametric bootstrapping]] used to detect [[non-linearity]] in a [[time series]]. The technique basically involves specifying a [[null hypothesis]] <math>H_0</math> describing a [[linear model|linear process]] and then generating several [[surrogate data]] sets according to <math>H_0</math> using [[Monte Carlo method|Monte Carlo]] methods. A discriminating statistic is then calculated for the original time series and all the surrogate set. If the value of the statistic is significantly different for the original series than for the surrogate set, the null hypothesis is rejected and non-linearity assumed.


The particular surrogate data method to be used is directly related to the null hypothesis. Usually this is similar to the following:
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''The data is a realization of a stationary linear system, whose output has been possibly measured by a monotonically increasing possibly nonlinear (but static) function''.<ref name=Theiler1 /> Here ''linear'' means that each value is linearly dependent on past values or on present and past values of some independent identically distributed (i.i.d.) process, usually also Gaussian. This is equivalent to saying that the process is [[ARMA model|ARMA]] type. In case of fluxes (continuous mappings), linearity of system means that it can be expressed by a linear differential equation. In this hypothesis, the ''static'' measurement function is one which depends only on the present value of its argument, not on past ones.
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Many algorithms to generate surrogate data have been proposed. They are usually classified in two groups:<ref name=Theiler2>
This model is staying viewed as as full deal for adult males and women of all ages.
{{cite journal
|author=J. Theiler, D. Prichard
|journal=Physica D
|volume=94
|issue=4
|year=1996
|doi=10.1016/0167-2789(96)00050-4
}}</ref>
* ''Typical realizations'': data series are generated as outputs of a well-fitted model to the original data.
* ''Constrained realizations'': data series are created directly from original data, generally by some suitable transformation of it.
 
The last surrogate data methods do not depend on a particular model, nor on any parameters, thus they are non-parametric methods. These surrogate data methods are usually based on preserving the linear structure of the original series (for instance, by preserving the [[autocorrelation function]], or equivalently the [[periodogram]], an estimate of the sample spectrum).
Among constrained realizations methods, the most widely used (and thus could be called the ''classical methods'') are:
 
# Algorithm 0, or RS (for ''Random Shuffle''):<ref name=Theiler1 /><ref name=Scheinkman>
{{cite journal
|author=J.A. Scheinkman, B. LeBaron
|journal=J. Business
|volume=62
|page=311
|year=1989
|url=http://ideas.repec.org/a/ucp/jnlbus/v62y1989i3p311-37.html
}}</ref> New data are created simply by random permutations of the original series. The permutations guarantee the same amplitude distribution than the original series, but destroy any linear correlation. This method is associated to the null hypothesis of the data being uncorrelated noise (possibly Gaussian and measured by a static nonlinear function).
# Algorithm 1, or RP (for ''Random Phases''; also known as FT, for [[Fourier Transform]]):<ref name=Theiler1 /><ref name=Osborne>
{{cite journal
|author=A.R. Osborne, A.D. Kirwan Jr., A. Provenzale, L. Bergamasco
|journal=Physica D
|volume=23
|page=75
|year=1986
}}</ref> In order to preserve the linear correlation (the periodogram) of the series, surrogate data are created by the inverse Fourier Transform of the modules of Fourier Transform of the original data with new (uniformly random) phases. If the surrogates must be real, the Fourier phases must be antisymmetric with respect to the central value of data.
# Algorithm 2, or AAFT (for ''Amplitude Adjusted Fourier Transform''):<ref name=Theiler1 /><ref name=Theiler2 /> This method has approximately the advantages of the two previous ones: it tries to preserve both the linear structure and the amplitude distribution. This method consists of these steps:
#* Scaling the data to a Gaussian distribution (''Gaussianization'').
#* Performing a RP transformation of the new data.
#* Finally doing a transformation inverse of the first one (''de-Gaussianization'').
#:The drawback of this method is precisely that the last step changes somewhat the linear structure.
# Iterative algorithm 2, or IAAFT (for ''Iterative Amplitude Adjusted Fourier Transform''):<ref name=Schreiber1>
{{cite journal
|author=T. Schreiber, A. Schmitz
|journal=Phys. Rev. Lett.
|volume=77
|issue=4
|year=1996
|doi=10.1103/PhysRevLett.77.635
|pmid=10062864
}}</ref> This algorithm is an iterative version of AAFT. The steps are repeated until the autocorrelation function is sufficiently similar to the original, or until there is no change in the amplitudes.
 
Many other surrogate data methods have been proposed, some based on optimizations to achieve an autocorrelation close to the original one,<ref name=Schreiber2>
{{cite journal
|author=T. Schreiber, A. Schmitz
|journal=Physica D
|volume=142
|page=346
|year=2000
|doi=10.1016/S0167-2789(00)00043-9
}}</ref><ref name=Schreiber3>
{{cite journal
|author=T. Schreiber
|journal=Phys. Rev. Lett.
|volume=80
|issue=4
|year=1998
|doi=10.1103/PhysRevLett.77.635
|pmid=10062864
}}</ref><ref name=Engbert>
{{cite journal
|author=R. Engbert
|journal=Chaos Solitons Fractals
|volume=13
|issue=1
|year=2002
|doi=10.1016/S0960-0779(00)00236-8
}}</ref> some based on wavelet transform<ref name=Breakspear>
{{cite journal
|author=M. Breakspear, M. Brammer, P.A. Robinson
|journal=Physica D
|volume=182
|issue=1
|year=2003
|doi=10.1016/S0167-2789(03)00136-2
}}</ref><ref name=Keylock1>
{{cite journal
|author=C.J. Keylock
|journal=Phys. Rev. E
|volume=73
|page=036707
|year=2006
|doi=10.1103/PhysRevE.73.036707
}}</ref><ref name=Keylock2>
{{cite journal
|author=C.J. Keylock
|journal=Physica D
|volume=225
|issue=2
|year=2007
|doi=10.1016/j.physd.2006.10.012
}}</ref> and some capable of dealing with some types of non-stationary data.<ref name=Nakamura>
{{cite journal
|author=T. Nakamura, M. Small, Y. Hirata
|journal=Phys. Rev. E
|volume=74
|page=026205
|year=2006
|doi=10.1103/PhysRevE.74.026205
}}</ref><ref name=Lucio>
{{cite journal
|author=J.H. Lucio, R. Valdés, L.R. Rodríguez
|journal=Phys. Rev. E
|volume=85
|page=056202
|year=2012
|doi=10.1103/PhysRevE.85.056202
}}</ref>
 
== References ==
{{reflist}}
 
[[Category:Nonlinear time series analysis]]
[[Category:Statistical tests]]

Latest revision as of 08:41, 24 August 2014

People today, who have been exploring for some real very good things to use and also want to help save some funds at the exact same time, then a stop by to Abercrombie outlet can satisfy their wishes in the safest and least complicated way.
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The trouble, which is remaining confronted by most of the passionate enthusiasts of this manufacturer, is that there are few retailers of this manufacturer throughout America. All people people, who stay considerably from the states, where by the retailers are situated, they are in celine bags will need of driving to the spot, by getting time out and only then they can make their lives attractive in a modish wa

Such programs will automatically serve you as your pleasing journeys, for you will get time to expend with your family members and at the very same time, you can store best items for oneself. When you will arrive back from searching, then you will get to realize that you have invested in the most effective locati

When you will place the genuine Abercrombie outlet, then you will come to know that this firm focuses on setting up straightforward stores. And such simplicity can confuse people, for they might believe it to be an inauthentic outlet, but when you will action in and will spy on the goods, then you will appear to know that everything shelved inside is primar
There is a sturdy explanation, thanks to which A&F has been focusing on maintaining the selection of its stores minimal. It has a unique way of offer

� Its minimal selection of retailers has built it these kinds of a prestigious and lavish brand. Just lately, it has commenced to open up its manufacturing unit shops, which have not been mentioned on its web site however. If you by likelihood, you appear throughout any one particular of them, then you should action in and but some very good things from the most up-to-date sele
i .� There are lots of people today, who find it complicated to identify Abercrombie outlet. For all such people today, it is superior to look through the web page of this brand. Make confident that the web page, which you have picked to area an buy for your sizzling preferred items, is genu

Do not celine bags get trapped by fraud internet sites. The pattern of searching on the internet has been escalating. When you have these kinds of an possibility, then why to squander time in looking for the
to s?� If you have not buy any products from closest Abercrombie outlet�yet, then you are lacking something useful in your closets. You ought to add worth to your clothes and equipment selection. And Abercrombie and Fitch is the only model, which can do all this for

.

This model is staying viewed as as full deal for adult males and women of all ages.

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