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The distribution of random variations of a signal ''s'' around its mean value need not be Gaussian. | The distribution of random variations of a signal ''s'' around its mean value need not be Gaussian. | ||
E.g., photon statistics are typically of the Poisson type, which is especially important for low signal levels. | E.g., photon statistics are typically of the [[:Wikipedia:Poisson distribution|Poisson]] type, which is especially important for low signal levels. | ||
<ref>[http://link.aip.org/link/?RSINAK/74/3998/1 B. van Milligen, I. Classen, and C. Barth, Rev. Sci. Instrum. 74, 3998 (2003)]</ref> | <ref>[http://link.aip.org/link/?RSINAK/74/3998/1 B. van Milligen, I. Classen, and C. Barth, Rev. Sci. Instrum. 74, 3998 (2003)]</ref> | ||
In other cases, the random component of the signal ''s'' is simply a non-linear function of a (Gaussian) noise source, causing the distribution to be skewed or distorted. | In other cases, the random component of the signal ''s'' is simply a non-linear function of a (Gaussian) noise source, causing the distribution to be skewed or distorted. | ||
Or the random component of the measured signal could correspond to the maximum or minimum value of a (Gaussian) random number, leading to extremal (Gumbel) distributions. | Or the random component of the measured signal could correspond to the maximum or minimum value of a (Gaussian) random number, leading to extremal ([[:Wikipedia:Gumbel distribution|Gumbel]]) distributions. | ||
<ref>[http://link.aip.org/link/?PHPAEN/12/052507/1 B. van Milligen, R. Sánchez, B. Carreras, V. Lynch, B. LaBombard, M. Pedrosa, C. Hidalgo, B. Gonçalves, and R. Balbín, Phys. Plasmas 12, 052507 (2005)]</ref> | <ref>[http://link.aip.org/link/?PHPAEN/12/052507/1 B. van Milligen, R. Sánchez, B. Carreras, V. Lynch, B. LaBombard, M. Pedrosa, C. Hidalgo, B. Gonçalves, and R. Balbín, Phys. Plasmas 12, 052507 (2005)]</ref> | ||
The log-normal distribution is also quite common (e.g. in potential fluctuations). | The [[:Wikipedia:Log-normal distribution|log-normal distribution]] is also quite common (e.g. in potential fluctuations). | ||
<ref>[http://link.aip.org/link/?PHPAEN/11/5032/1 F. Sattin, N. Vianello, and M. Valisa, Phys. Plasmas 11, 5032 (2004)]</ref> | <ref>[http://link.aip.org/link/?PHPAEN/11/5032/1 F. Sattin, N. Vianello, and M. Valisa, Phys. Plasmas 11, 5032 (2004)]</ref> | ||
However, all the previous distributions can be obtained by suitable manipulations of Gaussian random variables. | However, all the previous distributions can be obtained by suitable manipulations of Gaussian random variables. | ||
A totally different class of statistics is known as Lévy distributions (of which the Gaussian distribution is only a special case), which is the class of distributions satisfying the requirement that the sum of independent random variables with a distribution ''P'' again has a distribution ''P'' (generalisation of the Central Limit Theorem). | A totally different class of statistics is known as [[:Wikipedia:Lévy distribution|Lévy distributions]] (of which the Gaussian distribution is only a special case), which is the class of distributions satisfying the requirement that the sum of independent random variables with a distribution ''P'' again has a distribution ''P'' (generalisation of the Central Limit Theorem). | ||
Such distributions are expected to appear in self-organised systems (such as plasmas). | Such distributions are expected to appear in [[Self-Organised Criticality|self-organised systems]] (such as plasmas). | ||
In general, the detection of this type of non-Gaussian statistics is difficult. Some techniques are however available, such as renormalisation, rescaled-range analysis, | In general, the detection of this type of non-Gaussian statistics is difficult. Some techniques are however available, such as renormalisation, [[:Wikipedia:Rescaled range|rescaled-range]] analysis, | ||
<ref>[http://link.aip.org/link/?PHPAEN/6/1885/1 B. Carreras, B. van Milligen, M. Pedrosa, R. Balbín, C. Hidalgo, D. Newman, E. Sánchez, R. Bravenec, G. McKee, I. García-Cortés, et al., Phys. Plasmas 6, 1885 (1999)]</ref> | <ref>[http://link.aip.org/link/?PHPAEN/6/1885/1 B. Carreras, B. van Milligen, M. Pedrosa, R. Balbín, C. Hidalgo, D. Newman, E. Sánchez, R. Bravenec, G. McKee, I. García-Cortés, et al., Phys. Plasmas 6, 1885 (1999)]</ref> | ||
the detection of long-range time dependence, | the detection of long-range time dependence, | ||
<ref>[http://link.aip.org/link/?PHPAEN/6/485/1 B. Carreras, D. Newman, B. van Milligen, and C. Hidalgo, Phys. Plasmas 6, 485 (1999)]</ref> | <ref>[http://link.aip.org/link/?PHPAEN/6/485/1 B. Carreras, D. Newman, B. van Milligen, and C. Hidalgo, Phys. Plasmas 6, 485 (1999)]</ref> | ||
finite-size Lyapunov exponents, | finite-size [[:Wikipedia:Lyapunov exponent|Lyapunov exponents]], | ||
<ref>[http://link.aip.org/link/?PHPAEN/8/5096/1 B. Carreras, V. Lynch, and G. Zaslavski, Phys. Plasmas 8, 5096 (2001)]</ref> etc. | <ref>[http://link.aip.org/link/?PHPAEN/8/5096/1 B. Carreras, V. Lynch, and G. Zaslavski, Phys. Plasmas 8, 5096 (2001)]</ref> etc. | ||
Sometimes it is possible to obtain information on the nature of the errors by averaging experimental data (in space or time) - this is the renormalisation technique referred to above. | Sometimes it is possible to obtain information on the nature of the errors by averaging experimental data (in space or time) - this is the renormalisation technique referred to above. | ||
When averaging over ''N'' samples, the variation of the ''N''-averaged (or smoothed) data is less than that of the original data. | When averaging over ''N'' samples, the variation of the ''N''-averaged (or smoothed) data is less than that of the original data. | ||
The way in which the variance (and other statistical moments) decreases with ''N'' provides information both on the type of statistics involved (Gaussian or otherwise) and on the random or non-random nature of the data variability (random contributions decaying to zero as ''N'' & | The way in which the variance (and other statistical moments) decreases with ''N'' provides information both on the type of statistics involved (Gaussian or otherwise) and on the random or non-random nature of the data variability (random contributions decaying to zero as ''N'' → ∞). | ||
== Integrated data analysis == | == Integrated data analysis == | ||