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Error propagation: Difference between revisions
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This technique also provides a quick method to check for possible problems such as ill-conditioning, cited above. | This technique also provides a quick method to check for possible problems such as ill-conditioning, cited above. | ||
When the model relating ''s'' and ''p'' is known, as well as the error distributions (and the latter may either be Gaussian or not), a more systematic approach to error propagation is provided by a technique known as the [[:Wikipedia:Maximum likelihood|maximum likelihood method]]. | When the model relating ''s'' and ''p'' is known, as well as the error distributions (and the latter may either be Gaussian or not), a more systematic approach to error propagation is provided by a technique known as the [[:Wikipedia:Maximum likelihood|maximum likelihood method]]. | ||
<ref>[ | <ref>Particle Data Group, [[doi:10.1007/s10052-998-0104-x|Eur. Phys. J. C 3, 1 (1998)]]</ref> | ||
This technique is simply the generalisation of standard error propagation to general error distributions (i.e. not limited to Gaussians). | This technique is simply the generalisation of standard error propagation to general error distributions (i.e. not limited to Gaussians). | ||
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<ref>J. van den Berg, ed., Wavelets in Physics (Cambridge University Press, 1999) ISBN 978-0521593113</ref>, [[Biorthogonal decomposition]], | <ref>J. van den Berg, ed., Wavelets in Physics (Cambridge University Press, 1999) ISBN 978-0521593113</ref>, [[Biorthogonal decomposition]], | ||
determination of fractal dimension, mutual information, reconstruction of chaotic attractor, | determination of fractal dimension, mutual information, reconstruction of chaotic attractor, | ||
<ref> | <ref>H. Abarbanel, R. Brown, J. Sidorowich, and L. S. Tsimring, [[doi:10.1103/RevModPhys.65.1331|Rev. Mod. Phys. 65, 1331 (1993)]]</ref> ...). | ||
== Non-Gaussian statistics == | == Non-Gaussian statistics == | ||
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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 [[:Wikipedia:Poisson distribution|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> | <ref>B. van Milligen, I. Classen, and C. Barth, [[doi:10.1063/1.1597951|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 ([[:Wikipedia:Gumbel distribution|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> | <ref>B. van Milligen, R. Sánchez, B. Carreras, V. Lynch, B. LaBombard, M. Pedrosa, C. Hidalgo, B. Gonçalves, and R. Balbín, [[doi:10.1063/1.1884615|Phys. Plasmas 12, 052507 (2005)]]</ref> | ||
The [[:Wikipedia:Log-normal distribution|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> | <ref>F. Sattin, N. Vianello, and M. Valisa, [[doi:10.1063/1.1797671|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 [[: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). | 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 Criticality|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, [[:Wikipedia:Rescaled range|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> | <ref>B. Carreras, B. van Milligen, M. Pedrosa, R. Balbín, C. Hidalgo, D. Newman, E. Sánchez, R. Bravenec, G. McKee, I. García-Cortés, et al., [[doi:10.1063/1.873490|Phys. Plasmas 6, 1885 (1999)]]</ref> | ||
the detection of [[Long-range correlation|long-range time dependence]], | the detection of [[Long-range correlation|long-range time dependence]], | ||
<ref> | <ref>B. Carreras, D. Newman, B. van Milligen, and C. Hidalgo, [[doi:10.1063/1.873192|Phys. Plasmas 6, 485 (1999)]]</ref> | ||
finite-size [[:Wikipedia:Lyapunov exponent|Lyapunov exponents]], | finite-size [[:Wikipedia:Lyapunov exponent|Lyapunov exponents]], | ||
<ref> | <ref>B. Carreras, V. Lynch, and G. Zaslavski, [[doi:10.1063/1.1416180|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. | ||