Function parametrization: Difference between revisions

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Function Parametrization (FP) is a technique to provide fast (real-time) construction of system parameters from a set of diverse measurements.
Function Parametrization (also spelt Parameterization) or FP is a technique to provide fast (real-time) construction of system parameters from a set of diverse measurements. It consists of the numerical determination, by statistical regression on a database of simulated states, of  simple functional representations
<ref>B.J. Braams, W. Jilge, and K. Lackner, ''Fast determination of plasma parameters through function parametrization'', Nucl. Fusion '''26''' (1986) 699</ref>
of  parameters characterizing the state of a particular physical system, where the arguments of the functions are statistically independent combinations of diagnostic raw measurements of the system.  
The technique,  developed by H. Wind for the purpose of momentum determination from spark chamber data, <ref>Wind, H., ''Function Parametrization'',
in ''Proceedings of the 1972 CERN Computing and Data Processing School'', CERN 72-21, 1972, pp. 53-106</ref>  <ref>Wind, H., (a) ''Principal component analysis and its application to track finding'',  (b) ''interpolation and function representation'', in ''Formulae and Methods in Experimental Data Evaluation'', Vol. '''3''', European Physical Society, Geneva, 1984</ref> was introduced by B. Braams to plasma physics,
where it was  first applied to the analysis of equilibrium magnetic measurements on the
circular cross-section ASDEX tokamak. <ref>[http://iopscience.iop.org/0029-5515/26/6/001 B.J. Braams, W. Jilge, and K. Lackner, ''Fast determination of plasma parameters through function parametrization'', Nucl. Fusion '''26''' (1986) 699]</ref> It was later extended to the non-circular cross-section ASDEX Upgrade tokamak<ref>[http://www.physics.ucc.ie/~pjm/people/trachtas.htm  P.J. Mc Carthy,  ''An Integrated Data Interpretation System for Tokamak Discharges'', PhD thesis, University College Cork, 1992]</ref>
and the Wendelstein 7-AS stellarator.
<ref>[http://iopscience.iop.org/0029-5515/39/4/308  H.P. Callaghan, P.J. Mc Carthy, J. Geiger, ''Fast equilibrium interpretation on the W7-AS stellarator using Function Parameterization'', Nucl. Fusion '''39''' (1999) 509]</ref>
 


== Method ==
== Method ==


                                                                               
The method of function parameterization (FP) consists of the                   
numerical determination,                                                       
by statistical regression on a database of                                     
simulated states, of  simple functional representations of                     
parameters characterizing the state of a particular physical                 
system, where                                                                 
the arguments of the functions are statistically independent                   
combinations of diagnostic raw measurements                                   
of the system whose geometry is                                               
fixed.  The technique,  developed by H. Wind                                   
for the purpose of momentum determination from spark chamber data             
<ref>    Wind, H.,                                                               
    `Function Parametrization'                                               
in ``Proceedings of the 1972 CERN Computing and Data Processing School'',     
CERN 72--21, 1972, pp.~53--106.} </ref> , <ref>Wind, H.,                                                               
(a)`Principal component analysis and its application to track           
finding',  (b) `interpolation and function representation'                 
in ``Formulae and Methods in Experimental Data Evaluation'',                   
Vol.~3, European Physical Society, Geneva, 1984</ref>,
was                                             
introduced by B. Braams to plasma physics, where its  first                   
application (to the analysis of equilibrium magnetic                           
measurements on ASDEX) together with an admirably succinct                     
mathematical description, appeared in ref. <ref>B.J. Braams, W. Jilge, and K. Lackner, ''Fast determination of plasma parameters through function parametrization'', Nucl. Fusion '''26''' (1986) 699</ref>.


The application of the technique requires that a model exists to compute the response of the measurements (''q'') to variations of the system parameters (''p''), i.e. the mapping ''q = M(p)'' is known.
The application of the technique requires that a model exists to compute the response of the measurements (''q'') to variations of the system parameters (''p''), i.e. the mapping ''q = M(p)'' is known.
Line 36: Line 17:


The fast reconstruction of the system parameters is obtained by computing the inverse of the mapping ''M''. To do so, the parameters ''p'' are varied over a range corresponding to the expected variation in actual experiments, the corresponding ''q'' are obtained, and the set of ''(p,q)'' data are stored in a database. This database is then subjected to a statistical analysis in order to recover the inverse of ''M''. This analysis is typically a [[:Wikipedia:Principal Component Analysis|Principal Component Analysis]]. This procedure is also amenable to a rather detailed error analysis, so that errors in the recovered parameters ''p'' for the interpretation of actual data ''q'' can be obtained.
The fast reconstruction of the system parameters is obtained by computing the inverse of the mapping ''M''. To do so, the parameters ''p'' are varied over a range corresponding to the expected variation in actual experiments, the corresponding ''q'' are obtained, and the set of ''(p,q)'' data are stored in a database. This database is then subjected to a statistical analysis in order to recover the inverse of ''M''. This analysis is typically a [[:Wikipedia:Principal Component Analysis|Principal Component Analysis]]. This procedure is also amenable to a rather detailed error analysis, so that errors in the recovered parameters ''p'' for the interpretation of actual data ''q'' can be obtained.
<ref name=RTP>B.Ph. van Milligen, N.J. Lopes Cardozo, ''Function Parametrization: a fast inverse mapping method'', Comp. Phys. Commun. '''66''' (1991) 243</ref>
<ref name=RTP>[http://dx.doi.org/10.1016/0010-4655(91)90073-T B.Ph. van Milligen, N.J. Lopes Cardozo, ''Function Parametrization: a fast inverse mapping method'', Comp. Phys. Commun. '''66''' (1991) 243]</ref>


== Applications ==
== Applications ==


* RTP <ref name=RTP></ref>
* RTP <ref name=RTP></ref>
* TEXTOR <ref>B.Ph. van Milligen et al., ''Application of Function Parametrization to the analysis of polarimetry and interferometry data in TEXTOR'', Nucl. Fusion '''31''' (1991) 309</ref>
* TEXTOR <ref>[http://iopscience.iop.org/0029-5515/31/2/007 B.Ph. van Milligen et al., ''Application of Function Parametrization to the analysis of polarimetry and interferometry data in TEXTOR'', Nucl. Fusion '''31''' (1991) 309]</ref>
* ASDEX-UG <ref>[http://dx.doi.org/10.1016/S0920-3796(00)00109-5 W. Schneider, P.J. McCarthy, et al., ''ASDEX upgrade MHD equilibria reconstruction on distributed workstations'', Fusion Engineering and Design '''48''', Issues 1-2 (2000) 127-134]</ref>
* ASDEX-UG <ref>[http://dx.doi.org/10.1016/S0920-3796(00)00109-5 W. Schneider, P.J. Mc Carthy, et al., ''ASDEX upgrade MHD equilibria reconstruction on distributed workstations'', Fusion Engineering and Design '''48''', Issues 1-2 (2000) 127-134]</ref>
* Wendelstein 7-X <ref>[http://dx.doi.org/10.1088/0029-5515/44/11/003 A. Sengupta, P.J. McCarthy, et al., ''Fast recovery of vacuum magnetic configuration of the W7-X stellarator using function parametrization and artificial neural networks'', Nucl. Fusion '''44''' (2004) 1176]</ref>
* Wendelstein 7-X <ref>[http://dx.doi.org/10.1088/0029-5515/44/11/003 A. Sengupta, P.J. Mc Carthy, et al., ''Fast recovery of vacuum magnetic configuration of the W7-X stellarator using function parametrization and artificial neural networks'', Nucl. Fusion '''44''' (2004) 1176]</ref>


== Alternatives ==
== Alternatives ==


* [[Bayesian data analysis]], which allows non-Gaussian error distributions.
* Neural networks. Like with FP, most of the computational effort is concentrated in an analysis phase (network ''training''), before the actual application to data. Therefore, this method is fast and suited for real-time applications. With FP, non-linear dependencies are limited by the degree of the polynomial expansions used, whereas neural networks allow more general non-linear dependencies, in principle.
* Neural networks.
* [[Bayesian data analysis]], which allows non-Gaussian error distributions and complex data dependencies. A very powerful method but not fast due to the need for maximization (not suited for real-time applications).


== References ==
== References ==
<references />
<references />

Latest revision as of 13:44, 24 April 2011

Function Parametrization (also spelt Parameterization) or FP is a technique to provide fast (real-time) construction of system parameters from a set of diverse measurements. It consists of the numerical determination, by statistical regression on a database of simulated states, of simple functional representations of parameters characterizing the state of a particular physical system, where the arguments of the functions are statistically independent combinations of diagnostic raw measurements of the system. The technique, developed by H. Wind for the purpose of momentum determination from spark chamber data, [1] [2] was introduced by B. Braams to plasma physics, where it was first applied to the analysis of equilibrium magnetic measurements on the circular cross-section ASDEX tokamak. [3] It was later extended to the non-circular cross-section ASDEX Upgrade tokamak[4] and the Wendelstein 7-AS stellarator. [5]


Method

The application of the technique requires that a model exists to compute the response of the measurements (q) to variations of the system parameters (p), i.e. the mapping q = M(p) is known. In doing so, all functional dependencies are parametrized (hence the name of the technique), e.g., spatially dependent functions f(r) are given in terms of an parametric expansion (such as a polynomial), and the corresponding parameters are included in the vector p.

The fast reconstruction of the system parameters is obtained by computing the inverse of the mapping M. To do so, the parameters p are varied over a range corresponding to the expected variation in actual experiments, the corresponding q are obtained, and the set of (p,q) data are stored in a database. This database is then subjected to a statistical analysis in order to recover the inverse of M. This analysis is typically a Principal Component Analysis. This procedure is also amenable to a rather detailed error analysis, so that errors in the recovered parameters p for the interpretation of actual data q can be obtained. [6]

Applications

Alternatives

  • Neural networks. Like with FP, most of the computational effort is concentrated in an analysis phase (network training), before the actual application to data. Therefore, this method is fast and suited for real-time applications. With FP, non-linear dependencies are limited by the degree of the polynomial expansions used, whereas neural networks allow more general non-linear dependencies, in principle.
  • Bayesian data analysis, which allows non-Gaussian error distributions and complex data dependencies. A very powerful method but not fast due to the need for maximization (not suited for real-time applications).

References

  1. ↑ Wind, H., Function Parametrization, in Proceedings of the 1972 CERN Computing and Data Processing School, CERN 72-21, 1972, pp. 53-106
  2. ↑ Wind, H., (a) Principal component analysis and its application to track finding, (b) interpolation and function representation, in Formulae and Methods in Experimental Data Evaluation, Vol. 3, European Physical Society, Geneva, 1984
  3. ↑ B.J. Braams, W. Jilge, and K. Lackner, Fast determination of plasma parameters through function parametrization, Nucl. Fusion 26 (1986) 699
  4. ↑ P.J. Mc Carthy, An Integrated Data Interpretation System for Tokamak Discharges, PhD thesis, University College Cork, 1992
  5. ↑ H.P. Callaghan, P.J. Mc Carthy, J. Geiger, Fast equilibrium interpretation on the W7-AS stellarator using Function Parameterization, Nucl. Fusion 39 (1999) 509
  6. ↑ 6.0 6.1 B.Ph. van Milligen, N.J. Lopes Cardozo, Function Parametrization: a fast inverse mapping method, Comp. Phys. Commun. 66 (1991) 243
  7. ↑ B.Ph. van Milligen et al., Application of Function Parametrization to the analysis of polarimetry and interferometry data in TEXTOR, Nucl. Fusion 31 (1991) 309
  8. ↑ W. Schneider, P.J. Mc Carthy, et al., ASDEX upgrade MHD equilibria reconstruction on distributed workstations, Fusion Engineering and Design 48, Issues 1-2 (2000) 127-134
  9. ↑ A. Sengupta, P.J. Mc Carthy, et al., Fast recovery of vacuum magnetic configuration of the W7-X stellarator using function parametrization and artificial neural networks, Nucl. Fusion 44 (2004) 1176
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