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Function parametrization: Difference between revisions
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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. | ||
In doing so, all functional dependencies are parametrized (hence the name of the technique), | 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 | e.g., spatially dependent functions ''f(r)'' are given in terms of an parametric expansion (such as a [[:Wikipedia:Polynomial|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 [[: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. | ||