Given a set of n variables spanning an n-dimensional search space. The elimination of one of these variables corresponds to the projection of the n-dimensional search space onto an (n-1)-dimensional search space. RA analyzes, if the original n-dimensional search space can be reconstructed from any subset of the available (n-1)-dimensional search spaces. If this can be done, the (n-1)-dimensional subspaces that are being used in the reconstruction can be considered as equivalent to the original n-dimensional space.
In this way, it is possible to determine a suitable hypothesis for the internal structure of a system under study, a hypothesis that suggests, which variables should best be used in the identification of behavioral models of the subsystems.
Just like FIR, also RA methodology is derived from the concepts of general system theory. Unfortunately, the RA algorithms are characterized by a very high computational complexity. There is still much work to do, until RA can be applied successfully to large-scale systems in a fully automated fashion. In particular, suboptimal search techniques need to be defined that would keep the computational burden within acceptable limits.
RA assumes that the behavior of a given system has been characterized by means of inductive reasoning. In this process, each combination of discrete input variable values is linked to one or more discrete output variable values. These associations are thus mappings from the space of the input variables, the so-called input space, to the space of the output variable, the so-called output space.
If the mappings from the input space to the output space are not unique, each mapping is associated in addition with an observation frequency function that determines the relative frequency of occurrence of a given mapping from the input space to the output space.
RA attempts to project these mappings, mappings that characterize the behavior of the system, onto subspaces of the input space, whereby the mappings from the reconstructed input space to the output space should exhibit as few distortions as possible caused by the various projections and recombinations that were carried out, and this includes the reconstructed values of the observation frequencies.
In crisp reconstruction analysis (CRA), the observation frequencies are being interpreted as experimentally determined probability values in the sense of experimental statistics, and these values are propagated through the projections and recombinations in accordance with the laws of probability theory.
In 1993, Adelinde (Lin) Uhrmacher, during her postdoctoral research visit with the University of Arizona, developed and implemented a fuzzy extension of RA methodology. In fuzzy reconstruction analysis (FRA), the observation frequencies are being interpreted as confidence values in the sense of fuzzy logic, and these values are propagated through the projections and recombinations in accordance with the laws of possibility theory.
It was discovered later that the fuzzy interpretation of reconstruction analysis carries important advantages, since fuzzy reconstructions are much less plagued by distortions than their crisp cousins.