Some Considerations on Fuzzy Least-Squares

In Geodesy, parameter estimation based on the least-squares principle is a common tool for the solution of data analysis problems. It is assumed, at least implicitly, that the unavoidable observation errors are exclusively stochastic with zero expectation. The corresponding variance-covariance matrix (vcm) of the estimated parameters is then computed from the observations’ vcm just by means of variance propagation. However, the complete error budget of the observation process comprises additional, non-stochastic types of observation errors like, e.g., imprecision. Imprecision summarizes effects due to the imperfect knowledge about the observation setup. Fuzzy set theory and fuzzy data analysis supply adequate techniques to model and to handle imprecision. Since in geodetic data analysis both stochasticity and imprecision of the observations may be relevant, approaches for their combination are needed. Techniques from fuzzy-theory are introduced in this paper for the handling of observation imprecision. The joint treatment of observation stochasticity and imprecision is discussed. Numerical examples are given.