On the possibilistic approach to linear regression models involving uncertain, indeterminate or interval data

We consider linear regression models where both input data (the values of independent variables) and output data (the observations of the dependent variable) are affected by loss of information caused by uncertainty, indeterminacy, rounding or censoring. Some or all of the crisp data, which are unavailable, are replaced by known intervals. We introduce a possibilistic generalization of the least squares estimator, so called OLS-set for the interval model. Investigation of the OLS set allows us to quantify whether the replacement of crisp values by interval values can have a significant effect on the usual OLS estimator; or, in other words, whether the loss of information caused by replacement of crisp values by intervals can be considered as serious. We show that in the model with both interval input and output data in general there is no computationally feasible way to describe the OLS set reasonably (assuming P ̸= NP ). Nevertheless, we show and compare various approximation methods that could be useful in particular cases. We also focus on restricted versions of the general interval linear regression model. We show that in the crisp input – interval output model, the OLS set is a convex polyhedron of a special structure. We present both exact and approximate methods for description of the OLS set, in particular interval enclosures and ellipsoidal enclosures. We also present a meta-algorithm, called Reduction and Reconstruction Recursion, which can be used for computation of vertex and facet description of the OLS set. We discuss special cases of the regression model, e.g. a model with repeated observations and a model where we want to estimate a single regression parameter. We illustrate the approaches by examples.