Breakthrough in Interval Data Fitting II. From Ranges to Means and Standard Deviations

Interval analysis, when applied to the so called problem of experimental data fitting, appears to be still in its infancy. Sometimes, partly because of the unrivaled reliability of interval methods, we do not obtain any results at all. Worse yet, if this happens, then we are left in the state of complete ignorance concerning the unknown parameters of interest. This is in sharp contrast with widespread statistical methods of data analysis. In this paper I show the connections between those two approaches: how to process experimental data rigorously, using interval methods, and present the final results either as intervals (guaranteed, rigorous results) or in a more familiar probabilistic form: as a mean value and its standard deviation. This article is a companion paper to [1] and is meant to be its extension, but otherwise it is self-contained. This is why we don’t repeat everything here, except for the most important thing: a correct way to bound the distances between uncertain experimental values and the corresponding theoretical predictions of thereof. 1 The goals of experimental data processing The problem in front of us may be stated as follows. We have N experimental data points, labelled as m1, . . . ,mN (measurements), each one obtained in different conditions xj , j = 1, . . . , N , (called environments from now on), so that each mj = mj(xj). In addition, we have a theory, T , predicting the behavior of the investigated phenomenon in various environments. T is characterized by k (k < N) unknown parameters, p1, . . . pk, so formally we can write: T (p1, . . . ,pk,xj) = tj . In words: when the (yet) unknown parameters have values p1, . . . ,pk respectively, and the environment state is xj , the T predicts the observed outcome as tj . All quantities typeset in boldface are interval objects, usually just intervals, but they may be interval vectors as well. Contrary to the earlier theoretical attempts (for the relevant references see the literature cited in [1]) we no longer insist that experimental intervals mj are guaranteed, i.e. that they contain the true values with probability equal exactly to 1, nevertheless they may have this property. There are essentially two goals addressed by uncertain data processing: • to determine the values of interesting parameters, p1, . . . pk, best of all together with their uncertainties, or • to test whether a given model of phenomenon under study (theory T ) is adequate. We will not go into hypothesis testing but instead will concentrate on finding unknown parameters given the uncertain experimental information. 2 How do we find ‘best fitted’ parameters? In [1] we put forward the idea that the so called ‘best fits’ should be based on the distance between measured and theoretical values. In one dimension, when we compare a single result of measurement with the predicted one, and at least one of those quantities is an interval, the mathematically correct distance is the one valid in the interval space IR. Starting with the familiar Moore-Hausdorff distance [2], usually written as d (a, b) = max (