Imprecise Regression and Regression on Fuzzy Data - A Preliminary Discussion

Abstract

The paper provides a discussion of the possibilistic regression method originally proposed by H. Tanaka. This method has the advantage of allowing the learning of an imprecise model, in the form of an interval-valued function. It may lead to an imprecise model even in presence of precise data, which is satisfactory from a learning point of view. Indeed, finding a precise model that perfectly represents the concept to be learned is illusory, due to the existence of the bias caused by the choice of a modeling representation space, the limited amount of data, and the possibility of missing relevant data. However, what is obtained with possibilistic regression is more an imprecise model than a genuine fuzzy one. The paper illustrates and emphasizes this point on environmental data and suggest two different approaches for learning genuine fuzzy regression models from precise data.

Learning Fitts’ law with imprecise regression

“…We consider now a second example, where we want to compare a task to perform a series of 4 pointing of "easy" targets (difficulty degree 1, 4) with one consisting in pointing only one target (difficulty degree 6,9). As previously, crisp linear model predict equivalent times for the two configurations (around 1340ms).…”

“…Thus, imprecise regression is less sensitive to outliers than possibilistic regression and can be defined for any kind of fuzzy function (interval, triangular, trapezoid, ...). We now define imprecise regression for trapezoid fuzzy function, see [9] for more details. A regression data is a set of m pairs ( − → x i , y i ), 1 ≤ i ≤ m, where − → x i ∈ X is a vector of n input variables and y i ∈ IR is the real output variable.…”

“…The goal of a general framework for imprecise regression [9] is to overcome the learning biases by considering them as factors that have impact on the precision of the models rather than as a boundary to the effectiveness of the learning process. Knowing that the representation of the examples and the hypothesis correspond necessarily to an incomplete view of the world, we will search for imprecise hypotheses that take into account this incompleteness.…”

“…In this paper, we make the assumption that the link between time and difficulty is naturally imprecise. Thus, we propose to use fuzzy regression, and more specifically imprecise regression which has been described in [9], in order to learn a fuzzy function which associates to a difficulty degree a trapezoidal fuzzy set which describes the set of possible times. This allows us to describe both the general tendency of the data and the variation around it.…”

Loi de fitts

“…La première est la régression floue [4] à partir de données elles-mêmes floues, qui ne nous intéresse pas dans le cadre de l'apprentissage de la loi de Fitts. La seconde approche décrit la régression floue à partir de données classiques [14,11]. Cette méthode est nommée régression imprécise.…”