Fuzzy Modeling of Survival Function from Interval or Censored Observations

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In medicine, biology, or actuarial theory the so called survival function is often used, i.e., a probability P(X ＞ x) that the time X from the beginning of an initial event to a final one will not exceed x. The observed data are mostly right-censored what causes that the observed variable is T = min(X, Z) where Z is the so called censoring variable. The celebrated and widely used nonparametric estimator of survival function is the Kaplan-Meier estimator which suffers from a disadvantage that it is stepwise and therefore it may happen that in considerably distinct points the values of the estimator may be equal.Rossa and Zielinski proposed a local smoothing of the Kaplan-Meier estimator based on an approximation by means of the piecewise Weibull survival function. They have shown that Mean Square Error and Mean Absolute Deviation of the smoothed estimator have been significantly smaller. Moreover, the Weibull approximation method has appeared to be a quite simple algorithm based on logarithmic transformations of the data and by applying the standard estimating procedure of the simple regression model y=ax+b. However, the censoring variables introduce uncertainty, which can be treated as the source of fuzziness. Thus, the estimation problem can be transferred into the fuzzy analysis. Another estimator leading to fuzzy model is the semi-parametric model proposed by Rossa. Parametric part of the estimator contains formulae based on estimates of the Weibull parameters, whereas the censored observations yield uncertainty.The proposed approaches have a general character. As it has been pointed out in Rossa and Zielinski, the Kaplan-Meier survival function can be approximated with a prescribed level of accuracy by a piecewise Weibull survival function. Using double logarithms of that Weibull pieces we get the piecewise linear intervals. All the intervals can have the same slop parameters or can have changing points separating different slops. Thus, the basic tool in the approach is the fuzzy linear regression. For simplicity, it will be assumed the fuzzy regression coefficients have symmetric triangular membership functions.