Uncertainty of data, fuzzy membership functions, and multilayer perceptrons.

Probability that a crisp logical rule applied to imprecise input data is true may be computed using fuzzy membership function (MF). All reasonable assumptions about input uncertainty distributions lead to MFs of sigmoidal shape. Convolution of several inputs with uniform uncertainty leads to bell-shaped Gaussian-like uncertainty functions. Relations between input uncertainties and fuzzy rules are systematically explored and several new types of MFs discovered. Multilayered perceptron (MLP) networks are shown to be a particular implementation of hierarchical sets of fuzzy threshold logic rules based on sigmoidal MFs. They are equivalent to crisp logical networks applied to input data with uncertainty. Leaving fuzziness on the input side makes the networks or the rule systems easier to understand. Practical applications of these ideas are presented for analysis of questionnaire data and gene expression data.