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Many measures of similarity among fuzzy sets have been proposed in the literature, and some have been incorporated into linguistic approximation procedures. The motivations behind these measures are both geometric and set-theoretic. We briefly review 19 such measures and compare their performance in a behavioral experiment. For crudely categorizing pairs of fuzzy concepts as either "'similar" or "'dissimilar, "" all measures performed well. For distinguishing between degrees of similarity or dissimilarity, certain measures were clearly superior and others were clearly inferior; for a few subjects, however, none of the distance measures adequately modeled their similarity judgments. Measures that account for ordering on the base variable proved to be more highly correlated with subjects" actual similarity judgments. And, surprisingly, the best measures were ones that focus on only one "'slice" of the membership function. Such measures are easiest to compute and may provide insight into the way humans judge similarity among fuzzy concepts.

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The block bootstrap for time series consists in randomly resampling blocks of consecutive v alues of the given data and aligning these blocks into a bootstrap sample. Here we suggest improving the performance of this method by aligning with higher likelihood those blocks which match at their ends. This is achieved by resampling the blocks according to a Markov c hain whose transitions depend on the data. The matching algorithms we propose take some of the dependence structure of the data into account. They are based on a kernel estimate of the conditional lag one distribution or on a tted autoregression of small order. Numerical and theoretical analyses in the case of estimating the variance of the sample mean show that matching reduces bias and, perhaps unexpectedly, has relatively little e ect on variance. Our theory extends to the case of smooth functions of a vector mean.

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