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a b s t r a c tThis paper, introduces the nearest weighted interval and point approximations of a fuzzy number, and then suggests weighted possibilistic moments about these points of fuzzy numbers. The possibilistic moments play an important role in fuzzy sets and systems, specifically in physics, mathematics and statistics. We provide the definition of the moments of fuzzy numbers as well as the definition of moments in probability theory; some of their applications are mentioned.

Abstractobjectives Although hidden Markov model (HMM) is known as a powerful tool for the detection of epidemics based on the historical data, the frequent use of such a model poses some limitation especially when decision-making is required for new observations. This study was aimed to address a warning threshold for monitoring the weekly incidences of tuberculosis as an alternative to HMM.methods We extracted the weekly counts of newly diagnosed patients with sputum smear-positive pulmonary TB from 2005 to 2011 nationwide. To detect unexpected incidences of the disease, two approaches: Serfling and HMM, were applied in presence/absence of linear, seasonal and autoregressive components. Models were subsequently evaluated in terms of goodness of fit, and their results were compared in detection of the disease phases. Then, multiple hypothetical thresholds were constructed based on the estimate of models and the optimal one was revealed through ROC curve analysis.results Findings from both adjusted R-square (R 2 ) and Bayesian information criterion (BIC) presented a higher goodness of fit for periodic autoregressive HMM (BIC = À1323.6;R 2 ¼ 0:74) than other models. According to ROC analysis, better values for both Youden's index and area under curve (0. 96 and 0. 98 respectively) were obtained by the threshold based on the estimate of periodic autoregressive model.conclusions As the optimal threshold presented in this study is simple in concept and has no limitation in practice, especially for monitoring new observations, we would recommend such a threshold to be used for monitoring of TB incidence data in the surveillance system.

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