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Unformatted text preview: Fuzzy Set Theory by ShinYun Wang Before illustrating the fuzzy set theory which makes decision under uncertainty, it is important to realize what uncertainty actually is. Uncertainty is a term used in subtly different ways in a number of fields, including philosophy , statistics , economics , finance , insurance , psychology , engineering and science . It applies to predictions of future events, to physical measurements already made, or to the unknown . Uncertainty must be taken in a sense radically distinct from the familiar notion of risk, from which it has never been properly separated.... The essential fact is that 'risk' means in some cases a quantity susceptible of measurement, while at other times it is something distinctly not of this character; and there are far reaching and crucial differences in the bearings of the phenomena depending on which of the two is really present and operating.... It will appear that a measurable uncertainty, or 'risk' proper, as we shall use the term, is so far different from an immeasurable one that it is not in effect an uncertainty at all. What is relationship between uncertainty, probability, vagueness and risk? Risk is defined as uncertainty based on a well grounded (quantitative) probability . Formally, Risk = (the probability that some event will occur) X (the consequences if it does occur). Genuine uncertainty, on the other hand, cannot be assigned such a (well grounded) probability. Furthermore, genuine uncertainty can often not be reduced significantly by attempting to gain more information about the phenomena in question and their causes. Moreover the relationship between uncertainty, accuracy, precision, standard deviation, standard error, and confidence interval is that the uncertainty of a measurement is stated by giving a range of values which are likely to enclose the true value. This may be denoted by error bars on a graph, or as value uncertainty, or as decimal fraction (uncertainty). Often, the uncertainty of a measurement is found by repeating the measurement enough times to get a good estimate of the standard deviation of the values. Then, any single value has an uncertainty equal to the standard deviation. However, if the values are averaged and the mean is reported, then the averaged measurement has uncertainty equal to the standard error which is the standard deviation divided by the square root of the number of measurements. When the uncertainty represents the standard error of the measurement, then about 68.2% of the time, the true value of the measured quantity falls within the stated uncertainty range. Therefore no matter how accurate our measurements are, some uncertainty always remains. The possibility is the degree that thing happens, but the probability is the probability that things be happen or not. So the methods that we deal with uncertainty are to avoid the uncertainty, statistical mechanics and fuzzy set (Zadeh in 1965)....
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 Spring '11
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