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A confidence band is used in statistical analysis to represent the uncertainty in an estimate of a curve or function based on limited or noisy data. Similarly, a prediction band is used to represent the uncertainty about the value of a new data-point on the curve, but subject to noise. Confidence and prediction bands are often used as part of the graphical presentation of results of a regression analysis. Confidence bands are closely related to confidence intervals, which represent the uncertainty in an estimate of a single numerical value. "As confidence intervals, by construction, only refer to a single point, they are narrower (at this point) than a confidence band which is supposed to hold simultaneously at many points."〔p.65 in W. Härdle, M. Müller, S. Sperlich, A. Werwatz (2004), Nonparametric and Semiparametric Models, Springer, ISBN 3540207228 (),()〕 ==Pointwise and simultaneous confidence bands== Suppose our aim is to estimate a function ''f(x)''. For example, ''f(x)'' might be the proportion of people of a particular age ''x'' who support a given candidate in an election. If ''x'' is measured at the precision of a single year, we can construct a separate 95% confidence interval for each age. Each of these confidence intervals covers the corresponding true value ''f(x)'' with probability 0.95. Taken together, these confidence intervals constitute a ''95% pointwise confidence band'' for ''f(x)''. In mathematical terms, a pointwise confidence band with coverage probability 1−α satisfies the following condition separately for each value of ''x'': : where is the point estimate of ''f''(''x''). The ''simultaneous coverage probability'' of a collection of confidence intervals is the probability that all of them cover their corresponding true values simultaneously. In the example above, the simultaneous coverage probability is the probability that the intervals for x=18,19,... all cover their true values (assuming that 18 is the youngest age at which a person can vote). If each interval individually has coverage probability 0.95, the simultaneous coverage probability is generally less than 0.95. A ''95% simultaneous confidence band'' is a collection of confidence intervals for all values ''x'' in the domain of ''f(x)'' that is constructed to have simultaneous coverage probability 0.95. In mathematical terms, a simultaneous confidence band with coverage probability 1−α satisfies the following condition: : In nearly all cases, a simultaneous confidence band will be wider than a pointwise confidence band with the same coverage probability. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Confidence and prediction bands」の詳細全文を読む スポンサード リンク
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