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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hurst exponent ： ウィキペディア英語版 Hurst exponent The Hurst exponent is used as a measure of long-term memory of time series. It relates to the autocorrelations of the time series, and the rate at which these decrease as the lag between pairs of values increases. Studies involving the Hurst exponent were originally developed in hydrology for the practical matter of determining optimum dam sizing for the Nile river's volatile rain and drought conditions that had been observed over a long period of time. The name "Hurst exponent", or "Hurst coefficient", derives from Harold Edwin Hurst (1880–1978), who was the lead researcher in these studies; the use of the standard notation ''H'' for the coefficient relates to his name also. In fractal geometry, the generalized Hurst exponent has been denoted by ''H'' or ''Hq'' in honor of both Harold Edwin Hurst and Ludwig Otto Hölder (1859–1937) by Benoît Mandelbrot (1924–2010). ''H'' is directly related to fractal dimension, ''D'', and is a measure of a data series' "mild" or "wild" randomness. The Hurst exponent is referred to as the "index of dependence" or "index of long-range dependence". It quantifies the relative tendency of a time series either to regress strongly to the mean or to cluster in a direction.〔Torsten Kleinow (2002)( Testing Continuous Time Models in Financial Markets ), Doctoral thesis, Berlin 〕 A value ''H'' in the range 0.5–1 indicates a time series with long-term positive autocorrelation, meaning both that a high value in the series will probably be followed by another high value and that the values a long time into the future will also tend to be high. A value in the range 0 – 0.5 indicates a time series with long-term switching between high and low values in adjacent pairs, meaning that a single high value will probably be followed by a low value and that the value after that will tend to be high, with this tendency to switch between high and low values lasting a long time into the future. A value of ''H''=0.5 can indicate a completely uncorrelated series, but in fact it is the value applicable to series for which the autocorrelations at small time lags can be positive or negative but where the absolute values of the autocorrelations decay exponentially quickly to zero. This in contrast to the typically power law decay for the 0.5 < ''H'' < 1 and 0 < ''H'' < 0.5 cases. ==Definition== The Hurst exponent, ''H'', is defined in terms of the asymptotic behaviour of the rescaled range as a function of the time span of a time series as follows; :$\backslash operatorname\; \backslash left\; (\backslash frac\; \backslash right)=C\; n^H\; \backslash text\; n\; \backslash to\; \backslash infty\; \backslash ,\; ,$ where; * $R(n)$ is the range of the first $n$ values, and $S(n)$ is their standard deviation * $\backslash operatorname\; \backslash left\; (\backslash right)\; \backslash ,$ is the expected value * $n$ is the time span of the observation (number of data points in a time series) * $C$ is a constant. 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「Hurst exponent」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.