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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 Rescaled range ： ウィキペディア英語版 Rescaled range The rescaled range is a statistical measure of the variability of a time series introduced by the British hydrologist Harold Edwin Hurst (1880–1978). Its purpose is to provide an assessment of how the apparent variability of a series changes with the length of the time-period being considered. The rescaled range is calculated from dividing the range of the values exhibited in a portion of the time series by the standard deviation of the values over the same portion of the time series. For example, consider a time series which has a range, R, of 12 - 2 = 10. Its standard deviation, s, is 3.46, so the rescaled range is R/s = 2.89. If we consider the same time series, but increase the number of observations of it, the rescaled range will generally also increase. The increase of the rescaled range can be characterized by making a plot of the logarithm of R/s vs. the logarithm of n. The slope of this line gives the Hurst exponent, H. If the time series is generated by a random walk (or a Brownian motion process) it has the value of H =1/2. Many physical phenomena that have a long time series suitable for analysis exhibit a Hurst exponent greater than 1/2. For example, observations of the height of the Nile River measured annually over many years gives a value of H = 0.77. Several researchers (including Peters, 1991) have found that the prices of many financial instruments (such as currency exchange rates, stock values, etc.) also have H > 1/2. This means that they have a behavior that is distinct from a random walk, and therefore the time series is not generated by a stochastic process that has the nth value independent of all of the values before this. According to model of Fractional Brownian motion this is referred to as long memory of positive linear autocorrelation. However it has been shown that this measure is correct only for linear evaluation: complex nonlinear processes with memory need additional descriptive parameters. Several studies using Lo's modified rescaled range statistic have contradicted Peters' results as well. ==Calculation== :The Rescaled Range is calculated for a time series, $X=X\_1,X\_2,\backslash dots,\; X\_n\; \backslash ,$, as follows: # Calculate the mean #: $m=\backslash frac\; \backslash sum\_^\; X\_i\; \backslash ,$ # Create a mean adjusted series #: $Y\_t=X\_-m\; \backslash text\; t=1,2,\; \backslash dots\; ,n\; \backslash ,$ # Calculate the cumulative deviate series Z; #:$Z\_t=\; \backslash sum\_^\; Y\_\; \backslash text\; t=1,2,\; \backslash dots\; ,n\; \backslash ,$ # Create a range series R; #:$R\_t\; =\; \backslash max\backslash left\; (Z\_1,\; Z\_2,\; \backslash dots,\; Z\_t\; \backslash right\; )-\; \backslash min\backslash left\; (Z\_1,\; Z\_2,\; \backslash dots,\; Z\_t\; \backslash right\; )\; \backslash text\; t=1,2,\; \backslash dots,\; n\; \backslash ,$ # Create a standard deviation series S; #:$S\_=\; \backslash sqrt\; \backslash sum\_^\backslash left\; (\; X\_\; -\; m(t)\; \backslash right\; )^\}\; \backslash text\; t=1,2,\; \backslash dots\; ,n\; \backslash ,$ #:Where ''m(t)'' is the mean for the time series values through time $t$ $X\_1,X\_2,\; \backslash dots,\; X\_t\; \backslash ,$ # Calculate the rescaled range series (R/S) #:$\backslash left\; (\; R/S\; \backslash right\; )\_\; =\; \backslash frac\}\; \backslash text\; t=1,2,\; \backslash dots,\; n\; \backslash ,$ Lo (1991) advocates adjusting the standard deviation $S$ for the expected increase in range $R$ resulting from short-range autocorrelation in the time series.〔 This involves replacing $S$ by $\backslash hat$, which is the square root of $\backslash hat^2\; =\; S^2\; +\; 2\; \backslash sum\_^\; \backslash left(\; 1\; -\; \backslash frac\; \backslash right)\; C(j),$ where $q$ is some maximum lag over which short-range autocorrelation might be substantial and $C(j)$ is the sample autocovariance at lag $j$. Using this adjusted rescaled range, he concludes that stock market return time series show no evidence of long-range memory. 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「Rescaled range」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.