# Rescaled Range

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## Rescaled Range (R/S) Analysis

The analysis developed by H.E. Hurst to determine long-memory effects and fractional Brownian motion. Rescaled range analysis measures how the distance covered by a particle increases as we look at longer and longer time scales. For Brownian motion, the distance covered increases with the square root of time. A series which increases at a different rate is not random. See: Anti-persistence, Fractional Brownian Motion, Hurst Exponent, Persistence, Joseph Effect, Noah Effect.
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(1996) in a study of 22 international stock markets, which includes the French, Dutch and Belgian markets, and in which the modified rescaled range analysis of Lo (1991) and the rescaled variance ratio test of Lo & MacKinlay (1988) are employed to monthly returns from 1962 to 1990.
Using the standard deviation of the original return series within each box, one can calculate the ratio of ranges and standard rescaled range series (R/S) deviations.
In the absence of any compelling a priori reason to either exclude any available sectoral indices or any available price information on those indices, the present study applies three fractal analysis techniques, viz., the Classical Rescaled Range (R/S), Wavelets, and Roughness-Length relationship methods, to the complete range of price information available for all sectoral indices on the BSE and NSE, ending on August 31, 2017.
For the time series {[x.sub.k]} (k = 1,2, 3, ..., n), given a sample of observation, [x.sub.1], [x.sub.2], [x.sub.3], ..., xn, the definition of the modified rescaled range theory is as follows.
Band name Original scaling range Rescaled range Blue [0, 47] [0, 12] [47, 116] [12, 138] [116, 255] [138, 255] Hybrid green [0, 38] [0, 13] (F = 0.13) [38, 107] [13, 130] [107, 255] [130, 255] Red [0, 33] [1- 14] [33, 100] [14, 124] [100, 255] [124, 255]
Residuals of this estimated fundamental time series are then tested for possible speculative deviations using a Hamilton regime switching test and a rescaled range Hurst coefficient test, with a further test for nonlinearity beyond the ARCH effects using the BDS statistic.
In this study, we estimate the fractal dimension of price returns and test the Efficient Market Hypothesis (EMH), employing rescaled range analysis in order to use fewer assumptions about the underlying system.
A variation at around 1990 is then corroborated by wavelet variance analysis, which is used to detect changes in runoff periodicity [18-22], Kendall's rank test (a nonparametric method for detecting or testing trends), and rescaled range analysis (a nonlinear technique for evaluating the persistence of apparent trends), which are widely used in hydrology (e.g., [23-28]).
In the theory of chaotic systems different methods are used , among others such as: correlation dimension ; Kolmogorov entropy, Lapunov exponent, fractal dimension ; Brock-Dechert-Scheinkman's test, rescaled range analysis [8,9,10,12].
Hurst (1951) developed a rescaled range statistics (R/S) to study long-range dependence in river flows.
Some studies (such as those by Peters, 1989; 1996) using classical rescaled range (or R/S) analysis have found evidence of long-term memory.
To quantify the intensity of long range correlations, the local randomness (Hurst) exponents of P(t), as well as of each time record [V.sub.n](t), were determined by five methods: the rescaled range analysis (R/[sigma]), the roughness length (SD), the variogram (V), the power-spectrum (P), and the wavelet methods (W[X](a)).

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