Rescaled range

<h2 id="calculation">Calculation</h2>
The Rescaled Range is calculated for a time series,  
  
    
      
        X
        =
        
          X
          
            1
          
        
        ,
        
          X
          
            2
          
        
        ,
        …
        ,
        
          X
          
            n
          
        
        
      
    
    {\displaystyle X=X_{1},X_{2},\dots ,X_{n}\,}
  
, as follows:<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
<ol><li>Calculate the <a href="/facts/Mean/swcEd4Pg">mean</a>

m
        =
        
          
            1
            n
          
        
        
          ∑
          
            i
            =
            1
          
          
            n
          
        
        
          X
          
            i
          
        
        
      
    
    {\displaystyle m={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\,}
  
</li>
<li>Create a mean adjusted series

Y
          
            t
          
        
        =
        
          X
          
            t
          
        
        −
        m
        
           for 
        
        t
        =
        1
        ,
        2
        ,
        …
        ,
        n
        
      
    
    {\displaystyle Y_{t}=X_{t}-m{\text{  for }}t=1,2,\dots ,n\,}
  
</li>
<li>Calculate the cumulative deviate series Z;

Z
          
            t
          
        
        =
        
          ∑
          
            i
            =
            1
          
          
            t
          
        
        
          Y
          
            i
          
        
        
           for 
        
        t
        =
        1
        ,
        2
        ,
        …
        ,
        n
        
      
    
    {\displaystyle Z_{t}=\sum _{i=1}^{t}Y_{i}{\text{  for }}t=1,2,\dots ,n\,}
  
</li>
<li>Create a range series R;

R
          
            t
          
        
        =
        max
        
          (
          
            
              Z
              
                1
              
            
            ,
            
              Z
              
                2
              
            
            ,
            …
            ,
            
              Z
              
                t
              
            
          
          )
        
        −
        min
        
          (
          
            
              Z
              
                1
              
            
            ,
            
              Z
              
                2
              
            
            ,
            …
            ,
            
              Z
              
                t
              
            
          
          )
        
        
           for 
        
        t
        =
        1
        ,
        2
        ,
        …
        ,
        n
        
      
    
    {\displaystyle R_{t}=\max \left(Z_{1},Z_{2},\dots ,Z_{t}\right)-\min \left(Z_{1},Z_{2},\dots ,Z_{t}\right){\text{  for }}t=1,2,\dots ,n\,}
  
</li>
<li>Create a <a href="/facts/Standard_deviation/qSug9BRl">standard deviation</a> series S;

S
          
            t
          
        
        =
        
          
            
              
                1
                t
              
            
            
              ∑
              
                i
                =
                1
              
              
                t
              
            
            
              
                (
                
                  
                    X
                    
                      i
                    
                  
                  −
                  m
                  (
                  t
                  )
                
                )
              
              
                2
              
            
          
        
        
           for 
        
        t
        =
        1
        ,
        2
        ,
        …
        ,
        n
        
      
    
    {\displaystyle S_{t}={\sqrt {{\frac {1}{t}}\sum _{i=1}^{t}\left(X_{i}-m(t)\right)^{2}}}{\text{  for }}t=1,2,\dots ,n\,}

Where <i>m(t)</i> is the mean for the time series values through time 
  
    
      
        t
      
    
    {\displaystyle t}
  
 
  
    
      
        
          X
          
            1
          
        
        ,
        
          X
          
            2
          
        
        ,
        …
        ,
        
          X
          
            t
          
        
        
      
    
    {\displaystyle X_{1},X_{2},\dots ,X_{t}\,}
  
</li>
<li>Calculate the rescaled range series (R/S)

(
            
              R
              
                /
              
              S
            
            )
          
          
            t
          
        
        =
        
          
            
              R
              
                t
              
            
            
              S
              
                t
              
            
          
        
        
           for 
        
        t
        =
        1
        ,
        2
        ,
        …
        ,
        n
        
      
    
    {\displaystyle \left(R/S\right)_{t}={\frac {R_{t}}{S_{t}}}{\text{  for }}t=1,2,\dots ,n\,}
  
</li></ol>
<p>Lo (1991) advocates adjusting the standard deviation 
  
    
      
        S
      
    
    {\displaystyle S}
  
 for the expected increase in range 
  
    
      
        R
      
    
    {\displaystyle R}
  
 resulting from short-range <a href="/facts/Autocorrelation/dm4MqK0A">autocorrelation</a> in the time series.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> This involves replacing 
  
    
      
        S
      
    
    {\displaystyle S}
  
 by 
  
    
      
        
          
            
              S
              ^
            
          
        
      
    
    {\displaystyle {\hat {S}}}
  
, which is the square root of
</p><p>
  
    
      
        
          
            
              
                S
                ^
              
            
          
          
            2
          
        
        =
        
          S
          
            2
          
        
        +
        2
        
          ∑
          
            j
            =
            1
          
          
            q
          
        
        
          (
          
            1
            −
            
              
                j
                
                  q
                  +
                  1
                
              
            
          
          )
        
        C
        (
        j
        )
        ,
      
    
    {\displaystyle {\hat {S}}^{2}=S^{2}+2\sum _{j=1}^{q}\left(1-{\frac {j}{q+1}}\right)C(j),}

</p><p>where 
  
    
      
        q
      
    
    {\displaystyle q}
  
 is some maximum lag over which short-range autocorrelation might be substantial and 
  
    
      
        C
        (
        j
        )
      
    
    {\displaystyle C(j)}
  
 is the sample <a href="/facts/Autocovariance/3uM1sVVV">autocovariance</a> at lag 
  
    
      
        j
      
    
    {\displaystyle j}
  
. Using this adjusted rescaled range, he concludes that stock market return time series show no evidence of long-range memory.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Fractal/sXj38JeN">Fractal</a></li>
<li><a href="/facts/Fractional_Brownian_motion/KBzUMrNV">Fractional Brownian motion</a></li>
<li><a href="/facts/Fat_tail/ygvABPqH">Fat tail</a></li></ul>

<h3>Further reading</h3>

<ul><li>Hurst, H.E.; Black, R.P.; Simaika, Y.M. (1965). <i>Long-term storage: an experimental study</i>. London: Constable.</li>
<li>Beran, J. (1994). <i>Statistics for Long-Memory Processes</i>. Chapman & Hall. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-412-04901-9.</li>
<li>Thiele, T. A. (2014). "Multiscaling and Stock Market Efficiency in China". <i>Review of Pacific Basin Financial Markets and Policies</i>. 17 (4): 1450023. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2FS0219091514500234">10.1142/S0219091514500234</a>.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li>Matlab code for computing R/S, DFA, periodogram regression and wavelet estimates of the Hurst exponent and their corresponding confidence intervals is available from RePEc: <a href="https://ideas.repec.org/s/wuu/hscode.html">https://ideas.repec.org/s/wuu/hscode.html</a></li>
<li>Implementation in Python: <a href="https://github.com/Mottl/hurst">https://github.com/Mottl/hurst</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Hurst, H. E. (1951). "Long term storage capacity of reservoirs". Trans. Am. Soc. Eng. 116: 770–799. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Peters, E. E. (1991). Chaos and order in the capital markets. John Wiley and Sons. ISBN 978-0-471-53372-6. <a href="978-0-471-53372-6" target="_blank">978-0-471-53372-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Mandelbrot, B. (1968). "Fractional Brownian motions, fractional noises and applications". SIAM Review. 10 (4): 422–437. Bibcode:1968SIAMR..10..422M. doi:10.1137/1010093. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Kamenshchikov, S. (2014). "Transport Catastrophe Analysis as an Alternative to a Monofractal Description: Theory and Application to Financial Crisis Time Series". Journal of Chaos. 2014: 1–8. doi:10.1155/2014/346743. <a href="https://doi.org/10.1155%2F2014%2F346743" target="_blank">https://doi.org/10.1155%2F2014%2F346743</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Lo, A. (1991). "Long-Term Memory in Stock Market Prices" (PDF). Econometrica. 59 (5): 1279–1313. doi:10.2307/2938368. hdl:1721.1/2245. JSTOR 2938368. <a href="http://dspace.mit.edu/bitstream/1721.1/2245/1/SWP-3014-20126283.pdf" target="_blank">http://dspace.mit.edu/bitstream/1721.1/2245/1/SWP-3014-20126283.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Bo Qian; Khaled Rasheed (2004). HURST EXPONENT AND FINANCIAL MARKET PREDICTABILITY. IASTED conference on "Financial Engineering and Applications"(FEA 2004). pp. 203–209. CiteSeerX 10.1.1.137.207. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Lo, A. (1991). "Long-Term Memory in Stock Market Prices" (PDF). Econometrica. 59 (5): 1279–1313. doi:10.2307/2938368. hdl:1721.1/2245. JSTOR 2938368. <a href="http://dspace.mit.edu/bitstream/1721.1/2245/1/SWP-3014-20126283.pdf" target="_blank">http://dspace.mit.edu/bitstream/1721.1/2245/1/SWP-3014-20126283.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
</ol>

Rescaled range open-in-new

Rescaled range