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Hurst and Spectra of 5 Italian precipitation series

dm43.html;     cm92 site;   cm92.html;   cm93.html;    cm94.html;    dm42.html (The whole story in English)

References: (not available on the net)

  • long-memory.com
  • Baillie, 1996
  • HURST, H.E., 1951. Long-term storage capacity of reservoirs. Transactions of the American Society of Civil Engineers, Volume 116, Pages 770--799.;
  • Koutsoyiannis, 2002 ; •Koutsoyiannis, 2003; •Koutsoyiannis, 2006 ; •Mureika's 2015 WUWT article (Persistence).
  • Montanari et al., 1996; • Wilson et al., 2005 (Bavaria); • Simonsen et al, 1998 (wavelets); • Taqqu et al., 1995 (methods for H);
  • Teverovsky et al., 1998 (method for R/S by Lo: critics);
  • Mangianti & Leone, 2007? (Roma CR Pluviometry, in Italian, no pub.date);


    •••R/S Analysis in Finance (How to apply the R/S method for H);
    •••Estimating the Hurst Exponent

    • See here for derivatives/differences properties (at the bottom of the "Basic Properties of Derivative Signals" paragraph)
    "The derivative of a periodic signal containing several sine components of different frequency will still contain those same frequencies, but with altered amplitudes and phases."

    I compute the approximate Hurst exponent (H) reported here by the eq.(5) of the Koutsoyiannis', 2003 paper.
    Please note that eq.(5) refers to an "aggregated process", what our processes may not be; you should use the H-value only as a simple approach to the value the Hurst exponent really owns. Koutsoyiannis does not make use of this method, so caution is needed in too sharp statements about the H value!
    H=(1.+sqrt(1.+8.*acf(1)))/4

    from:
    acf(1)=H(2H-1) or eq.(5) when lag=1.

    From Baillie, 1996, pag.16: Kolmogorov (1940) and Mandelbrot and Van Ness (1968) show that the autocovariance function of a self-similar process γt, which is observed at discrete, regular intervals of time, is given by
    γk=C(1/2)•γ0•{|k+1|2H-2|k|2H+|k-1|2H},
    for C>0 and 1/2<H<1. For high lags k,
    γk≈CH(2H-1)|k|2H-2,
    as k-->∞    .
    So, we need a self-similar process, regular time intervals, high lags and C=1 in the equation above [acf(1)= ...] and these limits bring to an approximate process. The use of lag 1 implies a further approximation.
    Please note that the auto-correlation function acf assumes values only between 0 and 1, so if acf(1) has negative value the computed H exponent is given as NaN (not a number).
    We can consider the negative acf value as a fluctation around zero and assign zero-value to it: in such a case the Hurst exponent is 0.5 and identifies a not-autocorrelated series which can be managed with normal statistics.

    >
    Plots, Figures        
    Montanari et al., 1996 (pdf) (png) Estimates of H Table 3
    H for Milano via R/S (pdf) (png) Estimate of H (0.190) (fz)
    H for Parma via R/S (pdf) (png) Estimate of H (0.472) (fz)
    H for Genova via R/S (pdf) (png) Estimate of H (0.313) (fz)
    H for Sondrio via R/S (pdf) (png) Estimate of H (0.340) (fz)
    ACF/legge di potenza (pdf) (png) acf(k)=C*k^a 101 lags
    Milano rain annual Obs (pdf) (png) HISTALP 1800-2012
    Milano rain annual Diff (pdf) (png) 1800-2012
    Milano rain ACF (pdf) (png) Hobs=0.591; Hdiff=0.51800-2012
    Milano Lomb CCF (pdf) (png) CCF Obs/Der, ccf(0)=0.2501894-2005
    Milano rain Comp (txt) Spectral Comparison
    Parma rain annual Obs (pdf) (png) HISTALP 1800-2012
    Parma rain annual Diff (pdf) (png) 1800-2012
    Parma rain ACF (pdf) (png) Hobs=0.585; Hdiff=0.3821800-2012
    Parma rain ACF long (pdf) (png) Hobs=0.585; Hdiff=0.5±150 lags
    Parma Lomb CCF (pdf) (png) CCF Obs/Diff, ccf(0)=0.5391894-2005
    Parma rain Comp (txt) Spectral Comparison
    Genova rain annual Obs (pdf) (png) HISTALP 1833-2010
    Genova rain annual Diff (pdf) (png) 1833-2010
    Genova rain ACF (pdf) (png) Hobs=0.469; Hdiff=0.51833-2010
    Genova Lomb CCF (pdf) (png) CCF Obs/Der, ccf(0)=0.5391894-2005
    Genova rain Comp (txt) Spectral Comparison
    Firenze rain annual Obs (pdf) (png) HISTALP 1860-2011
    Firenze rain annual Diff (pdf) (png) 1860-2011
    Firenze rain ACF (pdf) (png) Hobs=0.5; Hdiff=0.4131860-2011
    Firenze Lomb CCF (pdf) (png) CCF Obs/Der, ccf(0)=0.5391894-2005
    Firenze rain Comp (txt) Spectral Comparison
    Sondrio rain annual Obs (pdf) (png) SCIA 1894-2005
    Sondrio rain annual Deriv (pdf) (png) 1894-2005
    Sondrio rain ACF (pdf) (png) Hobs=0.745; Hder=0.6141894-2005
    Sondrio Lomb CCF (pdf) (png) CCF Obs/Der, ccf(0)=0.5391894-2005
    Sondrio rain Comp (txt) Spectral Comparison
    Roma CR rain annual Obs (pdf) (png) HISTALP 1952-2002
    Roma CR rain annual Deriv (pdf) (png) 1952-2002
    Roma CR rain ACF (pdf) (png) Hobs=0.5; Hdiff=0.4131952-2002
    Roma CR Lomb CCF (pdf) (png) CCF Obs/Der, ccf(0)=0.5391952-2002
    Roma CR rain Comp (txt) Spectral Comparison
    Bavaria rain annual Obs (pdf) (png) NOAA Paleo 1480-1978
    Bavaria rain annual Diff (pdf) (png) 1480-1978
    Bavaria rain ACF (pdf) (png) Hobs=0.754; Hdiff=0.51894-2005
    Bavaria Lomb CCF (pdf) (png) CCF Obs/Diff, ccf(0)=0.5391894-2005
    Bavaria rain Comp (txt) Spectral Comparison
    --------------------------------------------
    Data        
    Milano Stat (out) rs-linrs-name.bon
    Parma Stat (out) rs-linrs-name.bon
    Genova Stat (out) rs-linrs-name.bon
    Sondrio Stat (out) rs-linrs-name.bon
    Milano rain Obs (txt) Brera1800-2012
    Milano rain ACF (out) Hobs=0.591; Hdiff=0.5
    Milano rain LOMB (out)
    Milano rain diff (out)
    Milano diff ACF (out)
    Milano rain CCF (out) CCF Obs/Diff
    Parma rain Obs (txt) Università 1833-2010
    Parma rain ACF (out) Hobs=0.585, Hdiff=0.382
    Parma rain ACF long (out) Hobs=0.585
    Parma rain LOMB (out)
    Parma rain Diff (out)
    Parma rain ACF Diff (out) Hobs=0.585, Hdiff=0.382
    Parma rain ACF Diff long (out) Hdiff=0.5
    Parma rain CCF (out) CCF Obs/Diff
    Genova rain Obs (txt) Università 1833-2010
    Genova rain ACF (out) Hobs=0.469, Hdiff=0.5
    Genova rain LOMB (out)
    Genova rain Diff (out)
    Genova Diff ACF (out) Hobs=0.469, Hdiff=0.5
    Genova rain CCF (out) CCF Obs/Diff
    Firenze rain Obs (txt) Ximeniano1860-2011
    Firenze rain ACF (out) Hobs=0.5, Hdiff=0.413
    Firenze rain LOMB (out)
    Firenze rain Diff (out)
    Firenze Diff ACF (out) Hobs=0.5, Hdiff=0.413
    Firenze rain CCF (out) CCF Obs/Der
    Sondrio rain Obs (txt) 1894-2005
    Sondrio rain ACF (out) Hobs=0.745, Hder=0.614
    Sondrio rain LOMB (out)
    Sondrio rain Deriv (out)
    Sondrio Deriv ACF (out) Hobs=0.745, Hder=0.614
    Sondrio rain CCF (out) CCF Obs/Der
    Roma CR rain Obs (txt) Collegio Romano1952-2002, SCIA
    Roma CR rain ACF (out) Hobs=0.444, Hder=0.529
    Roma CR rain LOMB (out)
    Roma CR rain Der (out)
    Roma CR der ACF (out) Hder=0.529
    Roma CR rain CCF (out) CCF Obs/Der
    Bavaria rain Obs (txt) Reconstructed, RCS1480-1978
    Bavaria rain ACF (out) Hobs=0.754, Hdiff=0.5
    Bavaria rain LOMB (out)
    Bavaria rain Diff (out)
    Bavaria diff ACF (out) Hobs=0.754, Hdiff=0.5
    Bavaria CCF (out)

    Code
    mily.bon, milyh.bon, mily-acf.bon, mily-ccf.bon,    parmy.bon, parmyh.bon, parmy-acf.bon, parmy-ccf.bon,
    geny.bon, genyh.bon, geny-acf.bon, geny-ccf.bon,    firy.bon, firyh.bon, firy-acf.bon, firy-ccf.bon,
    sondy.bon, sondyh.bon, sondy-acf.bon, sondy-ccf.bon,

    romacrr.bon, romacrrh.bon, romacrr-acf.bon, romacrr-ccf.bon,
    bavaria.bon, bavariah.bon, bavaria-acf.bon, bavaria-ccf.bon,
    rs-manuale.txt,
    rs-milano.bon, rs-parma.bon, rs-genova.bon, rs-sondrio.bon,
    h-milano.bon, h-parma.bon, h-genova.bon, h-sondrio.bon,

    Page Written: February 22, 2018,        Last Updated: