Approximate entropy applied to the Pi digits

From: Roger Bagula (tftn_at_earthlink.net)
Date: 11/23/04

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    Date: Tue, 23 Nov 2004 22:32:17 GMT

      In Ivars Peterson's MathTrek - Randomness, Risk, and Financial Markets:

    http://www.maa.org/mathland/mathtrek_10_11_04.html
    I found the reference to Steven M. Pincus' Approximate entropy.
    I applied it to Hofstader's sequence, the last digits of the primes and
    Pi's digits
    and the Approximate entropy came out larger in that order.
    My program is really slow, but it does seem to give the ApEn function as
    defined in the paper. It is a lot like a Lyapunov Largest exponent
    in the way I've calculated it, but it more a probability measure
    on the variables than a direct result of the variables. It is also much
    harder
    and takes longer than a Lyapunov since it has two distinct sums in it.
    It is more closely related to correlation dimension that Kaplan-York
    dimension
    in it's method of calculation.

    Clear [f,n,d,c,Phi,ApEn,a,i,j,k,r,m,g,digits]
    (*Steven M. Pincus,Approximate entropy as a measure of system complexity,
      PNAS,vol 88,pp2297-2301,March 1991,Mathematics*)
    digits=100
    $MaxExtraPrecision =digits
    f[n_]:=Floor[Mod[10^n*Pi,10]]
    (* approximate Entropy for Pi digits sequence*)
    d[i_,j_,m_,n_]:=Max[Table[Abs[f[i+k-1]-f[j+k-1]],{k,1,m-1}]]
    c[i_,r_,m_,n_]:=N[Sum[If[d[i,j,m,nTrue]
    y=Fit[a,{1,x},x]
    gb=Plot[y,{x,1,digits}]
    Show[{ga,gb}]

    Respectfully, Roger L. Bagula

    tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
    alternative email: rlbtftn@netscape.net
    URL : http://home.earthlink.net/~tftn

    Ap_En_Pi.gif
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