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      Return the information entropy of a time series


      This function will return S, the information entropy of a time series
      for a set of trial periods


      Time series analysis, period finding, astronomical utilities.

Calling Sequence

      Result = PENT(P, T, X, [N, M ] )


      P - array of trial period values.
      T - array of observation times (same units as P).
      X - array of observations.

Optional Inputs

      N - If four parameters are given then the 4th parameter is assumed
              to be N. Then NxN boxes are used to calculate S.
      M,N - If five parameters are given then parameter 4 is M and parameter
              5 is N. S is then calculated using MxN boxes - M partitions for the
              phase and N partitions for the data.


      This function returns S, the information entropy of the time series for
      the periods given in P as defined by Cincotta, Me'ndez & Nu'n~ez
      (Astrophysical Journal 449, 231-235, 1995). The minima of S occur at
      values of P where X shows periodicity.


      The procedure involves dividing the phase space into N^2 partitions
      (NxN boxes) and then calculating:
              __ N^2
        S = - \ mu_i . ln(mu_i) for all mu_i <> 0
                i = 1
      where mu_i is the number of data points in partition i normalised by
      the number of partitions.
      The option of using MxN boxes is an additional feature of this routine.


      To generate a similar synthetic data set to Cincotta et al. we
        do the following:
      IDL> P0 = 173.015 ; Fundamental period
      IDL> T = randomu(seed,400)*15000 ; 400 random observation times
      IDL> A0 = 14.0 ; Mean magnitude
      IDL> M0 = -0.5 * sin(2*!pi*T/P0) ; Fundamental mode
      IDL> M1 = -0.15 * sin(4*!pi*T/P0) ; 1st harmonic
      IDL> M2 = -0.05 * sin(6*!pi*T/P0) ; 2nd harmonic
      IDL> sig = randomu(seed,400)*0.03 ; noise
      IDL> U = A0 + M0 + M1 + M2 + sig ; Synthetic data
      IDL> Ptest = 100. + findgen(2000)/2. ; Trial periods
      IDL> S = pent(Ptest,T,U) ; Calculate S
              ... this takes a few seconds ...
      IDL> plot,Ptest,S,xtitle="P",ytitle="S" ; plot S v. P
      IDL> print,Ptest(where(S eq min(S))) ; Print best period (+/- 0.5)
      The plot produced should be similar to Fig. 2 of Cincotta et al.


      My own (limited) experience with this routine suggests that it is not
      as good as other techniques for finding weak, multi-periodic signals in
      poorly sampled data, but is good for establishing periods of eclipsing
      binary stars when M is quite large (try MxN = 64x16, 128x16 or even
      256x16). This suggests it may be good for other periodic light curves
      (Cepheids, RR Lyrae etc.).
      I would be glad to receive reports of other peoples experience with
      this technique (e-mail pflm@bro730.astro.ku.dk).

Modification History

      Written by: Pierre Maxted, 14Sep95
      Normalisation of S corrected, T-min(T) taken out of loop.
              - Pierre Maxted, 15Sep95
      Converted to IDL V5.0 W. Landsman September 1997

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