PENT

       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.

       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.

       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 sythetic 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).

       Written by:   Pierre Maxted, 14Sep95
       Modifications:
       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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