Astropython.org visitor "Morgan" contributed a Python implementation of Lomb-Scargle via the comments to [Question] period-finding packages in python. This script is based on:

Press, W. H. & Rybicki, G. B. 1989

ApJ vol. 338, p. 277-280.

Fast algorithm for spectral analysis of unevenly sampled data

bib code: 1989ApJ...338..277P

In order to make the script easier to access via cut and paste we are providing it here as a code snippet. This version repairs a couple of apparent issues with indentation in the original comment posting (near the top of the **__spread__** function).

#!/usr/bin/env python """ Fast algorithm for spectral analysis of unevenly sampled data The Lomb-Scargle method performs spectral analysis on unevenly sampled data and is known to be a powerful way to find, and test the significance of, weak periodic signals. The method has previously been thought to be 'slow', requiring of order 10(2)N(2) operations to analyze N data points. We show that Fast Fourier Transforms (FFTs) can be used in a novel way to make the computation of order 10(2)N log N. Despite its use of the FFT, the algorithm is in no way equivalent to conventional FFT periodogram analysis. Keywords: DATA SAMPLING, FAST FOURIER TRANSFORMATIONS, SPECTRUM ANALYSIS, SIGNAL PROCESSING Example: > import numpy > import lomb > x = numpy.arange(10) > y = numpy.sin(x) > fx,fy, nout, jmax, prob = lomb.fasper(x,y, 6., 6.) Reference: Press, W. H. & Rybicki, G. B. 1989 ApJ vol. 338, p. 277-280. Fast algorithm for spectral analysis of unevenly sampled data bib code: 1989ApJ...338..277P """ from numpy import * from numpy.fft import * def __spread__(y, yy, n, x, m): """ Given an array yy(0:n-1), extirpolate (spread) a value y into m actual array elements that best approximate the "fictional" (i.e., possible noninteger) array element number x. The weights used are coefficients of the Lagrange interpolating polynomial Arguments: y : yy : n : x : m : Returns: """ nfac=[0,1,1,2,6,24,120,720,5040,40320,362880] if m > 10. : print 'factorial table too small in spread' return ix=long(x) if x == float(ix): yy[ix]=yy[ix]+y else: ilo = long(x-0.5*float(m)+1.0) ilo = min( max( ilo , 1 ), n-m+1 ) ihi = ilo+m-1 nden = nfac[m] fac=x-ilo for j in range(ilo+1,ihi+1): fac = fac*(x-j) yy[ihi] = yy[ihi] + y*fac/(nden*(x-ihi)) for j in range(ihi-1,ilo-1,-1): nden=(nden/(j+1-ilo))*(j-ihi) yy[j] = yy[j] + y*fac/(nden*(x-j)) def fasper(x,y,ofac,hifac, MACC=4): """ function fasper Given abscissas x (which need not be equally spaced) and ordinates y, and given a desired oversampling factor ofac (a typical value being 4 or larger). this routine creates an array wk1 with a sequence of nout increasing frequencies (not angular frequencies) up to hifac times the "average" Nyquist frequency, and creates an array wk2 with the values of the Lomb normalized periodogram at those frequencies. The arrays x and y are not altered. This routine also returns jmax such that wk2(jmax) is the maximum element in wk2, and prob, an estimate of the significance of that maximum against the hypothesis of random noise. A small value of prob indicates that a significant periodic signal is present. Reference: Press, W. H. & Rybicki, G. B. 1989 ApJ vol. 338, p. 277-280. Fast algorithm for spectral analysis of unevenly sampled data (1989ApJ...338..277P) Arguments: X : Abscissas array, (e.g. an array of times). Y : Ordinates array, (e.g. corresponding counts). Ofac : Oversampling factor. Hifac : Hifac * "average" Nyquist frequency = highest frequency for which values of the Lomb normalized periodogram will be calculated. Returns: Wk1 : An array of Lomb periodogram frequencies. Wk2 : An array of corresponding values of the Lomb periodogram. Nout : Wk1 & Wk2 dimensions (number of calculated frequencies) Jmax : The array index corresponding to the MAX( Wk2 ). Prob : False Alarm Probability of the largest Periodogram value MACC : Number of interpolation points per 1/4 cycle of highest frequency History: 02/23/2009, v1.0, MF Translation of IDL code (orig. Numerical recipies) """ #Check dimensions of input arrays n = long(len(x)) if n != len(y): print 'Incompatible arrays.' return nout = 0.5*ofac*hifac*n nfreqt = long(ofac*hifac*n*MACC) #Size the FFT as next power nfreq = 64L # of 2 above nfreqt. while nfreq < nfreqt: nfreq = 2*nfreq ndim = long(2*nfreq) #Compute the mean, variance ave = y.mean() ##sample variance because the divisor is N-1 var = ((y-y.mean())**2).sum()/(len(y)-1) # and range of the data. xmin = x.min() xmax = x.max() xdif = xmax-xmin #extirpolate the data into the workspaces wk1 = zeros(ndim, dtype='complex') wk2 = zeros(ndim, dtype='complex') fac = ndim/(xdif*ofac) fndim = ndim ck = ((x-xmin)*fac) % fndim ckk = (2.0*ck) % fndim for j in range(0L, n): __spread__(y[j]-ave,wk1,ndim,ck[j],MACC) __spread__(1.0,wk2,ndim,ckk[j],MACC) #Take the Fast Fourier Transforms wk1 = ifft( wk1 )*len(wk1) wk2 = ifft( wk2 )*len(wk1) wk1 = wk1[1:nout+1] wk2 = wk2[1:nout+1] rwk1 = wk1.real iwk1 = wk1.imag rwk2 = wk2.real iwk2 = wk2.imag df = 1.0/(xdif*ofac) #Compute the Lomb value for each frequency hypo2 = 2.0 * abs( wk2 ) hc2wt = rwk2/hypo2 hs2wt = iwk2/hypo2 cwt = sqrt(0.5+hc2wt) swt = sign(hs2wt)*(sqrt(0.5-hc2wt)) den = 0.5*n+hc2wt*rwk2+hs2wt*iwk2 cterm = (cwt*rwk1+swt*iwk1)**2./den sterm = (cwt*iwk1-swt*rwk1)**2./(n-den) wk1 = df*(arange(nout, dtype='float')+1.) wk2 = (cterm+sterm)/(2.0*var) pmax = wk2.max() jmax = wk2.argmax() #Significance estimation #expy = exp(-wk2) #effm = 2.0*(nout)/ofac #sig = effm*expy #ind = (sig > 0.01).nonzero() #sig[ind] = 1.0-(1.0-expy[ind])**effm #Estimate significance of largest peak value expy = exp(-pmax) effm = 2.0*(nout)/ofac prob = effm*expy if prob > 0.01: prob = 1.0-(1.0-expy)**effm return wk1,wk2,nout,jmax,prob def getSignificance(wk1, wk2, nout, ofac): """ returns the peak false alarm probabilities Hence the lower is the probability and the more significant is the peak """ expy = exp(-wk2) effm = 2.0*(nout)/ofac sig = effm*expy ind = (sig > 0.01).nonzero() sig[ind] = 1.0-(1.0-expy[ind])**effm return sig