13.8 Spectral Analysis of Unevenly Sampled Data 577 and cosines,only at times t;that are actually measured.Suppose that there are N data pointshi=h(ti),i=1,...,N.Then first find the mean and variance of the data by the usual formulas. 1 =立a- (13.8.3) 1 Now,the Lomb normalized periodogram (spectral power as a function of angular frequency w=2f>0)is defined by Pw(u)≡ [区，仙-刀s6-，-列血刊 22 ∑c0s2w(4-T) ∑，sin2w(化，-T) (138.4) Here T is defined by the relation ∑jsin2wt tan(2WT)= (13.8.5) ∑，cos2w5 RECIPES The constant r is a kind of offset that makes P(w)completely independent of shifting 令 all the ti's by any constant.Lomb shows that this particular choice of offset has another, deeper,effect:It makes equation (13.8.4)identical to the equation that one would obtain if one estimated the harmonic content of a data set,at a given frequency w,by linear least-squares Press. fitting to the model h(t)=Acoswt+B sinwt (13.8.6) This fact gives some insight into why the method can give results superior to FFT methods:It weights the data on a"per point"basis instead of on a"per time interval"basis,when uneven sampling can render the latter seriously in error. SCIENTIFIC A very common occurrence is that the measured data points h are the sum of a periodic signal and independent (white)Gaussian noise.If we are trying to determine the presence 6 or absence of such a periodic signal,we want to be able to give a quantitative answer to the question,"How significant is a peak in the spectrum Py(w)?"In this question,the null hypothesis is that the data values are independent Gaussian random values.A very nice property of the Lomb normalized periodogram is that the viability of the null hypothesis can be tested fairly rigorously,as we now discuss. The word"normalized"refers to the factor o2 in the denominator of equation(13.8.4). 10621 Scargle [4]shows that with this normalization,at any particular w and in the case of the null Numerica hypothesis,PN(w)has an exponential probability distribution with unit mean.In other words, the probability that P(w)will be between some positive z and z+dz is exp(-z)dz.It uctio 43106 readily follows that,if we scan some M independent frequencies,the probability that none Recipes give values larger than z is (1-e-=)M.So P(>)≡1-(1-e)M (13.8.7) North Software. is the false-alarm probability of the null hypothesis,that is,the significance level of any peak in PN(w)that we do see.A small value for the false-alarm probability indicates a highly significant periodic signal. To evaluate this significance,we need to know M.After all,the more frequencies we look at,the less significant is some one modest bump in the spectrum.(Look long enough, find anything!)A typical procedure will be to plot Py(w)as a function of many closely spaced frequencies in some large frequency range.How many of these are independent? Before answering,let us first see how accurately we need to know M.The interesting region is where the significance is a small (significant)number,<1.There,equation (13.8.7) can be series expanded to give P(>z)≈Me- (13.8.8)13.8 Spectral Analysis of Unevenly Sampled Data 577 Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America). and cosines, only at times ti that are actually measured. Suppose that there are N data points hi ≡ h(ti), i = 1,...,N. Then first find the mean and variance of the data by the usual formulas, h ≡ 1 N N 1 hi σ2 ≡ 1 N − 1 N 1 (hi − h) 2 (13.8.3) Now, the Lomb normalized periodogram (spectral power as a function of angular frequency ω ≡ 2πf > 0) is defined by PN (ω) ≡ 1 2σ2     j (hj − h) cos ω(tj − τ ) 2  j cos2 ω(tj − τ ) +  j (hj − h) sin ω(tj − τ ) 2  j sin2 ω(tj − τ )    (13.8.4) Here τ is defined by the relation tan(2ωτ ) =  j sin 2ωtj  j cos 2ωtj (13.8.5) The constant τ is a kind of offset that makes PN (ω) completely independent of shifting all the ti’s by any constant. Lomb shows that this particular choice of offset has another, deeper, effect: It makes equation (13.8.4) identical to the equation that one would obtain if one estimated the harmonic content of a data set, at a given frequency ω, by linear least-squares fitting to the model h(t) = A cos ωt + B sin ωt (13.8.6) This fact gives some insight into why the method can give results superior to FFT methods: It weights the data on a “per point” basis instead of on a “per time interval” basis, when uneven sampling can render the latter seriously in error. A very common occurrence is that the measured data points hi are the sum of a periodic signal and independent (white) Gaussian noise. If we are trying to determine the presence or absence of such a periodic signal, we want to be able to give a quantitative answer to the question, “How significant is a peak in the spectrum PN (ω)?” In this question, the null hypothesis is that the data values are independent Gaussian random values. A very nice property of the Lomb normalized periodogram is that the viability of the null hypothesis can be tested fairly rigorously, as we now discuss. The word “normalized” refers to the factor σ2 in the denominator of equation (13.8.4). Scargle [4] shows that with this normalization, at any particular ω and in the case of the null hypothesis, PN (ω) has an exponential probability distribution with unit mean. In other words, the probability that PN (ω) will be between some positive z and z + dz is exp(−z)dz. It readily follows that, if we scan some M independent frequencies, the probability that none give values larger than z is (1 − e−z) M. So P(> z) ≡ 1 − (1 − e−z ) M (13.8.7) is the false-alarm probability of the null hypothesis, that is, the significance level of any peak in PN (ω) that we do see. A small value for the false-alarm probability indicates a highly significant periodic signal. To evaluate this significance, we need to know M. After all, the more frequencies we look at, the less significant is some one modest bump in the spectrum. (Look long enough, find anything!) A typical procedure will be to plot PN (ω) as a function of many closely spaced frequencies in some large frequency range. How many of these are independent? Before answering, let us first see how accurately we need to know M. The interesting region is where the significance is a small (significant) number,  1. There, equation (13.8.7) can be series expanded to give P(> z) ≈ Me−z (13.8.8)