Significance Testing for Periodic Component

This example shows how to assess the significance of a sinusoidal component in white noise using Fisher's g-statistic. Fisher's g-statistic is the ratio of the largest periodogram value to the sum of all the periodogram values over 1/2 of the frequency interval, (0,Fs/2). A detailed description of the g-statistic and exact distribution can be found in [1] and [2].

Create a signal consisting of a 100-Hz sine wave in white Gaussian noise with zero mean and variance 1. The amplitude of the sine wave is 0.25. The sampling rate is 1 kHz. Set the random number generator to the default settings for reproducible results.

Fs = 1e3;
t = 0:0.001:1-0.001;
rng default;
x = 0.25*cos(2*pi*100*t)+randn(size(t));

Obtain the periodogram of the signal using periodogram. Exclude 0 and the Nyquist frequency (Fs/2).

[Pxx,F] = periodogram(x,rectwin(length(x)),length(x),Fs);
Pxx = Pxx(2:length(x)/2);

Find the maximum value of the periodogram. Fisher's g-statistic is the ratio of the maximum periodogram value to the sum of all periodogram values.

[maxval,index] = max(Pxx);
fisher_g = Pxx(index)/sum(Pxx);

The maximum periodogram value occurs at 100 Hz, which you can verify by finding the frequency corresponding to the index of the maximum periodogram value.

F = F(2:end-1);
F(index)

Use the distributional results detailed in [1] and [2] to determine the significance level, pval, of Fisher's g-statistic. The following MATLAB® code implements equation 6 on page 7 in [2].

N = length(Pxx);
    upper  = floor(1/fisher_g);
    for nn = 1:3
        I(nn) = ...
            (-1)^(nn-1)*nchoosek(N,nn)*(1-nn*fisher_g)^(N-1);
    end
pval = sum(I);

The p-value is less than 0.00001, which indicates a significant periodic component at 100 Hz. The interpretation of Fisher's g-statistic is complicated by the presence of other periodicities. See [1] for a modification when multiple periodicities may be present.

References

[1] Percival, Donald B. and Andrew T. Walden. Spectral Analysis for Physical Applications. Cambridge, UK: Cambridge University Press, 1993, p. 491.

[2] Wichert, Sofia, Konstantinos Fokianos, and Korbinian Strimmer. "Identifying Periodically Expressed Transcripts in Microarray Time Series Data." Bioinformatics. Vol. 20, 2004, pp. 5–20.

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