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 the references.

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 sample rate is 1 kHz. Set the random number generator to the default settings for reproducible results.

```
rng default
Fs = 1e3;
t = 0:1/Fs:1-1/Fs;
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). Plot the periodogram.

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

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)

fisher_g = 0.0381

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)

ans = 100

Use the distributional results detailed in the references to determine the significance level, `pval`

, of Fisher's *g*-statistic. The following MATLAB® code implements equation (6) of [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)

pval = 2.0163e-06

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.

[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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