How does the $r^2$ test work ?

In Section we have derived the statistical properties of the $r^2$ test, and described it in mathematical terms. This is a bit deceptive, because this test was actually developed based on some simple physical intuition. We noticed with experience that many of the high SNR events that were not found by the outlier is_gaussian() test did not sound anything like chirps (when listened to with the audio() and sound() functions). It was clear from just listening that for these spurious signals did not have the low frequency signal arriving first, followed by the high frequency signal arriving last, in the same way as a chirp signal. So in fact the $r^2$ test was designed to discriminate the way in which the different frequencies arrived with time. In effect, the filter used to construct the signal $S_1$ passes only the lowest frequencies, the filter used to construct the signal $S_2$ passes the next-to-lowest frequencies, and so on. The filter which produces the signal $S_p$ passes the highest range of frequencies which would make a significant contribution (i.e. a fraction $1/p$) of the SNR for a true chirp.

If the signal is a true chirp, then the outputs of each of these different filters (the $S_i(t_0)$ may be thought of as functions of lag $t_0$) all peak at the same time-offset $t_0$, the same time-offset that maximizes the total signal $S(t_0)$. This is illustrated in Figure .

Figure: This figure shows the output of four single-phase filters for the $p=4$ case, for a ``true chirp" injected into a stream of real IFO data (left set of figures) and a transient noise burst already present in another stream of real IFO data (right set of figures). When a true chirp is present, the filters in the different frequency bands all peak at the same time offset $t_0$: the time offset which maximizes the SNR. At this instant in time, all of the $S_i$ are about the same value. However when the filter was triggered by a non-chirp signal, the filters in the different frequency bands peak at different times, and in fact at time $t_0$ they have very different values (some large, some small, and so on).

It is also instructive to compare the values of the filter outputs (single-phase test) for the two cases shown in Figure . For the injected chirp, the signal-to-noise ratio was 9.2, and the signal values in the different bands were

$$S_1 = 2.25$$

$$S_2 = 2.44$$

$$S_3 = 1.87$$

$$S_4 = 2.64$$

$$S = S_1 + S_2 + S_3 + S_4 = 9.2$$ (6.25.143)

$$r^2 = \sum_{i=1}^4 (S/4 - S_i)^2 = 0.324$$

$$P = Q(3/2, 2 r^2) = 0.730,$$

so there is a large probability $P$ of having $r^2$ this large.

For the spurious noise event shown in Figure the SNR was quite similar (8.97) but the value of $r^2$ is very different:

$$S_1 = 0.23$$

$$S_2 = 0.84$$

$$S_3 = 5.57$$

$$S_4 = 2.33$$

$$S = S_1 + S_2 + S_3 + S_4 = 8.97$$ (6.25.144)

$$r^2 = \sum_{i=1}^4 (S/4 - S_i)^2 = 17.1$$

$$P = Q(3/2, 2 r^2) =9.4 \times 10^{-15},$$

so the probability that this value of $r^2$ would be obtained for a chirp plus Gaussian noise is extremely small.