tokens: Fig. 9 favour periodic models more strongly when the light curve is sinusoidal and favour QPO models more strongly when the light curves are sawtooth like. We introduced the higher order p 4 QPO models with periods locked in a 2:1 ratio and a common coherence time to explore if this difference was driven by the assumption of sinusoidal waveforms in the one-component QPO model. We show in Fig. 10 that using the two-component QPO models halves the number of RRab and RRd variables that prefer a QPO compared to the one-component model. The increase in periodic preference going from a one-component to the two-component QPO models for the more sinusoidal RRc and RRe sources is marginal. An even larger shift is seen for the fundamental mode Cepheids. These results are in agreement with the behaviour we found in Section 3 for the experiments comparing artificial light curves with sawtooth or sinusoid waveforms. It is not surprising, therefore, that a higher order QPO model results in a greater preference for periodicity. The two-component QPO is a better approximation to the sawtooth-like light curves of RRab, RRd and fundamental mode Cepheids, so the coherence time can act more like the desired physical parameter, and less as a proxy for light-curve shape. Figure 10. Distribution of the RRab+RRd, RRc+RRe, fundamental Cepheids and first overtone Cepheids in the AIC likelihood difference for the QPO and periodic models when using either the standard QPO model (dotted) or the higher order model (solid). The changes are small for the more sinusoidal RRc+RRe and first overtone Cepheids, but shift strongly towards the periodic case for the more sawtooth-like RRab+RRd and fundamental mode Cepheids. In addition to light-curve shape, we also noted in Section 2 that mischaracterized errors can bias the results, irrespective of light-curve shape. If some of the noise level assumed in the model is less than the true noise, the extra variance will be interpreted as signal. The natural way to model this extra variance is to move the model away from periodic to model the mischaracterized noise as stochastic variability. If we take the best-fitting model and re-estimate the errors so that it has 2 /dof1, then the fraction of Cepheids and RRLyrae favouring the periodic model increase significantly, from 70 and 10percent to 90 and 65percent in the case of first overtone and fundamental Cepheids, respectively, and increasing from 80 and 15percent among RRc/RRe and RRab/RRd to 95 and 90percent, respectively. 4.3 Variable distributions in QPO parameters The lesson from Section 3 is that the QPO parameter cannot simply be interpreted as a measure of coherence because it is significantly affected by light-curve shape. This is, however, no barrier to using the QPO parameters to classify variables. Fig. 11 shows the distribution of all the sources in the P , P and QPO parameter spaces. We see that the P plane maps physically distinct variable types to nearly unique spaces.
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