with 99$\%$ and 95$\%$ confidence level for a red noise process and 99$\%$ confidence level for a white noise process.

The distribution for the local wavelet power spectrum is $$\frac{|\widetilde f(l_{j},t'_{i})|^2}{\sigma^{2}} \Rightarrow P_{j}\frac{\chi^{2}_{\nu}}{\nu}$$ where the arrow means “distributed as," $\sigma^{2}$ is the variance, and $\nu$ is the number of degrees of freedom, which is two here. The indices on the scale $l$ are $j$=1,2,...,$J$ where $J$ is the number of scales, and the indices on the translation $t'$ are $i$=1,2,...,$N_{data}$. We evaluate this equation at each scale to get 95$\%$ confidence contour lines and in this paper our continuous transforms are normalized to the 95$\%$ confidence level for the corresponding red noise process. Doing this allows one to see the strength of the wavelet coefficients relative to the 95$\%$ confidence level of a red noise process.

### The Cross-Wavelet Transform {#xwtsection}

Although the continuous wavelet transform is useful in examining how a time series varies in time and scale, it does not tell us how the time series varies in dilation over a range of scales when assigning a characteristic timescale. Since a quasi-periodic signal has no unique dilation we use the cross-wavelet transform (XWT) which filters out noise and 