for details), and $b$ can be estimated by minimizing the mean squared cross-correlations at lags longer than a particular decorrelation timescale. For instance, assuming a decorrelation time of 7 days, the estimated strength of eddy-jet feedback (hereafter $b_{LH}$) is around 0.13 day$^{-1}$, and the red curve in Figure 6 shows the corresponding cross-correlations between $\tilde{z}$ and $\tilde{m}$. Bootstrap confidence intervals (at 95$\%$ confidence levels) are plotted to indicate errors (black dashed curves in Figure 8a). A bootstrap ensemble of 5000 members is constructed by resampling from the original time series. Feedback strength is calculated for each of the bootstrap ensemble member, which provides the probability density function of $b_{LH}$ and thus the confidence intervals. $b_{LH}$ varies with the choices of decorrelation time. Note that it is difficult to determine an optimal decorrelation time [*a priori* due to the quasi-oscillatory behavior of $\tilde{m}$]{}, especially when the decorrelation timescale varies by season [e.g., @Sheshadri2016].

Lag regressions
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Lag regression is applied to find the feedback strength following S13. Denote the auto-covariance function of $z$ with lag $l$ as $\gamma_z (l)$, and write the cross-covariance function between $z$ and $m$ as $\gamma_{zm} (l)$ when $z$ leads $m$ by $l$ days. Consider the lag 