Performance

In Figure 5.4, I compare the performance of three different timescales, all derived by finding the time lag at which the autocorrelation function first drops below a particular threshold. In the top two panels, the timescale calculated from simulations is divided by the corresponding timescale from section 4.5; the dotted line across the middle of the plot represents behavior consistent with analytical theory.

For sinusoidal signals with periods shorter than 16 days, the calculated timescale is an order of magnitude too large; for longer periods, the bias is much smaller, a few tens of percent. All three thresholds have similar performance in this case. For damped random walks, the calculated timescale varies little with the true timescale, so the ratio is high at short timescales and low at long timescales. The timescale from cutting the autocorrelation function at 1/2 shows a slightly shallower slope in this plot, indicating it scales more strongly with the true timescale.

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(a) Damping Time 0.5 days (b)Damping Time 2.0 days

(c) Damping Time 5 days (d)Damping Time 16 days

(e)Damping Time 64 days (f )Damping Time 256 days

Figure 5.3: The mean interpolated autocorrelation function from 1,000 simulations of a damped random walk at several representative timescales. Dotted lines represent the standard deviation of the ACF at each time lag. The red curve shows the ACF predicted from subsubsection 4.5.3.4, after correcting for noise and for the finite observing window as described insubsection 5.3.1.

(a)Output Timescale for Sine (b)Output Timescale for Squared Exponential Gaus- sian Process

(c)Output for Damped Random Walk

(d)Timescale Repeatability for Sine

(e)... for Squared Exponential Gaussian Process

(f )... for Damped Random Walk

(g) Timescale Discrimination for Sine

(h)... for Squared Exponential Gaussian Process

(i)... for Damped Random Walk

Figure 5.4: The timescale calculated from the autocorrelation function, plotted as a function of the true underlying timescale. Only simulations with no measurement noise are shown. Top panels show the ratio of the output timescale to the value predicted insection 4.5. Middle panels show the ratio of the standard deviation to the mean output timescale. Bottom panels show the degree by which the input timescale has to change to signifi- cantly affect the output timescale. In all plots, blue represents the time at which the autocorrelation function first falls below 1/9, orange the time at which it falls below 1/4, and green the time at which it falls below 1/2.

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5.3.3.2 Precision

In the middle two panels ofFigure 5.4, the scatter in the estimated timescale across multiple simu- lation runs is plotted for three different timescale metrics. For both sinusoidal and damped random walk lightcurves, cutting the autocorrelation function at 1/2 generally yields less scatter than cutting at 1/4 or 1/9.

5.3.3.3 Discrimination

Discrimination is a measure of how well the inferred timescale correlates with the true lightcurve timescale. Good discrimination requires both a strong dependence of output on input timescale and high repeatability for individual timescale measurements. Poor discrimination means a timescale metric cannot be used to separate short- and long-timescale lightcurves with any fidelity.

In the bottom two panels of Figure 5.4, the scatter in the estimated timescale across multiple simulation runs is plotted for three different timescale metrics. The discrimination is not defined at short sine periods because the output timescaledecreases with the true period (cf. Figure 5.2). At longer sine periods, the inferred timescale correlates well with the period and the discriminatory power is comparable to the scatter in the individual measurements. For the damped random walk, the inferred timescale correlates poorly with the true timescale, so the discrimination is much poorer than the scatter alone would imply. While the true timescale of a long-period sinusoidal signal can be inferred to 10% or better, the accuracy of the same estimate for a damped random walk is only 70-80% at best.

5.3.3.4 Sensitivity to Noise

The average value of the autocorrelation timescale changes very little between an effectively infinite signal-to-noise and a signal-to-noise ratio of 10. At a signal-to-noise ratio of 4, on the other hand, the calculated timescale shows much less variation with the true timescale. This is particularly dramatic for the sine wave (upper left panel), since for signal-to-noise of 10 or greater, the calculated timescale is proportional to the period for periods of 16 days or longer, while for signal-to-noise of 4 the proportionality disappears.

The scatter in individual measurements is more sensitive to signal-to-noise. The degree to which noise degrades measurement precision is independent of the true timescale. Again, signal-to-noise of 4 is significantly different from signal-to-noise of 10 or higher.

The discriminating power of timescales based on autocorrelation functions shows a similar de- pendence on signal-to-noise as the scatter. Discrimination gets poorer at low signal-to-noise in a manner independent of the true timescale.

(a)Output Timescale for Sine (b)Output Timescale for Squared Exponential Gaus- sian Process

(c)Output Timescale for Damped Random Walk

(d)Timescale Repeatability for Sine

(e)Timescale Repeatability for Squared Exponential Gaus- sian Process

(f )Timescale Repeatability for Damped Random Walk

(g) Timescale Discrimination for Sine

(h)Timescale Discrimination for Squared Exponential Gaus- sian Process

(i)Timescale Discrimination for Damped Random Walk

Figure 5.5: As Figure 5.4, but plotting only the timescale at which the autocorrelation function first falls below 1/2. Blue represents zero noise, orange represents a signal-to-noise ratio of 20, green a signal-to-noise ratio of 10, and black a signal-to-noise ratio of 4. All lightcurves have an expected 5-95% amplitude of 0.5 magnitudes.

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5.3.3.5 Sensitivity to Cadence

The behavior of the interpolated autocorrelation function is similar for the two PTF cadences and the YSOVAR cadence (Figure 5.6), except in their characteristic scales. Timescales calculated for lightcurves sampled with the full PTF cadence are systematically higher than those calculated for the PTF cadence in 2010, which are in turn higher than those for lightcurves sampled at the YSOVAR cadence.

For sinusoidal lightcurves sampled at either PTF cadence, the autocorrelation timescale correlates with the period for periods larger than or comparable to 16 days. For the YSOVAR sampling, the timescale is reliable for periods of 5 days or longer. The autocorrelation timescale is never well correlated with the period for a damped random walk, regardless of sampling. The fractional scatter shows no strong trends with the input timescale for either sampling or for either type of lightcurve. The discriminating power is best at long timescales for a sinusoidal signal, but is best at intermediate timescales (relative to the cadence and time baseline of the observations) for a damped random walk.

In general, the interpolated autocorrelation function performs best at the YSOVAR cadence, worse at the 2010-only PTF cadence, and worst of all at the full PTF cadence, as measured by output timescale bias or discriminating power. InTable 5.1, the YSOVAR cadence has the smallest maximum gap in the data, the smallest time base line, and the fewest points of the three cadences; the full PTF cadence has the largest maximum gap, the longest base line, and the most points. Of these properties, the maximum gap is most likely the one that controls ACF performance: after interpolation, the gap will be filled with a perfectly linear time series that an autocorrelation solver cannot distinguish from real data.

Neither characteristic cadence nor, surprisingly, linearity of the time series correlates with ACF performance; the YSOVAR cadence is intermediate between the two PTF cadences in both metrics.