Change point detection

To determine whether a substantive change in visitation is observed in each time series, we use the Bayesian Estimator of Abrupt Change, Seasonality, and Trend (BEAST) [30]. This method decomposes a time series into a seasonal (harmonic) component, and trend (linear) component, and uses Bayesian Inference to fit a model which estimates the location of change points in either of the components. BEAST was chosen because the underlying model acknowledges the seasonal nature of most park visitation time series (more visits in summer). By specifying a 52 week season length, we were able to train the model to the annual cycle shape of the data.

Parametric methods applied without the seasonal decomposition are susceptible to under estimating change points in these particular time series because of the combination of seasonality and the proximity in the series of the data collection change in December 2019 to the onset of the pandemic in March 2020. The initial event represents a sharp increase in visitation volume (roughly 150 pct), while the second appears, for most regions, as a sharp decline. When fit with a single model, these two features appear together as a change in variance, and a parametric model can be nicely fit using a single change point in December 2019.

By decomposing the time series and forcing a decoupling of the two events by specification of seasonal, length we make each event visible as a unique discontinuity in the linear component.

The December 2019 discontinuity could then be accommodated with a trend change point, which incorporates a discontinuity into the linear component. In this way the model was fit while accounting for seasonality, and the abrupt change in data volume.

Allowing a trend change point to be used as described above, the model was effectively limited to selecting a single trend change point, which enabled it to identify the most likely change point in the data. It is possible for the algorithm to detect no change point, reducing concern that one would be identified artificially.

Regions which had a change point occurring in between mid March and mid April 2020 were considered to have had an abrupt change in park visitation coinciding with the onset of the pandemic and social distancing measures. If a region was found not to have had a change point in this window, it can be assumed that either no change point was found in the time series, or any change occurring in the specified window was not as significant as a change at another time.

Changes induced by seasonality are in most cases more gradual than those that occur in the window of interest, and these changes are accounted for by the harmonic component of the model. The harmonic component is fit using both 2019 and 2020 data, which informs the model of the expected seasonal shape. Since these changes are accounted for in the model fitting, it is unlikely that change points identified in the window of interest are due only to seasonal variation. Because the length of the time series only included two seasons (park visitation demonstrates a yearly cycle), it was not pertinent to search for changes in the seasonal structure.

BEAST is less effective in identifying change points in time series with high variance. The recorded park visits in some of the counties were low enough that the behavior of only a few individuals could have large impacts on the time series itself. To ensure that BEAST was only considering counties for which there was enough data we used a mean normalized visitation threshold of 10^−5.5 (this corresponds to about 120 visits per week in the month with the least DAUs) in 2020. A total of 322 counties did not meet this criteria and were excluded from further analysis. The remaining 711 counties that contain parks in our dataset met this criteria. The counties included in the analysis are roughly 21% of all the counties in the United States, and span all of the states. Details on the selection of the visitation threshold can be found in the Supplementary Materials (See S1 Fig).