One of the ultimate questions of my work is comparing what factors drive bull kelp populations in northern California versus in Oregon. With this exercise I wanted to examine whether patches exhibited temporal synchrony from year to year. If they are in sync, this may suggest that the two populations are being driven by large-scale, coastwide factors (e.g. ENSO phase, wind strength). If not, this gives evidence that the populations are being influenced more so by local factors than shared, coast-wide factors. One way to look at temporal synchrony of two areas is via cross correlation.

To conduct cross correlation, I used both the randomly generated data and my actual data. The randomly generated data should be representative of the null hypothesis that there is no synchrony between patches. I can then compare this to what patterns of cross correlation I get for my actual data, to see if it differs measurably or conforms to a similar pattern as my null hypothesis.

To interpret cross correlation you first need to look the autocorrelation for each variable. I used the acf function in the ‘ncf’ package in R to conduct autocorrelation on the time-series for two patches, Orford Reef and Rogue Reef (See Figure 1). With the randomly generated data, neither of the patches shows any kind of significant autocorrelation.

**Figure 1: Autocorrelation of maximum annual kelp coverage for Orford Reef (left) and Rogue Reef (right) with randomly generated data. Lag is in terms of years.**

I also ran autocorrelation on two kelp patches from my real data. I would expect to see some kind of autocorrelation in real populations. It makes intrinsic sense that with a real population, the size of the population now should influence how many there are one step into the future. However, the patch at Orford Reef had no autocorrelation other than at time=0 (see Figure 2). The patch at Rogue Reef was somewhat auto-correlated at a time lag of a year, but otherwise had no significant autocorrelation. This indicates that the size of the kelp canopy one year tells us very little about what it will look like in the future, although at Rogue Reef, canopy size in the current year will have some positive correlation with the canopy size next year.

**Figure 2: Autocorrelation of maximum annual kelp coverage for Orford Reef (left) and Rogue Reef (right). Lag time is in years.**

Once I understood what autocorrelation looked like for each of these reefs, I then moved on to looking at cross correlation between them. I did this using the ccf function in the R package ‘ncf’. For the randomly generated data, I expected no cross correlation, and this is what I saw (see Figure 3). The ccf results for the random data of Rogue Reef and Orford Reef did not go beyond the confidence envelope (except for one small point at a 10 year leading lag). While there is this small bit of cross correlation, since the data is randomly generated we can assume this is coincidental.

**Figure 3: Cross correlation between maximum annual kelp cover at Rogue Reef and Orford Reef over a 35 year time series. The lag is in years.**

With the real data, I would have expected some kind of cross-correlation between the reefs. The two are less than 20 miles apart and should be influenced by similar oceanographic conditions. However, the ccf graph for the real data was very similar to that of the random data. Other than a small amount of correlation at 0 years and -18 years, these two reefs were essentially uncorrelated.

**Figure 4: Cross correlation between annual maximum kelp cover at Rogue Reef and Orford Reef. Lag is in years.**

Overall, these results were surprising. For one, I was expecting some level of autocorrelation for the reefs. However, because bull kelp is an annual species, it is possible that each year the population resets and recruitment next spring is determined by density independent factors. Furthermore, since bull kelp have a bipartite life cycle, alternating between gametophytes and sporophytes, its possible that the transition over the winter from spores to gametophytes to gametes to baby kelp next spring may further decouple maximum canopy size in the fall from population size the next year.

I was also somewhat surprised that there was no cross correlation between the two patches. The lack of cross correlation suggests that two reefs are not correlated on an annual or multi-annual scale. Therefore, despite the fact that the patches are within 20 miles of one another, apparently there is enough local variation in the factors controlling population size to create substantially different sizes and patterns between the two.

I found this technique to be very useful. If I am interpreting the results correctly, then this technique is already helping me uncover some surprising results. One caveat about these results is that my data may not be stationary. However, there may be some reasons that this test is not the most appropriate for my data. Cross-correlation assumes stationarity, and given the intense inter-annual variability, it is not clear whether there are any long term changes in the mean of the population. Therefore, my data may not fulfill this assumption. I would welcome any feedback on how to better assess stationarity in my time series.