Hello Benjamin,

The two standard approaches to description of multiple time series are in time domain (autocorrelation) and the frequency domain (spectral analysis). The former works with time as in the ticking of a clock, the later with time as a frequency, such as heart rate. In my experience the former is more informative at short times scales. The latter gives a better picture at large scales (low frequencies). The information is the same. It depends on how you want to present it.

Attached is a figure showing both approaches -- in the distance domain (correlation) and in the frequency domain (coherence). In the attached example synchrony was high at low frequencies (large scale averages), poor at small scales.

As to multiple sites, you will have n*(n-1)/2 pairs, where n is number of sites. This can be handled graphically by showing each pair arranged according to some other variable, such as separation among sites.

Hope that helps,

~David S

Hi Benjamin,

It is a challenge to improve on the methods you present in your "Ecology Letters" paper but there are a number of things you might try that would give some different perspectives on the data.

First of all, why not use correlations instead of covariances? As you note, the covariance values are a bit arbitrary as different parameters have different absolute concentrations. Using correlations could help to get around this issue. You could also try z-transforming the data, that should give similar results to an analysis based on correlations.

Have you explored the use of multivariate statistics? A PCA or NMDS analysis could give you a nice graphical representation of between-series similarity.You could combine this with a Mantel test to compare distance between series in chemical ordination space with their geographic distance.

Sincerely,

Martyn

Thanks so much David and Martyn. We will try all of these approaches. We really appreciate your thoughts!

If you have several hundred sites then it might also be interesting to ask whether the degree of covariance is related to spatial separation. Check out a Mantel test. You basically have two matrices, one with all pairwise correlations and another with pairwise distances between streams (not sure how to measure that). The Mantel test asks whether these two matrices are random with respect to one another.

If you have hundreds of sites, perhaps different watersheds, different drainage sizes, etc., a straight sequencing using time series is unlikely to produce desired results for comparisons. Here are a couple ideas if you have detailed streamflow data on these sites. 1. Normalize flow for the corresponding water quality sample, such as by drainage size, to units of CSM (cubic feet per second per square mile). Or if you have calculated bankfull flow for each site, normalize by bankfull flow. 2. Identify for each reading if hydrograph is increasing, decreasing or relative steady as baseflow. 3. Use land use to help differentiate watershed inputs to each site. Since there are so many potential combinations, you might combine forests and grasslands in one unit, agriculture, urban, roads, industry in another to help reduce the number of factors. 4. Characterize measurement site by channel type (eg, Rosgen) as there are differences in energy, substrate, entrenchment, sinuousity-response time, etc. 5. Time series may work better in snow dominates terrain, but in rainfall dominated areas over broad landscapes, topographic change, etc., as rainfall pushes over area, some areas are reached before others, and rainfall is not well distributed by time or amount, so time series of comparing so many sites for water quality needs adjustment. The hydrology of each site is not in sync with the others, so it is helpful to look for ways to help qualify comparisons such as suggested by the normalizing by drainage area or bankfull flow. You still may need an element of time series included to help differentiate dormant from growing season, snow melt from thunderstorms, etc. You might syncronize the sites by using the storm peaks as references in time, from which adjustments can be made for each storm. If these sites happen to be USGS sites, you might discuss with them to see if they have programs or routines that might augment this desire for comparison by syncronizing.

We faced with the same problem. Once we used segmented regression to determine the breaking point in time series. But later, we need to compare the similarity between several time series trends and found statistical cointegration tests, for example, Engle-Granger two- step method (Xlstat).

You might want to consider apply Bayesian correlation tests, which accommodate different type of data with no priori assumption on their distributions.

Wetzels, R., & Wagenmakers, E.-J. (2012). A default Bayesian hypothesis

test for correlations and partial correlations. Psychonomic Bulletin &

Review, 19, 1057–1064

Good luck! Xavi