# Quantifying the relationship between two disparate time series

I have two time series that have a roughly similar trend, though both variables are noisy. This graph shows means and standard errors throughout a season of measurements. I'd like to be able to make a quantitative statement about the relationship between these two data sets.

While the two data sets were collected from the same experimental plots, the individual samples from which the means and standard errors were calculated are not meaningfully paired with one another, and you can see that the carbohydrate data set was measured more frequently.

By taking a subset of the carbohydrate measurements that are closest to the microbial biomass measurement dates, I can make a scatterplot showing the means and standard errors that I think gives a fair visual representation of the relationship (TRS.ml is the carbohydrates): This is where I am stuck. I'm not sure how to estimate regression coefficients or calculate an r2 value for a regression of this sort where I have estimates of uncertainty for both variables. Here are some approaches I have been considering:

1. Deming regression. I'm not sure that this would be the right approach. It seems to be more for data sets in which the same technique was used for both variables. If it is, my question is how would I calculate the variance ratio based on the information I have?

2. Regression of all underlying data points. This doesn't really work because the data are not meaningfully paired, so of the 80 or so microbial biomass measurements that underlie the data shown in the graphs here, I can't directly match them to individual measurements of carbohydrates. Matching them arbitrarily seems bad.

3. Regression of carbohydrate means by date against microbial biomass means by date. Basically regress the points in my scatterplot above but throw out the information about the uncertainty. This gives a high r2 driven by the coinciding peaks on July 1st, but to me, seems to overestimate the strength of the relationship.

4. Regression of all microbial biomass values against carbohydrate means by date or vice versa. This allows more of the underlying uncertainty to be incorporated while not forcing the pairing of unrelated data points in an arbitrary way. Again though, it does not incorporate the uncertainty in both variables.

My question is which of these approaches, or any other unlisted approaches, would you recommend for quantifying the relationship between these two time series?