# Compare trends in time series

I have two time series that represent yearly population estimates from different sources, for some species. In the first species ('Forc') the two series are definetly not consistent despite the overall trend is similar, in the second ('Cot') both trend and fluctuations are somewhat consistent.

Data should be autocorrelated since they represent population abundance, but fitting an ARIMA model (with auto.arima from forecast package) results in some 0 in the AR(p) parameters.

 Series: Forc_Abb
ARIMA(0,1,0)

Series: Forc_ntot_est
ARIMA(0,0,0) with zero mean

Series: Cot_Abb
ARIMA(1,0,0) with zero mean

Series: Cot_ntot_est
ARIMA(2,0,0) with zero mean


The question is how to assess consistency of the two proxies for each species.

That is, not only a test for difference of the slopes but something that accounts also for the yearly consistency (e.g. in 'Forc' species, slope is similar but definitely the proxies are not consistent).

Some species (not shown here) show an evident negative or positive trend.

I'm not interested in forecasting.

Are any of these ideas suitable?

• GLS with autocorrelation at lag 1 with proxy and year as indipendent variables, plus the interaction term? Do I need to add some polynomial term in 'Cot' species in this case to better fit the peak in 2006-2007?

• remove the trend by differencing and compare the differenced series (how?)

• adding a changepoint analysis? (e.g. cpt.meanvar from changepoint package)? Do I need apply some smoothing function (e.g. LOESS, moving average etc.)?

• The question is how to assess consistency of the two proxies for each species. How do you define consistency? I understand this may depend on your application, that is why I am asking. (It does not seem you are talking about the property of estimators known as consistency.) How different can the proxies be that you would still call them consistent? Or how different should they be so that you could not longer call them consistent? – Richard Hardy Nov 28 '16 at 18:35
• Thank you Richard for pointing that out, and I apologize for being naive in my question. I do not have a define threshold. A first goal would be having an "index" (a cross correlation value at lag 0 and 1?) that quantify the (mutual) "goodness-of-fit" of the two series. Possibly, I'm also looking for an approach that defines itself a threshold (confidence intervals? P-value?). In my case, the aim is to define whether bag sizes ('Abb') and counts ('ntot_est') describe the same population trend for certain gamebirds, at long term (the overall slope) and at shorter term (the "goodness-of-fit"). – Quechua Nov 29 '16 at 8:33