# how to compare short-term trend with a long-term trend?

Let's asume I have this annual temperature data from 1966 to 2018 from one region:

temp = [11.9, 11.7, 12.2 11.6 11.4 11.6 12.2 11.4 11.4 11.7 11.8 12.6 12.5 12.0 12.6 12.6 12.7 12.3 12.0 12.4 12.5 12.9 12.1 12.5 12.1 12.4 12.4 12.6 12.7 12.3 12.8 13.2 12.5 12.5 12.4 12.0 11.8 12.5 12.6 12.4 12.5 11.5 12.7 12.3 12.2 12.2 12.7 12.5 12.85 12.8 12.7 12.6 12.5].

This looks like

Where in blue is the data, black line is mean, dashed black line is mean+-std, in red is the long-term trend and I put a transparent green rectangle in the last 5 years (2013-2018) to show an opposite trend in the last 5 years (purple line). How can I be sure that this short-trend is significant compared to the long-trend? Is there a method to compare these two trends and that can tell me "ok, this change in the last five years is something..." ? I'm trying to replicate this paper: Zador et al., 2016. Ecosystem considerations in Alaska: the value of qualitative assessments. ICES Journal of Marine Science (2017), 74(1), 421–430. doi:10.1093/icesjms/fsw144

Best regards,

Marco

This an example of what I have referred to as "The dummy trend line trap" where a supposedly innocuous assumption is made and unchallenged. See Forecasting recurring orders for an online subscription business using Facebook Prophet and R for another example of this using Prophet which assumes growth is pervasive.

Discerning between a step/level shift and a deterministic time trend is not "a nice to have feature" BUT "a must have feature" for both humans and software.

In this case the long-term trend and the short-term trend are the same ....They do not exist ! . Your data suggests a LEVEL SHIFT ( that is not a trend ! ) and a number of one-time anomalies. Note the "trend" is the name that I gave to the series.

If one were willing to set 4 consecutive numbers as "proof of trend change" then these results might be different as the last 3 are downwards trending ! but with 3 values downwards this is hardly proof of "proof of trend" .

witk equation here and an ACF of the residuals suggesting randomness