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 likeenter image description here

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,



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" .

enter image description here witk equation here enter image description here and an ACF of the residuals suggesting randomness enter image description here

SOME COMMENTS AFTER op commented...

the concept of a long-term trend and a short term trend .can be restated as the idea of a number of trends and thusly break-points in trends. One has to decide the minimum # of values that they wish to define a new trend AND a level of confidence that be in place. If you wish you can take an actual series or a fabricated one in a new question and ask the group to respond . By the way your data was far from random as the first 11 values were statistically significantly different from the last 40 . In terms of your comments here , if a series has a trend and no other trend is detected then you have an answer. If another trend is detected then you have a second trend at some level of confidence. the minimum # of values in a group has to be pre-specified. The software (AUTOBOX which I helped to develop) that I used uses a fixed percentage of the total number of observations BUT it this can be can be manually specified with the software.

  • $\begingroup$ Hello IrishStat, Thanks for your response. I think I get what you say, about that a level shift occured in 1976-1977 and that is not a trend. But it is just a random annual time series of temperature, an example to show what I really want: to have an statistical method to compare two trends and that can tell me if the second short-trend is mean something or not. I'm trying to replicate something like 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 $\endgroup$ Jul 28 '19 at 1:03
  • $\begingroup$ If you are happy with my response please accept it to close the question $\endgroup$
    – IrishStat
    Aug 15 '19 at 8:18

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