# How to show trend line is typical

I have sets of trend line data - values by year 2005-2017. I want to show one set is 'the same' as the other sets, with 95% probability - or something like that. They look similar - I want to put a numerical value on the degree of similarity.

The idea is how to get a quantitative measure of how similar Coventry is to the others - or how unlikely it is that Coventry is different

• Post your data and I will try and help as there can be multiple trends and/or level shifts and.or outliers ....all of which can complicate answering a simple question like this one. – IrishStat Jul 1 '19 at 17:39
• and of course possibly different arima structure ..... – IrishStat Jul 1 '19 at 18:10
• Link to data added above – Walter William Milner Jul 2 '19 at 13:59
• Potentially, a great deal could be said about this spatio-temporal dataset. But for your stated purpose, why not just plot the time series in a way that supports visual comparison? If you haven't any idea a priori of how to assess "degree of similarity," then attempting to devise some such measure after a review of the data could be deceptive, suggesting you ought to limit your analysis to description and exploration, anyway. – whuber Jul 2 '19 at 14:03

As a working alternative to my "tired old eyes" what I would do is as follows. Statistics longa, Eyesight brevis .

I took the 9 annual values for Coventry and obtained and with stats here I would then estimate this model separately on each of the other 25 candidates and perform a test of significance between Coventry and each of the candidates to assess which one was the MOST DIFFERENT so to speak OR the MOST SIMILAR.

You only shared data from 2009 ..... not 2005 . Also all models are wrong but this one seems useful .

EDITED TO EXPOUND ON THE TEST FOR SIMILARITY BETWEEN COVENTRY AND 1 ONE OTHER CITY:

FOLLOWING GREGORY CHOW 1960 : https://www.jstor.org/stable/1910133?seq=1#page_scan_tab_contents AND here https://en.wikipedia.org/wiki/Chow_test

I took the 9 years from CITY 1 Birmingham and estimate the COVNENTRY MODEL and obtained and and . We add the two error sums of squares and obtain .000263+.004391=.00464.

I then used AUTOBOX ( a piece of software that I had helped to develop ) to estimate 1 set of parameters for the combined time series and obtained and and . Note that the first fitted/expected value for the second series has nothing whatsoever to do with the last observation in the first series.

Now to test the hypothesis of a common set of parameters we construct an F test with 2 and 14 degrees of freedom

NUMERATOR (.015283-.0004654 )/2 = .005315 DENOMINATOR .004654/14 = .000332

Yielding an F value > 16+ which is quite significnt suggesting that COVRNTRY & BIRMINGHAM differ at a very high level probability (which is confirmed by eye)

This can then be performed for each of the other 24 cities .

• This post does not seem to address the question that was asked, which is (in a very general sense) requesting a way to compare the historical data among different time series, not to forecast future data for one series. Could you please make explicit, then, how you interpret the question and establish for your readers how you answer it? – whuber Jul 2 '19 at 15:59
• comparing time series can be accomplished by comparing models/coefficients . The error sum of squares from the same model estimated on each of the other 26 is a way of assesing similarity or lack of similarity and deriving a probability as the OP requested. I should have disabled forecasting when I analyzed the data as it has no direct role in role in answering the question as I couldn't imagine anyone reflecting negatively on this. – IrishStat Jul 2 '19 at 19:11
• I'm not reflecting negatively on your analysis: I'm only asking why this post is even here, since it doesn't respond to the question. – whuber Jul 2 '19 at 19:52
• The OP asked " I want to put a numerical value on the degree of similarity.. I answered: ":The error sum of squares from the same model estimated on each of the other 26 is a way of assessing similarity or lack of similarity and deriving a probability as the OP requested.: .... Thus I did respond to his question IMHO . – IrishStat Jul 2 '19 at 21:49
• Given that you propose fitting 26 models, exactly what test of "significance" do you propose applying and why do you claim that statistical significance could be a meaningful measure of "similarity"? There is no evident choice of a test here. – whuber Jul 2 '19 at 22:08