1
The -legacy

By Kostas I. Nikolopoulos and Dimitrios D. Thomakos

The laws of nature are but the mathematical thoughts of God

Euclid 

1.1 The Origins.

(Written in first person by Kostas)

Once upon a time

There was a little boy keen on his (Euclidean) geometry.! It seems like yesterday.but actually it was 22 years ago - winter of 1996 - February of 1996. Me (Kostas), an eighth semester young and promising engineer-to-be, having completed a year ago my elective module in "Forecasting Techniques" and keen to start the (compulsory) dissertation in the National Technical University of Athens in Greece.

This is an endeavor students usually engage in during the 10 and last semester of MEng - but given that sufficient progress in my studies had been achieved to that point - I opted for an early start: this without being aware that it would mean moooore work; so a dissertation that usually takes 6 months to complete ended up an 18-month-long journey.

The topic "Nonparametric regression smoothing" or, in lay terms, time series smoothing (or filtering) with kernel methods - most notably nearest-neighbor approaches - would later on prove useful in my academic career as well, for forecasting time series. For a non-statistician - but math-bombarded engineer - this was quite a spin-off, but one that very much paid off intellectually and career-wise in the years to come.

And, yes, it took 18 months - no discounts there from Vassilis - damping my hopes for a relaxed have-nothing-to-do 10 semester. The reason was my Professor Vassilis Assimakopoulos and the fact that I was lucky enough to join him in the best part of his academic journey, while he was still an Assistant Professor and had this hunger to climb the academic ladder as soon as possible - just like any other academic, although none of them will never admit it, so keen to publish and supervise young promising students like me - and modest (ha ha!). Hope he has not regretted it over the years, but ours is a relationship that is still alive and kicking. I still get the odd phone call wherever I am in the world, and I always find it very hard to say no to any of his academic requests. Vassilis is an extremely bright guy with amazing ideas, all of which you have to pursue until one or none flourishes; but this is a model that I was and still am happy to follow and live by. And, thus began my DEng journey.

The task was simple and bluntly disclosed - in the very words of my supervisor: build a new univariate forecasting method that would win the M3 competition (that was about to be launched). In Greece in the year 1997, in an Engineering school, outperform the entire academic forecasting community and win a blind empirical forecasting competition: probably Mission Impossible! The quest for a new method started with much experimentation - that is out of the scope of this book - and resulted in the "The Theta Model: A Decomposition Approach to Forecasting."

Earlier participatory attempts in the M3 competition along the lines of Theta-sm had been far simpler, non-versatile, and, as such, had never stood the test of time. But the big one had been achieved, and it was a brand new method. A method so versatile that it allowed for many new series to be created from the original data and each one of them could be extrapolated with any forecasting method, and the derived forecasts could be combined in all possible ways: the sky was the limit.

Any action brings on a reaction, inevitably so; in this case, it came from the same International Journal of Forecasting () with the article "Unmasking the Theta method" from Professor Rob J. Hyndman, who later became Editor-in-chief of the journal, and his student Baki Billah in Monash. I have extensively elaborated on this story in the Preface, but I reiterate here that despite that article not doing justice to the method, it did, however, add up to the discussion and kept the interest in the method alive. Looking back, it was an important thing because the next big set of results on the method was not published till 2008 and 2009 (in a series of working papers1 and in the respective presentations in the International Symposium on Forecasting, 2 ), where attempts to mix the method with neural networks were made, results in financial time series were presented, as well as a new theory for unit root data-generating processes (s) in 2009 - the early version of a paper published in the Institute of Mathematics and its Applications () and the superiority of the method was reconfirmed in the NN3 competition.

1.1.1 The Quest for Causality

Bring in then an econometrician . Then came Dimitrios - switch to the year 2005.spring; I was well situated in North West United Kingdom doing my postdoc in Lancaster. What a guy.I have never seen such capacity in my life. A bright and hardworking academic, educated in Columbia in theoretical econometrics - an ideal scholar to throw light on the why and when the method works; a valuable friend always willing to give sharp and short advice, exactly what at that stage a not-so-young-any-more academic needed. Somehow he managed to find me in the Daedalian jungle of the School of Electrical and Computer Engineering building in the National Technical University of Athens, where the Forecasting and Strategy unit - that I was still loosely affiliated with and regularly visiting - was based. After that we worked closely, with him leading the research on the model from 2006 to late 2009, coauthoring a series of working papers.

The academic year 2009-2010 was the turning point: 10 years after publishing the method in a descriptive format, Dimitrios finally laid the foundations of the quest why and when it works - starting with how the method works on unit root DGPs. For two years the paper was in review for IJF without a single error being found in the analysis. In the end, we had to employ a third reviewer only to have him say - to still say that despite all the analysis done as requested by the reviewers these results were not interesting enough for the audience of IJF. Academic judgment is academic judgment.and as such we respected it, but this delayed the whole process by two years - the paper was already out as a working paper since 2009.

The paper took two more years to see the light of day, and in 2014 we finally had why the Theta method works for unit roots (that many of the M3 data series actually are), with the article being published in the Oxford IMA Journal of Management Mathematics. We also had a series of more results from local behaviors of the models, weight optimizations, single theta-line extrapolations, and many more. It was clear among the academic community that we were looking at a method that allowed for a series of models, to be developed within the same set of principles, more robust and accurate than the ETS or ARIMA family, but still equally or more versatile.

With the statement made and the road paved, the journey that followed was much easier. The year 2015 was another milestone, as the first multivariate extension was proposed: a bivariate Theta method that works very well. The results were presented in the JoF; special thanks go out to Editor Professor Derek Bunn for handling this submission so efficiently and for his personal attention to and handling of the paper.

The year 2016 was another important milestone as it was the first year that saw new results being presented by researchers other than Vassilis (and his team), me, and Dimitrios, or Rob (and his students). Fotios - that had in the past worked on the method with me and Vassilis - engaged in joint work with colleagues in Brazil led by Jose Fiorucci, as well as in the later stages of the project with Anne Koehler, provided further extensions and optimizations of the method and a link to state space models. This was the first time that a team - other than the aforementioned three - was using the method not just as a benchmark for evaluation purposes but was also developing a genuinely new theory.

That was the moment we decided it was about time for this book, and Wiley grabbed the opportunity. A book capturing progress in data but, more importantly, proposing a new and more complete theory on the method and many practical applications of it; with the dual scope of capturing the work done so far and emphatically and more prominently inspiring the next generations of forecasters to evolve the method onwards and upwards .

Reflecting on the journey so far, I believe it is truly an extraordinary story - especially given that it started in the cemented basement of an engineering school in Greece and not in a luxurious Ivy League business school; forecasting is a fragmented field where the ability to improve accuracy per se is very small and differences in performance in between models very very small, at the limits of statistical error. So when out of the blue a new method comes out organically from an academic group and has been evolved and is still evolving after 20 years. it must be something.

1.2 The Original Concept: THETA as in...