These are very good suggestions above from people who know some very interesting techniques. I think they have a lot to add to your question.

I do not know your particular application or situation.. So the required degree of complexity could range from very simple to highly technical.

Technicality, for its own sake, may not be needed if your application is fairly pedestrian. In fact, if your situation does not call for advanced techniques, I would not suggest trying something that may alienate a less proficient reader of your work.

For example, if you are dealing with simple time-of-day, day-of-week, or lower-frequency calendar seasonality, you may be better off using basic seasonal dummy variables.

However, if you are dealing with seasonality of a nature that is not accommodated by basic methods, then you should consider using greater sophistication. SARIMA models might be a good next choice. But if nonlinearity is also an issue, then maybe take it to the next level, as suggested above.

I have found myself in situations where I am interested in learning and applying sophisticated techniques but that my application does not require such complexity. In such cases I find it valuable to learn about the technically sophisticated methods, so that I am able to address why or why not I am using them. But if the application doesn't require it, I'd suggest keeping it simple.

I hope this is helpful.

Take a look to Ghysels and Osborn (2002) The Econometric Analysis of Seasonal Time Series CUP. I do not think that filtering the data is good idea. if you plant to do inference in the filtered data see the JofE paper by Maravall

Gabriele, excellent question. Please take a look at the paper we published at CMG in 2006: https://www.researchgate.net/profile/Alex_Gilgur/contributions/?ev=prf_act There are a few things I would (and I do) do differently now than I did back then, but the idea is the same: you measure the time intervals between your peaks or between your valleys, and check if the new intervals are in the same distribution. Use nonparametric methods though: you are dealing with a Poisson process (the distribution will not be normal)

Hi Gabriele:

There are several ways to solve this problem. One way to start with is the analysis of ARIMA models of Box and Jenkins. See the contents of this book in:

http://books.google.es/books?hl=es&lr=&id=jyrCqMBW_owC&oi=fnd&pg=PP1&dq=time+series+box+jenkins+book&ots=LPK1adFJP3&sig=R5n_zwWo7QxlTor27CYdIygljB4#v=onepage&q=time%20series%20box%20jenkins%20book&f=false

Cheers.

One way is to use SARIMA models; another way is use time-varying models, under the Bayesian perspective (see papers by Michael West) or frequentist (see papers by Dahlhaus et al, 1999 or Chiann and Morettin (1999,2000).

For seasonal change detection, I recommend the BFAST algorithm:

Detecting trend and seasonal changes in satellite image time series

J Verbesselt, R Hyndman, G Newnham… - Remote sensing of …, 2010 - Elsevier

Also implemented in R. Good luck.

I would do the same a Morettin, with some extensions. The SARIMA option is quite attractive if you aim to forecast the series. The model is quite simple:

Ps(L)xP(L) Yt =Hs(L)x H(L)Ut, where Ut is white noise, P and Q are the standard autoregressive and moving average polynomials, and Ps and Hs are seasonal polynomials (with typical lag lengths of 12, 24,... for a monthly series). The advantage of inserting the seasonal polynomials this way is that you obtain "free addtional" parameters. For example (1 - Ps L^12) (1 - P L) [SAR(12)xAR(1)] is 1-Ps ^12-P L + Ps x P L^13. If you are not trying to make forecasts, I would instead remove the seasonal pattern. You could use X-12 ARIMA or TRAMO SEATS. Both are available at the free software GRETL. If you do that, there is an additional advantage. The X-12 ARIMA output provides 2 or 3 statistical tests that test whether there is evindence of seasonality or not.

If the system you are modelling has a strong non-linear behaviour then one approch I have used to good effect is to start with a structural model of the system and then get the parameters for this model using the Marquardt method. Using Solver in Excel is often good enough.

You may extend the well know simple ARIMA models to SARISMA that take care of dynamic seasonality. For more visit:

http://home.ubalt.edu/ntsbarsh/stat-data/Forecast.htm

http://home.ubalt.edu/ntsbarsh/Business-stat/otherapplets/SeasonalTools.htm

Another possibility is to consider unobserved components models; see http://epp.eurostat.ec.europa.eu/portal/page/portal/euroindicators_conferences/documents_seasons/KOOPMAN_EUROSTAT_STAMPPRES.pdf

Hi Gabriele,

The question is what are you looking for? Models are mathematical theories and in this sense they provide statistical evidences about your thinking. I understood of your question that you are looking for evidences of changes over time of the constant deterministic patterns present in the time series y(t) = trend(t) + seasonal(t) + error(t).

I disagree with the use of detrend the time series because you don't know what part of the changes in the observed values of the time series are due to variations in the level or slope of the time series and what part is due to genuine variations in the seasonal pattern. In this sense, I will use structural time series models (see the book "forecasting, structural time series models and the kalman filter" by Harvey, 1989) with the aim of isolating the components of the time series and prove in a statistical sense if there are variations in the seasonal pattern.

In this sense, you will be able to isolate deterministic changes in the seasonal pattern and build some model for forecasting such changes.

good luck.

Another possibility is to consider unobserved components models.

http://epp.eurostat.ec.europa.eu/portal/page/portal/euroindicators_conferences/documents_seasons/KOOPMAN_EUROSTAT_STAMPPRES.pdf

The previous answer i thinks is the best ones.

You may think of using Fourier series, you can read about this in the link below,

http://en.wikipedia.org/wiki/Fourier_series

The equation you show looks like a standard decomposition as anaive method you could either do an additive(constant variance) or multiplicative(changing variance) decomposition here is a template of some decomposition methods in excel.

These are very good suggestions above from people who know some very interesting techniques. I think they have a lot to add to your question.

I do not know your particular application or situation.. So the required degree of complexity could range from very simple to highly technical.

Technicality, for its own sake, may not be needed if your application is fairly pedestrian. In fact, if your situation does not call for advanced techniques, I would not suggest trying something that may alienate a less proficient reader of your work.

For example, if you are dealing with simple time-of-day, day-of-week, or lower-frequency calendar seasonality, you may be better off using basic seasonal dummy variables.

However, if you are dealing with seasonality of a nature that is not accommodated by basic methods, then you should consider using greater sophistication. SARIMA models might be a good next choice. But if nonlinearity is also an issue, then maybe take it to the next level, as suggested above.

I have found myself in situations where I am interested in learning and applying sophisticated techniques but that my application does not require such complexity. In such cases I find it valuable to learn about the technically sophisticated methods, so that I am able to address why or why not I am using them. But if the application doesn't require it, I'd suggest keeping it simple.

I hope this is helpful.

Giving some further though to your description: "The goal of the analysis is to detect a possible non linearity in the temporal evolution of the seasonal pattern." I would suggest starting with a linear model that either includes a time trend or is based on the de-trended data that employs simple calendar seasonal dummy variables.

Then you can test using Ramsay's RESET test and also test the normality of your residuals. If your residuals are ~NIID, and RESET does not suggest specification error, then I don't think you need more advanced methods.

I am not sure what type of process would have non-linearity in the seasonal factors. I've never come across such an issue in my applications. But it could exist outside my area of expertise.

James, great posts. Re. your question "what type of process would have non-linearity in the seasonal factors", I think Gabriele is either referring to a gradual change (like a drift) in the seasonality, in which case my earlier post today is applicable (what happened to it I have no clue) or to the change in the amplitude of the seasonal variation, in which case it's a matter of choosing between additive and multiplicative seasonality. Gabriele, could you please clarify?

Thanks for the clarification Alex. Very interesting.

Some interesting recent work on model selection in a non-linear environment and testing nonlinearity. I know one paper was just recently published in Journal of Econometrics. Here is a site with the free working paper versions:

http://ideas.repec.org/f/pca273.html

This type of figure might help on the visual aspect, and then you can try different combinations of patterns, how past conditions can influence current ones. or fit different models to the onset of the "season".

Thank you very much for all your answers. I have a lot to read this week-end!

For Alex and James: I have a time series containing the number of marriages celebrated in each month for 150 years for Italy. As first step, I calculated a very simple descriptive seasonal indicator , i.e. The Henry index, and plotting this index for all the 150 years, I noticed, not surprinsingly, that as time passes, the number of marriage in the agricoltural work intensive months (generally the summer months) increases at a convex rate and the same time the not work intense months steeply decrease in their importance after the second world war. I think that this is clearly the effect produced by the industrialization process on the seasonality of marriage and I was beginning to think to an empirical strategy for this type of analysis.

Fascinating subject! You might want to throw in an "agricultural_season" dummy variable then and build a regression to time and to that variable, or use an AR + dummy variable... For simplicity, you can start out with two levels 0 and 1, but you can also make it continuous for how far into the season your data point is.

Thank you again Alex. The reason for which I want to apply also the m.a. filter to smooth my data is that there are clearly some perturbation events in my time series, e.g. the two wars. Furthermore, from my descriptive analysis it seems that also the 70's oil crisis has produced some effects on marriages. In general these extraordinary events hit especially the months of maximun concentration (in normal years), than decreasing the degree of seasonality for the interested years.

If it is an increase in child births, I don't think it is the oil crisis that is explaining the result. You might consider it a second wave of the baby boom, as baby boomers began to reach child bearing age.

I also think that birth control pills have an effect on this. I can think of no logical argument for a connection between oil crisis and birth rates.

Also, what appears to possibly be a nonlinear seasonal could be just an interaction between the normal seasonal pattern and the one-off event of demobilization after WWII. Not sure what frequency you are using (probably quarterly) but data may exist on the "supply of men" returning home.

Any study on birth rates (and I'm not a sociologist) should include factors that affect the "supply" of "child creating counterparties."

Here is an extreme example. After the US Civil War, I've read that there were places where the white male population in the age bracket roughly between 16 and 46 had dropped by 80%.

My suggestion is to try and model war factors that might appear to be nonlinear seasonals. If you can make estimates of changes in male population based on conscription (draft) records, as well as demobilization factors.

I'd also suggest including dynamics by distributed lags. This simply for the reason that it takes 9 months to make a baby. So there should be a three-quarter lag between an adult population shock and a resulting child aged population.

Here's another important factor. The influenza pandemic during WWI killed more people than the war. There are some non-trivial structural changes going on.

It's very interesting topic.

Cheers,

Jim

Gabriele, I am not sure I follow. MA filter will not help you with the effect of the two wars.

I suspect that you will observe that the overall birth rate will have dropped during the wars, but the seasonality during the wars will probably not change much (but the signal to noise ratio will be reduced significantly so that seasonality may be unnoticeable). In other words, the wars will have a confounding effect on your data.

On another thought, I second James' suggestions, especially regarding the use of distributed lags.

Perhaps the fact that I mentioned two works that deal with the seasonality of births, as example for similar empirical strategy, has generated confusion. I'm studying the time series of marriage.

However your suggestions are all very interesting and helpful.

The de.trending is an excellent first step. I would then visualise it with the following method. Make a double plot only of those time series values that are above the trend (i.e., only of the positive values). Double plots for seasonal phenomena are based on 365 day chunks plotted below each other and the doubled. See examples in Roenneberg, T. & Aschoff, J. Annual rhythm of human reproduction: I. Biology, sociology, or both? J. Biol. Rhythms 5, 195–216 (1990). I can send you a reprint if you send me an email ([email protected]).

Once you see the data plotted this way, additional analysis steps will become apparent.

Hope that helps

TR

I forgot to say that you have to normalise the monthly rates, since you would otherwise overestimate February versus March and so on. A moving average 2-harmonic cosine fit would also serve as a model (then no smoothing would be necessary and you would have a goodness of fit for your stats, I also attach some papers I mentioned earlier.

TR

Thank you very much Till for your suggestions and for the paper.

P.s. I forgot to say that my very first step was to equalize all the months to 30 days.

Try a gamm (library mgcv / gamm4) using different splines and including corAr1,

see Gavin Simpson's block about this

Sven

Thanks also to Southrik and Sven. I'm taking a look to Wood (2006).

If you are a fan of R like me try this nice package: http://bfast.r-forge.r-project.org/

Dear Gabriel,

I think there are different models to capture evolving seasonal patterns. However, if you are interested in modelling an evolving seasonal pattern whose seasonal period does not remain the same over time (for example, the number of days in a month), more flexible models can be needed. Perhaps, the following paper can be useful for you.

Martín-Rodríguez, G. y Cáceres-Hernández, J.J. (2012) Forecasting pseudo-periodic seasonal patterns in agricultural prices. Agricultural Economics, 43(5), 531-544.

Jose

Thank you very much Josè.

Dear Gabriele

I think in order to detect signals in a time series for which a linear trend is also present, you can use the Least Squares Harmonic Estimation. Please see the following references:

Amiri-Simkooei, A.R., J. Asgari, (2012) “Harmonic analysis of total electron contents time series: methodology and results”, GPS Solutions, 16 (1): 77-88

Amiri-Simkooei, A.R. C.C.J.M. Tiberius, P.J.G. Teunissen (2007). “Assessment of noise in GPS coordinate time series: methodology and results”, Journal of Geophysical Research, 112, B07413, doi: 10.1029/2006JB004913

Best regards

Alireza

Dear all, the Tinbergen institute PhD thesis of Irma Hindrayanto at www.tinbergen.nl has all the recent references and most linear state space model techniques if you like modelling after the testing. Choosing trending or multiplicative seasonality is key. For hourly data parsimony is key too. We have an overview of state space model software in the OA JSS, www.jstatsoft.org, search for Koopman, Ooms, Commandeur. Lots of free, peer reviewed code!

Thank you very much Marius.