For the systematic way of going about time series work please read Gary King's Time Series Fundamentals (2010), If you are sure of the seasonality in the data, then SARIMA models will be just good for you.

Use the R website to further read on the SARIMA and time series in general so that you can handle the techniques of it as well. The R software does time series better. You must be good at calculus to be able to programme the differential equation models since it needs a lot of mathematical interpretations.

Stick to the time series one for that is straight forward and the literature is in abundance and does not need great expertise to understand, apply and interpret.

Good luck.

1) To systematically learn about time series analysis you should start with linear series representations. Which is fairly complicated itself. However, from a more practical point of view, you should learn how to apply decomposition and exponential smoothing methods such as Brown's, Holt's, Winters'. This is amply used in business (meaning that apparently good results can be achieved without going into math too much).

3) The differential method approach tries to make a model that would simulate the behaviour - such as a number of hens at some time given the number vixen who ate some hens earlier when the were chicks and could not lay eggs, however the reminder of them now can etc. Whereas time series modelling is more like regression and statistical methods however applied in the time domain, with lag autocorrelations etc.

If you have to start from first principles what you need is a good book with a good software package. I recommend 'Time Series Analysis with Applications in R (Cryer and Chan (Springer 2008) )', (http://homepage.stat.uiowa.edu/~kchan/TSA.htm). That way you can start working on your problem whilst you are reading the book. In the long term you achieve both objectives: learning about time series and solving your problem.

I would recommend you to talk to a physicist studying statistical mechanics of magnetic models, such as Ising model and so on. There are a plenthora of models, named cellular automata, to study the propagation of infectious diseases. Including a colleague of mine, Daniel Girardi

http://scholar.google.com/citations?user=ShTADw0AAAAJ&hl=en

Hi Matej,

I went through the data that you have attached and as I can see, you have provided the time series of N(case) for 180 months. I tried to run simple Auto regressive models on your data and found that the time series has very less correlation with its lags. This means that if you try to model future N(case) with its previous values, may be the results won't be that encouraging.

In my opinion, it would be wise to try and involve some more explanatory variables in the model (explanatory variables are those which tend to explain the cause behind the occurrence in the time series we are trying to model). And then first step may be to try and fit regression based models on the data in the beginning(as they are the simplest ones and can be done in MS- Excel itself).Simple illustration for Regression models is in this video:

Regression (linear modeling of data with its explanatory variables): http://www.youtube.com/watch?v=QWhsH1vcIaw

Auto-regression (modeling with lags of only one variable): http://www.youtube.com/watch?v=hlR1D0DOg3o

Once you are done with your data preparation, you may then go for ARIMA, ANN or ANFIS methodology for modeling. I can help with codes for the same :)

Best wishes.

For the systematic way of going about time series work please read Gary King's Time Series Fundamentals (2010), If you are sure of the seasonality in the data, then SARIMA models will be just good for you.

Use the R website to further read on the SARIMA and time series in general so that you can handle the techniques of it as well. The R software does time series better. You must be good at calculus to be able to programme the differential equation models since it needs a lot of mathematical interpretations.

Stick to the time series one for that is straight forward and the literature is in abundance and does not need great expertise to understand, apply and interpret.

Good luck.

Hi Matej,

The time series analysis of the data is a diagnostic approach to learn the characteristics of the data set. I would suggest to apply an auto-regression and cross-correlation (if you have an idea of appropriate predictors) analysis.

If you have an idea of predictors then you can do the next step and set a simple model like the classic SI or SIR model to do the step from an analytic to a prognostic approach.

Greetings

Gunther

Thank you all for the great replies. The advises are really helpful for everyone who is a beginner in this area of science, because of the systematic approach you've suggested. As soon as possible I'll start learning from the scratch. Sooner or later you I hope you'll see some of my publications about time series in the epidemiology of infectious diseases! :)

Hi Matej

My main area of research is time series but I have been asked to give a training on time series in a school of public health. That means that I can help you and you can help me too. I have discussed this with a colleague in public health who gave me some recommendations.

I agree with most of the answers, especially that ARIMA or SARIMA (seasonal ARIMA) models would allow you to predict your series although. I am not sure that your predictions will be very good, because of the nature of the data. If you contact me personally I can give you more indications and even possibly a full analysis for your example. Tell me what statistical software packages you have. But, the most important as this stage is to know about the infectious disease, and in particular existence of vectors. Indeed the seasonality can be related to temperature, rainfall, ..., unfortunately as difficult to predict. The only information given is that your data are number of cases, not prevalence. This is also important to know because, for some other diseases, your number of cases can be small integers and in then the normality assumption usually made when providing forecast intervals will not be adequate.

Guy

As Pandelis suggested, I tried to do the X11 method and got some pretty nice graphs. I'll try to do the same with the prevalence values, like Guy said and if possible, I'll try to get more weather data to get a more deeper understanding about the seasonality.

Dear Marek,

Very useful to attach the data of your analysis !

A simple inspection shows that your data are (non large) counts .

Hence Gaussian or continuous distribution assumptions do not hold.

In particular, as happens very often in epidemiology, a discrete time series approach is recommended.

The basic point is a Poisson time series model, which allows you to detect trends and/or cycles in the average number of cases.

R libraries can help you.

Good job !

Alessandro

Of course Matej not Marek caused by orrible T9 !

The interval of 180 months is kind of short. The solar cycle is 11/22 years. And seasonality might require at last two solar cycles. Then there are longer cycles for climate condition (weather is about one two weeks).

Dear Boris,

are you sure about that? I would like to hear a detailed exlplanation. I don't agree with you, but as I've said, I am not a professional in time series analysis.

180 months is 15 years, and for a single bacteria population causing the disease, it's like forever, and reading some other articles I've seen that people are using even smaller time intervals.

Alessandro,

you've caught my attention with the Poisson time series model. Thanks for the suggestion!

Dear Matej,

may I ask if the data are typically always of the type you attached? I mean rather low values, monthly basis, many values? Or are there also weekly, quarterly or yearly time series that are of your interest? Are there also short time series? What is the forecasting horizon, how many values you usually need to forecast?

Otherwise I agree with previous answers that (S)ARIMA may ve bery useful. If your goal is not to become an expert in time series models and you just need to model the series, I actually do not think you have to heavily study the mathematical background, the tools are rather standardized. (Which does not mean thewre is nothing to improve but it is a taks for somebody else.)

Regarding your will to "learn" something from the time series, of course, there are standard tools to detect e.g. seasonality, they are implemented in many SW packages. Most of the tools you will find in R, which is for free. But I think that mining information from data, no matter that the data come from a time series, is a task for data-mining techniques. The goal you foreshadowed seems to be very appropriate for associations mining.

Dear Martin,

the data is always like in the attachment, because it's count numbers of reported patients and even if I standardize them into incidence numbers, I will get low values

.

The way I reported it, monthly based, is just based on my intuition, although I can make it on daily basis, but then there would be lots of zero values.

The main reason I used monthly values is because of the easiness in visualizing and interpreting the seasonal effects. if i were to form data on a yearly basis, i would lose the ability to interpret the seasonal effects.

I'm aware of the lack of the temperature, humidity or other environmental data, but until I get it, I am interested in exploring what I have. I would need about 12 months.

Thanks for the response :)

It is ok with the monthly basis, I was only asking if it is always like that. I am no expert in medical stuff so, I can imagine some type of quick epidemy that is monitored e.g. weekly, therefore I ask, but if it is not the case, it is even better.

Regarding the other data, e.g. environmental data, you hit the point but then you leave the are of time series and enter either multiple time series and/or data-mining. I will try to look at the data you posted and get in touch with you later on.

The time interval has to be significant for the system of consideration. Actually the time series analysis has to start from approximation of the conceptual model.

Who is getting seek during epidemic? What is changing in the interacting system (human and bacteria populations)?

It was theory about a century ego that everything in biosphere effected by Sun: epidemics, revolutions...

(Alexander Chizhevsky).

Dear Boris,

the word epidemiological, doesn't necessary mean I am investigating a real epidemic. An epidemic doesn't have characteristics like my dataset, and with regard to the sun theroy, I am not sure if the idea is has any plausible applications in my work.

Best regards,

Matej

Hi Matej. We have made simple predictions by ARIMA, Exponential smoothing and Theta methods form the R package which is for free. Depending if the results were rounded (to avoid prediction such as 8,76 patients :-) ) or not, the results slightly changed however, theyr were taking RMSE values from ca 2.9 to 3.3 with ES as the best followed by ARIMA and THETA. The differences were not big as you may see. We tried also are ensemble that combines all three methods also with random walk. Without roundening, our approach was between ES and ARIMA, with roundening, it was the best, but again, with a small difference to the others.

However, all methods failed in the prediction of seasonal effects. All predictions were too flat. That is the influence of the fact that the season is not alwasy in the same months, one year, the epidemy is in 10-12 month, another year, it is 12-03 etc. Therefore, also the ensemble of flat predictions was flat. The positive influence of the ensemble could be seen on more time series were each methods sometimes wins and sometimes loses, which is warning for you: do not rely on a single method. Unless it is a specific one developed for your problem, but do not rely on a single general method for any time series.

For the analysis of some aspects of such time series, some data-mining technique would be desirable. I would prefer associations mining.

Nice weekend

Marrtin

What do you think, would a 2x2 ANOVA analysis be acceptable to see if there is a difference within years and between months?

@ Martin, thank you for taking the time to analyze the data. I appreciate your explanation and things are getting more clear than before :) After getting more environmental data, I hope we could better understand the process behind the potential seasonality :)

Dear Matej,

I am not a statistician but I conducted studies on environmental modelling. Since the data you upload here is is only the year and cases, we don't know your parameters. ANOVA analysis can be carried out if you want to find the relationship/ association of your parameters. You also can conduct ANOVA to find out relation between data. For modelling the time series, I have tried the normal ARIMA since my data is nots seasonal. May be you can try. All the best.

Hi Matej,

in my mind the problem of this time series is that the overall linear trend of the series is decreasing all the way. Time series is too short (not in time interval, but too few points), and kind of non-stationary. That is why, at least, linear models predict decreasing values with twelve month peak. The confidence levels are wider that the predicted value itself. See my prediction result with matlab.

br,

Jouni Takalo

Are you looking for seasonal triggers or attempting to identify interannual cycles? This will determine what approach to use. For instance, if you are looking for cycles then wavelet analysis is a useful method which copeswell with non-stationary time series. A 1998 paper by Torrence and Compo called "An Introduction to Wavelet Analysis" is a good place to start. They also provided a website to perform preliminary interactive analyses (http://paos.colorado.edu/research/wavelets/).

Let me recommend a paper that has a great visualization for calendar data. It is called "Cluster and Calendar based Visualization of Time Series Data" by van Wijk and van Selow. They had daily data so it was easier to cluster and to visualize, but thought you might like it.

Conference Paper Cluster and calendar based visualization of time series data

It depends what you really want to do with your data. Mathematical modelling in epidemiology is a large field of inquiry. You can either take the a posteriori statistical bio-demographic models (e.g., statistical approach) or dynamic models to quantify the time course of epidemics. If you used the dynamic models, you might be forced to use the model to fit to your data. As a dynamicist, I might be biased toward the use of dynamic models (e.g., ODE, PDE, etc...), for they can sometime provide both mechanistic insights and dynamic trends.

Since your data is a time series, running ANOVA is not really an option. You should be fitting a time series regression models, as follows:

1) Plot the time series data - I saw the variance of y (N_cases) is non-constant over time, so use log or square root transform of y to get second order stationarity.

2) Create 11 dummy columns for 12 months (use any one of the 12 months as base),

the fit a linear regression model to y as a function of time t and 11 dummy variables.

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 3.1382762 0.1231695 25.479 < 2e-16 ***

t -0.0039986 0.0006199 -6.451 1.16e-09 ***

month_2 -0.5001675 0.1574464 -3.177 0.001774 **

month_3 -0.3231007 0.1574500 -2.052 0.041723 *

month_4 -0.5594037 0.1574561 -3.553 0.000496 ***

month_5 -0.4504513 0.1574647 -2.861 0.004768 **

month_6 -0.4323790 0.1574757 -2.746 0.006700 **

month_7 -0.7479012 0.1574891 -4.749 4.38e-06 ***

month_8 -0.4314013 0.1575049 -2.739 0.006833 **

month_9 -0.4694249 0.1575232 -2.980 0.003313 **

month_10 -0.5458512 0.1575440 -3.465 0.000675 ***

month_11 -0.8541136 0.1575671 -5.421 2.05e-07 ***

month_12 -0.9761414 0.1575928 -6.194 4.41e-09 ***

Residual standard error: 0.4312 on 167 degrees of freedom

(17 observations deleted due to missingness)

Multiple R-squared: 0.3831, Adjusted R-squared: 0.3387

Above model shows strong seasonality and a significant downward trend.

Compute Auto-correlation and Partial Auto-correlation functions, then try to pick ARIMA terms.

Next fit a time series regression model - all of this is easily done in R.

You can get help on this topic from stats24x7.com site.

You may also want to check Dataplore (http://www.ixellence.com/index.php/en/products/dataplore), an analysis tool for stationary and non-stationary, linear and non-linear time series. There is a free demo version that is probably sufficient for your (short) time series. The package also contains macros for forecasting based on ARMA modeling. In particular, the macros ExtrapolARMA, APostForecast, and AnalyseForecast might be interesting for you. It also provides several non-linear correlation functions.

Thanks Ralf, I will certainly look into this - it might be helpful when linear time series regression fails.