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Time series Plot

I have the time series plot shown above. Is is possible to know which model I should use solely by looking at this plot?

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  • $\begingroup$ What is this a time series of? $\endgroup$ – The Laconic Mar 12 '19 at 18:24
  • $\begingroup$ This looks very much like a difference of a series rather than an original series of data. As always, it would help to know what the data actually mean and how they were measured: this kind of valuable information is one thing that separates data analysis from robotic application of mathematical formulas. $\endgroup$ – whuber Mar 12 '19 at 19:50
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    $\begingroup$ @whuber That’s what I was getting at, laconically. I suspect it’s something that was differenced but should have been logged first. And it’s growing, creating the mysterious increase in volatility. $\endgroup$ – The Laconic Mar 12 '19 at 21:20
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    $\begingroup$ All of these answers are premature. $\endgroup$ – The Laconic Mar 12 '19 at 21:21
  • $\begingroup$ @Laconic Yep. And to press your point home, there might be simple cures for the problem--such as a Box-Cox transformation--that cannot be carried out using the differences alone. I know such a proposal might raise objections in certain quarters, so I'll state right now that I won't respond to them--that would be taking us too far off topic and we haven't any relevant information, anyway. $\endgroup$ – whuber Mar 12 '19 at 21:23
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In order to make a good model-selection you should always do some statistical tests and evaluate the accuracy of your model with a training and a test set (or via forward chaining, because cross-validation is not possible in time series.


1. Basics of the time series

According to your plot you have a univariate time series with ~13 years of data (from 2007 to 2019). However I do not know the frequency of your data. Do you have monthly or weekly or even daily observations? ARIMA and SARIMA are often good models for monthly data. Nonetheless in time series with weekly and daily and hourly seasonalities you face distinct problems. There might exist multiple seasonalities and uneven numbers of seasons per year (a year has ~52,1429 weeks). In this case you should rather stick to a tbats model.

2. Stationarity

A time series is (strongly) stationary if is has a constant mean and its higher moments are also constant over time. A time series is weak stationary if it has constant mean and constant variance over time. In order to simplify I will discuss whether the data underlying your plot is weakly stationary. While ARIMA can be applied on non-stationary time series it cannot capture all types of missing stationarity.

2.1 Constant mean (no trend)

Your data apparently has no underlying trend. That makes model selection easier.

2.2 Constant variance

The variance of your data is increasing over time. As far as I know increasing variance cannot be captured with a standard ARIMA model.

2.3 Unit root and structural change

Until 2017 the variance of your data is steadily increasing, but then the variance gets suddenly very small. This can be either due to noise (and lack of data after 2017) or due to a structural change. If it is just noise an ARIMA/SARIMA model can be appropriate, but ARIMA and SARIMA cannot deal with structural breaks.

3. Seasonality

You obviously have a certain type of seasonality. Once per year you have a very high number and the subsequent number is far below the average. Therefore I would stick to a model which takes seasonalities into account, e.g. SARIMA.

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  • $\begingroup$ Thanks! Your answer cleared a lot of my doubts. However, I still have some confusion relating to SARIMA. This plot is for daily data. Does it make sense to use SARIMA with parameters (P, D, Q, 365)? How do I handle series where variance is increasing? You also mention there are structural breaks in the data. How do I handle them? I have just started learning time series. Help would be much appreciated. $\endgroup$ – Siddhi Kiran Bajracharya Mar 12 '19 at 17:03
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    $\begingroup$ I personally don't see a seasonal pattern. This TS correlation structure seems similar to the one you may see in many ARMA models. $\endgroup$ – Stats Mar 12 '19 at 17:23
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    $\begingroup$ @Ferdi increasing variance can be handled by power transforms if it is proven that the variance of the errors is proportional to the expected value (via Box-Cox see my answer to stats.stackexchange.com/questions/18844/… ) or changes deterministically at 1 or more points in time via Tsay's procedure docplayer.net/… $\endgroup$ – IrishStat Mar 12 '19 at 19:54
  • $\begingroup$ @Ferdi I don't see the error variance (constantly) changing over time I see it changing at perhaps only 1 point in time. Did you mean to imply constantly changing or a discrete change a la Tsay. $\endgroup$ – IrishStat Mar 12 '19 at 20:41
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First step in time series modelling is the visual inspection. Eye can see things a model can't necessarily see (but be aware: may also see things of no existence).

Hence what can our eye definitely see? Mean does no seem to change. The process seems to be zero-mean, hence there is no unit root. We can also see that the process is too erratic to be Gaussian. It is time series driven by (mildly) fat tailed noise (like a t-student 3 degrees of freedom). And correlations seem to be compatible to some ARMA models. It could be an ARMA(2,2) driven by t-noise. But we can't definitely know that just by visual inspection.

Hence note, even though the eye can reject classes of models and speculate, it can not estimate and run diagnostics. You can attempt and fit what you may think are the most likely models (and avoid fitting unlikely ones) but statistical estimation and diagnostics is what would allow you get to a model closer (in certain respect at least) to the data generating process.

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  • $\begingroup$ Thank you! Do you mean white noise when you say t-noise? I read somewhere that white noise is random and cannot be predicted. Is there a way to remove the white noise from the time series? $\endgroup$ – Siddhi Kiran Bajracharya Mar 12 '19 at 17:07
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    $\begingroup$ No I don't mean white noise. I mean i.i.d. t-noise and yes there are ways to remove it. Denoising can be done in many different ways e.g. splines, regressions, kernels, wavelets. $\endgroup$ – Stats Mar 12 '19 at 17:10
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The mean clearly shifts at the end of 2009 to a lower mean .

Model error variance increases at about the same time suggesting the need for weighted estimation. see Negative values in time series forecast and high fluctuations in input data AND
How to improve this time series model? improve-this-time-series-model/396423#396423

Seasonal Pulses or Pulses seem to emerge in 2011 . Would have to actually have the data to diagnose which.

The underlying ARIMA model appears to be autoregressive showing a degree of persistancy

There may also be a change in parameters over time detectable by the CHOW test for constancy of ALL parameters in the model https://en.wikipedia.org/wiki/Chow_test NOT just the mean as is implemented in the free R program.

Signed by HAWKEYE ...

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