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Here I have a time series (the dots represent measured data, and the line connecting the dots was just added by excel)

https://i.imgur.com/j6NDowH.png

It clearly goes across 3 different states

  • First is approximately linear, with some noise
  • Then goes oscillatory
  • Then turns into a smooth curve (all with noise)

The question is, what kind of mathematical tools would be appropriate to detect the changes of state?

I want to automate (code) the detection of the points were the changes of state occur.

I do not want to assume previos knowledge about what the time series represent, to avoid bias.

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"changes of state" to me means a change in model/parameters at one or more points in time. This requires ARIMA model identification and Intervention Detection with possible variance change points weights culminating in a sequential search process for break-points or "changes of state" . Why don't you post a csv file (one column excel file) of the data you graphed and I will try and help further.

For example data like this often arrive on our door step ..

and here

enter image description here

enter image description here

In order to automate an answer to this you would have to incorporate automatic procedures for the next four items.

  1. ARIMA modelling
  2. Intervention Detection
  3. Model/Parameter change point detection
  4. Error Variance change point detection

It might take a while to figure this out but if you are a skilled statistician and a skilled programmer it is definitely doable.

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  • $\begingroup$ Sorry for the delay. It took me some time to digest it. I'm a programmer, but not a statistician. The most important piece of data I get from your answer is "Intervention Detection" and "Model/Parameter change point detection" $\endgroup$ – lixutopes Mar 12 '18 at 20:17
  • $\begingroup$ why don't you post your data .. If u don't wish then you can contact me and I will try and help. $\endgroup$ – IrishStat Mar 12 '18 at 20:19
  • $\begingroup$ I still have not received real data, so I made a synthetic example, hoping to have some work started before I receive the data. I just added noise to a line, a sinusoidal and a parabola. Real data should look like this: scielo.br/img/revistas/riem/v6n5/a03fig10.jpg $\endgroup$ – lixutopes Mar 12 '18 at 20:26
  • $\begingroup$ It appears that the frequency of measurements (time interval) changes .. Is that so? $\endgroup$ – IrishStat Mar 12 '18 at 20:41
  • $\begingroup$ I guess that at constant intervals $\Delta t$, the series gets 2 lectures x(t) and y(t), but $\Delta t$ may not really be constant, because is experimental data. $\endgroup$ – lixutopes Mar 12 '18 at 21:20

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