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Below I'm showing just a small subset of a larger set of measurements of a process that I'm using to in turn predict something else. The part of the process that is my signal of interest is the random-walk. I've posted the data in csv format for those especially interested, but it is not necessary to look at this to answer my question.

I've fitted an ARIMA(1,1,2) model to my signal (after log tranforming it). It was the best by AIC/SBC model selection, and the prediction is overlaid on the original (after log transform) below:

Predicted Overlaid on Actual

And residuals look like white noise to me (no test yet performed for that though): enter image description here

  • In general, how do I get the prediction at each time step for the random walk portion of the ARIMA model?
  • The outlier with a value around -5 is a bad data point and I'd like to exclude it. If it's not broadening the question too much I'd like to know how to exclude data points that fall outside of pre-determined limits during an online prediction.
  • I did notice my residuals show a change in variance, so if that violates some kind of ARIMA model assumptions or something let me know.
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    $\begingroup$ Interesting Q's but @Glen_b makes good point. Ideas that might help rephrase the Q follow. ARIMA is a set of models. Box-Jenkins is a methodology, which requires (inducing) stationarity. State-space form (SSF) is a representation. E.g. all ARIMA models can be written in SSF. SSF is very flexible, does not require differencing, and an associated methodology is Harvey. Estimation of SS params can be done with KF via prediction error decomposition. IMO they kind of main aspects. Good luck! $\endgroup$ – Graeme Walsh Oct 27 '15 at 2:24
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    $\begingroup$ I will do that Tom. Honestly probably the best idea to help me get it! $\endgroup$ – JPJ Oct 27 '15 at 14:50
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    $\begingroup$ @Glen_b shoot, JMP just ignored them and I didn't catch it! That's the problem with using the higher level software packages. But I guess for discussion sake I would treat those the same as any other outlier since I know those are bad values in this case. $\endgroup$ – JPJ Oct 28 '15 at 0:48
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    $\begingroup$ You might want a couple of introductory sentences at the start of your post describing what your data are and what you're trying to achieve with this model. Anyway, I'm reopening on the basis that it's no longer too broad, but it may still run into trouble on other criteria (requiring a few additional edits). $\endgroup$ – Glen_b Oct 28 '15 at 0:59
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    $\begingroup$ You're fitting a constant variance model to non-constant variance process. The implications could be various depending on the goal of forecasting. For instance, in log-difference model the variance impacts the expected drift, as you probably know, so your forecast could become biased. Your estimates of the parameter covariance matrix will be messed, it may or not matter, depending on what your goal is. $\endgroup$ – Aksakal Oct 28 '15 at 20:09
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Thanks to the help on this forum i was also able to ask this consolidated form of the original question to one of the profs at my university who is teaching a time-series class this semester....which i should have taken :-)

I thought his answer was pretty good, and also has some echos of other comments posted for this question.

I'm not so sure I should accept this answer officially...but at least want to share it here.

Prof Answer:

You have some strange patterns in your data… it looks like there is some type of “structural break” (the form or pattern of the time process changes around time point 3000). It might make sense to break the time series into two pieces and analyze each separately, if everything else is not working so well.

If covariances are indeed changing over time, then you would be violating the Box-Jenkins model.

I would try some intermediate steps. First just difference your time series: compute Y_t= X_t - X_{t-1} (after log transformation) and see if it looks like there’s some similar pattern throughout. If so, then you could try fitting a ARMA model to the difference. Are you using a software package that computes ARMA models? Can you fit such models on the differences Y_t?

Sometimes software packages will compute the k-step forecasts for the future… if you had those, then you could undo the differencing to get your predictions on-line. So for example, you get predictions in the future for Y_{n+1},…,Y_{n+k} and you know Y_{n+i} = X_{n+i}- X_{n+i-1} so you can find X_{n+i} = Y_{n+i}+X_{n+i-1} for i=1,…,k by the predictions for the differences Y_{n+i} and the observation you have for X_n. Otherwise, you might have to resort to Kalman filtering, which can be ok too.

If you have any outliers you could just delete them in fitting the model. Essentially, you would have no information at that time point, but that’s better than misleading information.

I guess that I just wonder (based on your initial graph) about how the fit of the time series model might improve if you considered breaking the time series into two pieces. Maybe you would do a better job of capturing extremes in the 2nd portion of the series.

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" did notice my residuals show a change in variance, so if that violates some kind of ARIMA model assumptions or something let me know."

Your residuals suggest non-constant error variance and thusly you should employ a Generalized Least Squares (GLS0) model as suggested by http://www.unc.edu/~jbhill/tsay.pdf . Your selection of a log transform is probably unwarranted . AUTOBOX a piece of software that I have helped develop can seamlessly put together a solution that includes a minimally sufficient ARIMA model , level shifts/local time trends , outliers and the GLS that you apparently need. A seconD thought is that perhaps your ARIMA coefficients are time-varying which can also induce/create the appearance of non-constant errors . This facility/capability is also available

If you share this with your Professor he may have something to say about Tsay .

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  • $\begingroup$ Recommending fitting differences is a very dangerous option as is assuming any other form of a filter ( like power transformations) . Data such as ours an reflect a myriad of causes based upon the symptoms that you observe (compute). I would wager that you have a time series with a deterministic change in error variance strongly suggesting GLS ( weighted regression/ARIMA) with a number of pulses. Since most software packages don't have this feature most analysts ( even professors ) have no experience in this regard. $\endgroup$ – IrishStat Nov 17 '16 at 14:30

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