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I have several different time series with monthly values for 8 years, where I fit an ARIMA model.

And the purpose is to forecast the next year and indicate possible outliers in a fancy way.

Is the simplest way of doing this just to check the graph and see which values are outside the confidence intervals of the suggested forecast by the ARIMA model?

For e.g (random pic from google)

enter image description here

Or can one, in say R, use some more fancy tools that would tell you the outliers aswell as the ARIMA forecasts?

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    $\begingroup$ Isn't the purpose of the confidence bounds (the gray shaded regions) to show outliers? If you want something more elaborate, I've taken the commercial tool Autobox for a very brief test-drive. It identifies "interventions" (components of a time series that are rather extreme to safely attribute to a model without additional explanatory variables). I'm sure there is a more technical definition, but that's my intuitive take on interventions. I browed through Pankratz's "Forecasting with Dynamic Regression Models" to get an idea of what's behind it. Of course, there may be other textbooks. $\endgroup$ – StatSmartWannaB Jul 7 '15 at 14:49
  • $\begingroup$ I'm confused by question, do you want to identify outliers in historical data or how would the forecast behave if we induce an outlier in the forecast itself. Can you clarify ? Future predictions will be devoid of outliers. You have to simulate/add outliers to your forecast. $\endgroup$ – forecaster Jul 7 '15 at 15:05
  • $\begingroup$ Hmm, so by the provided historical data that we use to train the ARIMA model I want to present forecasted outliers in a nice way. Could one get the actual datapoints of the values that is not in the suggested confidence interval by the ARIMA forecast? $\endgroup$ – Isbister Jul 7 '15 at 15:09
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You need to use all the data to identify a combined ARIMA model and the anomalous points. Anomalous points can be pulses,seasonal pulses,level shifts and/or local time trends. A possible inadequate procedure is one that identifies an arima model and then identifies the anomalies . An alternative is to identify the anomalies first and then identify the arima model. Commercial software exists to perform both strategies suggesting an optimal approach.

just an update ....

The latest version of AUTOBOX includes the potential of pulse outliers when computing forecast confidence intervals. This is done in conjunction with Probability Management http://probabilitymanagement.org/ which enables uncertainties in the drivers to provide more realistic uncertainties in the output series.

enter image description here

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  • $\begingroup$ Thanks for answering. So my question is still, how would I present the outliers in a nice way to someone else? Would I just say, "check the data that's outside the confidence interval". Or could I get some indication from R that selects / marks the actual datapoints, and outputs them to me? It would be nice not to mark them manually since there is a possibility of doing mistakes! $\endgroup$ – Isbister Jul 7 '15 at 15:06
  • $\begingroup$ To begin with the outlier is detected as part of the modelling process thus there are no confidence limits .If you elect to partition the data into old and new and then generate a forecast form the end of the old the confidence bounds are useless as they assume that you model parameters and model are the population/perfect combination. AUTOBOX has feature to deliver an empirical probability distribution of forecasts enabling users to assess the probability of observing different values in the forecast horizon.This facility is not within the reach of free software without extensive programming. $\endgroup$ – IrishStat Jul 7 '15 at 15:24
  • $\begingroup$ Regarding your answer. What if I dont have all the data? I mean, as I said, I have data for 8 years, and the 9'th is to be predicted, why could I not use the confidence intervals made from the ARIMA model generated from the 8 year collection and point out possible outliers in the upcoming data as in the picture? $\endgroup$ – Isbister Jul 7 '15 at 15:48
  • $\begingroup$ Like I said the confidence limits for an arima model are based solely on the psi weights (derivable from the model) AND the presumption that the estImated parameters of the ARIMA model are identical to the population values. This unwarranted assumption makes the confidence limits too narrow causing lase positives about outliers.. The fix is to decelop a probability distribution of possible forecasts which can then be used by you in the manner you want, $\endgroup$ – IrishStat Jul 7 '15 at 16:14

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