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I have two and half years of the weekly time series data. The seasonally period is 52 weeks. I differed the data with log transformation and feed the data into the MATLAB arima model.

ARIMA('Constant', 0 ,'D', 0 ,'Seasonality', 52, 'MALags',1 ,'SMALags',52);

The first plot is loged origin data and second plot is residual data.

Log Transformed auto and partial correction plots

residual auto and partial correlation plots

My questions:

  1. How to interpret residual data partial and auto correlations plots especillay at lag 52?

  2. The partial corr plot shows getting larger, does it indicate system going to be unstable?

  3. Does the model adequate?

  4. What can be done?

Any insight will be very helpful. Thanks in advance.

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  • $\begingroup$ Why haven't you posted your data ? If it is proprietary then scale it ... $\endgroup$ – IrishStat Dec 9 '16 at 15:31
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  1. why take logs
  2. why difference

by unnecessarily differencing you can inject structure into the residuals.

did you take into account that the seasonality might be deterministic ?

did you identify anomalies (interventions) in your data ... otherwise the acf is dampened

I guess my answer is that you might be better served by advanced model identification diagnostics ala http://www.unc.edu/~jbhill/tsay.pdf

EDITED TO RESPOND TO A COMMENT FROM OP REGARDING PREDICTING FUTURE ANOMALIES

We recently have incorporated a (very unique )feature to predict anomalies using monte carlo procedures http://www.autobox.com/cms/index.php/blog/entry/you-should-have-50-confidence-in-your-confidence-limits . Essentially the history of one-time anomalies is used to propagate forecast scenarios which include future expected anomalies. –

An excerpt is here reflecting some previous stack-exchange dialogue detailing this very important issue...

In the forecasting context, removing outliers is can be very dangerous. If you are forecasting sales of a product and let’s assume that there was a shortage of supply thus there are periods of time with zero sales. Recall that sales data is not demand data. The observed flawed time series then contains a number of outliers/pulses. Good analysis detects the outliers, removes them or in effect replaces the observed values with estimates and then proceeds to model and then forecast. You assumed that no supply shortage like this will happen in future. In practical sense, you compressed your observed variance and estimated error variance. So, if you show the confidence bands for your forecast they will be tighter/narrower than they would have been if you did not remove the outliers. Of course, you could keep the outliers, and proceed as usual, but this is not a good approach either since these outliers will distort the model identification process and of course the resultant model coefficients.

A better approach in this case is to continue to require an error distribution that is normal (no fat tails) while allowing for forecast uncertainties to be based upon an error distribution with fat tails. In this case, your outlier will not skew the coefficients too much. They'll be close to the coefficients with an outlier removed. However, the outlier will show up in the forecast error distribution. Essentially, you'll end up with wider and more realistic forecast confidence bands.

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  • $\begingroup$ Thank you for your comments. You are right. The difference didn't make the difference. The seasonality is domination factor in the time series. Any suggestion for improving the current ARIMA model? $\endgroup$ – mazeandball Nov 6 '14 at 17:19
  • $\begingroup$ The purpose of my ARIMA model is to predict the anomalies in the data. I will educate myself by reading the paper you listed. Do you know any commercial software which is capable of solving this type of problem? $\endgroup$ – mazeandball Nov 6 '14 at 17:41
  • $\begingroup$ Yes one that I had helped develop . It is called AUTOBOX and you can get more information from autobox.com/cms and sales@autobox,com $\endgroup$ – IrishStat Nov 10 '14 at 3:13
  • $\begingroup$ why don't you post your original data and I may be able to give you some pointers $\endgroup$ – IrishStat Sep 1 '15 at 19:58
  • $\begingroup$ We recently have incorporated a (very unique )feature to predict anomalies using monte carlo procedures autobox.com/cms/index.php/blog/entry/… . Essentially the history of one-time anomalies is used to propagate forecast scenarios which include future expected anomalies. $\endgroup$ – IrishStat Sep 2 '15 at 0:12

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