Diagnose ARIMA seasonality model residual auto and partial correlation plots 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.


My questions:


*

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

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

*Does the model adequate? 

*What can be done? 
Any insight will be very helpful.
Thanks in advance.
 A: *

*why take logs

*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.
