6.
Process or Product Monitoring and Control
6.6. Case Studies in Process Monitoring 6.6.2. Aerosol Particle Size


Residuals 
After fitting the model, we should check whether the model is
appropriate.
As with standard nonlinear least squares fitting, the primary tool for model diagnostic checking is residual analysis. 

4Plot of Residuals from ARIMA(2,1,0) Model 
The 4plot
is a convenient graphical technique for model validation in that it tests
the assumptions for the residuals on a single graph.


Interpretation of the 4Plot 
We can make the following conclusions based on the above 4plot.


Autocorrelation Plot of Residuals from ARIMA(2,1,0) Model 
In addition, the
autocorrelation plot
of the residuals from the ARIMA(2,1,0) model was generated.


Interpretation of the Autocorrelation Plot  The autocorrelation plot shows that for the first 25 lags, all sample autocorrelations except those at lags 7 and 18 fall inside the 95 % confidence bounds indicating the residuals appear to be random.  
Test the Randomness of Residuals From the ARIMA(2,1,0) Model Fit  We apply the BoxLjung test to the residuals from the ARIMA(2,1,0) model fit to determine whether residuals are random. In this example, the BoxLjung test shows that the first 24 lag autocorrelations among the residuals are zero (pvalue = 0.080), indicating that the residuals are random and that the model provides an adequate fit to the data.  
4Plot of Residuals from ARIMA(0,1,1) Model 
The 4plot
is a convenient graphical technique for model validation in that it tests
the assumptions for the residuals on a single graph.


Interpretation of the 4Plot from the ARIMA(0,1,1) Model 
We can make the following conclusions based on the above 4plot.


Autocorrelation Plot of Residuals from ARIMA(0,1,1) Model 
The autocorrelation plot of the residuals from ARIMA(0,1,1) was
generated.


Interpretation of the Autocorrelation Plot  Similar to the result for the ARIMA(2,1,0) model, it shows that for the first 25 lags, all sample autocorrelations expect those at lags 7 and 18 fall inside the 95% confidence bounds indicating the residuals appear to be random.  
Test the Randomness of Residuals From the ARIMA(0,1,1) Model Fit 
The BoxLjung test is also applied to the residuals from the
ARIMA(0,1,1) model. The test indicates that there is at least one
nonzero autocorrelation amont the first 24 lags. We conclude that there
is not enough evidence to claim that the residuals are random
(pvalue = 0.026).


Summary  Overall, the ARIMA(0,1,1) is an adequate model. However, the ARIMA(2,1,0) is a little better than the ARIMA(0,1,1). 