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

## Model Validation

Residuals After fitting the model, we should check whether the model is appropriate.

As with standard non-linear least squares fitting, the primary tool for model diagnostic checking is residual analysis.

4-Plot of Residuals from ARIMA(2,1,0) Model The 4-plot is a convenient graphical technique for model validation in that it tests the assumptions for the residuals on a single graph. Interpretation of the 4-Plot We can make the following conclusions based on the above 4-plot.
1. The run sequence plot shows that the residuals do not violate the assumption of constant location and scale. It also shows that most of the residuals are in the range (-1, 1).
2. The lag plot indicates that the residuals are not autocorrelated at lag 1.
3. The histogram and normal probability plot indicate that the normal distribution provides an adequate fit for this model.
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 Box-Ljung test to the residuals from the ARIMA(2,1,0) model fit to determine whether residuals are random. In this example, the Box-Ljung test shows that the first 24 lag autocorrelations among the residuals are zero (p-value = 0.080), indicating that the residuals are random and that the model provides an adequate fit to the data.
4-Plot of Residuals from ARIMA(0,1,1) Model The 4-plot is a convenient graphical technique for model validation in that it tests the assumptions for the residuals on a single graph. Interpretation of the 4-Plot from the ARIMA(0,1,1) Model We can make the following conclusions based on the above 4-plot.
1. The run sequence plot shows that the residuals do not violate the assumption of constant location and scale. It also shows that most of the residuals are in the range (-1, 1).
2. The lag plot indicates that the residuals are not autocorrelated at lag 1.
3. The histogram and normal probability plot indicate that the normal distribution provides an adequate fit for this model.
This 4-plot of the residuals indicates that the fitted model is adequate for the data.
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 Box-Ljung test is also applied to the residuals from the ARIMA(0,1,1) model. The test indicates that there is at least one non-zero autocorrelation amont the first 24 lags. We conclude that there is not enough evidence to claim that the residuals are random (p-value = 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). 