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

## Model Estimation

 AR(2) Model Parameter Estimates The following parameter estimates were computed for the AR(2) model based on the differenced data.  Parameter Standard 95 % Confidence Source Estimate Error Interval ------ --------- -------- ---------------- Intercept -0.0050 0.0119 AR1 -0.4064 0.0419 (-0.4884, -0.3243) AR2 -0.1649 0.0419 (-0.2469, -0.0829) Number of Observations: 558 Degrees of Freedom: 558 - 3 = 555 Residual Standard Deviation: 0.4423  Both AR parameters are significant since the confidence intervals do not contain zero. The model for the differenced data, $$Y_t$$, is an AR(2) model, $$Y_{t} = -0.4064 Y_{t-1} - 0.1649 Y_{t-2} - 0.0050 \, ,$$ with $$\sigma = 0.4423$$. It is often more convenient to express the model in terms of the original data, $$X_t$$, rather than the differenced data. From the definition of the difference, $$Y_t = X_t - X_{t-1}$$, we can make the appropriate substitutions into the above equation, $$X_{t} - X_{t-1} = -0.4064 (X_{t-1} - X_{t-2}) - 0.1649 (X_{t-2} - X_{t-3}) - 0.0050 \, ,$$ to arrive at the model in terms of the original series, $$X_{t} = 0.5936 X_{t-1} + 0.2415 X_{t-2} + 0.1649 X_{t-3} - 0.0050 \, .$$ MA(1) Model Parameter Estimates Alternatively, the parameter estimates for an MA(1) model based on the differenced data are the following.  Parameter Standard 95 % Confidence Source Estimate Error Interval ------ --------- -------- ---------------- Intercept -0.0051 0.0114 MA1 -0.3921 0.0366 (-0.4638, -0.3205) Number of Observations: 558 Degrees of Freedom: 558 - 2 = 556 Residual Standard Deviation: 0.4434  The model for the differenced data, $$Y_t$$, is an ARIMA(0,1,1) model, $$Y_{t} = a_{t} - 0.3921 a_{t-1} - 0.0051 \, ,$$ with $$\sigma = 0.4434$$. It is often more convenient to express the model in terms of the original data, $$X_t$$, rather than the differenced data. Making the appropriate substitutions into the above equation, $$X_{t} - X_{t-1} = a_{t} - 0.3921 a_{t-1} - 0.0051 \, ,$$ we arrive at the model in terms of the original series, $$X_{t} = X_{t-1} + a_{t} - 0.3921 a_{t-1} - 0.0051 \, .$$