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6.
Process or Product Monitoring and Control
6.6. Case Studies in Process Monitoring 6.6.2. Aerosol Particle Size
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| 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, $$ X_{t} = X_{t-1} + a_{t} - 0.3921 a_{t-1} - 0.0051 \, . $$ |
