4.
Process Modeling
4.6. Case Studies in Process Modeling 4.6.4. Thermal Expansion of Copper Case Study


Q/Q Rational Function Model Starting Values 
Based on the procedure described in
4.6.4.2,
we fit the model:
$$ y = A_0 + A_1 x + A_2 x^2  B_1 x  B_2 x^2 + \varepsilon \, ,$$
using the following five representative points to generate the
starting values for the Q/Q rational function.
Temp THERMEXP   10 0 50 5 120 12 200 15 800 20The coefficients from the preliminary linear fit of the five points are: A0 = 3.005450 A1 = 0.368829 A2 = 0.006828 B1 = 0.011234 B2 = 0.000306 

Nonlinear Fit Results 
The results for the nonlinear fit are shown below.
Parameter Estimate Stan. Dev t Value A0 8.028e+00 3.988e01 20.13 A1 5.083e01 1.930e02 26.33 A2 7.307e03 2.463e04 29.67 B1 7.040e03 5.235e04 13.45 B2 3.288e04 1.242e05 26.47 Residual standard deviation = 0.5501 Residual degrees of freedom = 231The regression yields the following estimated model. $$ \hat{y} = \frac{8.028 + 0.508x  0.007307x^{2}} {1  0.00704x  0.0003288x^{2}} $$ 

Plot of Q/Q Rational Function Fit 
We generate a plot of the fitted rational
function model with the raw data.
Looking at the fitted function with the raw data appears to show a reasonable fit. 

6Plot for Model Validation 
Although the plot of the fitted function with the raw
data appears to show a reasonable fit, we need to
validate the
model assumptions.
The 6plot
is an effective tool for this purpose.
The plot of the residuals versus the predictor variable temperature (row 1, column 2) and of the residuals versus the predicted values (row 1, column 3) indicate a distinct pattern in the residuals. This suggests that the assumption of random errors is badly violated. 

Residual Plot 
We generate a fullsized residual plot in order to
show more detail.
The fullsized residual plot clearly shows the distinct pattern in the residuals. When residuals exhibit a clear pattern, the corresponding errors are probably not random. 