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

4.6.4.4.

Quadratic/Quadratic Rational Function Model

Q/Q Rational Function Model

We used Dataplot to fit the Q/Q rational function model. Dataplot first uses the EXACT RATIONAL FIT command to generate the starting values and then the FIT command to generate the nonlinear fit.

We used the following 5 points to generate the starting values.

 TEMP        THERMEXP
 ----        --------
  10             0
  50             5
 120            12
 200            15
 800            20
       

Exact Rational Fit Output

Dataplot generated the following output from the EXACT RATIONAL FIT command. The output has been edited for display.

EXACT RATIONAL FUNCTION FIT
NUMBER OF POINTS IN FIRST SET    =        5
DEGREE OF NUMERATOR              =        2
DEGREE OF DENOMINATOR            =        2
  
NUMERATOR  --A0  A1  A2          =      -0.301E+01       0.369E+00      -0.683E-02
DENOMINATOR--B0  B1  B2          =       0.100E+01      -0.112E-01      -0.306E-03
 
APPLICATION OF EXACT-FIT COEFFICIENTS
TO SECOND PAIR OF VARIABLES--
  
NUMBER OF POINTS IN SECOND SET           =      236
NUMBER OF ESTIMATED COEFFICIENTS         =        5
RESIDUAL DEGREES OF FREEDOM              =      231

RESIDUAL STANDARD DEVIATION (DENOM=N-P)  =  0.17248161E+01
AVERAGE ABSOLUTE RESIDUAL   (DENOM=N)    =  0.82943726E+00
LARGEST (IN MAGNITUDE) POSITIVE RESIDUAL =  0.27050836E+01
LARGEST (IN MAGNITUDE) NEGATIVE RESIDUAL = -0.11428773E+02
LARGEST (IN MAGNITUDE) ABSOLUTE RESIDUAL =  0.11428773E+02
  
       

The important information in this output are the estimates for A0, A1, A2, B1, and B2 (B0 is always set to 1). These values are used as the starting values for the fit in the next section.

Nonlinear Fit Output

Dataplot generated the following output for the nonlinear fit. The output has been edited for display.

LEAST SQUARES NON-LINEAR FIT
SAMPLE SIZE N =      236
MODEL--THERMEXP =(A0+A1*TEMP+A2*TEMP**2)/(1+B1*TEMP+B2*TEMP**2)
REPLICATION CASE
REPLICATION STANDARD DEVIATION =     0.8131711930D-01
REPLICATION DEGREES OF FREEDOM =           1
NUMBER OF DISTINCT SUBSETS     =         235
  
FINAL PARAMETER ESTIMATES           (APPROX. ST. DEV.)    T VALUE
1  A0                  -8.12326       (0.3908    )         -21.
2  A1                  0.513233       (0.5418E-01)          9.5
3  A2                 -0.736978E-02   (0.1705E-02)         -4.3
4  B1                 -0.689864E-02   (0.3960E-02)         -1.7
5  B2                 -0.332089E-03   (0.7890E-04)         -4.2

RESIDUAL    STANDARD DEVIATION =         0.5501883030
RESIDUAL    DEGREES OF FREEDOM =         231
REPLICATION STANDARD DEVIATION =         0.0813171193
REPLICATION DEGREES OF FREEDOM =           1
LACK OF FIT F RATIO =      45.9729 = THE  88.2878% POINT OF THE
F DISTRIBUTION WITH    230 AND      1 DEGREES OF FREEDOM

       

The above output yields the following estimated model.

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.

6-Plot 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 6-plot 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 full-sized residual plot in order to show more detail.

The full-sized residual plot clearly shows the distinct pattern in the residuals. When residuals exhibit a clear pattern, the corresponding errors are probably not random.