4.6.4.5. Cubic/Cubic Rational Function Model

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

4.6.4.5. Cubic/Cubic Rational Function Model

Since the Q/Q model did not describe the data well, we next fit a cubic/cubic (C/C) rational function model.

We used Dataplot to fit the C/C rational function model with the following 7 subset points to generate the starting values.

 TEMP        THERMEXP
 ----        --------
  10             0
  30             2
  40             3
  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    =        7
DEGREE OF NUMERATOR              =        3
DEGREE OF DENOMINATOR            =        3
  
NUMERATOR  --A0  A1  A2  A3      =
   -0.2322993E+01  0.3528976E+00 -0.1382551E-01  0.1765684E-03
DENOMINATOR--B0  B1  B2  B3      = 
    0.1000000E+01 -0.3394208E-01  0.1099545E-03  0.7905308E-05
  
APPLICATION OF EXACT-FIT COEFFICIENTS
TO SECOND PAIR OF VARIABLES--
  
NUMBER OF POINTS IN SECOND SET           =      236
NUMBER OF ESTIMATED COEFFICIENTS         =        7
RESIDUAL DEGREES OF FREEDOM              =      229
  
RESIDUAL SUM OF SQUARES                  =  0.78246452E+02
RESIDUAL STANDARD DEVIATION (DENOM=N-P)  =  0.58454049E+00
AVERAGE ABSOLUTE RESIDUAL   (DENOM=N)    =  0.46998626E+00
LARGEST (IN MAGNITUDE) POSITIVE RESIDUAL =  0.95733070E+00
LARGEST (IN MAGNITUDE) NEGATIVE RESIDUAL = -0.13497944E+01
LARGEST (IN MAGNITUDE) ABSOLUTE RESIDUAL =  0.13497944E+01

The important information in this output are the estimates for A0, A1, A2, A3, B1, B2, and B3 (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+A3*TEMP**3)/
                 (1+B1*TEMP+B2*TEMP**2+B3*TEMP**3)
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                   1.07913       (0.1710    )          6.3
2  A1                 -0.122801       (0.1203E-01)         -10.
3  A2                  0.408837E-02   (0.2252E-03)          18.
4  A3                 -0.142848E-05   (0.2610E-06)         -5.5
5  B1                 -0.576111E-02   (0.2468E-03)         -23.
6  B2                  0.240629E-03   (0.1060E-04)          23.
7  B3                 -0.123254E-06   (0.1217E-07)         -10.

RESIDUAL    STANDARD DEVIATION =         0.0818038210
RESIDUAL    DEGREES OF FREEDOM =         229
REPLICATION STANDARD DEVIATION =         0.0813171193
REPLICATION DEGREES OF FREEDOM =           1
LACK OF FIT F RATIO =       1.0121 = THE  32.1265% POINT OF THE
F DISTRIBUTION WITH    228 AND      1 DEGREES OF FREEDOM

The above output yields the following estimated model.

$y = \frac{1.079 - 0.122x + 0.004097x^{2} - 0.00000143x^{3}} {1 - 0.00576x + 0.000241x^{2} - 0.000000123x^{3}}$

Plot of C/C Rational Function Fit

We generate a plot of the fitted rational function model with the raw data.

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 6-plot indicates no significant violation of the model assumptions. That is, the errors appear to have constant location and scale (from the residual plot in row 1, column 2), seem to be random (from the lag plot in row 2, column 1), and approximated well by a normal distribution (from the histogram and normal probability plots in row 2, columns 2 and 3).

Residual Plot

We generate a full-sized residual plot in order to show more detail.

The full-sized residual plot suggests that the assumptions of constant location and scale for the errors are valid. No distinguishing pattern is evident in the residuals.

Conclusion

We conclude that the cubic/cubic rational function model does in fact provide a satisfactory model for this data set.