This example illustrates how a complex model can be readily solved by attacking the problem in a series of simple steps. The system being studied is the first order conversion of A to B, which is subsequently converted to C by another first order reaction. We begin by selecting File / Open Model from the main menu and navigating to Example 2 ABC Kinetics.

The model is displayed in the model interface as follows:

Next we open the ABC Kinetics spreadsheet containing the data to be fit. The spreadsheet is located in the Models / Examples folder in the Scientist installation directory.

The four data columns, TIME, A, B, and C, represent values for the independent variable, TIME, and each of the three dependent variables. We then select File / Open Parameter set from the menu in the model interface. Select the Variable and Parameter set ABC Kinetics.mmp.

We are now ready to fit the data by selecting the Least Squares Fit option from the Calculate Menu in the Variable and Parameter Interface. The value for A0, obtained by inspection from the data, is likely to be fairly good. Since A depends only on A0 and KAB, we shall first fit these two parameters using only the data for A. In the dependent variable section of the Variable and Parameter interface we deselect B and C. We also fix the value of KBC in the Parameter window by checking the Fix box next to the name of that variable as shown.

We select the Least Squares Fit option and find that the best values are:

Next, we will hold these two parameters constant and fit KBC, using the data for B. We indicate that we wish to fit KBC alone by clicking in the Fix check box for A0 and KAB in the Parameters section. We uncheck A and select B as the dependent variable to be used in fitting. A few simulations suggest that the best value of KBC is between 0.03 and 0.10 and we specify 0.03 as the initial estimate. Scientist then determines, when the other parameter values are fixed and only B data is used, that the best value for KBC is 0.048968.

Finally, we ask that all three parameters be fit simultaneously, using all the available data. After the parameter estimates are found, we perform a simulation to get the following results:

We quickly run a Statistics Report for this fit by selecting Calculate / Statistics from the menu. We run the Statistics Report using the default options and obtain the following results:

Micromath Scientist Statistics Report | ||||

Input Information | ||||

Model: | ABC Kinetics.eqn | |||

Data Set: | ABC Kinetics | |||

Parameter Set: | Parameter Set 3 | |||

Report Options | ||||

Descriptive Statistics | N | |||

Goodness-of-Fit Statistics | Y | |||

Confidence Intervals | Y | |||

Variance-Covariance Matrix | Y | |||

Correlation Matrix | Y | |||

Rigorous Limits | N | |||

Residual Analysis | Y | |||

Confidence Interval | 95% | 50 | ||

Goodness-of-Fit Statistics | ||||

Data Column Name: | A | |||

Weighted | Unweighted | |||

Sum of squared observations: | 0.26339 | 0.26339 | ||

Sum of squared deviations: | 5.6341E-005 | 5.6341E-005 | ||

Standard deviation of data: | 0.0026538 | 0.0026538 | ||

R-squared: | 0.99979 | 0.99979 | ||

Coefficient of determination: | 0.99919 | 0.99919 | ||

Correlation: | 0.99977 | 0.99977 | ||

Data Column Name: | B | |||

Weighted | Unweighted | |||

Sum of squared observations: | 0.023946 | 0.023946 | ||

Sum of squared deviations: | 8.5958E-006 | 8.5958E-006 | ||

Standard deviation of data: | 0.0010366 | 0.0010366 | ||

R-squared: | 0.99964 | 0.99964 | ||

Coefficient of determination: | 0.99767 | 0.99767 | ||

Correlation: | 0.99928 | 0.99928 | ||

Data Column Name: | C | |||

Weighted | Unweighted | |||

Sum of squared observations: | 0.23976 | 0.23976 | ||

Sum of squared deviations: | 2.9212E-005 | 2.9212E-005 | ||

Standard deviation of data: | 0.0019109 | 0.0019109 | ||

R-squared: | 0.99988 | 0.99988 | ||

Coefficient of determination: | 0.99959 | 0.99959 | ||

Correlation: | 0.99984 | 0.99984 | ||

Data Set Name: | ABC Kinetics | |||

Weighted | Unweighted | |||

Sum of squared observations: | 0.5271 | 0.5271 | ||

Sum of squared deviations: | 9.4149E-005 | 9.4149E-005 | ||

Standard deviation of data: | 0.0017715 | 0.0017715 | ||

R-squared: | 0.99982 | 0.99982 | ||

Coefficient of determination: | 0.99953 | 0.99953 | ||

Correlation: | 0.99979 | 0.99979 | ||

Model Selection Criterion: | 7.4719 | 7.4719 | ||

Confidence Intervals | ||||

Parameter Name: | A0 | |||

Estimated Value: | 0.29838 | |||

Standard Deviation: | 0.00076227 | |||

95% Range (Univariate): | 0.29682 | 0.29994 | ||

95% Range (Support Plane): | 0.29612 | 0.30064 | ||

Parameter Name: | KAB | |||

Estimated Value: | 0.020057 | |||

Standard Deviation: | 9.5348E-005 | |||

95% Range (Univariate): | 0.019862 | 0.020252 | ||

95% Range (Support Plane): | 0.019775 | 0.020339 | ||

Parameter Name: | KBC | |||

Estimated Value: | 0.050832 | |||

Standard Deviation: | 0.00052765 | |||

95% Range (Univariate): | 0.049754 | 0.051909 | ||

95% Range (Support Plane): | 0.049269 | 0.052394 | ||

Variance-Covariance Matrix | ||||

5.8105E-007 | ||||

1.1359E-010 | 9.0912E-009 | |||

-1.1171E-007 | -1.761E-008 | 2.7842E-007 | ||

Correlation Matrix | ||||

1 | ||||

0.0015629 | 1 | |||

-0.27774 | -0.35002 | 1 | ||

Residual Analysis | ||||

Expected Value: | The following are normalized parameters with an expected value of 0.0. Values are in units of standard deviations from the expected value. | |||

Serial Correlation: | 1.0161 | indicates a systematic, non-random trend in the residuals | ||

Skewness | 2.1848 | indicates the likelihood of a few large positive residuals having an unduly large effect on the fit. | ||

Kurtosis | 1.3297 | is probably not significant. | ||

Weighting Factor: | 0 | |||

Heteroscedacticity: | 1.4482 | |||

Optimal Weighting Factor: | 1.4482 |