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Scientist: Examples: Kinetics

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:

Micromath Scientist ABC Kinetics Model

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.

Micormath Scientist Spreadsheet ABC Kinetics

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.

Micromath Scientist Parameter Set ABC Kinetics

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.

Micromath Scientist Parameter Set ABC Kinetics

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

Micromath Scientist Parameter Set ABC Kinetics

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.

Micromath Scientist Parameter Set ABC Kinetics Model

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:

Micromath Scientist Parameter Set ABC Kinetics Model

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


1.1359E-010 9.0912E-009

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

Correlation Matrix


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