Analytical vs. Numerical Minimization Each experimental data point, l, has an error, ε l, associated with it ‣ Difference between the experimentally measured value of the response variable, ŷ l, and the value that the model predicts for the response variable, y l ‣ The objective function, Φ, is a measure of the overall error for all the data points collectively ‣ One objective is to find parameters values that minimize Φ ‣ Analytical approach is to take derivatives and set equal to zero  Then solve the resulting set of equations for the θ values  Only works if the model can be solved explicitly for the response ‣ Alternative is to minimize Φ numerically  Model does not have to be solved explicitly for the response Solve the model numerically for y within the code for numerically minimizing Φ  Allows using sets of non-linear equations or sets of initial-value ordinary differential equations as the model The response variable might be one of the unknowns in the set of equation or a quantity calculated from those unknowns

MATLAB Template Files for Single Response Data FitNonlinSR.m (fit a non-linear equation to single response data) ‣ Model: FitNumAlgSR.m (numerically fit a set of non-linear (algebraic) equations to single response data) ‣ Model: ‣ Response:

FitNumDifSR.m (numerically fit a set of initial-value ordinary differential equations to single response data) ‣ Model: ‣ Response: Output ‣ Best value for each parameter ‣ Correlation coefficient, r 2 ‣ 95% confidence interval for each parameter ‣ Plot(s)  Model plot if there is only one set variable  Parity plot and residuals plots if there are more than one set variables The template files must be modified each time they are applied to a new problem

Required Information and Data FitNonlinSR.m ‣ a set of experimental data points, each of which consists of a value for the response variable, ŷ, and corresponding values for each of the set variables x i ‣ a guess for the value of each unknown parameter that appears in the model ‣ code that calculates the model-predicted value of the response variable, given values for the parameters and values for the set variables FitNumAlgSR.m ‣ first two items above ‣ code that evaluates the set of non-linear equations to be solved, given values for the unknowns ‣ code that provides a guess for the values of the unknowns FitNumDifSR.m ‣ first two items above ‣ code that evaluates the derivatives in the set of initial value ordinary differential equations given values of the independent and dependent variables ‣ code that provides the initial values of the independent and dependent variables ‣ code that provides the final value of the independent variable

A Note on Numerical Minimization An objective function may have more than one “local” minimum and maximum The analytical approach (taking the derivatives analytically and solving the resulting equations) will find all minima and maxima of the objective function (red arrows) The numerical algorithm, as described here, will only find one minimum ‣ It could be the global minimum ‣ It could be a local minimum It is advisable to repeat a regression analysis using very different guesses for the parameters to see whether the algorithm finds the same minimum or different ones

Fitting Models to Multiple Response Data The objective function differs depending upon the nature of the data ‣ Here only consider a complete data set where every response variable was measured in every experiment ‣ In this case, MATLAB template files (requiring complete data set) ‣ Analogous to FitNumAlgSR.m and FitNumDifSR.m  filenames end with MR.m instead of SR.m ‣ Correlation coefficient and 95% confidence intervals for parameters are not computed