Fitness Functions

Introduction

MCalibration compares experimental data and model predictions using a “fitness function”, also called a error function. The fitness function returns a scalar value that is minimized during parameter optimization. The following fitness functions are supported.

Normalized Mean Absolute Difference (NMAD)

The Normalized Mean Absolute Difference (NMAD) fitness value represents the average error in percent between the experimental data and the model predictions. The NMAD value is defined by:

\(\text{NMAD }= 100 \cdot \displaystyle\frac{\langle| \mathbf{e} – \mathbf{p}| \rangle}{\text{max}( \langle| \mathbf{e}| \rangle, \langle|\mathbf{p}|\rangle)}\)

where:

  • e is the vector of experimental stress values (or if strain control, the vector of experimental strain values)
  • p is the vector of experimental stress values (or if strain control, the vector of experimental strain values)
  • | . | is the absolute value of the provided vector components
  • < . > is the average of the provided vector

Mean Square Difference (MSD)

The Mean Square Difference (MSD) is defined by:

\(\text{MSD}=\frac{1}{n}\displaystyle\sum_{i=1}^n \left( e_i – p_i\right)^2\)

where:

  • ei is an experimental stress value (or strain value if stress control)
  • pi is a predicted stress value (of strain value if stress control)
  • n is the total number of points

Coefficient of Determination

The Coefficient of Determination (R2) is defined by:

\(R^2 = 1- \displaystyle\frac{\sum_i \left(e_i – p_i\right)^2}{\left( e_i – e_m\right)^2}\)

where

  • ei is an experimental stress value (or strain value if stress control)
  • pi is a predicted stress value (or strain value if stress control)
  • em is the mean of the experimental values