“…The estimation of the rates can be accomplished in two different ways, a differential or integral way. [ 41–44 ] Since the interest was in analyzing changes in the specific rates over time, the differential way was adopted here, following the best practice, [ 45,46,28 ] that is, - Starting from the integrate version of the material balance (Equation (5)), the rate related terms were isolated on the right‐hand side since they cannot be measured:
- Fit arbitrary time dependent functions (e.g., cubic smoothing splines, gaussian process models, polynomials or others), f ( t , w ), to approximate the measured quantities, , such that the residual ɛ was small, though the function also does not overfit the data.
- Build the derivative of f ( t , w ) analytically with respect to time, that is,
- Evaluate the derivative at the time instance t i at which the concentrations have been measured (assuming that the concentrations have the lowest measurement frequency) and divide by the approximated biomass ( x m ( t ) · V m ( t ) = g ( t , ω...
…”