Hello,
I want to generate a correlation between multiple variables using the nonlinear curve fitting. I want to use the formula y=C*x1^a*x2^b. Here, x1 and x2 are the independent variables and y is the dependent variable. How can I create such a user-defined function and use the relevant experimental data for generating the equation? In this case, the parameters would be a, b and C. I have experimental data for y, x1, and x2. Any kind of help would be highly appreciated. thanks
I. Least Squares (LS)
i) The correlation can first be linearized as ln(|y/C|) = a·ln(|x1|) + b·ln(|x2|); from which the parameters C, a, and b, can be fitted by least-squares linear regression (after intercept and the two slopes). These are unbiased estimates for the linearized correlation ln{|y - C|} = ln{|A|} + B·x; but not for the original correlation
ii) Previous estimates of the parameters can be refined, by iterative nonlinear least-squares regression, to find unbiased estimates for the original correlation.
II. Recursive Least Squares (RLS)
The recursive least squares algorithm (RLS) allows for (real-time) dynamical application of least squares (LS) regression to a time series of time-stamped continuously acquired data points. As with LS, there may be several correlation equations with the corresponding set of dependent (observed) variables. For the recursive least squares algorithm with forgetting factor (RLS-FF), acquired data is weighted according to its age, with increased weight given to the most recent data. This is often convenient for adaptive control and/or real-time optimization.
Application example for a similar linearized correlation ― the KLa correlation, used to predict the O2 gas-liquid mass-transfer ― In the signalled work (MSc Thesis), while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS-FF algorithm to estimate the parameters from that correlation. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The power dissipated by agitation was accessed by a torque meter. Simulations, carried in Excel 5.0 with Visual Basic for Applications (VBA) macros, assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay:
Thesis Controlo do Oxigénio Dissolvido em Fermentadores para Minimi...
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