Let's assume that we have independently assessed two quantities in relation (like an adsorption isotherm for surfactant). Both measurements are characterized by the complex probability distribution (say, trapezoid - sum of the uniform distribution of scattered observations and the instrumental uncertainty). How should we fit the non-linear model to these data points? Our aim is to obtain the probability distributions of the parameters of the model. I guess we should interpret each point as rectangle with borders defined by errors by x and y since the measurements are independent, and then fit the model by upper and lower boundaries. But how should we get the probability distributions for parameters then, after we will know the upper and lower boundary values for them?
you can try transform quantities by exponentiation, logarithm to obtain linear relation
Dear Andrzej,
real isotherms do not linearize easily. However, let us assume that we succeded in linearization. Boundary estimates can be easily found for the linear function. But what about the probability distributions for parameters?
If izoterm is in form p=nRT/V and nonlinear relation between p and V is analized then just transform V to 1/V. and you have linear relation between p and 1/V. p is y and 1/V is x so y=ax, a represent nRT and boundary estimates for y can be easily found. Bounduary for x may be estimated using propagation of uncertainty. If U(V) ( uncertainty. for V) is known then U(x) =U(V)*V^(-2).
Hope it will help
Dear Andrzej,
I meant the adsorption or surface tension isotherm, like the von Szyszkowski one: sigma = sigma0 - a*ln(1+bc). The most interesting are the probability distributions for parameters, since most transformations lead to the changes in the distributions' form, they can become asymmetrical, for example.
Dear Andrei
I see that your task is not trivial, especially if you need high accuracy. If some transformation of variable led to linear relation and you get distribution of transformed data then you can transform it back. Transformation and back should not change nature of phenomenon.
Maybe divide data to parts with linear relation beetwen variables.
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