If you open the Negative Exponential.jmp nonlinear sample data example, the actual formula for the Nonlinear column looks something like this:
Parameter({b0=0.5, b1=0.5}, b0*(1-Exp(-b1*X)))
The Parameter block in the formula is hidden if you use the formula editor. That is how it is stored in the column and how it appears in the Nonlinear Launch dialog. Two parameters named b0 and b1 are given initial values and used in the formula to be fit.
The Nonlinear platform makes a separate copy of the formula, and edits it to extract the parameters from the expression. Then it maps the references to them to the place where they are estimated. Nonlinear takes the analytic derivatives of the prediction formula with respect to the parameters. If you use the Show Derivatives command, you get the resulting formulas listed in the log, like this:
b0 * First(T#1=1-(T#2=Exp(-b1*X)), T#3=-(-1*T#2*X))
The Derivative of Model with respect to the parameters is:
{T#1, T#3*b0}
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When the prediction model needs additional subexpressions evaluated, it uses the First function, which returns the value of the first argument expression, and also evaluates the other arguments. In this case additional assignments are needed for derivatives.
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The derivative table itself is a list of expressions, one expression for each parameter to be fit. For example, the derivative of the model with respect to b0 is T#1; its thread in the prediction model is 1–(Exp(-b1*X)). The derivative with respect to b1 is T#3*b0, which is –(–1*Exp(-b1*X)*X)*b0 if you substitute in the assignments above. Although many optimizations are made, it does not always combine the operations optimally. You can see this by the expression for T#3, which does not remove a double negation.
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If the derivative mechanism does not know how to take the analytic derivative of a function, then it takes numerical derivatives, using the NumDeriv function. If this occurs, the platform shows the delta that it used to evaluate the change in the function with respect to a delta change in the arguments. You might need to experiment with different delta settings to obtain good numerical derivatives.
There are always many ways to represent a given model, and some ways behave much better than other forms. Ratkowsky (1990) covers alternative forms in his text.
If(Y==0, Log(1/(1+Exp(model))), Log(1 - 1/(1 + Exp(model))));
temp=1/(1+Exp(model));
If(Y==0, Log(temp), Log(1-temp));