Approaches for multivariate nonlinear parametric regression?

#1
I have some data that is nonlinear, but is curvilinear. It is two factors and a dependent variable. Each factor has a pretty good fit when applying a growth curve when compared against (this is biological growth). I know based on what I'm looking at, that there is an interaction between the two factors (time and temp are both factors in how much quantity needs to be used).

The data is:
x1: time
x2: temp
y: quantity (dependent)

Each row of my y,x1,x2 data represents a known 'good' combination for a desired state. The goal of the model is to predict the quantity needed based on time and time to achieve a desired known good outcome. (i.e. at x2 temp for x2 amount of time, if I use y quantity I will achieve my desired outcome).

I'd like to be able to take the data and build a predictive model on it, but I am struggling with the right approach given that I know I need to account for an interaction and the curvilinear relationship.

I tried to use nlm in R, but the model seems too complex to be able to guess the starting values. I've tried giving it various values, but it seems to runaway to 0 or infinity. I also tried non-linear regressions in SPSS with the same approach.

I am thinking that the end model would look something like:
Code:
    Y ~ e^(A+B*X1) + e^(C+D*X2) + e^(E+F*X1*X2)
          Var-1        Var-2          Interaction
I've tried to remove the interaction from the model above, but it doesn't simplify it enough to get correct starting values.

I tried to transform the data by doing curve fits on each variable to the growth model (e^(A+B*X1)) and then doing a multivariate linear regression on the transformed data, but the result doesn't seem right since I can't account for the interaction in the transformed data.

I'm new to non-linear multivariate regressions. I don't want to go the route of a decision tree, since my end goal is to build a predictive calculator on this, and I need a formula that I can program into software.

I have SPSS and R, but open to other tools. Any thoughts or guidance? I have my data here if anyone wants to take a look: https://drive.google.com/file/d/0B3Lw4Ug7AvqwUTg2dWROQ2k3bjA/view?usp=sharing
 

CowboyBear

Super Moderator
#2
I am thinking that the end model would look something like:
Code:
    Y ~ e^(A+B*X1) + e^(C+D*X2) + e^(E+F*X1*X2)
          Var-1        Var-2          Interaction
Can you tell us anything about how you ended up picking this model? If we knew something about the relationships, there might be some appropriate linear model we could suggest to simplify things a bit. It's possible to use a linear model (i.e., a model that is linear in the parameters) to model some types of non-linear relationships between variables.