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Decision tree functions are discontinuous functions of the predictors. Have continuous decision trees with smooth transitions been studied? For example, a decision tree in two variables

f(x1,x2) = c1 if x1 <  t1 and x2 <  t2
           c2 if x1 <  t1 and x2 >= t2
           c3 if x1 >= t1 and x2 <  t3
           c4 if x1 >= t1 and x2 >= t3

could be replaced by

    f(x1,x2) = c1*lg(t1-x1)*lg(t2-x2) + c2*lg(t1-x1)*lg(x2-t2) + 
               c3*lg(x1-t1)*lg(t3-x2) + c4*lg(x1-t1)*lg(x2-t3)

where lg is the logistic function.

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