Nonlinear Logistic Regression Model Based On Simplex Basis Function

In this paper a novel nonlinear logistic regression model based on a simplex basis function neural network is introduced that outputs probability of categorical variables in response to multiple predictors. It is shown that since a linear combination of the simplex basis functions can be represented as a piecewise linear model, the proposed nonlinear logistic regression model retains the main advantage of linear logistic regression model, that is, allowing probabilistic interpretation of the data sets from an identified model. The associated estimation problem is treated based on the principle of maximum likelihood by alternating over two algorithms; the iteratively reweighted least squares algorithm for linear parameters, while the simplex basis functions are fixed; then nonlinear parameters in each simplex basis function are adapted in turn based on gradient descent of the negative likelihood. The proposed algorithm is then extended to estimation of nonlinear multinomial logistic model. Numerical experiments are initially carried out to illustrate the advantage of nonlinear logistic regression model versus its linear counterpart in terms of approximation capability. Then we apply the proposed method for a difficult computer vision example of land-cover real data set.