Figures & data
Table 1. Comparison of the homogeneous model (null model), the logistic regression model, the early stopping FNN and the GLM improved/regularised FNN.
Figure 1. Preliminary analysis exploring the early stopping rule: model fitting on the training data (in upper graph) and tracking over-fitting on the validation data
(in lower graph); note that this is a standard output in Keras which (unfortunately) drops the factor 2 from the loss function (Equation7
(7)
(7) ), thus, the y-axis needs to be scaled with 2.
![Figure 1. Preliminary analysis exploring the early stopping rule: model fitting on the training data D0 (in upper graph) and tracking over-fitting on the validation data V (in lower graph); note that this is a standard output in Keras which (unfortunately) drops the factor 2 from the loss function (Equation7(7) (w,θ)↦L(w,θ;D)=−2nℓ(w,θ;D).(7) ), thus, the y-axis needs to be scaled with 2.](/cms/asset/9c09a631-49cb-4a8a-b2dd-1d60c1f88c55/tstf_a_1877960_f0001_oc.jpg)
Figure 2. (lhs) Balance property (Equation4(4)
(4) ) of the early stopping FNN over 50 different seeds (starting points), the orange horizontal line shows the balance property of 5.007276%; (middle) in-sample learning losses on
and (rhs) out-of-sample test losses on
of the early stopping FNN (left box plots in graphs) and the GLM improved/regularised FNN (right box plots in graphs) over the 50 different seeds (starting values of the SGD algorithm).
![Figure 2. (lhs) Balance property (Equation4(4) 1n∑i=1nYi=1n∑i=1npˆMLE(xi)=1n∑i=1nσwˆ0MLE+∑j=1qwˆjMLExi,j,(4) ) of the early stopping FNN over 50 different seeds (starting points), the orange horizontal line shows the balance property of 5.007276%; (middle) in-sample learning losses on D and (rhs) out-of-sample test losses on T of the early stopping FNN (left box plots in graphs) and the GLM improved/regularised FNN (right box plots in graphs) over the 50 different seeds (starting values of the SGD algorithm).](/cms/asset/5e50747d-0883-4a4a-aa46-96a7c16d7030/tstf_a_1877960_f0002_oc.jpg)
Table 2. Comparison of the homogeneous model (null model), the logistic regression model, the early stopping FNN and the GLM improved/regularised FNN, shrinkage regularised versions for different tuning parameters .