## Model selection

The total number of possible GMDH models is increasing exponentially by increasing the number of layers. The model selection can be formulated in the following way. Given a validation data set

$D_{v}&space;=&space;\left&space;\{&space;(\mathbf{x_{vi}},y_{vi})&space;\right&space;\}_{i=1}^{N}$

of data vectors $\small&space;\mathbf{x_{vi}}=\left&space;(&space;x_{vi1},&space;x_{vi2},&space;\cdots&space;,&space;x_{viK}&space;\right&space;),&space;x_{vi}\in&space;\Re^{K}$ and the corresponding output values $\small&space;y_{vi}\in&space;\Re$, evaluate the derived model $\small&space;p(x)$ by using the performance measure $\small&space;E=E(p(x))$ and align it accordingly. A maximum of $\small&space;P_{\lambda}$ best qualified models are selected and retained at each layer. In this way the models on the next layer combine only the best models from the previous layers.