**4. Model validity tests**

In order to compare the modeling performances in both cases with all the number of submodels in the library and in the reduction submodel case, there are three different performance criteria considered: The normalized mean-square error (NMSE), the best fit (FIT) and the variance accounted for (VAF) criterion defined respectively as follows:

The normalized root mean squared error (NRMSE) is defined as:

$$\text{NRMSE} = \mathbb{1}\_{\text{max}\left(y\right)} \sqrt{\frac{1}{M} \sum\_{k} \left(y - y\_{s}\right)}\tag{26}$$

The best fit is defined as:

$$FIT = \left(1 - \frac{||\boldsymbol{y}(k) - \boldsymbol{y}\_{\mathrm{S}}||}{||\boldsymbol{y}(k) - \boldsymbol{y}\_{\mathrm{mean}}||}\right) . \mathbf{100}\text{\%}\tag{27}$$

The variance accounted for is defined as:

$$\text{VAF} = \max\left\{ 1 - \frac{\text{var}\left(y(k) - y\_{\text{S}}\right)}{\text{var}(y(k))}, 0 \right\}.\text{100\%}\tag{28}$$

Where *y* denotes the measured output and *yS* is the estimated output of the multimodel.
