**3. Reduction of the model's base**

Based on the new validity calculation of each submodel, the need of each one in the model's base is identified via the numerical values *α* and *β*. Thus, the impact of each submodel above the global model is also determined by the numerical values *α* and *β*. Each submodel can contribute in one way or another via its new validity and one can thus determine the adequate number of submodels where their contribution in modeling is important compared to the other submodels. A comparison is made on to have a good performance by comparing each submodel to the global model. The submodel whose contribution is important will be retained. To solve this problem, the following matrix *X* defined by Eq. (23) have to be determined first and an analysis using singular value decomposition analysis should be carried out.

$$X = \begin{bmatrix} \left( a\_1 \boldsymbol{\upsilon}\_1^{simp} + \beta\_1 \boldsymbol{\upsilon}\_1^{ref} \right) \boldsymbol{\nu}\_1 \circ \boldsymbol{\nu} & \left( a\_2 \boldsymbol{\upsilon}\_2^{simp} + \beta\_2 \boldsymbol{\upsilon}\_2^{ref} \right) \boldsymbol{\nu}\_2 \circ \boldsymbol{\nu} \\\\ \left( a\_i \boldsymbol{\upsilon}\_i^{simp} + \beta\_i \boldsymbol{\upsilon}\_i^{ref} \right) \boldsymbol{\nu}\_i \circ \boldsymbol{\nu} \end{bmatrix} \tag{23}$$

SVD analysis is a numerical algorithm that decomposes an *Nm* � *m* matrix into three unique component matrices:

$$X = U\Sigma V^T\tag{24}$$

Where *U* is left singular vectors with *Nm* � *Nm* matrix, *V* is right singular vectors with *Nm* � *m* matrix and Σ is singular value with *m* � *m* matrix. Among these component matrices, *U* vectors provide the information of retained submodel in an orthogonal form. *U*<sup>1</sup> and *U*<sup>2</sup> which is the first two rows in the *U* vectors

indicate the two most combinations of required submodel. Another approach that can facilitate this analysis is modified version of principal component analysis (PCA). In modified PCA analysis, the differences between *U*<sup>1</sup> and *U*<sup>2</sup> are also calculated by:

$$Z\_i = |U\_{1i}| - |U\_{2i}|\tag{25}$$

Where *Zi* denotes the absolute value of difference between the best two *U* vectors. The result of this analysis is easier to interpret as only one function is plotter versus submodel.
