# Non-Deterministic Methods and Surrogates in the Design of Rockfill Dams
## 4 The Surrogate-Assisted Non-Deterministic Algorithm
### Polynomial Regression Model  
A regression model identifies the relationship between a dependent and one or more independent variables [25]. In this investigation, the estimated displacements along the INV01 inclinometer are the dependent variables, and the set of five parameters: the shear modulus (G), Poisson coefficient (v), friction angle (\(\varphi\)), cohesion (c), and specific weight (\(\gamma\)) of the material/soil represent the independent variables. Assuming that the true displacement at a point in the dam (over INV01) due to the influence of vector \(\mathbf{X}\) composed of the set of independent variables is y, the relationship between the two is defined as:  
\[\mathrm{y}=\mathrm{f}(\mathbf{X})+\epsilon \tag{10}\]  
where \(\epsilon\) is an error term and f(\(\mathbf{X}\))denotes the response. In the real world, f(), in a search space, is unknown, and this necessitates the use of some form of approximation. A simple approach that performs this task is linear regression, which can be expressed in the following manner:  
\[\widetilde{y}=\mathrm{\ }\beta_{0}+\sum\limits_{i=1}^{\mathrm{m}}\beta_{i} \mathrm{x}_{i} \tag{11}\]  
where \(\beta_{0}\) and \(\beta_{i}\) are the coefficients, m is the dimension (five in the present scenario), and \(\mathrm{x}_{i}\) implies the individual parameters. The regression coefficients are calculated by minimizing the residual sum of squares \(\sum\limits_{k=1}^{\mathrm{n}}\left(y_{k}-\widetilde{y}_{k}\right)^{2}\), where n is the number of observations. In situations where there is nonlinearity involved in the design space, linear regression models may not give satisfactory results. Polynomial regression can be a suitable option in such scenarios; it succeeds by extending the ideas applied to fitting a linear regression model. Instead of combining input variables linearly, they are combined by raising them to degrees greater than one, as shown below.  
\[\widetilde{y}=\mathrm{\ }\beta_{0}+\sum\limits_{i=1}^{\mathrm{m}}\beta_{i} \mathrm{x}_{i}+\sum\limits_{i=1}^{\mathrm{m}}\sum\limits_{j=1,\mathrm{\ i}\leq j}^{ \mathrm{m}}\beta_{\bar{i}}\mathrm{x}_{i}\mathrm{x}_{j} \tag{12}\]  
Equation (12) depicts the case of a second-order polynomial model with \(\beta_{0}\), \(\beta_{i}\) and \(\beta_{\bar{i}}\) denoting the coefficients; the input variables are represented by \(\mathrm{x}_{i}\) and \(\mathrm{x}_{j}\). The number of coefficients to be estimated increases as the order of the polynomial increases, which in turn requires more sample points and adds to the complexity. Instabilities and unwanted fluctuations may also arise, all of which are major drawbacks of polynomial regression. This work does not delve further into the limitations; however, neural networks have proven to be an effective approach for problems that contain a large number of input variables and display highly nonlinear behavior. [26] provide a good insight into model approximation methods, and try to relate the experimental design, choice of model, number of dimensions and sample points using different examples. Scikit-learn, an open-source machine-learning library in Python, has been used to implement the polynomial-based surrogate model.