.. _method-polynomial_chaos-expansion_order-collocation_ratio:
"""""""""""""""""
collocation_ratio
"""""""""""""""""
Set the number of points used to build a PCE via regression to be proportional to the number of terms in the expansion.
.. toctree::
:hidden:
:maxdepth: 1
method-polynomial_chaos-expansion_order-collocation_ratio-least_squares
method-polynomial_chaos-expansion_order-collocation_ratio-orthogonal_matching_pursuit
method-polynomial_chaos-expansion_order-collocation_ratio-basis_pursuit
method-polynomial_chaos-expansion_order-collocation_ratio-basis_pursuit_denoising
method-polynomial_chaos-expansion_order-collocation_ratio-least_angle_regression
method-polynomial_chaos-expansion_order-collocation_ratio-least_absolute_shrinkage
method-polynomial_chaos-expansion_order-collocation_ratio-cross_validation
method-polynomial_chaos-expansion_order-collocation_ratio-ratio_order
method-polynomial_chaos-expansion_order-collocation_ratio-response_scaling
method-polynomial_chaos-expansion_order-collocation_ratio-use_derivatives
method-polynomial_chaos-expansion_order-collocation_ratio-tensor_grid
method-polynomial_chaos-expansion_order-collocation_ratio-reuse_points
method-polynomial_chaos-expansion_order-collocation_ratio-max_solver_iterations
**Specification**
- *Alias:* None
- *Arguments:* REAL
**Child Keywords:**
+-------------------------+--------------------+---------------------------------+-----------------------------------------------+
| Required/Optional | Description of | Dakota Keyword | Dakota Keyword Description |
| | Group | | |
+=========================+====================+=================================+===============================================+
| Optional (Choose One) | Regression | `least_squares`__ | Compute the coefficients of a polynomial |
| | Algorithm | | expansion using least squares |
| | +---------------------------------+-----------------------------------------------+
| | | `orthogonal_matching_pursuit`__ | Compute the coefficients of a polynomial |
| | | | expansion using orthogonal matching pursuit |
| | | | (OMP) |
| | +---------------------------------+-----------------------------------------------+
| | | `basis_pursuit`__ | Compute the coefficients of a polynomial |
| | | | expansion by solving the Basis Pursuit |
| | | | :math:`\ell_1` -minimization problem using |
| | | | linear programming. |
| | +---------------------------------+-----------------------------------------------+
| | | `basis_pursuit_denoising`__ | Compute the coefficients of a polynomial |
| | | | expansion by solving the Basis Pursuit |
| | | | Denoising :math:`\ell_1` -minimization |
| | | | problem using second order cone optimization. |
| | +---------------------------------+-----------------------------------------------+
| | | `least_angle_regression`__ | Compute the coefficients of a polynomial |
| | | | expansion by using the greedy least angle |
| | | | regression (LAR) method. |
| | +---------------------------------+-----------------------------------------------+
| | | `least_absolute_shrinkage`__ | Compute the coefficients of a polynomial |
| | | | expansion by using the LASSO problem. |
+-------------------------+--------------------+---------------------------------+-----------------------------------------------+
| Optional | `cross_validation`__ | Use cross validation to choose the 'best' |
| | | polynomial order of a polynomial chaos |
| | | expansion. |
+----------------------------------------------+---------------------------------+-----------------------------------------------+
| Optional | `ratio_order`__ | Specify a non-linear the relationship between |
| | | the expansion order of a polynomial chaos |
| | | expansion and the number of samples that will |
| | | be used to compute the PCE coefficients. |
+----------------------------------------------+---------------------------------+-----------------------------------------------+
| Optional | `response_scaling`__ | Perform bounds-scaling on response values |
| | | prior to surrogate emulation |
+----------------------------------------------+---------------------------------+-----------------------------------------------+
| Optional | `use_derivatives`__ | Use derivative data to construct surrogate |
| | | models |
+----------------------------------------------+---------------------------------+-----------------------------------------------+
| Optional | `tensor_grid`__ | Use sub-sampled tensor-product quadrature |
| | | points to build a polynomial chaos expansion. |
+----------------------------------------------+---------------------------------+-----------------------------------------------+
| Optional | `reuse_points`__ | This describes the behavior of reuse of |
| | | points in constructing polynomial chaos |
| | | expansion models. |
+----------------------------------------------+---------------------------------+-----------------------------------------------+
| Optional | `max_solver_iterations`__ | Maximum iterations in determining polynomial |
| | | coefficients |
+----------------------------------------------+---------------------------------+-----------------------------------------------+
.. __: method-polynomial_chaos-expansion_order-collocation_ratio-least_squares.html
__ method-polynomial_chaos-expansion_order-collocation_ratio-orthogonal_matching_pursuit.html
__ method-polynomial_chaos-expansion_order-collocation_ratio-basis_pursuit.html
__ method-polynomial_chaos-expansion_order-collocation_ratio-basis_pursuit_denoising.html
__ method-polynomial_chaos-expansion_order-collocation_ratio-least_angle_regression.html
__ method-polynomial_chaos-expansion_order-collocation_ratio-least_absolute_shrinkage.html
__ method-polynomial_chaos-expansion_order-collocation_ratio-cross_validation.html
__ method-polynomial_chaos-expansion_order-collocation_ratio-ratio_order.html
__ method-polynomial_chaos-expansion_order-collocation_ratio-response_scaling.html
__ method-polynomial_chaos-expansion_order-collocation_ratio-use_derivatives.html
__ method-polynomial_chaos-expansion_order-collocation_ratio-tensor_grid.html
__ method-polynomial_chaos-expansion_order-collocation_ratio-reuse_points.html
__ method-polynomial_chaos-expansion_order-collocation_ratio-max_solver_iterations.html
**Description**
Set the number of points used to build a PCE via regression to be proportional to the number of terms in the expansion. To avoid requiring the user to calculate N from n and p, the collocation_ratio allows for specification of a constant factor applied to N (e.g., collocation_ratio = 2. produces samples = 2N). In addition, the default linear relationship with N can be overridden using a real-valued exponent specified using ratio_order. In this case, the number of samples becomes :math:`cN^o` where :math:`c` is the collocation_ratio and :math:`o` is the ratio_order. The use_derivatives flag informs the regression approach to include derivative matching equations (limited to gradients at present) in the least squares solutions, enabling the use of fewer collocation points for a given expansion order and dimension (number of points required becomes :math:`\frac{cN^o}{n+1}` ).