.. _method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-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-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-collocation_points_sequence
   method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-least_squares
   method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-orthogonal_matching_pursuit
   method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-basis_pursuit
   method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-basis_pursuit_denoising
   method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-least_angle_regression
   method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-least_absolute_shrinkage
   method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-cross_validation
   method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-ratio_order
   method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-response_scaling
   method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-use_derivatives
   method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-tensor_grid
   method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-reuse_points
   method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-max_solver_iterations


**Specification**

- *Alias:* None

- *Arguments:* REAL


**Child Keywords:**

+-------------------------+--------------------+---------------------------------+-----------------------------------------------+
| Required/Optional       | Description of     | Dakota Keyword                  | Dakota Keyword Description                    |
|                         | Group              |                                 |                                               |
+=========================+====================+=================================+===============================================+
| Optional                                     | `collocation_points_sequence`__ | Sequence of collocation point counts used in  |
|                                              |                                 | a multi-stage expansion                       |
+-------------------------+--------------------+---------------------------------+-----------------------------------------------+
| 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-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-collocation_points_sequence.html
__ method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-least_squares.html
__ method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-orthogonal_matching_pursuit.html
__ method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-basis_pursuit.html
__ method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-basis_pursuit_denoising.html
__ method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-least_angle_regression.html
__ method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-least_absolute_shrinkage.html
__ method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-cross_validation.html
__ method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-ratio_order.html
__ method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-response_scaling.html
__ method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-use_derivatives.html
__ method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-tensor_grid.html
__ method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-collocation_ratio-reuse_points.html
__ method-bayes_calibration-dream-emulator-ml_pce-expansion_order_sequence-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}` ).


