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# 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.

Specification

- Alias: None

- Arguments: REAL

Child Keywords:

Required/Optional | Description of Group | Dakota Keyword | Dakota Keyword Description
--- | --- | --- | ---
Optional | collocation_points_sequence | Sequence of collocation point counts used in a multi-stage expansion
Optional (Choose One) | Regression Algorithm | least_squares | Compute the coefficients of a polynomial 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 \(\ell_1\) -minimization problem using linear programming.
basis_pursuit_denoising | Compute the coefficients of a polynomial expansion by solving the Basis Pursuit Denoising \(\ell_1\) -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

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 \(cN^o\) where \(c\) is the collocation_ratio and \(o\) 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 \(\frac{cN^o}{n+1}\) ).

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