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 \(\frac{cN^o}{n+1}\) ).