get_M0_and_M1_matrices¶
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pyapprox.optimal_experimental_design.get_M0_and_M1_matrices(homog_outer_prods, design_prob_measure, noise_multiplier, regression_type)[source]¶ Compute the matrices \(M_0\) and \(M_1\) used to compute the asymptotic covariance matrix \(C(\mu) = M_1^{-1} M_0 M^{-1}\) of the linear model
\[y(x) = F(x)\theta+\eta(x)\epsilon.\]For least squares
\[M_0 = \sum_{i=1}^M\eta(x_i)^2f(x_i)f(x_i)^Tr_i\]\[M_1 = \sum_{i=1}^Mf(x_i)f(x_i)^Tr_i\]and for quantile regression
\[M_0 = \sum_{i=1}^M\frac{1}{\eta(x_i)}f(x_i)f(x_i)^Tr_i\]\[M_1 = \sum_{i=1}^Mf(x_i)f(x_i)^Tr_i\]- Parameters
- homog_outer_prodsnp.ndarray(num_factors,num_factors,num_design_pts)
The outer products \(f(x_i)f(x_i)^T\) for each design point \(x_i\)
- design_prob_measurenp.ndarray (num_design_pts)
The weights \(r_i\) for each design point
- noise_multipliernp.ndarray (num_design_pts)
The design dependent noise function \(\eta(x)\)
- regression_typestring
The method used to compute the coefficients of the linear model. Currently supported options are
lstsqandquantile.
- Returns
- M0np.ndarray (num_factors,num_factors)
The matrix \(M_0\)
- M1np.ndarray (num_factors,num_factors)
The matrix \(M_1\)