get_M0_and_M1_matrices

pyapprox.optimal_experimental_design.get_M0_and_M1_matrices(homog_outer_prods, design_prob_measure, noise_multiplier, regression_type)

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 lstsq and quantile.

Returns

M0np.ndarray (num_factors,num_factors)

The matrix $M_0$

M1np.ndarray (num_factors,num_factors)

The matrix $M_1$