def test_exponential_growth_model_bayesian_d_optimal_design(self):
"""
See Table 2 in Dietrich Braess and Holger Dette.
On the number of support points of maximin and Bayesian optimal designs,
2007 dx.doi.org/10.1214/009053606000001307
"""
num_design_pts = 50
optimal_design_samples = {
10: ([0.182], [1.0]),
40: ([0.048, 0.354], [0.981, 0.019]),
50: ([0.038, 0.318], [0.973, 0.027]),
100: ([0.019, 0.215], [.962, 0.038]),
200: ([0.010, 0.134], [0.959, 0.041]),
300: ([0.006, 0.084, 0.236], [0.957, 0.037, 0.006]),
3000: ([0.0006, 0.009, 0.055, 1.000], [0.951, 0.039, 0.006, 0.004])
}
lb2 = 1
for ub2 in optimal_design_samples.keys():
# assuming middle of parameter domain is used to find local design
design_samples = np.linspace(0, 1, num_design_pts)
design_samples = np.sort(
np.unique(
np.concatenate(
[design_samples, optimal_design_samples[ub2][0]])))
noise_multiplier = None
local_design_factors = \
lambda p,x: exponential_growth_model_grad_parameters(p,x).T
xx2, ww2 = pya.gauss_jacobi_pts_wts_1D(40, 0, 0)
xx2 = (xx2 + 1) / 2 * (
ub2 - lb2) + lb2 # transform from [-1,1] to [lb2,ub2]
parameter_samples = xx2[np.newaxis, :]
opt_problem = AlphabetOptimalDesign('D',
local_design_factors,
opts=None)
mu, res = opt_problem.solve_nonlinear_bayesian(
parameter_samples,
design_samples[np.newaxis, :],
sample_weights=ww2,
options={
'iprint': 0,
'ftol': 1e-14,
'disp': True,
'tol': 1e-12
},
return_full=True)
I = np.where(mu > 1e-5)[0]
J = np.nonzero(design_samples == np.array(
optimal_design_samples[ub2][0])[:, None])[1]
mu_paper = np.zeros(design_samples.shape[0])
mu_paper[J] = optimal_design_samples[ub2][1]
# published designs are not optimal for larger values of ub2
if I.shape == J.shape and np.allclose(I, J):
assert np.allclose(mu[I],
optimal_design_samples[ub2][1],
rtol=3e-2)
assert (res.obj_fun(mu) <= res.obj_fun(mu_paper) + 1e-6)
def test_heteroscedastic_quantile_bayesian_doptimal_design(self):
"""
Create D-optimal designs, for least squares regression with
homoscedastic noise, and compare to known analytical solutions.
See Theorem 4.3 in Dette & Trampisch, Optimal Designs for Quantile
Regression Models https://doi.org/10.1080/01621459.2012.695665
"""
poly_degree = 2
num_design_pts = 100
x_lb, x_ub = 1e-3, 2000
design_samples = np.linspace(x_lb, x_ub, num_design_pts)
design_samples = np.sort(np.concatenate([design_samples, [754.4]]))
n = 1 # possible values -2,1,0,1
def link_function(z):
return 1 / z**n
def noise_multiplier(p, x):
return link_function(michaelis_menten_model(p, x))
local_design_factors = \
lambda p, x: michaelis_menten_model_grad_parameters(p, x).T
xx1 = np.array([10]) # theta_1 does not effect optimum
#xx2 = np.linspace(100,2000,50)
#parameter_samples = cartesian_product([xx1,xx2])
p_lb, p_ub = 100, 2000
local_design_factors = \
lambda p, x: michaelis_menten_model_grad_parameters(p, x).T
xx2, ww2 = pya.gauss_jacobi_pts_wts_1D(20, 0, 0)
# transform from [-1,1] to [p_lb,p_ub]
xx2 = (xx2 + 1) / 2 * (p_ub - p_lb) + p_lb
parameter_samples = cartesian_product([xx1, xx2])
opt_problem = AlphabetOptimalDesign('D',
local_design_factors,
noise_multiplier=noise_multiplier,
regression_type='quantile')
mu, res = opt_problem.solve_nonlinear_bayesian(
parameter_samples,
design_samples[np.newaxis, :],
sample_weights=ww2,
options={
'iprint': 0,
'ftol': 1e-8,
'disp': True
},
return_full=True)
# vals = []
# for ii in range(design_samples.shape[0]):
# xopt = np.zeros(design_samples.shape[0])+1e-8;
# #xopt[-1]=.5; xopt[ii]=.5
# vals.append(res.obj_fun(xopt))
#plt.plot(design_samples,vals); plt.show()
I = np.where(mu > 1e-5)[0]
assert np.allclose(design_samples[I], [754.4, x_ub])
assert np.allclose(mu[I], [0.5, 0.5])
def univariate_quadrature_rule(n):
x, w = pya.gauss_jacobi_pts_wts_1D(n, 0, 0)
x *= 2
return x, w