def get_polynomial_indices_of_subspace_index( subspace_index,
quad_rule ):
num_dims = quad_rule.num_dims
if subspace_index.is_root():
return [PolynomialIndex(None)]
indices_1d = []
for d in xrange( subspace_index.num_effect_dims() ):
dim = subspace_index.dimension_from_array_index( d )
level = subspace_index.level_from_array_index( d )
# nodel point indices will only work for nested rules
# if no nested rule is used we will have to find the
# number of unique points in all subspaces with level l < level
num_unique_points = \
quad_rule.quadrature_rule_objects[quad_rule.indices[dim]].num_nodal_points( level )
indices_1d.append( numpy.hstack( ( dim * numpy.ones( ( num_unique_points, 1 ), numpy.int32 ), numpy.arange( num_unique_points, dtype=numpy.int32 ).reshape( num_unique_points, 1 ) ) ) )
indices_1d[d] = \
indices_1d[d].reshape(indices_1d[d].shape[0]*indices_1d[d].shape[1])
poly_indices_data = cartesian_product( indices_1d, 2 )
poly_indices = []
num_poly_indices = poly_indices_data.shape[0]
for i in xrange( num_poly_indices ):
poly_indices.append( PolynomialIndex( poly_indices_data[i,:].reshape( poly_indices_data[i,:].shape[0] / 2, 2 ) ) )
return poly_indices
def test_cartesian_product( self ):
# test when num elems = 1
s1 = numpy.arange( 0, 3 )
s2 = numpy.arange( 3, 5 )
sets = numpy.array( [[0,3], [1,3], [2,3], [0,4],
[1,4], [2,4]], numpy.int )
output_sets = cartesian_product( [s1,s2], 1 )
assert numpy.array_equal( output_sets, sets )
# test when num elems > 1
s1 = numpy.arange( 0, 6 )
s2 = numpy.arange( 6, 10 )
sets = numpy.array( [[ 0, 1, 6, 7], [ 2, 3, 6, 7],
[ 4, 5, 6, 7], [ 0, 1, 8, 9],
[ 2, 3, 8, 9], [ 4, 5, 8, 9]], numpy.int )
output_sets = cartesian_product( [s1,s2], 2 )
assert numpy.array_equal( output_sets, sets )
def xtest_grid_search_cross_validation( self ):
f_1d = lambda x: x**10
build_pts = numpy.linspace(-.85,.9,14)
build_pts = numpy.atleast_2d( build_pts )
build_vals = f_1d( build_pts ).T
# Test grid search cross validation when applied to Gaussian Process
num_dims = 1
func_domain = TensorProductDomain( num_dims, [[-1,1]] )
GP = GaussianProcess()
GP.set_verbosity( 0 )
GP.function_domain( func_domain )
loo_cv_iterator = LeaveOneOutCrossValidationIterator()
CV = GridSearchCrossValidation( loo_cv_iterator, GP )
CV.run( build_pts, build_vals )
I = numpy.arange( build_pts.shape[1] )
for i in xrange( build_pts.shape[1] ):
if i == 0 : J = I[1:]
elif i == build_pts.shape[1]-1 : J = I[:-1]
else: J = numpy.hstack( ( I[:i], I[i+1:] ) )
train_pts = build_pts[:,J]
train_vals = build_vals[J,:]
GP.build( train_pts, train_vals )
pred_vals = GP.evaluate_set( build_pts )
assert numpy.allclose( build_vals[i,0]-pred_vals[i],
CV.residuals[0][i] )
# Test grid search cross validation when applied to polynomial chaos
# expansions that are built using ridge regression
# The vandermonde matrix is built from scratch every time by the pce
num_dims = 1
order = 3
build_vals = f_1d( build_pts ).T
poly_1d = [ LegendrePolynomial1D() ]
basis = TensorProductBasis( num_dims, poly_1d )
pce = PCE( num_dims, basis, order, func_domain )
loo_cv_iterator = LeaveOneOutCrossValidationIterator()
CV = GridSearchCrossValidation( loo_cv_iterator, pce )
CV.run( build_pts, build_vals )
I = numpy.arange( build_pts.shape[1] )
V = pce.vandermonde( build_pts ).T
for i in xrange( V.shape[0] ):
if i == 0 : J = I[1:]
elif i == build_pts.shape[1]-1 : J = I[:-1]
else: J = numpy.hstack( ( I[:i], I[i+1:] ) )
A = V[J,:]
b = build_vals[J,:]
x = ridge_regression( A, b )
assert numpy.allclose( (build_vals[i,0]-numpy.dot( V, x ))[i],
CV.residuals[0][i] )
# Test grid search cross validation when applied to polynomial chaos
# expansions that are built using ridge regression
# Specifying parse_cross_validation_data = True will ensure that
# the vandermonde matrix is not built from scratch every time by
# the pce
num_dims = 1
order = 3
build_vals = f_1d( build_pts ).T
poly_1d = [ LegendrePolynomial1D() ]
basis = TensorProductBasis( num_dims, poly_1d )
pce = PCE( num_dims, basis, order, func_domain )
loo_cv_iterator = LeaveOneOutCrossValidationIterator()
CV = GridSearchCrossValidation( loo_cv_iterator, pce,
use_predictor_cross_validation = True)
CV.run( build_pts, build_vals )
I = numpy.arange( build_pts.shape[1] )
V = pce.vandermonde( build_pts ).T
for i in xrange( V.shape[0] ):
if i == 0 : J = I[1:]
elif i == build_pts.shape[1]-1 : J = I[:-1]
else: J = numpy.hstack( ( I[:i], I[i+1:] ) )
A = V[J,:]
b = build_vals[J,:]
x = ridge_regression( A, b )
assert numpy.allclose( (build_vals[i,0]-numpy.dot( V, x ))[i],
CV.residuals[0][i] )
# Test grid search cross validation when applied to polynomial chaos
# expansions that are built using ridge regression
# A closed form for the cross validation residual is used
num_dims = 1
order = 3
build_vals = f_1d( build_pts ).T
poly_1d = [ LegendrePolynomial1D() ]
basis = TensorProductBasis( num_dims, poly_1d )
pce = PCE( num_dims, basis, order, func_domain )
loo_cv_iterator = LeaveOneOutCrossValidationIterator()
CV = GridSearchCrossValidation( loo_cv_iterator, pce,
use_predictor_cross_validation = True,
use_fast_predictor_cross_validation = True )
CV.run( build_pts, build_vals )
I = numpy.arange( build_pts.shape[1] )
V = pce.vandermonde( build_pts ).T
for i in xrange( V.shape[0] ):
if i == 0 : J = I[1:]
elif i == build_pts.shape[1]-1 : J = I[:-1]
else: J = numpy.hstack( ( I[:i], I[i+1:] ) )
A = V[J,:]
b = build_vals[J,:]
x = ridge_regression( A, b )
assert numpy.allclose( (build_vals[i,0]-numpy.dot( V, x ))[i],
CV.residuals[0][i] )
# Test grid search cross validation when applied to polynomial chaos
# expansions that are built using ridge regression
num_dims = 1
order = 3
build_vals = f_1d( build_pts ).T
poly_1d = [ LegendrePolynomial1D() ]
basis = TensorProductBasis( num_dims, poly_1d )
pce = PCE( num_dims, basis, order, func_domain )
max_order = build_pts.shape[1]
orders = numpy.arange( 1, max_order )
lamda = numpy.array( [0.,1e-3,1e-2,1e-1] )
# note cartesian product takes type from first array in 1d sets
# so if I use orders first lamda will be rounded to 0
cv_params_grid_array = cartesian_product( [lamda,orders] )
cv_params_grid = []
for i in xrange( cv_params_grid_array.shape[0] ):
cv_params = {}
cv_params['lambda'] = cv_params_grid_array[i,0]
cv_params['order'] = numpy.int32( cv_params_grid_array[i,1] )
cv_params_grid.append( cv_params )
loo_cv_iterator = LeaveOneOutCrossValidationIterator()
CV = GridSearchCrossValidation( loo_cv_iterator, pce,
use_predictor_cross_validation = True,
use_fast_predictor_cross_validation = False )
CV.run( build_pts, build_vals, cv_params_grid )
k = 0
I = numpy.arange( build_pts.shape[1] )
for cv_params in cv_params_grid:
order = cv_params['order']
lamda = cv_params['lambda']
pce.set_order( order )
V = pce.vandermonde( build_pts ).T
for i in xrange( V.shape[0] ):
if i == 0 : J = I[1:]
elif i == build_pts.shape[1]-1 : J = I[:-1]
else: J = numpy.hstack( ( I[:i], I[i+1:] ) )
A = V[J,:]
b = build_vals[J,:]
x = ridge_regression( A, b, lamda = lamda )
assert numpy.allclose( ( build_vals[i,0]-
numpy.dot( V, x ) )[i],
CV.residuals[k][i] )
k += 1
print 'best',CV.best_cv_params
# Test grid search cross validation when applied to
# expansions that are built using a step based method
# ( LARS )
num_dims = 1
order = 3
build_vals = f_1d( build_pts ).T
poly_1d = [ LegendrePolynomial1D() ]
basis = TensorProductBasis( num_dims, poly_1d )
pce = PCE( num_dims, basis, order, func_domain )
max_order = build_pts.shape[1]
orders = numpy.arange( 1, max_order )
lamda = numpy.array( [0.,1e-3,1e-2,1e-1] )
# note cartesian product takes type from first array in 1d sets
# so if I use orders first lamda will be rounded to 0
cv_params_grid_array = cartesian_product( [lamda,orders] )
cv_params_grid = []
for i in xrange( cv_params_grid_array.shape[0] ):
cv_params = {}
cv_params['solver'] = 4 # LARS
cv_params['order'] = numpy.int32( cv_params_grid_array[i,1] )
cv_params_grid.append( cv_params )
print cv_params_grid
loo_cv_iterator = LeaveOneOutCrossValidationIterator()
#loo_cv_iterator = KFoldCrossValidationIterator( 3 )
CV = GridSearchCrossValidation( loo_cv_iterator, pce,
use_predictor_cross_validation = True,
use_fast_predictor_cross_validation = False )
CV.run( build_pts, build_vals, cv_params_grid )
k = 0
I = numpy.arange( build_pts.shape[1] )
for cv_params in cv_params_grid:
order = cv_params['order']
pce.set_order( order )
V = pce.vandermonde( build_pts ).T
for i in xrange( V.shape[0] ):
if i == 0 : J = I[1:]
elif i == build_pts.shape[1]-1 : J = I[:-1]
else: J = numpy.hstack( ( I[:i], I[i+1:] ) )
A = V[J,:]
b = build_vals[J,:]
b = b.reshape( b.shape[0] )
x, metrics = least_angle_regression( A, b, 0., 4, 0., 1000,
0 )
assert numpy.allclose( ( build_vals[i,0]-
numpy.dot( V, x ) )[i],
CV.residuals[k][i] )
k += 1
#for i in xrange( len( CV.cv_params_set ) ):
# print CV.cv_params_set[i], CV.scores[i]
print 'best param', CV.best_cv_params
print 'best score', CV.best_score
print build_pts.shape[1]
# ( OMP )
num_dims = 1
order = 3
build_vals = f_1d( build_pts ).T
poly_1d = [ LegendrePolynomial1D() ]
basis = TensorProductBasis( num_dims, poly_1d )
pce = PCE( num_dims, basis, order, func_domain )
max_order = build_pts.shape[1]
orders = numpy.arange( 1, max_order )
lamda = numpy.array( [0.,1e-3,1e-2,1e-1] )
# note cartesian product takes type from first array in 1d sets
# so if I use orders first lamda will be rounded to 0
cv_params_grid_array = cartesian_product( [lamda,orders] )
cv_params_grid = []
for i in xrange( cv_params_grid_array.shape[0] ):
cv_params = {}
cv_params['solver'] = 2 # OMP
cv_params['order'] = numpy.int32( cv_params_grid_array[i,1] )
cv_params_grid.append( cv_params )
print cv_params_grid
loo_cv_iterator = LeaveOneOutCrossValidationIterator()
#loo_cv_iterator = KFoldCrossValidationIterator( 3 )
CV = GridSearchCrossValidation( loo_cv_iterator, pce,
use_predictor_cross_validation = True,
use_fast_predictor_cross_validation = False )
CV.run( build_pts, build_vals, cv_params_grid )
k = 0
I = numpy.arange( build_pts.shape[1] )
for cv_params in cv_params_grid:
order = cv_params['order']
pce.set_order( order )
V = pce.vandermonde( build_pts ).T
for i in xrange( V.shape[0] ):
if i == 0 : J = I[1:]
elif i == build_pts.shape[1]-1 : J = I[:-1]
else: J = numpy.hstack( ( I[:i], I[i+1:] ) )
A = V[J,:]
b = build_vals[J,:]
b = b.reshape( b.shape[0] )
x, metrics = orthogonal_matching_pursuit( A, b, 0., 1000, 0 )
assert numpy.allclose( ( build_vals[i,0]-
numpy.dot( V, x ) )[i],
CV.residuals[k][i] )
k += 1
#for i in xrange( len( CV.cv_params_set ) ):
# print CV.cv_params_set[i], CV.scores[i]
print 'best param', CV.best_cv_params
print 'best score', CV.best_score
print build_pts.shape[1]
# x.reshape( ( 1, x.shape[0] ) ) ) ) ).T
print pts.shape
level = [7,6]
abscissa_1d = []
barycentric_weights_1d = []
for l in level:
nodes, tmp = clenshaw_curtis( l )
nodes = hypercube_map_1d( nodes, -1., 1., a, b )
abscissa_1d.append( nodes.copy() )
barycentric_weights_1d.append(clenshaw_curtis_barycentric_weights( l ).copy() )
from utilities.math_utils import cartesian_product
abscissa = cartesian_product( abscissa_1d ).T
fn_vals = f( abscissa )
print fn_vals.shape
print len( barycentric_weights_1d )
print len( abscissa_1d )
a = True
#a = False
if ( a ):
#poly_vals = multivariate_lagrange_interpolation(
from interpolation_cpp import multivariate_barycentric_lagrange_interpolation as multivariate_barycentric_lagrange_interpolation_cpp
poly_vals = multivariate_barycentric_lagrange_interpolation_cpp(
pts,
abscissa_1d,
barycentric_weights_1d,
fn_vals.reshape( 1, fn_vals.shape[0] ),
def xtest_gaussian_process( self ):
#try:
# 1D test
#build_pts, tmp = clenshaw_curtis( 2 )
build_pts = numpy.linspace(-.85,.9,7)
build_pts = numpy.atleast_2d( build_pts )
#build_pts = numpy.atleast_2d([-.8, -.4, 0., .2, .4, .6])
build_fn_vals = self.f_1d( build_pts ).T
num_dims = 1
func_domain = TensorProductDomain( num_dims, [[-1,1]] )
GP = GaussianProcess()
GP.function_domain( func_domain )
GP.build( build_pts, build_fn_vals )
a = -1.0; b = 1.0
eval_pts = numpy.linspace( a, b, 301 ).reshape(1,301)
pred_vals = GP.evaluate_set( eval_pts )
#reshape matrices for easier plotting
build_pts = build_pts[0,:]
build_fn_vals = build_fn_vals[:,0]
pred_var = GP.evaluate_surface_variance( eval_pts )
eval_pts = eval_pts[0,:]
fig = plot.pylab.figure()
plot.pylab.plot( eval_pts, self.f_1d(eval_pts),
'r:', label=u'$f(x) = x\,\sin(x)$' )
plot.pylab.plot( build_pts, build_fn_vals, 'r.',
markersize=10, label=u'Observations' )
plot.pylab.plot( eval_pts, pred_vals,
'b-', label=u'Prediction' )
# do I need to multiply pred_var by 1.96 to get 95% confidence interval
plot.pylab.fill( numpy.concatenate([eval_pts, eval_pts[::-1]]),
numpy.concatenate([pred_vals-1.96*pred_var,
(pred_vals+1.96*pred_var)[::-1]]),
alpha=.5, fc='b', ec='None',
label='95\% confidence interval')
plot.pylab.xlabel('$x$')
plot.pylab.ylabel('$f(x)$')
plot.pylab.ylim(-10, 20)
plot.pylab.legend(loc='upper left')
#plot.pylab.show()
# 2D test
level = [3,3]
nodes_0, tmp = clenshaw_curtis( level[0] )
nodes_1, tmp = clenshaw_curtis( level[1] )
build_pts_1d = [nodes_0,nodes_1]
build_pts = cartesian_product( build_pts_1d ).T
build_pts = numpy.array([[-4.61611719, -6.00099547],
[4.10469096, 5.32782448],
[0.00000000, -0.50000000],
[-6.17289014, -4.6984743],
[1.3109306, -6.93271427],
[-5.03823144, 3.10584743],
[-2.87600388, 6.74310541],
[5.21301203, 4.26386883]])
build_pts = hypercube_map( build_pts.T, [-8,8,-8,8],[-1,1,-1,1])
build_fn_vals = self.g( build_pts )
build_fn_vals = build_fn_vals.reshape( ( build_fn_vals.shape[0], 1 ) )
num_dims = 2
func_domain = TensorProductDomain( num_dims, [[-1,1]] )
GP = GaussianProcess()
GP.function_domain( func_domain )
GP.build( build_pts, build_fn_vals )
a = -1.0; b = 1.0
x = numpy.linspace( a, b, 50 )
[X,Y] = numpy.meshgrid( x, x )
eval_pts = numpy.vstack((X.reshape((1,X.shape[0]*X.shape[1] ) ),
Y.reshape( ( 1, Y.shape[0]*Y.shape[1]))))
pred_vals = GP.evaluate_set( eval_pts )
pred_var = GP.evaluate_surface_variance( eval_pts )
true_vals = self.g( eval_pts )
pred_vals = pred_vals.reshape( ( X.shape[0], X.shape[1] ) )
pred_var = pred_var.reshape( ( X.shape[0], X.shape[1] ) )
true_vals = true_vals.reshape( ( X.shape[0], X.shape[1] ) )
fig = plot.pylab.figure(2)
ax = fig.add_subplot(111)
ax.axes.set_aspect('equal')
plot.pylab.xticks([])
plot.pylab.yticks([])
ax.set_xticklabels([])
ax.set_yticklabels([])
plot.pylab.xlabel('$x_1$')
plot.pylab.ylabel('$x_2$')
# Standard normal distribution functions
phi = stats.distributions.norm().pdf
PHI = stats.distributions.norm().cdf
PHIinv = stats.distributions.norm().ppf
lim = b # set plot limits
pred_var = numpy.sqrt(pred_var)# * 1.96
build_fn_vals = build_fn_vals[:,0]
#normalize prediction variance
pred_var = pred_var / numpy.max( numpy.max( pred_var ) )
cax = plot.pylab.imshow(numpy.flipud(pred_var),
cmap=cm.gray_r, alpha=0.8,
extent=(- lim, lim, - lim, lim))
norm = plot.pylab.matplotlib.colors.Normalize(vmin=0., vmax=0.9)
cb = plot.pylab.colorbar(cax, ticks=[0., 0.2, 0.4, 0.6, 0.8, 1.], norm=norm)
cb.set_label(r'$\varepsilon[f(x)]$')
plot.pylab.plot(build_pts[0, build_fn_vals <= 0], build_pts[1, build_fn_vals <= 0], 'r.', markersize=12)
plot.pylab.plot(build_pts[0, build_fn_vals > 0], build_pts[1, build_fn_vals > 0 ], 'b.', markersize=12)
cs = plot.pylab.contour(X, Y, true_vals, [0.9], colors='k', linestyles='dashdot')
cs = plot.pylab.contour(X, Y, pred_var, [0.25], colors='b',
linestyles='solid')
plot.pylab.clabel(cs, fontsize=11)
cs = plot.pylab.contour(X, Y, pred_var, [0.5], colors='k',
linestyles='dashed')
plot.pylab.clabel(cs, fontsize=11)
cs = plot.pylab.contour(X, Y, pred_var, [0.75], colors='r',
linestyles='solid')
plot.pylab.clabel(cs, fontsize=11)
#plot.pylab.show()
plot.plot_surface( X, Y, true_vals,
show = False, fignum = 3,
axislabels = None,
title = None )