def test_polynomial_chaos_expansion( self ):
# test 1D bounded domain
num_dims = 1
func_domain = TensorProductDomain( num_dims )
poly_1d = [ JacobiPolynomial1D( 0., 0. ) ]
basis = TensorProductBasis( num_dims, poly_1d )
predictor = PCE( num_dims, basis, order = 2 )
predictor.set_coefficients( numpy.ones( predictor.coeff.shape ) )
predictor.function_domain( func_domain )
num_test_points = 20
test_points = \
numpy.linspace( 0., 1., num_test_points ).reshape(1,num_test_points)
pred_vals = predictor.evaluate_set( test_points )
x = 2. * test_points - 1.
test_vals = numpy.ones( num_test_points ) + x[0,:] + \
0.5 * ( 3.*x[0,:]**2 - 1. )
assert numpy.allclose( test_vals, pred_vals )
test_mean = 1.
test_variance = 1./3. + 1./5.
assert numpy.allclose( predictor.mean(), test_mean )
assert numpy.allclose( predictor.variance(), test_variance )
# test 2D bounded domain
num_dims = 2
func_domain = TensorProductDomain( num_dims )
poly_1d = [ JacobiPolynomial1D( 0., 0. ) ]
basis = TensorProductBasis( num_dims, poly_1d )
predictor = PCE( num_dims, basis, order = 2 )
predictor.set_coefficients( numpy.ones( predictor.coeff.shape ) )
predictor.function_domain( func_domain )
num_test_points = 20
test_points = numpy.random.uniform( 0., 1., ( num_dims, num_test_points))
pred_vals = predictor.evaluate_set( test_points )
x = 2. * test_points - 1.
test_vals = numpy.ones( num_test_points ) + x[0,:] + x[1,:] + \
0.5 * ( 3.*x[0,:]**2 - 1. ) + 0.5 * ( 3.*x[1,:]**2 - 1. ) + \
x[0,:] * x[1,:]
assert numpy.allclose( test_vals, pred_vals )
test_mean = 1.
test_variance = 2. * 1./3. + 1./9. + 2. * 1./5.
assert numpy.allclose( predictor.mean(), test_mean )
assert numpy.allclose( predictor.variance(), test_variance )
# test when domain is unbounded in one dimension
num_dims = 2
func_domain = TensorProductDomain( num_dims,
ranges = [[0.,1.],
[-numpy.inf, numpy.inf]] )
poly_1d = [ JacobiPolynomial1D( 0., 0. ), HermitePolynomial1D() ]
basis = TensorProductBasis( num_dims, poly_1d )
predictor = PCE( num_dims, basis, order = 2 )
predictor.set_coefficients( numpy.ones( predictor.coeff.shape ) )
predictor.function_domain( func_domain )
num_test_points = 20
x_1 = numpy.random.uniform( 0., 1., ( 1, 20 ) )
x_2 = numpy.random.normal( 0., 1., ( 1, 20 ) )
test_points = numpy.vstack( ( x_1, x_2 ) )
pred_vals = predictor.evaluate_set( test_points )
x = test_points
x[0,:] = 2. * x[0,:] - 1.
test_vals = numpy.ones( num_test_points ) + x[0,:] + x[1,:] + \
0.5 * ( 3.*x[0,:]**2 - 1. ) + ( x[1,:]**2 - 1. ) + \
x[0,:] * x[1,:]
assert numpy.allclose( test_vals, pred_vals )
test_mean = 1.
test_variance = 2. * 1./3. + 1./5. + 2. + 1.
assert numpy.allclose( predictor.mean(), test_mean )
assert numpy.allclose( predictor.variance(), test_variance )
def cv_vs_error_study( build_pts, build_vals, domain,
test_pts, test_vals,
results_file = None,
cv_file = None, solver_type = 2 ):
num_dims = build_pts.shape[0]
if ( num_dims == 10 ):
max_order = 5
elif ( num_dims == 15 ):
max_order = 4
else:
max_order = 3
poly_1d = [ LegendrePolynomial1D() ]
basis = TensorProductBasis( num_dims, poly_1d )
pce = PCE( num_dims, order = 0, basis = basis, func_domain = domain )
orders = numpy.arange( 1, max_order + 1 )
solvers = numpy.array( [solver_type], numpy.int32 )
cv_params_grid_array = cartesian_product( [solvers,orders] )
cv_params_grid = []
for i in xrange( cv_params_grid_array.shape[0] ):
cv_params = {}
cv_params['solver'] = numpy.int32( cv_params_grid_array[i,0] )
cv_params['order'] = numpy.int32( cv_params_grid_array[i,1] )
num_pce_terms = polynomial_space_dimension( num_dims,
cv_params['order'] )
if ( cv_params['solver'] <= 1 and
num_pce_terms >= build_pts.shape[1] ):
cv_params['lambda'] = 1.e-12
cv_params_grid.append( cv_params )
# print cv_params_grid
# cv_iterator = LeaveOneOutCrossValidationIterator()
cv_iterator = KFoldCrossValidationIterator( num_folds = 20 )
CV = GridSearchCrossValidation( cv_iterator, pce,
use_predictor_cross_validation = True,
use_fast_predictor_cross_validation = True )
t0 = time.time()
CV.run( build_pts, build_vals, cv_params_grid )
time_taken = time.time() - t0
print 'cross validation took ', time_taken, ' seconds'
print "################"
print "Best cv params: ", CV.best_cv_params
print "Best cv score: ", CV.best_score
print "################"
for order in orders:
residual_norms = numpy.empty( len( CV.cv_params_set ), numpy.double )
scores = numpy.empty( len( CV.cv_params_set ), numpy.double )
k = 0
for i in xrange( len( CV.cv_params_set ) ):
if ( CV.cv_params_set[i]['order'] == order ):
residual_norms[k] = CV.cv_params_set[i]['norm_residual']
scores[k] = CV.scores[i]
k += 1
residual_norms.resize( k )
scores.resize( k )
pce = PCE( num_dims,
order = order,
basis = basis,
func_domain = domain )
V = pce.vandermonde( build_pts ).T
pce.set_solver( CV.best_cv_params['solver'] )
# pce.linear_solver.max_iterations = 3
sols, sol_metrics = pce.linear_solver.solve( V, build_vals )
from sklearn.linear_model import orthogonal_mp
l2_error = numpy.empty( ( sols.shape[1] ), numpy.double )
residuals = numpy.empty( ( sols.shape[1] ), numpy.double )
test_pts = numpy.random.uniform( 0., 1., ( num_dims, 1000 ) )
f = GenzModel( domain, 'oscillatory' )
# f.set_coefficients( 4.5, 'no-decay' )
f.set_coefficients( 4.5, 'quadratic-decay' )
test_vals = f( test_pts ).reshape( ( test_pts.shape[1], 1 ) )
for i in xrange( sols.shape[1] ):
coeff = sols[:,i]
pce.set_coefficients( coeff )
residuals[i] = numpy.linalg.norm( build_vals -
pce.evaluate_set( build_pts ) )
num_test_pts = test_pts.shape[1]
pce_vals_pred = pce.evaluate_set( test_pts ).T
error = test_vals.squeeze() - pce_vals_pred
l2_error[i] = numpy.linalg.norm( error ) / numpy.sqrt( num_test_pts )
import pylab
print residuals, l2_error
print residual_norms, scores
pylab.loglog( residuals, l2_error, label = str( order ) + 'true' )
pylab.loglog( residual_norms, scores, label = str( order )+'-cv' )
pylab.xlim([1e-3,10])
pylab.legend()
pylab.show()
def pce_study( build_pts, build_vals, domain,
test_pts, test_vals,
results_file = None,
cv_file = None, solver_type = 2 ):
num_dims = build_pts.shape[0]
index_generator = IndexGenerator()
poly_1d = [ LegendrePolynomial1D() ]
basis = TensorProductBasis( num_dims, poly_1d )
pce = PCE( num_dims, order = 0, basis = basis, func_domain = domain )
if ( solver_type == 1 ):
num_folds = build_pts.shape[1]
else:
num_folds = 20
index_norm_orders = numpy.linspace( 0.4, 1.0, 4 )
#if (solver_tupe == 1):
# index_norm_orders = [.4,.5,.6,.7,.8,.9,1.]
#solvers = numpy.array( [solver_type], numpy.int32 )
#cv_params_grid_array = cartesian_product( [solvers,orders] )
cv_params_grid = []
for index_norm_order in index_norm_orders:
level = 2
# determine what range of orders to consider.
# spefically consider any order that results in a pce with terms <= 3003
while ( True ):
#index_generator.set_parameters( num_dims, level,
# index_norm_order = index_norm_order)
indices = index_generator.get_isotropic_indices( num_dims, level,
index_norm_order )
num_indices = len( indices )
print level, index_norm_order, len ( indices )
if ( num_indices > 3003 ):
break
cv_params = {}
cv_params['solver'] = solver_type
cv_params['order'] = level
cv_params['index_norm_order'] = index_norm_order
if ( cv_params['solver'] > 1 or
num_indices <= build_pts.shape[1] ):
# only do least squares on over-determined systems
cv_params_grid.append( cv_params )
level += 1
print cv_params_grid
# cv_iterator = LeaveOneOutCrossValidationIterator()
cv_iterator = KFoldCrossValidationIterator( num_folds = num_folds )
CV = GridSearchCrossValidation( cv_iterator, pce,
use_predictor_cross_validation = True,
use_fast_predictor_cross_validation = True )
t0 = time.time()
CV.run( build_pts, build_vals, cv_params_grid )
time_taken = time.time() - t0
print 'cross validation took ', time_taken, ' seconds'
print "################"
print "Best cv params: ", CV.best_cv_params
print "Best cv score: ", CV.best_score
print "################"
#for i in xrange( len( CV.cv_params_set ) ):
# print CV.cv_params_set[i], CV.scores[i]
best_order = CV.best_cv_params['order']
best_index_norm_order = CV.best_cv_params['index_norm_order']
best_pce = PCE( num_dims,
order = best_order,
basis = basis,
func_domain = domain,
index_norm_order = best_index_norm_order)
V = best_pce.vandermonde( build_pts ).T
best_pce.set_solver( CV.best_cv_params['solver'] )
if cv_params['solver'] != 1 and cv_params['solver'] != 5:
best_res_tol = CV.best_cv_params['norm_residual']
best_pce.linear_solver.residual_tolerance = best_res_tol
sols, sol_metrics = best_pce.linear_solver.solve( V, build_vals )
coeff = sols[:,-1]
best_pce.set_coefficients( coeff )
error = abs( build_vals - best_pce.evaluate_set( build_pts ) )
print max( error )
print 'Evaluating best pce at test points'
num_test_pts = test_pts.shape[1]
pce_vals_pred = best_pce.evaluate_set( test_pts ).T
print test_vals.shape, pce_vals_pred.shape
error = test_vals.squeeze() - pce_vals_pred
linf_error = numpy.max( numpy.absolute( error ) )
l2_error = numpy.sqrt( numpy.dot( error.T, error ) / num_test_pts )
mean = numpy.mean( pce_vals_pred )
var = numpy.var( pce_vals_pred )
pce_mean = best_pce.mean()
pce_var = best_pce.variance()
if results_file is not None:
results_file.write( '%1.15e' %linf_error + ',' + '%1.15e' %l2_error +
',' + '%1.15e' %mean + ',' + '%1.15e' %var +
',%1.15e' %pce_mean + ',' + '%1.15e' %pce_var + '\n')
print "linf error: ", linf_error
print "l2 error: ", l2_error
print "mean: ", mean
print "var: ", var
print "pce mean: ", pce_mean
print "pce var: ", pce_var
def pce_study( build_pts, build_vals, domain,
test_pts, test_vals,
results_file = None,
cv_file = None, solver_type = 2 ):
num_dims = build_pts.shape[0]
index_generator = IndexGenerator()
poly_1d = [ LegendrePolynomial1D() ]
basis = TensorProductBasis( num_dims, poly_1d )
pce = PCE( num_dims, order = 0, basis = basis, func_domain = domain )
if ( solver_type == 1 ):
num_folds = build_pts.shape[1]
else:
num_folds = 20
index_norm_orders = numpy.linspace( 0.4, 1.0, 4 )
#solvers = numpy.array( [solver_type], numpy.int32 )
#cv_params_grid_array = cartesian_product( [solvers,orders] )
cv_params_grid = []
for index_norm_order in index_norm_orders:
level = 2
# determine what range of orders to consider.
# spefically consider any order that results in a pce with terms <= 3003
while ( True ):
index_generator.set_parameters( num_dims, level,
index_norm_order = index_norm_order )
index_generator.build_isotropic_index_set()
print level, index_norm_order, index_generator.num_indices
if ( index_generator.num_indices > 3003 ):
break
cv_params = {}
cv_params['solver'] = solver_type
cv_params['order'] = level
cv_params['index_norm_order'] = index_norm_order
if ( cv_params['solver'] > 1 or
index_generator.num_indices <= build_pts.shape[1] ):
# only do least squares on over-determined systems
cv_params_grid.append( cv_params )
else:
break
level += 1
print cv_params_grid
# cv_iterator = LeaveOneOutCrossValidationIterator()
cv_iterator = KFoldCrossValidationIterator( num_folds = num_folds )
CV = GridSearchCrossValidation( cv_iterator, pce,
use_predictor_cross_validation = True,
use_fast_predictor_cross_validation = True )
t0 = time.time()
CV.run( build_pts, build_vals, cv_params_grid )
time_taken = time.time() - t0
print 'cross validation took ', time_taken, ' seconds'
print "################"
print "Best cv params: ", CV.best_cv_params
print "Best cv score: ", CV.best_score
print "################"
#for i in xrange( len( CV.cv_params_set ) ):
# print CV.cv_params_set[i], CV.scores[i]
best_order = CV.best_cv_params['order']
best_index_norm_order = CV.best_cv_params['index_norm_order']
best_pce = PCE( num_dims,
order = best_order,
basis = basis,
func_domain = domain,
index_norm_order = best_index_norm_order)
V = best_pce.vandermonde( build_pts ).T
best_pce.set_solver( CV.best_cv_params['solver'] )
if cv_params['solver'] > 1 :
best_res_tol = CV.best_cv_params['norm_residual']
best_pce.linear_solver.residual_tolerance = best_res_tol
sols, sol_metrics = best_pce.linear_solver.solve( V, build_vals )
coeff = sols[:,-1]
best_pce.set_coefficients( coeff )
error = abs( build_vals - best_pce.evaluate_set( build_pts ) )
print max( error )
print 'Evaluating best pce at test points'
num_test_pts = test_pts.shape[1]
pce_vals_pred = best_pce.evaluate_set( test_pts ).T
print test_vals.shape, pce_vals_pred.shape
error = test_vals.squeeze() - pce_vals_pred
linf_error = numpy.max( numpy.absolute( error ) )
l2_error = numpy.sqrt( numpy.dot( error.T, error ) / num_test_pts )
mean = numpy.mean( pce_vals_pred )
var = numpy.var( pce_vals_pred )
pce_mean = best_pce.mean()
pce_var = best_pce.variance()
if results_file is not None:
results_file.write( '%1.15e' %linf_error + ',' + '%1.15e' %l2_error +
',' + '%1.15e' %mean + ',' + '%1.15e' %var +
',%1.15e' %pce_mean + ',' + '%1.15e' %pce_var + '\n')
print "linf error: ", linf_error
print "l2 error: ", l2_error
print "mean: ", mean
print "var: ", var
print "pce mean: ", pce_mean
print "pce var: ", pce_var
me, te, ie = best_pce.get_sensitivities()
interaction_values, interaction_terms = best_pce.get_interactions()
show = False
fignum = 1
filename = 'oscillator-individual-interactions.png'
plot_interaction_values( interaction_values, interaction_terms, title = 'Sobol indices', truncation_pct = 0.95, filename = filename, show = show,
fignum = fignum )
fignum += 1
filename = 'oscillator-dimension-interactions.png'
plot_interaction_effects( ie, title = 'Dimension-wise joint effects', truncation_pct = 0.95, filename = filename, show = show,fignum = fignum )
fignum += 1
filename = 'oscillator-main-effects.png'
plot_main_effects( me, truncation_pct = 0.95, title = 'Main effect sensitivity indices', filename = filename, show = show, fignum = fignum )
fignum += 1
filename = 'oscillator-total-effects.png'
plot_total_effects( te, truncation_pct = 0.95, title = 'Total effect sensitivity indices', filename = filename, show = show, fignum = fignum )
fignum += 1
from scipy.stats.kde import gaussian_kde
pylab.figure( fignum )
pce_kde = gaussian_kde( pce_vals_pred )
pce_kde_x = numpy.linspace( pce_vals_pred.min(), pce_vals_pred.max(), 100 )
pce_kde_y = pce_kde( pce_kde_x )
pylab.plot( pce_kde_x, pce_kde_y,label = 'pdf of surrogate' )
true_kde = gaussian_kde( test_vals )
true_kde_x = numpy.linspace( test_vals.min(), test_vals.max(), 100 )
true_kde_y = true_kde( true_kde_x )
pylab.plot( true_kde_x, true_kde_y, label = 'true pdf' )
pylab.legend(loc=2)
pylab.show()
