# Todo: must scale x to -1,1 when evaluating pce
pce_vals = pce.value( pts )
exact_vals = TestFunctions.matlab_peaks( pts )
error = abs( exact_vals-pce_vals )
print numpy.mean( pce_vals ), numpy.var( pce_vals )
print numpy.linalg.norm( error ) / numpy.sqrt( error.shape[0] )
print numpy.max( error )
error = numpy.reshape( error, ( X.shape[0], X.shape[1]) )
pce_vals = numpy.reshape( pce_vals, ( X.shape[0], X.shape[1]) )
exact_vals = numpy.reshape( exact_vals, ( X.shape[0], X.shape[1]) )
plt.plot_surface( X, Y, exact_vals,
show = False, fignum = 1,
axislabels = None, title = None )
plt.plot_surface( X, Y, pce_vals,
show = False, fignum = 2,
axislabels = None, title = None )
plt.plot_surface( X, Y, numpy.log( error ),
show = False, fignum = 3,
axislabels = None, title = None )
pylab.show()
basis = basis,
func_domain = domain )#,
#linear_solver = LeastInterpolation() )
pce_least.basis_indices = basis_indices
pce_least.set_coefficients( coeff )
error = abs( vals - pce_least.evaluate_set( pts ) )
print 'Max error at the interpolation nodes is %1.3e'% max( error );
print 'Absolute error in the PCE mean is %1.3e' %( abs( pce_least.mean()-0.0 ) )
print 'Absolute error in the PCE variance is %1.3e' %( abs( pce_least.variance() - ( 7.+numpy.cos( 4. ) )/4./4. ) )
#import pylab
#test_pts = numpy.linspace( -1., 1. , 100 ).reshape( 1, 100 )
#pylab.plot( pts.squeeze(), vals, 'ro' )
#pylab.plot( test_pts.squeeze(), g(test_pts), 'r' )
#pylab.plot( test_pts.squeeze(), pce_least.evaluate_set( test_pts ) )
#pylab.show()
x = numpy.linspace( -1., 1., 100 );
[X,Y] = numpy.meshgrid( x, x )
pts = numpy.vstack( ( X.reshape( ( 1, X.shape[0]*X.shape[1] ) ),
Y.reshape( ( 1, Y.shape[0]*Y.shape[1] ) ) ) )
vals = g( pts )
pce_vals = pce_least.evaluate_set( pts )
print 'l2 (RMSE) error in the PCE surface is %1.3e' %( numpy.linalg.norm( vals-pce_vals ) / numpy.sqrt( pts.shape[1] ) )
pce_vals = numpy.reshape( pce_vals, ( X.shape[0], X.shape[1]) )
plt.plot_surface( X, Y, pce_vals, show = True, fignum = 1,
axislabels = None, title = None )
# must use numpy.int32 (c++ int)
numpy.array([0,1],numpy.int32) )
else:
poly_vals = multivariate_lagrange_interpolation(
pts,
abscissa_1d,
fn_vals.reshape( 1, fn_vals.shape[0] ),
# cannot use numpy.int (c++ long) for IntVectors
# must use numpy.int32 (c++ int)
numpy.array([0,1],numpy.int32) )
"""
poly_vals = multivariate_lagrange_interpolation( pts.T,
abscissa_1d,
fn_vals,
[0,1] )
"""
print numpy.linalg.norm( poly_vals.squeeze() - f( pts ) )
assert -1==1
#poly_vals = f( pts )
poly_vals = poly_vals.reshape( ( X.shape[0], X.shape[1] ) )
plt.plot_surface( X, Y, poly_vals,
show = False, fignum = 2,
axislabels = None,
title = None )
pylab.show()
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 )
