def __init__(self, num_dims, ranges=None, indices=None):
self.num_dims = num_dims
if ranges is None:
self.ranges = get_symmetric_hypercube(0, 1, num_dims)
self.ranges = self.ranges.reshape((num_dims, 2))
else:
if indices is None:
if len(ranges) == 1:
self.ranges = numpy.empty((num_dims, 2), numpy.double)
self.ranges[:, 0] = ranges[0][0]
self.ranges[:, 1] = ranges[0][1]
else:
msg = "ranges are inconsistent with num_dims. "
msg += "Specify indices or provide a range for each "
msg += "dimension"
assert len(ranges) == num_dims, msg
self.ranges = numpy.array(ranges, numpy.double)
else:
msg = "indices and ranges are inconsistent"
assert len(indices) == len(ranges), msg
self.ranges = numpy.empty((num_dims, 2), numpy.double)
dims = set() # used for error checking
for i in xrange(len(indices)):
for j in xrange(len(indices[i])):
msg = "the range of the " + str(indices[i][j])
msg += "dimension is not in ascending order"
assert ranges[i][0] < ranges[i][1], msg
self.ranges[indices[i][j], 0] = ranges[i][0]
self.ranges[indices[i][j], 1] = ranges[i][1]
dims.add(indices[i][j])
msg = "ranges and indices are inconsistent with num_dims"
assert len(dims) == self.num_dims, msg
if '__main__' in __name__:
import polynomial_chaos.polynomial_chaos_expansion as PCE
from interpolation.lagrange_interpolation import multivariate_lagrange_interpolation
import test_functions
import utilities.visualisation as plt
import numpy
import scipy
import pylab
from utilities.math_utils import get_symmetric_hypercube, hypercube_map
num_dims = 2
max_order = 30
domain = get_symmetric_hypercube( -3., 3., num_dims )
pce = PCE.PolynomialChaosExpansion( num_dims, order = max_order,
basis = 'legendre',
func_domain = domain )
x = numpy.linspace( -3., 3., 50 )
[X,Y] = numpy.meshgrid( x, x )
samples = numpy.vstack( ( X.reshape( ( 1, X.shape[0]*X.shape[1] ) ),
Y.reshape( ( 1, Y.shape[0]*Y.shape[1]) ) ) )
#num_samples=int(numpy.round(scipy.misc.comb(max_order+num_dims,num_dims)))
#num_samples = 2500
#samples = numpy.random.uniform( -3., 3., size = num_samples * num_dims )
#samples = numpy.resize( samples, ( num_dims, num_samples ) )
sample_vals = test_functions.matlab_peaks( samples )
#sample_vals = numpy.sum( samples**2, axis = 0 )
print sample_vals.shape
def test_tensor_product_basis( self ):
# test orthogonal basis functions
num_dims = 2
#domain = TensorProductDomain( num_dims, ranges = [[-1.,1.]] )
domain = get_symmetric_hypercube( -1., 1., num_dims )
poly_1d = OrthogPolyVector();
poly_1d.resize( 1 )
poly_1d[0] = JacobiPolynomial1D( 0.5, 0.5 )
#basis = TensorProductBasis( num_dims, poly_1d )
basis = TensorProductBasis()
basis.set_domain( domain )
basis.set_bases( poly_1d, [numpy.arange( num_dims,dtype=numpy.int32 )],
num_dims )
#assert basis.num_dims == num_dims
x = numpy.random.uniform( -1., 1., ( num_dims, 20 ) )
index = PolynomialIndex( numpy.array( [[1, 1]], numpy.int32 ) )
true_val = poly_1d[0].value_set( x[0,:], index.level(0), -1., 1. ) * \
poly_1d[0].value_set( x[1,:], index.level(1), -1., 1. )
assert numpy.allclose( basis.value_set( x, index), true_val )
#true_grad = poly_1d[0].gradient_set( x[0,:], index.level( 0 ), -1., 1. ) * \
# poly_1d[0].value_set( x[1,:], index.level(1), -1., 1. )
#assert numpy.allclose( basis.gradient( x, index, 0),
# true_grad )
#true_grad = poly_1d[0].value_set( x[0,:], index.level(0), -1., 1. ) * \
# poly_1d[0].gradient_set( x[1,:], index.level(1), -1., 1. )
#assert numpy.allclose( basis.gradient( x, index, 1),
# true_grad )
num_dims = 2
poly_1d = OrthogPolyVector();
poly_1d.resize( 2 )
poly_1d[0] = JacobiPolynomial1D( 0.5, 0.5 )
poly_1d[1] = JacobiPolynomial1D( 0.5, 1.5 )
#basis = TensorProductBasis( num_dims, poly_1d )
basis = TensorProductBasis()
basis.set_domain( domain )
basis.set_bases( poly_1d, [numpy.array([0],numpy.int32),
numpy.array([1],numpy.int32)],
num_dims )
x_1 = numpy.random.uniform( -1., 1., ( 1, 20 ) )
x_2 = numpy.random.normal( 0., 1., ( 1, 20 ) )
x = numpy.vstack( ( x_1, x_2 ) )
index = PolynomialIndex( numpy.array( [[1,1]], numpy.int32 ) )
true_val = poly_1d[0].value_set( x[0,:], index.level(0), -1., 1. ) * \
poly_1d[1].value_set( x[1,:], index.level(1), -1., 1. )
assert numpy.allclose( basis.value_set( x, index ), true_val )
#true_grad = poly_1d[0].gradient_set( x[0,:], index.level(0), -1., 1. ) * \
# poly_1d[1].value_set( x[1,:], index.level(1), -1., 1. )
#assert numpy.allclose( basis.gradient( x, index, 0),
# true_grad )
#true_grad = poly_1d[0].value_set( x[0,:], index.level(0), -1., 1. ) * \
# poly_1d[1].gradient_set( x[1,:], index.level(1), -1., 1. )
#assert numpy.allclose( basis.gradient( x, index, 1),
# true_grad )
num_dims = 3
domain = get_symmetric_hypercube( -1., 1., num_dims )
basis = TensorProductBasis()
basis.set_domain( domain )
basis.set_bases( poly_1d, [numpy.array([0,2],numpy.int32),
numpy.array([1],numpy.int32)],
num_dims )
indices = PolyIndexVector()
indices.resize( 3 )
indices[0] = PolynomialIndex( numpy.array( [[1,1]], numpy.int32 ) )
indices[1] = PolynomialIndex( numpy.array( [[2,1]], numpy.int32 ) )
indices[2] = PolynomialIndex( numpy.array( [[0,2],[2,3]], numpy.int32 ) )
indices[2] = PolynomialIndex( numpy.array( [[0,3],[1,1],[2,2]],
numpy.int32 ) )
x_1 = numpy.random.uniform( -1., 1., ( 1, 20 ) )
x_2 = numpy.random.normal( 0., 1., ( 1, 20 ) )
x_3 = numpy.random.normal( 0., 1., ( 1, 20 ) )
x = numpy.vstack( ( x_1, x_2, x_3 ) )
vals_m = basis.value_set_multiple( x, indices )
for i, index in enumerate( indices ):
true_vals = poly_1d[0].value_set( x[0,:], index.level(0),
domain[0], domain[1] ) * poly_1d[1].value_set( x[1,:], index.level(1), domain[2], domain[3] ) * poly_1d[0].value_set( x[2,:], index.level(2),
domain[4], domain[5] )
vals = basis.value_set( x, index )
assert numpy.allclose( true_vals, vals )
assert numpy.allclose( true_vals, vals_m[:,i] )
def test_polynomial_chaos_expansion( self ):
# test 1D bounded domain
num_dims = 1
domain = get_symmetric_hypercube( 0., 1., num_dims )
poly_1d = OrthogPolyVector();
poly_1d.resize( 1 )
poly_1d[0] = LegendrePolynomial1D()
basis = TensorProductBasis()
basis.set_domain( domain )
basis.set_bases( poly_1d, [numpy.array([0],numpy.int32)],
num_dims )
pce = PolynomialChaosExpansion()
pce.domain( domain )
pce.basis( basis )
pce.define_isotropic_expansion( 2, 1. );
pce.set_coefficients( numpy.ones( ( pce.num_terms(), 1 ) ) )
num_test_points = 20
test_points = \
numpy.linspace( 0., 1., num_test_points ).reshape(1,num_test_points)
pred_vals = pce.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( pce.mean(), test_mean )
assert numpy.allclose( pce.variance(), test_variance )
# test 2D bounded domain
num_dims = 2
domain = get_symmetric_hypercube( 0., 1., num_dims )
poly_1d = OrthogPolyVector();
poly_1d.resize( 1 )
poly_1d[0] = LegendrePolynomial1D()
basis = TensorProductBasis()
basis.set_domain( domain )
basis.set_bases( poly_1d, [numpy.array([0,1],numpy.int32)],
num_dims )
pce = PolynomialChaosExpansion()
pce.domain( domain )
pce.basis( basis )
order = 2
pce.define_isotropic_expansion( order, 1. );
pce.set_coefficients( numpy.ones( ( pce.num_terms(), 1 ) ) )
num_test_points = 20
test_points = numpy.random.uniform( 0., 1., ( num_dims, num_test_points))
pred_vals = pce.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( pce.mean(), test_mean )
assert numpy.allclose( pce.variance(), test_variance )
