def modelLine(self, theta, line=0, order=-1):
"""
Get only a single line. Update this to plot a continuous line rather than discretely at
the same values of wavelengths.
"""
noise, center, scale, herm = self.unpackTheta(theta)
# Shift and scale lambdas to evaluation points
lamEv = (self.lam[:, np.newaxis] - center) / scale
lamEdgeEv = (self.lamEdge[:, np.newaxis] - center) / scale
# Evaluate gaussian functions
gauss = np.exp(-lamEv**2 / 2.)
gaussEdge = np.exp(-lamEdgeEv**2 / 2.)
hn = np.zeros(lamEv.shape)
hnEdge = np.zeros(lamEdgeEv.shape)
i = line
if order > len(herm[i]):
order = len(herm[i])
hn[:, i] = hermeval(lamEv[:, i], herm[i]) * gauss[:, i]
hnEdge[:, i] = hermeval(lamEdgeEv[:, i], herm[i]) * gaussEdge[:, i]
# Compute integral over finite pixel size
hnEv = (4 * hn + hnEdge[1:] +
hnEdge[:-1]) / 6.0 * self.dlam / scale[np.newaxis, :]
return np.sum(hnEv, axis=1)
def test_hermemul(self) :
# check values of result
for i in range(5) :
pol1 = [0]*i + [1]
val1 = herme.hermeval(self.x, pol1)
for j in range(5) :
msg = "At i=%d, j=%d" % (i,j)
pol2 = [0]*j + [1]
val2 = herme.hermeval(self.x, pol2)
pol3 = herme.hermemul(pol1, pol2)
val3 = herme.hermeval(self.x, pol3)
assert_(len(pol3) == i + j + 1, msg)
assert_almost_equal(val3, val1*val2, err_msg=msg)
def test_hermemul(self):
# check values of result
for i in range(5):
pol1 = [0] * i + [1]
val1 = herme.hermeval(self.x, pol1)
for j in range(5):
msg = "At i=%d, j=%d" % (i, j)
pol2 = [0] * j + [1]
val2 = herme.hermeval(self.x, pol2)
pol3 = herme.hermemul(pol1, pol2)
val3 = herme.hermeval(self.x, pol3)
assert_(len(pol3) == i + j + 1, msg)
assert_almost_equal(val3, val1 * val2, err_msg=msg)
def q(i, j):
'''
copmute coefficient q_{ij}
Set up Gauss-Hermite quadrature, weighting function is exp^{-x^2}
'''
x, w = H.hermegauss(20)
Q = sum([
w[ldx] * sum([
w[kdx] * Q_FEM_quad[ldx * 20 + kdx] * H.hermeval(x[kdx], Phi(i))
for kdx in range(20)
]) * H.hermeval(x[ldx], Phi(j)) for ldx in range(20)
])
q = Q / (2 * np.pi * factorial(i) * factorial(j))
return q
def test_hermevander(self) :
# check for 1d x
x = np.arange(3)
v = herme.hermevander(x, 3)
assert_(v.shape == (3,4))
for i in range(4) :
coef = [0]*i + [1]
assert_almost_equal(v[...,i], herme.hermeval(x, coef))
# check for 2d x
x = np.array([[1,2],[3,4],[5,6]])
v = herme.hermevander(x, 3)
assert_(v.shape == (3,2,4))
for i in range(4) :
coef = [0]*i + [1]
assert_almost_equal(v[...,i], herme.hermeval(x, coef))
def weights(dim, degree):
# 1D sigma-points (x) and weights (w)
x, w = hermegauss(degree)
# hermegauss() provides weights that cause posdef errors
w = factorial(degree) / (degree**2 * hermeval(x, [0] *
(degree - 1) + [1])**2)
return np.prod(cartesian([w] * dim), axis=1)
def test_hermevander(self):
# check for 1d x
x = np.arange(3)
v = herme.hermevander(x, 3)
assert_(v.shape == (3, 4))
for i in range(4):
coef = [0] * i + [1]
assert_almost_equal(v[..., i], herme.hermeval(x, coef))
# check for 2d x
x = np.array([[1, 2], [3, 4], [5, 6]])
v = herme.hermevander(x, 3)
assert_(v.shape == (3, 2, 4))
for i in range(4):
coef = [0] * i + [1]
assert_almost_equal(v[..., i], herme.hermeval(x, coef))
def MollifyQuad(theta, c_j, a_n, x):
"""
Mollify the spectral reconstruction of a discontinuous function
to reduce the effect of Gibbs phenomenon. Perform a real-space convolution
of a spectral reconstruction with an adaptive unit-mass mollifier.
Parameters
----------
theta : float
free parameter to vary in the range 0 < theta < 1
c_j : 1D array,
Array containing x positions of the discontinuities in the data
a_n : 1D array, shape = (N+1,)
N - highest order in the Chebyshev expansion
vector of Chebyshev expansion coefficients
x : 1D array,
1D grid of points in real space, where the function is evaluated
Returns
----------
convolution : 1D array, shape = (len(x),)
real space mollified representation of a discontinuous function
"""
N = a_n.shape[0] - 1
offset = x
convolution = np.empty(len(x))
I_N = lambda y: T.chebval(y, a_n)
I_Nf = lambda y: I_N(y) if -1 <= y <= 1 else 0
for idx, off_x in enumerate(offset):
c_jx = c_j - off_x
dx = lambda y: sqrt(theta * N * min(abs(y - c) for c in c_jx))
var = lambda y: N / (2 * theta * dx(y))
p_N = lambda y: (theta**2) * dx(y) * N
j_max = int(np.amax(np.frompyfunc(p_N, 1, 1)(x)))
h_j = np.zeros(2 * (j_max + 1))
for j in range(j_max + 1):
h_j[2 * j] = ((-1)**j) / ((4**j) * factorial(j))
hermite = lambda y: H.hermeval(sqrt(var(y)) * y, h_j)
expon = lambda y: (1. / sqrt(theta * N * dx(y))) * exp(-var(y) *
(y**2))
phi = lambda y: hermite(y) * expon(y)
phif = lambda y: phi(y - off_x)
norm = quad(phi, -1.0, 1.0)[0]
convfunc = lambda y: (phif(y) * I_Nf(y))
convolution[idx] = (1 / norm) * quad(convfunc, -1.0, 1.0)[0]
return convolution
def __call__(self):
## Generate values of a hermite polynomial of the given order at the values
## of a fractional Gaussian Noise with the specified hurst index.
increments = hermeval(fgn.__call__(self), self.__coef)
## The renorm-group transformation, without the renormalisation by the n^{-H}
increments = np.cumsum(increments)[
self.__K - 1::self.__K] ## / ( self.__K ** self.__H )
return self.__t, np.concatenate(
([0], increments / np.max(np.abs(increments))))
def test_hermefit(self):
def f(x):
return x * (x - 1) * (x - 2)
# Test exceptions
assert_raises(ValueError, herme.hermefit, [1], [1], -1)
assert_raises(TypeError, herme.hermefit, [[1]], [1], 0)
assert_raises(TypeError, herme.hermefit, [], [1], 0)
assert_raises(TypeError, herme.hermefit, [1], [[[1]]], 0)
assert_raises(TypeError, herme.hermefit, [1, 2], [1], 0)
assert_raises(TypeError, herme.hermefit, [1], [1, 2], 0)
assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[[1]])
assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[1, 1])
# Test fit
x = np.linspace(0, 2)
y = f(x)
#
coef3 = herme.hermefit(x, y, 3)
assert_equal(len(coef3), 4)
assert_almost_equal(herme.hermeval(x, coef3), y)
#
coef4 = herme.hermefit(x, y, 4)
assert_equal(len(coef4), 5)
assert_almost_equal(herme.hermeval(x, coef4), y)
#
coef2d = herme.hermefit(x, np.array([y, y]).T, 3)
assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
# test weighting
w = np.zeros_like(x)
yw = y.copy()
w[1::2] = 1
y[0::2] = 0
wcoef3 = herme.hermefit(x, yw, 3, w=w)
assert_almost_equal(wcoef3, coef3)
#
wcoef2d = herme.hermefit(x, np.array([yw, yw]).T, 3, w=w)
assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
# test scaling with complex values x points whose square
# is zero when summed.
x = [1, 1j, -1, -1j]
assert_almost_equal(herme.hermefit(x, x, 1), [0, 1])
def test_hermefit(self) :
def f(x) :
return x*(x - 1)*(x - 2)
# Test exceptions
assert_raises(ValueError, herme.hermefit, [1], [1], -1)
assert_raises(TypeError, herme.hermefit, [[1]], [1], 0)
assert_raises(TypeError, herme.hermefit, [], [1], 0)
assert_raises(TypeError, herme.hermefit, [1], [[[1]]], 0)
assert_raises(TypeError, herme.hermefit, [1, 2], [1], 0)
assert_raises(TypeError, herme.hermefit, [1], [1, 2], 0)
assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[[1]])
assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[1,1])
# Test fit
x = np.linspace(0,2)
y = f(x)
#
coef3 = herme.hermefit(x, y, 3)
assert_equal(len(coef3), 4)
assert_almost_equal(herme.hermeval(x, coef3), y)
#
coef4 = herme.hermefit(x, y, 4)
assert_equal(len(coef4), 5)
assert_almost_equal(herme.hermeval(x, coef4), y)
#
coef2d = herme.hermefit(x, np.array([y,y]).T, 3)
assert_almost_equal(coef2d, np.array([coef3,coef3]).T)
# test weighting
w = np.zeros_like(x)
yw = y.copy()
w[1::2] = 1
y[0::2] = 0
wcoef3 = herme.hermefit(x, yw, 3, w=w)
assert_almost_equal(wcoef3, coef3)
#
wcoef2d = herme.hermefit(x, np.array([yw,yw]).T, 3, w=w)
assert_almost_equal(wcoef2d, np.array([coef3,coef3]).T)
# test scaling with complex values x points whose square
# is zero when summed.
x = [1, 1j, -1, -1j]
assert_almost_equal(herme.hermefit(x, x, 1), [0, 1])
def test_hermeval(self):
#check empty input
assert_equal(herme.hermeval([], [1]).size, 0)
#check normal input)
x = np.linspace(-1, 1)
y = [polyval(x, c) for c in Helist]
for i in range(10):
msg = "At i=%d" % i
tgt = y[i]
res = herme.hermeval(x, [0]*i + [1])
assert_almost_equal(res, tgt, err_msg=msg)
#check that shape is preserved
for i in range(3):
dims = [2]*i
x = np.zeros(dims)
assert_equal(herme.hermeval(x, [1]).shape, dims)
assert_equal(herme.hermeval(x, [1, 0]).shape, dims)
assert_equal(herme.hermeval(x, [1, 0, 0]).shape, dims)
def test_hermefromroots(self) :
res = herme.hermefromroots([])
assert_almost_equal(trim(res), [1])
for i in range(1,5) :
roots = np.cos(np.linspace(-np.pi, 0, 2*i + 1)[1::2])
pol = herme.hermefromroots(roots)
res = herme.hermeval(roots, pol)
tgt = 0
assert_(len(pol) == i + 1)
assert_almost_equal(herme.herme2poly(pol)[-1], 1)
assert_almost_equal(res, tgt)
def test_hermeval(self):
#check empty input
assert_equal(herme.hermeval([], [1]).size, 0)
#check normal input)
x = np.linspace(-1, 1)
y = [polyval(x, c) for c in Helist]
for i in range(10):
msg = "At i=%d" % i
tgt = y[i]
res = herme.hermeval(x, [0] * i + [1])
assert_almost_equal(res, tgt, err_msg=msg)
#check that shape is preserved
for i in range(3):
dims = [2] * i
x = np.zeros(dims)
assert_equal(herme.hermeval(x, [1]).shape, dims)
assert_equal(herme.hermeval(x, [1, 0]).shape, dims)
assert_equal(herme.hermeval(x, [1, 0, 0]).shape, dims)
def test_hermefromroots(self):
res = herme.hermefromroots([])
assert_almost_equal(trim(res), [1])
for i in range(1, 5):
roots = np.cos(np.linspace(-np.pi, 0, 2 * i + 1)[1::2])
pol = herme.hermefromroots(roots)
res = herme.hermeval(roots, pol)
tgt = 0
assert_(len(pol) == i + 1)
assert_almost_equal(herme.herme2poly(pol)[-1], 1)
assert_almost_equal(res, tgt)
def gabor_k(k, grid_x, xi0=np.pi * 3 / 4, sigma0=.5, scale=0):
xi = xi0 / 2**scale
sigma = sigma0 * 2**scale
k_choose_j = binom(k, np.arange(k + 1))
minus_one_power = (-1)**np.arange(k + 1)
i_xi_power = (1j * xi * sigma)**(k - np.arange(k + 1))
herm_coef = k_choose_j * minus_one_power * i_xi_power
hermite_eval = hermeval(grid_x / sigma, herm_coef)
gauss = np.exp(-.5 * (grid_x / sigma)**2) / np.sqrt(
2 * np.pi * sigma**2) / sigma**k
wave = np.exp(1j * xi * grid_x)
return hermite_eval * gauss * wave
def _column_feats(self, X, shift):
"""
Apply Hermite function evaluations of degrees 0..`degree` differentiated `shift` times.
When applied to the column `X` of shape(n,), the resulting array has shape(n, (degree + 1)).
"""
assert ndim(X) == 1
# this will have dimension (d,) + shape(X)
coeffs = np.identity(self._degree + shift + 1)[:, shift:]
feats = ((-1)**shift) * hermeval(X, coeffs) * np.exp(-X * X / 2)
# send the first dimension to the end
return transpose(feats)
def get_quadrature_weights(
n_features: int,
degree: int = 3,
) -> Array:
"""Generate normalizers for MCMC samples"""
# 1D sigma-points (x) and weights (w)
x, w = hermegauss(degree)
# hermegauss() provides weights that cause posdef errors
w = factorial(degree) / (degree**2 * hermeval(x, [0] *
(degree - 1) + [1])**2)
return jnp.prod(cartesian([w] * n_features), axis=1)
def test_hermeval(self) :
def f(x) :
return x*(x**2 - 1)
#check empty input
assert_equal(herme.hermeval([], [1]).size, 0)
#check normal input)
for i in range(10) :
msg = "At i=%d" % i
ser = np.zeros
tgt = self.y[i]
res = herme.hermeval(self.x, [0]*i + [1])
assert_almost_equal(res, tgt, err_msg=msg)
#check that shape is preserved
for i in range(3) :
dims = [2]*i
x = np.zeros(dims)
assert_equal(herme.hermeval(x, [1]).shape, dims)
assert_equal(herme.hermeval(x, [1,0]).shape, dims)
assert_equal(herme.hermeval(x, [1,0,0]).shape, dims)
def test_consistent_eval(self):
degree = 10
n_points = 10
quad = hm.Quad.gauss_hermite(n_points)
nodes_scipy, _ = herm.hermegauss(n_points)
for i in range(degree):
coeffs = np.zeros(degree + 1)
coeffs[i] = 1
factor = math.sqrt(math.factorial(i))
hi_nodes_scipy = herm.hermeval(nodes_scipy, coeffs)
hi_nodes = quad.eval(coeffs)
diff = sum(abs(hi_nodes_scipy / factor - hi_nodes))
self.assertAlmostEqual(diff, 0)
def modelPredict(self, theta):
"""
Full prediction given theta
"""
# issue: scale is sometimes coming out as ~9e-5 in PolyChord and this gives overflows...
noise, center, scale, herm = self.unpackTheta(theta)
# Shift and scale lambdas to evaluation points
lamEv = (self.lam[:, np.newaxis] - center) / scale
lamEdgeEv = (self.lamEdge[:, np.newaxis] - center) / scale
gauss = np.exp(-lamEv**2 / 2.)
gaussEdge = np.exp(-lamEdgeEv**2 / 2.)
hn = np.zeros(lamEv.shape)
hnEdge = np.zeros(lamEdgeEv.shape)
# Compute hermite functions to model lineData
for i in np.arange(self.nfit):
hn[:, i] = hermeval(lamEv[:, i], herm[i]) * gauss[:, i] ##
hnEdge[:, i] = hermeval(lamEdgeEv[:, i], herm[i]) * gaussEdge[:, i]
# Compute integral over finite pixel size
hnEv = (4 * hn + hnEdge[1:] +
hnEdge[:-1]) / 6.0 * self.dlam / scale[np.newaxis, :]
# Evaluate noise as 2nd order Legendre fit
if self.noiseParams == 1:
noiseEv = noise[0]
elif self.noiseParams == 3:
noiseEv = noise[0] + noise[1] * self.lamNorm + noise[2] * (
3 * self.lamNorm**2 - 1) / 2
# Sum over all lineData
pred = noiseEv + np.sum(hnEv, axis=1)
return pred
def hermite_measures(_points, _degree):
"""
Evaluation of the first p=_dimension Hermite polynomials `[He_0(y), He_1(y), ..., He_{p-1}(y)]` on each point `y` in `_points`.
"""
num_samples, num_assets, num_steps = _points.shape
tmp = hermeval(
_points, np.eye(_degree +
1)) #NOTE: The Hermite polynomials are NOT normalized.
assert tmp.shape == (_degree + 1, num_samples, num_assets, num_steps)
mIs = list(multiIndices(_degree, num_assets))
assert len(mIs) == comb(_degree + num_assets, num_assets)
j = np.arange(num_assets)
ret = np.empty((len(mIs), num_samples, num_steps))
for i, mI in enumerate(mIs):
ret[i] = np.prod(tmp[mI, :, j, :], axis=0)
return ret.T
def Hermite(data, basis_order):
from numpy.polynomial.hermite_e import hermeval #importing hermeval from hermite class
if data.ndim == 2:
data = data[:, :, np.newaxis]
n, m, l = data.shape
psi_data = data
if basis_order > 1:
psi_data = np.append(psi_data, np.ones((1, m, l)), axis=0)
for i in np.arange(2, basis_order + 1):
c = np.zeros(i + 1) # vector of coefficients of size i+1
c[i] = 1 # 1 at index n+1 for extracting H^i
psi_data = np.append(psi_data, hermeval(data, c),
axis=0) #appending H^i to psi
# psi_data = np.append(psi_data,herm(data,i),axis = 0)
return psi_data
def weights(dim, degree=3):
"""
Gauss-Hermite quadrature weights.
Parameters
----------
dim : int
Dimension of the input random variable.
degree : int, optional
Degree of the integration rule.
Returns
-------
: (num_points, ) ndarray
GH quadrature weights of given degree.
"""
# 1D sigma-points (x) and weights (w)
x, w = hermegauss(degree)
# hermegauss() provides weights that cause posdef errors
w = factorial(degree) / (degree**2 * hermeval(x, [0] *
(degree - 1) + [1])**2)
return np.prod(cartesian([w] * dim), axis=1)
def test_hermefit(self):
def f(x):
return x * (x - 1) * (x - 2)
def f2(x):
return x**4 + x**2 + 1
# Test exceptions
assert_raises(ValueError, herme.hermefit, [1], [1], -1)
assert_raises(TypeError, herme.hermefit, [[1]], [1], 0)
assert_raises(TypeError, herme.hermefit, [], [1], 0)
assert_raises(TypeError, herme.hermefit, [1], [[[1]]], 0)
assert_raises(TypeError, herme.hermefit, [1, 2], [1], 0)
assert_raises(TypeError, herme.hermefit, [1], [1, 2], 0)
assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[[1]])
assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[1, 1])
assert_raises(ValueError, herme.hermefit, [1], [1], [
-1,
])
assert_raises(ValueError, herme.hermefit, [1], [1], [2, -1, 6])
assert_raises(TypeError, herme.hermefit, [1], [1], [])
# Test fit
x = np.linspace(0, 2)
y = f(x)
#
coef3 = herme.hermefit(x, y, 3)
assert_equal(len(coef3), 4)
assert_almost_equal(herme.hermeval(x, coef3), y)
coef3 = herme.hermefit(x, y, [0, 1, 2, 3])
assert_equal(len(coef3), 4)
assert_almost_equal(herme.hermeval(x, coef3), y)
#
coef4 = herme.hermefit(x, y, 4)
assert_equal(len(coef4), 5)
assert_almost_equal(herme.hermeval(x, coef4), y)
coef4 = herme.hermefit(x, y, [0, 1, 2, 3, 4])
assert_equal(len(coef4), 5)
assert_almost_equal(herme.hermeval(x, coef4), y)
# check things still work if deg is not in strict increasing
coef4 = herme.hermefit(x, y, [2, 3, 4, 1, 0])
assert_equal(len(coef4), 5)
assert_almost_equal(herme.hermeval(x, coef4), y)
#
coef2d = herme.hermefit(x, np.array([y, y]).T, 3)
assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
coef2d = herme.hermefit(x, np.array([y, y]).T, [0, 1, 2, 3])
assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
# test weighting
w = np.zeros_like(x)
yw = y.copy()
w[1::2] = 1
y[0::2] = 0
wcoef3 = herme.hermefit(x, yw, 3, w=w)
assert_almost_equal(wcoef3, coef3)
wcoef3 = herme.hermefit(x, yw, [0, 1, 2, 3], w=w)
assert_almost_equal(wcoef3, coef3)
#
wcoef2d = herme.hermefit(x, np.array([yw, yw]).T, 3, w=w)
assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
wcoef2d = herme.hermefit(x, np.array([yw, yw]).T, [0, 1, 2, 3], w=w)
assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
# test scaling with complex values x points whose square
# is zero when summed.
x = [1, 1j, -1, -1j]
assert_almost_equal(herme.hermefit(x, x, 1), [0, 1])
assert_almost_equal(herme.hermefit(x, x, [0, 1]), [0, 1])
# test fitting only even Legendre polynomials
x = np.linspace(-1, 1)
y = f2(x)
coef1 = herme.hermefit(x, y, 4)
assert_almost_equal(herme.hermeval(x, coef1), y)
coef2 = herme.hermefit(x, y, [0, 2, 4])
assert_almost_equal(herme.hermeval(x, coef2), y)
assert_almost_equal(coef1, coef2)
def test_hermeint(self):
# check exceptions
assert_raises(TypeError, herme.hermeint, [0], .5)
assert_raises(ValueError, herme.hermeint, [0], -1)
assert_raises(ValueError, herme.hermeint, [0], 1, [0, 0])
assert_raises(ValueError, herme.hermeint, [0], lbnd=[0])
assert_raises(ValueError, herme.hermeint, [0], scl=[0])
assert_raises(TypeError, herme.hermeint, [0], axis=.5)
# test integration of zero polynomial
for i in range(2, 5):
k = [0] * (i - 2) + [1]
res = herme.hermeint([0], m=i, k=k)
assert_almost_equal(res, [0, 1])
# check single integration with integration constant
for i in range(5):
scl = i + 1
pol = [0] * i + [1]
tgt = [i] + [0] * i + [1 / scl]
hermepol = herme.poly2herme(pol)
hermeint = herme.hermeint(hermepol, m=1, k=[i])
res = herme.herme2poly(hermeint)
assert_almost_equal(trim(res), trim(tgt))
# check single integration with integration constant and lbnd
for i in range(5):
scl = i + 1
pol = [0] * i + [1]
hermepol = herme.poly2herme(pol)
hermeint = herme.hermeint(hermepol, m=1, k=[i], lbnd=-1)
assert_almost_equal(herme.hermeval(-1, hermeint), i)
# check single integration with integration constant and scaling
for i in range(5):
scl = i + 1
pol = [0] * i + [1]
tgt = [i] + [0] * i + [2 / scl]
hermepol = herme.poly2herme(pol)
hermeint = herme.hermeint(hermepol, m=1, k=[i], scl=2)
res = herme.herme2poly(hermeint)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with default k
for i in range(5):
for j in range(2, 5):
pol = [0] * i + [1]
tgt = pol[:]
for k in range(j):
tgt = herme.hermeint(tgt, m=1)
res = herme.hermeint(pol, m=j)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with defined k
for i in range(5):
for j in range(2, 5):
pol = [0] * i + [1]
tgt = pol[:]
for k in range(j):
tgt = herme.hermeint(tgt, m=1, k=[k])
res = herme.hermeint(pol, m=j, k=list(range(j)))
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with lbnd
for i in range(5):
for j in range(2, 5):
pol = [0] * i + [1]
tgt = pol[:]
for k in range(j):
tgt = herme.hermeint(tgt, m=1, k=[k], lbnd=-1)
res = herme.hermeint(pol, m=j, k=list(range(j)), lbnd=-1)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with scaling
for i in range(5):
for j in range(2, 5):
pol = [0] * i + [1]
tgt = pol[:]
for k in range(j):
tgt = herme.hermeint(tgt, m=1, k=[k], scl=2)
res = herme.hermeint(pol, m=j, k=list(range(j)), scl=2)
assert_almost_equal(trim(res), trim(tgt))
def product3_herm(i,j,l):
#compute \Phi_i*\Phi_j*\Phi_l
return lambda x: H.hermeval(x, H.hermemul(H.hermemul(Phi(i),Phi(j)),Phi(l)))
def Q(i, x):
return H.hermeval(x, coef[:(i + 1)])
def __call__(self, xi, coefficients):
return hermite_e.hermeval(xi, coefficients)
def test_hermeint(self) :
# check exceptions
assert_raises(ValueError, herme.hermeint, [0], .5)
assert_raises(ValueError, herme.hermeint, [0], -1)
assert_raises(ValueError, herme.hermeint, [0], 1, [0,0])
# test integration of zero polynomial
for i in range(2, 5):
k = [0]*(i - 2) + [1]
res = herme.hermeint([0], m=i, k=k)
assert_almost_equal(res, [0, 1])
# check single integration with integration constant
for i in range(5) :
scl = i + 1
pol = [0]*i + [1]
tgt = [i] + [0]*i + [1/scl]
hermepol = herme.poly2herme(pol)
hermeint = herme.hermeint(hermepol, m=1, k=[i])
res = herme.herme2poly(hermeint)
assert_almost_equal(trim(res), trim(tgt))
# check single integration with integration constant and lbnd
for i in range(5) :
scl = i + 1
pol = [0]*i + [1]
hermepol = herme.poly2herme(pol)
hermeint = herme.hermeint(hermepol, m=1, k=[i], lbnd=-1)
assert_almost_equal(herme.hermeval(-1, hermeint), i)
# check single integration with integration constant and scaling
for i in range(5) :
scl = i + 1
pol = [0]*i + [1]
tgt = [i] + [0]*i + [2/scl]
hermepol = herme.poly2herme(pol)
hermeint = herme.hermeint(hermepol, m=1, k=[i], scl=2)
res = herme.herme2poly(hermeint)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with default k
for i in range(5) :
for j in range(2,5) :
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j) :
tgt = herme.hermeint(tgt, m=1)
res = herme.hermeint(pol, m=j)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with defined k
for i in range(5) :
for j in range(2,5) :
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j) :
tgt = herme.hermeint(tgt, m=1, k=[k])
res = herme.hermeint(pol, m=j, k=range(j))
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with lbnd
for i in range(5) :
for j in range(2,5) :
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j) :
tgt = herme.hermeint(tgt, m=1, k=[k], lbnd=-1)
res = herme.hermeint(pol, m=j, k=range(j), lbnd=-1)
assert_almost_equal(trim(res), trim(tgt))
# check multiple integrations with scaling
for i in range(5) :
for j in range(2,5) :
pol = [0]*i + [1]
tgt = pol[:]
for k in range(j) :
tgt = herme.hermeint(tgt, m=1, k=[k], scl=2)
res = herme.hermeint(pol, m=j, k=range(j), scl=2)
assert_almost_equal(trim(res), trim(tgt))
def q(i,j):
x, w=H.hermegauss(20)
Q=sum([w[ldx]*sum([w[kdx] * Q_FEM_quad[ldx*20+kdx] * H.hermeval(x[kdx],Phi(i)) for kdx in range(20)])*H.hermeval(x[ldx],Phi(j)) for ldx in range(20)])
q= Q/(2*np.pi*factorial(i)*factorial(j))
return q
def test_hermefit(self):
def f(x):
return x*(x - 1)*(x - 2)
def f2(x):
return x**4 + x**2 + 1
# Test exceptions
assert_raises(ValueError, herme.hermefit, [1], [1], -1)
assert_raises(TypeError, herme.hermefit, [[1]], [1], 0)
assert_raises(TypeError, herme.hermefit, [], [1], 0)
assert_raises(TypeError, herme.hermefit, [1], [[[1]]], 0)
assert_raises(TypeError, herme.hermefit, [1, 2], [1], 0)
assert_raises(TypeError, herme.hermefit, [1], [1, 2], 0)
assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[[1]])
assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[1, 1])
assert_raises(ValueError, herme.hermefit, [1], [1], [-1,])
assert_raises(ValueError, herme.hermefit, [1], [1], [2, -1, 6])
assert_raises(TypeError, herme.hermefit, [1], [1], [])
# Test fit
x = np.linspace(0, 2)
y = f(x)
#
coef3 = herme.hermefit(x, y, 3)
assert_equal(len(coef3), 4)
assert_almost_equal(herme.hermeval(x, coef3), y)
coef3 = herme.hermefit(x, y, [0, 1, 2, 3])
assert_equal(len(coef3), 4)
assert_almost_equal(herme.hermeval(x, coef3), y)
#
coef4 = herme.hermefit(x, y, 4)
assert_equal(len(coef4), 5)
assert_almost_equal(herme.hermeval(x, coef4), y)
coef4 = herme.hermefit(x, y, [0, 1, 2, 3, 4])
assert_equal(len(coef4), 5)
assert_almost_equal(herme.hermeval(x, coef4), y)
# check things still work if deg is not in strict increasing
coef4 = herme.hermefit(x, y, [2, 3, 4, 1, 0])
assert_equal(len(coef4), 5)
assert_almost_equal(herme.hermeval(x, coef4), y)
#
coef2d = herme.hermefit(x, np.array([y, y]).T, 3)
assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
coef2d = herme.hermefit(x, np.array([y, y]).T, [0, 1, 2, 3])
assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
# test weighting
w = np.zeros_like(x)
yw = y.copy()
w[1::2] = 1
y[0::2] = 0
wcoef3 = herme.hermefit(x, yw, 3, w=w)
assert_almost_equal(wcoef3, coef3)
wcoef3 = herme.hermefit(x, yw, [0, 1, 2, 3], w=w)
assert_almost_equal(wcoef3, coef3)
#
wcoef2d = herme.hermefit(x, np.array([yw, yw]).T, 3, w=w)
assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
wcoef2d = herme.hermefit(x, np.array([yw, yw]).T, [0, 1, 2, 3], w=w)
assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
# test scaling with complex values x points whose square
# is zero when summed.
x = [1, 1j, -1, -1j]
assert_almost_equal(herme.hermefit(x, x, 1), [0, 1])
assert_almost_equal(herme.hermefit(x, x, [0, 1]), [0, 1])
# test fitting only even Legendre polynomials
x = np.linspace(-1, 1)
y = f2(x)
coef1 = herme.hermefit(x, y, 4)
assert_almost_equal(herme.hermeval(x, coef1), y)
coef2 = herme.hermefit(x, y, [0, 2, 4])
assert_almost_equal(herme.hermeval(x, coef2), y)
assert_almost_equal(coef1, coef2)
def weights(dim, degree):
# 1D sigma-points (x) and weights (w)
x, w = hermegauss(degree)
# hermegauss() provides weights that cause posdef errors
w = factorial(degree) / (degree ** 2 * hermeval(x, [0] * (degree - 1) + [1]) ** 2)
return np.prod(cartesian([w] * dim), axis=1)
def test_hermefit(self) :
def f(x) :
return x*(x - 1)*(x - 2)
# Test exceptions
assert_raises(ValueError, herme.hermefit, [1], [1], -1)
assert_raises(TypeError, herme.hermefit, [[1]], [1], 0)
assert_raises(TypeError, herme.hermefit, [], [1], 0)
assert_raises(TypeError, herme.hermefit, [1], [[[1]]], 0)
assert_raises(TypeError, herme.hermefit, [1, 2], [1], 0)
assert_raises(TypeError, herme.hermefit, [1], [1, 2], 0)
assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[[1]])
assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[1,1])
# Test fit
x = np.linspace(0,2)
y = f(x)
#
coef3 = herme.hermefit(x, y, 3)
assert_equal(len(coef3), 4)
assert_almost_equal(herme.hermeval(x, coef3), y)
#
coef4 = herme.hermefit(x, y, 4)
assert_equal(len(coef4), 5)
assert_almost_equal(herme.hermeval(x, coef4), y)
#
coef2d = herme.hermefit(x, np.array([y,y]).T, 3)
assert_almost_equal(coef2d, np.array([coef3,coef3]).T)
# test weighting
w = np.zeros_like(x)
yw = y.copy()
w[1::2] = 1
y[0::2] = 0
wcoef3 = herme.hermefit(x, yw, 3, w=w)
assert_almost_equal(wcoef3, coef3)
#
wcoef2d = herme.hermefit(x, np.array([yw,yw]).T, 3, w=w)
assert_almost_equal(wcoef2d, np.array([coef3,coef3]).T)
#test NA
y = f(x)
y[10] = 100
xm = x.view(maskna=1)
xm[10] = np.NA
res = herme.hermefit(xm, y, 3)
assert_almost_equal(res, coef3)
ym = y.view(maskna=1)
ym[10] = np.NA
res = herme.hermefit(x, ym, 3)
assert_almost_equal(res, coef3)
y2 = np.vstack((y,y)).T
y2[10,0] = 100
y2[15,1] = 100
y2m = y2.view(maskna=1)
y2m[10,0] = np.NA
y2m[15,1] = np.NA
res = herme.hermefit(x, y2m, 3).T
assert_almost_equal(res[0], coef3)
assert_almost_equal(res[1], coef3)
wm = np.ones_like(x, maskna=1)
wm[10] = np.NA
res = herme.hermefit(x, y, 3, w=wm)
assert_almost_equal(res, coef3)
def PiecewiseMollify(theta, c_j, a_n, x):
"""
Piecewise mollify the spectral reconstruction of a discontinuous function
to reduce the effect of Gibbs phenomenon. Perform a real-space convolution
of a spectral reconstruction with an adaptive unit-mass mollifier.
Parameters
----------
theta : float
free parameter to vary in the range 0 < theta < 1
c_j : 1D array,
Array containing x positions of the discontinuities in the data
a_n : 1D array, shape = (N+1,)
N - highest order in the Chebyshev expansion
vector of Chebyshev expansion coefficients
x : 1D array,
1D grid of points in real space, where the function is evaluated
Returns
----------
mollified : 1D array, shape = (len(x),)
real space mollified representation of a discontinuous function
mollified_err : 1D array, shape = (len(x),)
error estimate for each point in the convolution, derived from
scipy.integrate.quad
"""
N = a_n.shape[0] - 1
sanity_check = np.empty(len(x))
mollified = np.array([])
mollified_err = np.array([])
I_N = lambda y: T.chebval(y, a_n)
chi_top = lambda y, f: f(y) if -1 <= y <= 1 else 0
I_N_top = lambda y: chi_top(y, I_N)
c_jplus = np.append(c_j, 1.0)
for idx_c, pos in enumerate(c_jplus):
if idx_c == 0:
lim_left = -1.0
lim_right = pos
offset = np.ma.masked_where(x > lim_right, x)
offset = np.ma.compressed(offset)
else:
lim_left = c_jplus[idx_c - 1]
lim_right = pos
offset = np.ma.masked_where(x > lim_right, x)
offset = np.ma.masked_where(x <= lim_left, offset)
offset = np.ma.compressed(offset)
chi_cut = lambda y, f: f(y) if lim_left <= y <= lim_right else 0
convolution = np.empty(len(offset))
convolution_err = np.empty(len(offset))
for idx_o, off_x in enumerate(offset):
c_jx = c_j - off_x
dx = lambda y: sqrt(theta * N * min(abs(y - c) for c in c_jx))
var = lambda y: N / (2 * theta * dx(y))
p_N = lambda y: (theta**2) * dx(y) * N
j_max = int(np.amax(np.frompyfunc(p_N, 1, 1)(x)))
h_j = np.zeros(2 * (j_max + 1))
for j in range(j_max + 1):
h_j[2 * j] = ((-1)**j) / ((4**j) * factorial(j))
hermite = lambda y: H.hermeval(sqrt(var(y)) * y, h_j)
expon = lambda y: (1. / sqrt(theta * N * dx(y))) * exp(-var(y) *
(y**2))
phi0 = lambda y: hermite(y) * expon(y)
phi_off = lambda y: phi0(y - off_x)
phi = lambda y: chi_cut(y, phi_off)
norm = quad(phi, lim_left, lim_right)[0]
convfunc = lambda y: (1 / norm) * (phi(y) * I_N_top(y))
convolution[idx_o], convolution_err[idx_o] = quad(
convfunc, lim_left, lim_right)
mollified = np.append(mollified, convolution)
mollified_err = np.append(mollified_err, convolution_err)
assert mollified.shape == sanity_check.shape, "Piecewise mollification inconsistent with regular one"
assert mollified_err.shape == sanity_check.shape, "Piecewise mollification inconsistent with regular one"
return mollified, mollified_err
def test_hermefit(self):
def f(x):
return x * (x - 1) * (x - 2)
# Test exceptions
assert_raises(ValueError, herme.hermefit, [1], [1], -1)
assert_raises(TypeError, herme.hermefit, [[1]], [1], 0)
assert_raises(TypeError, herme.hermefit, [], [1], 0)
assert_raises(TypeError, herme.hermefit, [1], [[[1]]], 0)
assert_raises(TypeError, herme.hermefit, [1, 2], [1], 0)
assert_raises(TypeError, herme.hermefit, [1], [1, 2], 0)
assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[[1]])
assert_raises(TypeError, herme.hermefit, [1], [1], 0, w=[1, 1])
# Test fit
x = np.linspace(0, 2)
y = f(x)
#
coef3 = herme.hermefit(x, y, 3)
assert_equal(len(coef3), 4)
assert_almost_equal(herme.hermeval(x, coef3), y)
#
coef4 = herme.hermefit(x, y, 4)
assert_equal(len(coef4), 5)
assert_almost_equal(herme.hermeval(x, coef4), y)
#
coef2d = herme.hermefit(x, np.array([y, y]).T, 3)
assert_almost_equal(coef2d, np.array([coef3, coef3]).T)
# test weighting
w = np.zeros_like(x)
yw = y.copy()
w[1::2] = 1
y[0::2] = 0
wcoef3 = herme.hermefit(x, yw, 3, w=w)
assert_almost_equal(wcoef3, coef3)
#
wcoef2d = herme.hermefit(x, np.array([yw, yw]).T, 3, w=w)
assert_almost_equal(wcoef2d, np.array([coef3, coef3]).T)
#test NA
y = f(x)
y[10] = 100
xm = x.view(maskna=1)
xm[10] = np.NA
res = herme.hermefit(xm, y, 3)
assert_almost_equal(res, coef3)
ym = y.view(maskna=1)
ym[10] = np.NA
res = herme.hermefit(x, ym, 3)
assert_almost_equal(res, coef3)
y2 = np.vstack((y, y)).T
y2[10, 0] = 100
y2[15, 1] = 100
y2m = y2.view(maskna=1)
y2m[10, 0] = np.NA
y2m[15, 1] = np.NA
res = herme.hermefit(x, y2m, 3).T
assert_almost_equal(res[0], coef3)
assert_almost_equal(res[1], coef3)
wm = np.ones_like(x, maskna=1)
wm[10] = np.NA
res = herme.hermefit(x, y, 3, w=wm)
assert_almost_equal(res, coef3)
