def test_Q():
assert_raises(ValueError, rft.Q, -1)
assert_raises(ValueError, rft.Q, 0)
x = np.arange(-9, 10)
for dim in range(1, 4):
res = rft.Q(dim)
assert_almost_equal(res(x), hermitenorm(dim - 1)(x))
def test_Q():
assert_raises(ValueError, rft.Q, -1)
assert_raises(ValueError, rft.Q, 0)
x = np.arange(-9, 10)
for dim in range(1, 4):
res = rft.Q(dim)
assert_almost_equal(res(x), hermitenorm(dim - 1)(x))
def test_polynomial2():
# EC density of chi^2(1) is 2 * EC density of Gaussian so polynomial part is
# a factor of 2 as well.
for dim in range(1,10):
q = rft.ChiSquared(dfn=1).quasi(dim)
h = hermitenorm(dim-1)
yield assert_almost_equal, q.c, 2*h.c
def Q(dim, dfd=np.inf):
""" Q polynomial
If dfd == inf (the default), then
Q(dim) is the (dim-1)-st Hermite polynomial
H_j(x) = (-1)^j * e^{x^2/2} * (d^j/dx^j e^{-x^2/2})
If dfd != inf, then it is the polynomial Q defined in
Worsley, K.J. (1994). 'Local maxima and the expected Euler
characteristic of excursion sets of \chi^2, F and t fields.'
Advances in Applied Probability, 26:13-42.
"""
m = dfd
j = dim
if j > 0:
poly = hermitenorm(j - 1)
poly = np.poly1d(np.around(poly.c))
if np.isfinite(m):
for l in range((j - 1) / 2 + 1):
f = np.exp(
gammaln((m + 1) / 2.0)
- gammaln((m + 2 - j + 2 * l) / 2.0)
- 0.5 * (j - 1 - 2 * l) * (np.log(m / 2.0))
)
poly.c[2 * l] *= f
return np.poly1d(poly.c)
else:
raise ValueError, "Q defined only for dim > 0"
def gaussflux(pixbound, cen, sig, h_order=0):
"""
For monotonically increasing pixel boundaries specified by 'pixbound'
in some abscissa units, consider a Gaussian with unit integrated
amplitude that expresses a density per those same abscissa units,
centered on 'cen' and with sigma parameter 'sig', and return
the average value of that Gaussian between the boundaries
(i.e., its pixel-averaged value for possibly non-uniform pixels.)
Can specify a Gauss-Hermite order (a la scipy.special.hermitenorm)
via the 'h_order' argument, which defaults to zero.
(bolton@utah@iac 2014mayo)
(Added Hermite orders: bolton@utah@iac 2014junio)
"""
# Calculate the pixel widths and test for monotonicity:
pixdiff = pixbound[1:] - pixbound[:-1]
if (pixdiff.min <= 0):
print 'pixbound must be monotonically increasing!'
return 0
# Make sure scalar arguments are scalars:
if (n.asarray(cen).size != 1):
print 'cen argument must be scalar!'
return 0
if (n.asarray(sig).size != 1):
print 'sig argument must be scalar!'
return 0
# Compute and return:
if h_order > 0:
u = (pixbound - cen) / sig
int_term = - spc.hermitenorm(h_order-1)(u) * n.exp(-0.5 * u**2) / \
n.sqrt(2. * n.pi)
else:
int_term = 0.5 * spc.erf((pixbound - cen) / (n.sqrt(2.) * sig))
return (int_term[1:] - int_term[:-1]) / pixdiff
def test_polynomial2():
# EC density of chi^2(1) is 2 * EC density of Gaussian so polynomial part is
# a factor of 2 as well.
for dim in range(1, 10):
q = rft.ChiSquared(dfn=1).quasi(dim)
h = hermitenorm(dim - 1)
yield assert_almost_equal, q.c, 2 * h.c
def gen_grad_dict_dot_f(self, Xdot, X):
"""
compute the gradient of phi dot product with f
:type Xdot: np.ndarray
:param Xdot: time derivative of state
:return: generated_gradPhi_dot_f_array
:rtype: np.ndarray
"""
num_sample, num_components = Xdot.shape
if self.dict == 'hermite':
# normalized hermite polynomial
## compute [ [d[H0(x1).. H0(xn)]/dx1,...,d[HN(x1).. HN(xn)]/dx1 ],
# ...
# [d[H0(x1).. H0(xn)]/dxn,...,d[HN(x1).. HN(xn)]/dxn ] ]
generated_feature_array_list = []
feature_list_ddx_list = []
for i_component in range(num_components):
feature_list = []
for order in range(self.hermite_order + 1):
phi_i = hermitenorm(order)
phi_i_dx = np.poly1d.deriv(phi_i)
phi_i_X = np.polyval(phi_i, X)
# update i_component with the derivative one
phi_i_X[:,
i_component] = np.polyval(phi_i_dx, X[:,
i_component])
feature_list.append(phi_i_X)
feature_list_ddx_list.append(feature_list)
# generate feature array from feature list for each i_component
generated_feature_array = self.gen_cross_component_features(
feature_list=feature_list,
num_sample=num_sample,
num_components=num_components)
# dot product f with the gradient
Xdot_i_component = Xdot[:, i_component]
Xdot_i_matrix = np.diag(Xdot_i_component)
generated_feature_array_list.append(
np.matmul(Xdot_i_matrix, generated_feature_array))
# summing up the dot product for each component
generated_gradPhi_dot_f_array = np.sum(
generated_feature_array_list, axis=0)
elif self.dict == 'rff_gaussian':
generated_gradPhi_dot_f_array = self.gen_rff_features_dot(Xdot, X)
elif self.dict == 'nystrom':
pass
else:
raise NotImplementedError("the type of " + self.dict +
" is not implemented yet!")
return generated_gradPhi_dot_f_array
def gaussflux(pixbound, cen, sig, h_order=0):
"""
For monotonically increasing pixel boundaries specified by 'pixbound'
in some abscissa units, consider a Gaussian with unit integrated
amplitude that expresses a density per those same abscissa units,
centered on 'cen' and with sigma parameter 'sig', and return
the average value of that Gaussian between the boundaries
(i.e., its pixel-averaged value for possibly non-uniform pixels.)
Can specify a Gauss-Hermite order (a la scipy.special.hermitenorm)
via the 'h_order' argument, which defaults to zero.
(bolton@utah@iac 2014mayo)
(Added Hermite orders: bolton@utah@iac 2014junio)
"""
# Calculate the pixel widths and test for monotonicity:
pixdiff = pixbound[1:] - pixbound[:-1]
if (pixdiff.min <= 0):
print('pixbound must be monotonically increasing!')
return 0
# Make sure scalar arguments are scalars:
if (n.asarray(cen).size != 1):
print('cen argument must be scalar!')
return 0
if (n.asarray(sig).size != 1):
print('sig argument must be scalar!')
return 0
# Compute and return:
if h_order > 0:
u = (pixbound - cen) / sig
int_term = - spc.hermitenorm(h_order-1)(u) * n.exp(-0.5 * u**2) / \
n.sqrt(2. * n.pi)
else:
int_term = 0.5 * spc.erf((pixbound - cen) / (n.sqrt(2.) * sig))
return (int_term[1:] - int_term[:-1]) / pixdiff
def get_polys(self, x):
"""
Calculates the normalized Hermite polynomials evaluated at sample points.
**Inputs:**
* **x** (`ndarray`):
`ndarray` containing the samples.
**Outputs:**
(`list`):
Returns a list of 'ndarrays' with the design matrix and the
normalized polynomials.
"""
a, b = -np.inf, np.inf
mean_ = Polynomials.get_mean(self)
std_ = Polynomials.get_std(self)
x_ = Polynomials.standardize_normal(x, mean_, std_)
norm = Normal(0, 1)
pdf_st = norm.pdf
p = []
for i in range(self.degree):
p.append(special.hermitenorm(i, monic=False))
return Polynomials.normalized(self.degree, x_, a, b, pdf_st, p)
def __init__(self, filename):
"""
Initialize GaussHermitePSF from input file
"""
#- Check that this file is a current generation Gauss Hermite PSF
fx = fits.open(filename, memmap=False)
self._polyparams = hdr = fx[1].header
if 'PSFTYPE' not in hdr:
raise ValueError, 'Missing PSFTYPE keyword'
if hdr['PSFTYPE'] != 'GAUSS-HERMITE2':
raise ValueError, 'PSFTYPE %s is not GAUSS-HERMITE' % hdr['PSFTYPE']
if 'PSFVER' not in hdr:
raise ValueError, "PSFVER missing; this version not supported"
if hdr['PSFVER'] < '1':
raise ValueError, "Only GAUSS-HERMITE versions 1.0 and greater are supported"
#- Calculate number of spectra from FIBERMIN and FIBERMAX (inclusive)
self.nspec = hdr['FIBERMAX'] - hdr['FIBERMIN'] + 1
#- Other necessary keywords
self.npix_x = hdr['NPIX_X']
self.npix_y = hdr['NPIX_Y']
#- Load the parameters into self.coeff dictionary keyed by PARAM
#- with values as TraceSets for evaluating the Legendre coefficients
data = fx[1].data
self.coeff = dict()
for p in data:
domain = (p['WAVEMIN'], p['WAVEMAX'])
for p in data:
name = p['PARAM'].strip()
self.coeff[name] = TraceSet(p['COEFF'], domain=domain)
#- Pull out x and y as special tracesets
self._x = self.coeff['X']
self._y = self.coeff['Y']
#- Create inverse y -> wavelength mapping
self._w = self._y.invert()
self._wmin = np.min(self.wavelength(None, 0))
self._wmin_all = np.max(self.wavelength(None, 0))
self._wmax = np.max(self.wavelength(None, self.npix_y-1))
self._wmax_all = np.min(self.wavelength(None, self.npix_y-1))
#- Filled only if needed
self._xsigma = None
self._ysigma = None
#- Cache hermitenorm polynomials so we don't have to create them
#- every time xypix is called
self._hermitenorm = list()
maxdeg = max(hdr['GHDEGX'], hdr['GHDEGY'], hdr['GHDEGX2'], hdr['GHDEGY2'])
for i in range(maxdeg+1):
self._hermitenorm.append( sp.hermitenorm(i) )
fx.close()
def test_polynomial1():
# Polynomial part of Gaussian densities are Hermite polynomials.
for dim in range(1,10):
q = rft.Gaussian().quasi(dim)
h = hermitenorm(dim-1)
yield assert_almost_equal, q.c, h.c
def get_psi_sq(self):
"""
Calculate the term <psii,psij>
Returns
-------
psi_sq : array
the term <psii,psij>
"""
dim = self.my_experiment.dimension
multindices = self.basis['multi-indices']
n_terms = len(multindices)
psi_sq = np.ones(n_terms, )
for i in range(n_terms):
for j in range(dim):
deg = multindices[i][j]
if self.my_experiment.polytypes[j] == 'Legendre':
x_i, w_i = special.p_roots(deg + 1)
'''
Integrate exactly the SQUARE
of the Legendre polynomial. For example,
if the Legendre polynomial is of order (deg),
the numerical integration must be exact
till order (deg**2). Thus, we need at least
(deg+1) abscissas' and weights.
'''
poly = special.legendre(deg)**2
psi_sq[i] *= 1.0 / 2 * sum(w_i * poly(x_i))
elif self.my_experiment.polytypes[j] == 'Hermite':
x_i, w_i = special.he_roots(deg + 1)
'''
special.he_roots(deg) and
np.polynomial.hermite_e.hermegauss(deg)
returns the same abscissas'
but different weights (!). There is a factor 2
between the two. Given the fact that the integral of
the standard Gaussian must be 1,
np.polynomial.hermite_e.hermegauss(deg)
provides the right weights.
'''
poly = special.hermitenorm(deg)**2
# 2*w_i*poly(x_i)
psi_sq[i] *= 1.0 / np.sqrt(2 * np.pi) * sum(
w_i * poly(x_i))
return psi_sq
def shapelet1d(n, x0=0, s=1):
def sqfac(k):
fac = 1.
for i in xrange(k):
fac *= np.sqrt(i + 1)
return fac
u = lambda x: (x - x0) / s
fn = lambda x: (1. / (2 * np.pi)**0.25) * (1. / sqfac(n)) * hermitenorm(n)(
u(x)) * np.exp(-0.25 * u(x)**2)
return fn
def test_custom_hermitenorm(self):
#need to compare our custom hermite norm function to the scipy one
m = 3 #degree of the polynomial
u = np.random.uniform(0,10,size=10)
#generate scipy polynomial
scipy_poly = sp.hermitenorm(m)
#evalulate at point u
scipy_out = scipy_poly(u)
#now try our custom fuction
custom_out = custom_hermitenorm(m,u)
#check if they're the same
self.assertTrue(np.all(scipy_out == custom_out))
def gen_dict_feature(self, X):
"""
generate nonlinear feature given data + nonlinear transformation given
X = | x_1,.....,x_n @ time_step = 1|
| x_1,.....,x_n @ time_step = M|
X.shape = (num_samples, num_components)
.. note::
Computational time:
* The computational time scales with the number of features mainly, which is determined by number of components and polynomial order
:type X: np.ndarray
:param X: input data with shape = (num_samples, num_components)
:return: generated_feature_array with shape (num_samples, num_features)
:rtype: np.ndarray
"""
num_sample, num_components = X.shape
generated_feature_array = None
if self.dict == 'hermite':
# normalized hermite polynomial
## compute feature list = [[H0(x1).. H0(xn)],...,[HN(x1).. HN(xn)] ]
feature_list = []
for order in range(self.hermite_order + 1):
phi_i = hermitenorm(order)
phi_i_X = np.polyval(phi_i, X)
feature_list.append(phi_i_X)
# create feature array from feature list
generated_feature_array = self.gen_cross_component_features(
feature_list=feature_list,
num_sample=num_sample,
num_components=num_components)
elif self.dict == 'rff_gaussian':
# isotropic gaussian kernel exp(-||x||^2/2\sigma)
generated_feature_array = self.gen_rff_features(X=X)
elif self.dict == 'rff_gaussian_state':
generated_feature_array = self.gen_rff_features_include_state(X=X)
elif self.dict == 'nystrom':
pass
else:
raise NotImplementedError("we haven't implemented that!")
return generated_feature_array
def shapelet1d(n,x0=0,s=1):
"""Make a 1D shapelet template to be used in the construction of 2D shapelets
n : Integer representing energy quantum number
x0 : Integer defining the centroid
s : Float which is the same as beta parameter in refregier
"""
def sqfac(k):
fac = 1.
for i in xrange(k):
fac *= np.sqrt(i+1)
return fac
u = lambda x: (x-x0)/s
fn = lambda x: (1./(2*np.pi)**0.25)*(1./sqfac(n))*hermitenorm(n)(u(x))*np.exp(-0.25*u(x)**2)
return fn
def __init__(self,dim = 1, dist_type = None, var = None, order = None):
self.dim = dim
self.order = order
self.nbpoly = order + 1
self.var = var
self.dist_type = dist_type
#self.coef = np.zeros(self.nbpoly, self.order + 1)
if dist_type =='norm':
self.funcs = np.array([sp.hermitenorm(i)for i in range (0, self.order+1)])
for i in range(0, self.order+1):
normalized_const = np.sqrt(sp.factorial(i))
self.funcs[i] = poly1d(self.funcs[i].c/normalized_const)
if dist_type =='uniform':
#The polynomials are orthogonal over [0,1] with weight function 1.
self.funcs = np.array([sp.sh_legendre(i)for i in range (0, self.order+1)])
for i in range(0, self.order+1):
normalized_const = np.sqrt(1./(2.*i+1.))
self.funcs[i] = poly1d(self.funcs[i].c/normalized_const)
self.modelcoef = np.zeros((self.nbpoly,))
def basis(index, eps):
"""
BASIS is the multivariate basis
Parameters
----------
index (int) :
index of the basis (from 0 to nt)
eps (int) :
evaluation points:
* from -1 to +1 for a uniform distribution,
* from -inf to onf for a normal distribution.
Return
------
y (float) :
value of the basis in eps --> y = psi(eps)
AUTHOR: Luca Giaccone ([email protected])
DATE: 17.12.2018
HISTORY:
"""
# check input
if index > (nt - 1):
raise ValueError(f'max index possible is nt-1={self.nt-1}')
if isinstance(eps, (list, tuple)):
eps = np.array(eps)
# get multi-index
i = self.multi_index[index]
# initialize output
y = np.ones(eps.shape[0])
# cycle on distrubutions (nt element)
for k, dist in enumerate(distrib):
if dist.upper() == 'U':
y = y * np.polyval(legendre(i[k]), eps[..., k])
elif dist.upper() == 'N':
y = y * np.polyval(hermitenorm(i[k]), eps[..., k])
return y
def Q(dim, dfd=np.inf):
""" Q polynomial
If `dfd` == inf (the default), then Q(dim) is the (dim-1)-st Hermite
polynomial:
.. math::
H_j(x) = (-1)^j * e^{x^2/2} * (d^j/dx^j e^{-x^2/2})
If `dfd` != inf, then it is the polynomial Q defined in [Worsley1994]_
Parameters
----------
dim : int
dimension of polynomial
dfd : scalar
Returns
-------
q_poly : np.poly1d instance
References
----------
.. [Worsley1994] Worsley, K.J. (1994). 'Local maxima and the expected Euler
characteristic of excursion sets of \chi^2, F and t fields.' Advances in
Applied Probability, 26:13-42.
"""
m = dfd
j = dim
if j <= 0:
raise ValueError('Q defined only for dim > 0')
poly = hermitenorm(j - 1)
poly = np.poly1d(np.around(poly.c))
if np.isfinite(m):
for l in range((j - 1) // 2 + 1):
f = np.exp(
gammaln((m + 1) / 2.) - gammaln((m + 2 - j + 2 * l) / 2.) -
0.5 * (j - 1 - 2 * l) * (np.log(m / 2.)))
poly.c[2 * l] *= f
return np.poly1d(poly.c)
def get_polys(self, x):
"""
Calculates the normalized Hermite polynomials evaluated at sample points.
:param x: :class:`numpy.ndarray` containing the samples.
:return: Α list of :class:`numpy.ndarray` with the design matrix and the
normalized polynomials.
"""
a, b = -np.inf, np.inf
mean_ = Polynomials.get_mean(self)
std_ = Polynomials.get_std(self)
x_ = Polynomials.standardize_normal(x, mean_, std_)
norm = Normal(0, 1)
pdf_st = norm.pdf
p = []
for i in range(self.degree):
p.append(special.hermitenorm(i, monic=False))
return Polynomials.normalized(self.degree, x_, a, b, pdf_st, p)
def Q(dim, dfd=np.inf):
r""" Q polynomial
If `dfd` == inf (the default), then Q(dim) is the (dim-1)-st Hermite
polynomial:
.. math::
H_j(x) = (-1)^j * e^{x^2/2} * (d^j/dx^j e^{-x^2/2})
If `dfd` != inf, then it is the polynomial Q defined in [Worsley1994]_
Parameters
----------
dim : int
dimension of polynomial
dfd : scalar
Returns
-------
q_poly : np.poly1d instance
References
----------
.. [Worsley1994] Worsley, K.J. (1994). 'Local maxima and the expected Euler
characteristic of excursion sets of \chi^2, F and t fields.' Advances in
Applied Probability, 26:13-42.
"""
m = dfd
j = dim
if j <= 0:
raise ValueError('Q defined only for dim > 0')
coeffs = np.around(hermitenorm(j - 1).c)
if np.isfinite(m):
for L in range((j - 1) // 2 + 1):
f = np.exp(gammaln((m + 1) / 2.)
- gammaln((m + 2 - j + 2 * L) / 2.)
- 0.5 * (j - 1 - 2 * L) * (np.log(m / 2.)))
coeffs[2 * L] *= f
return np.poly1d(coeffs)
def gen_dict_feature(self, X):
"""
generate nonlinear feature given data + nonlinear transformation given
X = | x_1,.....,x_n @ time_step = 1|
| x_1,.....,x_n @ time_step = M|
X.shape = (num_samples, num_components)
.. note::
Computational time:
* The computational time scales with the number of features mainly, which is determined by number of components and polynomial order
:type X: np.ndarray
:param X: input data with shape = (num_samples, num_components)
:return: generated_feature_array with shape (num_samples, num_features)
:rtype: np.ndarray
"""
num_sample, num_components = X.shape
generated_feature_array = None
if self.dict == 'hermite':
# normalized hermite polynomial
## compute feature list = [[H0(x1).. H0(xn)],...,[HN(x1).. HN(xn)] ]
feature_list = []
for order in range(self.hermite_order + 1):
phi_i = hermitenorm(order)
phi_i_X = np.polyval(phi_i, X)
feature_list.append(phi_i_X)
# create feature array from feature list
generated_feature_array = self.gen_cross_component_features(
feature_list=feature_list,
num_sample=num_sample,
num_components=num_components
)
return generated_feature_array
def __init__(self, filename):
"""
Initialize GaussHermitePSF from input file
"""
#- Check that this file is a current generation Gauss Hermite PSF
fx = fits.open(filename, memmap=False)
#- Read primary header
phdr = fx[0].header
if 'PSFTYPE' not in phdr:
raise ValueError('Missing PSFTYPE keyword')
if phdr['PSFTYPE'] != 'GAUSS-HERMITE':
raise ValueError('PSFTYPE {} is not GAUSS-HERMITE'.format(phdr['PSFTYPE']))
if 'PSFVER' not in phdr:
raise ValueError("PSFVER missing; this version not supported")
PSFVER = float(phdr["PSFVER"])
if PSFVER<3 :
psf_hdu = 1
else :
psf_hdu = "PSF"
self._polyparams = hdr = dict(fx[psf_hdu].header)
if 'PSFTYPE' not in hdr:
raise ValueError('Missing PSFTYPE keyword')
if hdr['PSFTYPE'] != 'GAUSS-HERMITE':
raise ValueError('PSFTYPE {} is not GAUSS-HERMITE'.format(hdr['PSFTYPE']))
if 'PSFVER' not in hdr:
raise ValueError("PSFVER missing; this version not supported")
if hdr['PSFVER'] < '1':
raise ValueError("Only GAUSS-HERMITE versions 1.0 and greater are supported")
#- Calculate number of spectra from FIBERMIN and FIBERMAX (inclusive)
self.nspec = hdr['FIBERMAX'] - hdr['FIBERMIN'] + 1
#- Other necessary keywords
self.npix_x = hdr['NPIX_X']
self.npix_y = hdr['NPIX_Y']
#- PSF model error
if 'PSFERR' in hdr:
self.psferr = hdr['PSFERR']
else:
self.psferr = 0.01
#- Load the parameters into self.coeff dictionary keyed by PARAM
#- with values as TraceSets for evaluating the Legendre coefficients
data = fx[psf_hdu].data
self.coeff = dict()
if PSFVER<3 : # old format
for p in data:
domain = (p['WAVEMIN'], p['WAVEMAX'])
for p in data:
name = p['PARAM'].strip()
self.coeff[name] = TraceSet(p['COEFF'], domain=domain)
#- Pull out x and y as special tracesets
self._x = self.coeff['X']
self._y = self.coeff['Y']
else : # new format
domain = (hdr['WAVEMIN'], hdr['WAVEMAX'])
for p in data:
name = p['PARAM'].strip()
self.coeff[name] = TraceSet(p['COEFF'], domain=domain)
self._x = TraceSet(fx["XTRACE"].data, domain=(fx['XTRACE'].header["WAVEMIN"], fx['XTRACE'].header['WAVEMAX']))
self._y = TraceSet(fx["YTRACE"].data, domain=(fx['YTRACE'].header["WAVEMIN"], fx['YTRACE'].header['WAVEMAX']))
#- Create inverse y -> wavelength mapping
self._w = self._y.invert()
#- Cache min/max wavelength per fiber at pixel edges
self._wmin_spec = self.wavelength(None, -0.5)
self._wmax_spec = self.wavelength(None, self.npix_y-0.5)
self._wmin = np.min(self._wmin_spec)
self._wmin_all = np.max(self._wmin_spec)
self._wmax = np.max(self._wmax_spec)
self._wmax_all = np.min(self._wmax_spec)
#- Filled only if needed
self._xsigma = None
self._ysigma = None
#create dict to hold legval cached data
self.legval_dict = None
#- Cache hermitenorm polynomials so we don't have to create them
#- every time xypix is called
self._hermitenorm = list()
maxdeg = max(hdr['GHDEGX'], hdr['GHDEGY'])
for i in range(maxdeg+1):
self._hermitenorm.append( sp.hermitenorm(i) )
fx.close()
import numpy as np
import matplotlib.pyplot as plt
from scipy.special import chebyt, hermitenorm
x = np.linspace(-1, 1, 100)
xx = np.linspace(-5, 5, 100)
t1 = np.polyval(chebyt(1), x)
t2 = np.polyval(chebyt(2), x)
t3 = np.polyval(chebyt(3), x)
t4 = np.polyval(chebyt(4), x)
h1 = np.polyval(hermitenorm(1), x)
h2 = np.polyval(hermitenorm(2), x)
h3 = np.polyval(hermitenorm(3), x)
h4 = np.polyval(hermitenorm(4), x)
plt.rc('mathtext', fontset='stix')
fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(9, 6))
ax1.plot(x, t1, label=r'$T_{1}(x)$')
ax1.plot(x, t2, label=r'$T_{2}(x)$')
ax1.plot(x, t3, label=r'$T_{3}(x)$')
ax1.plot(x, t4, label=r'$T_{4}(x)$')
ax1.legend(loc='best')
ax1.set_xlabel(r'$x$')
ax1.set_title('First four Chebyshev polynomials of the first kind')
ax1.grid()
ax2.plot(xx, h1, label=r'$H_{1}(x)$')
ax2.plot(xx, h2, label=r'$H_{2}(x)$')
ax2.plot(xx, h3, label=r'$H_{3}(x)$')
def __init__(self, filename):
"""
Initialize GaussHermitePSF from input file
"""
#- Check that this file is a current generation Gauss Hermite PSF
fx = fits.open(filename, memmap=False)
self._polyparams = hdr = fx[1].header
if 'PSFTYPE' not in hdr:
raise ValueError('Missing PSFTYPE keyword')
if hdr['PSFTYPE'] != 'GAUSS-HERMITE2':
raise ValueError('PSFTYPE {} is not GAUSS-HERMITE'.format(hdr['PSFTYPE']))
if 'PSFVER' not in hdr:
raise ValueError("PSFVER missing; this version not supported")
if hdr['PSFVER'] < '1':
raise ValueError("Only GAUSS-HERMITE versions 1.0 and greater are supported")
#- Calculate number of spectra from FIBERMIN and FIBERMAX (inclusive)
self.nspec = hdr['FIBERMAX'] - hdr['FIBERMIN'] + 1
#- Other necessary keywords
self.npix_x = hdr['NPIX_X']
self.npix_y = hdr['NPIX_Y']
#- PSF model error
if 'PSFERR' in hdr:
self.psferr = hdr['PSFERR']
else:
self.psferr = 0.01
#- Load the parameters into self.coeff dictionary keyed by PARAM
#- with values as TraceSets for evaluating the Legendre coefficients
data = fx[1].data
self.coeff = dict()
for p in data:
domain = (p['WAVEMIN'], p['WAVEMAX'])
for p in data:
name = p['PARAM'].strip()
self.coeff[name] = TraceSet(p['COEFF'], domain=domain)
#- Pull out x and y as special tracesets
self._x = self.coeff['X']
self._y = self.coeff['Y']
#- Create inverse y -> wavelength mapping
self._w = self._y.invert()
#- Cache min/max wavelength per fiber at pixel edges
self._wmin_spec = self.wavelength(None, -0.5)
self._wmax_spec = self.wavelength(None, self.npix_y-0.5)
self._wmin = np.min(self._wmin_spec)
self._wmin_all = np.max(self._wmin_spec)
self._wmax = np.max(self._wmax_spec)
self._wmax_all = np.min(self._wmax_spec)
#- Filled only if needed
self._xsigma = None
self._ysigma = None
#- Cache hermitenorm polynomials so we don't have to create them
#- every time xypix is called
self._hermitenorm = list()
maxdeg = max(hdr['GHDEGX'], hdr['GHDEGY'], hdr['GHDEGX2'], hdr['GHDEGY2'])
for i in range(maxdeg+1):
self._hermitenorm.append( sp.hermitenorm(i) )
fx.close()
def test_polynomial1():
# Polynomial part of Gaussian densities are Hermite polynomials.
for dim in range(1, 10):
q = rft.Gaussian().quasi(dim)
h = hermitenorm(dim - 1)
yield assert_almost_equal, q.c, h.c
