# Source code for sas.sascalc.invariant.invariant

# pylint: disable=invalid-name
#####################################################################
#This software was developed by the University of Tennessee as part of the
#Distributed Data Analysis of Neutron Scattering Experiments (DANSE)
#project funded by the US National Science Foundation.
######################################################################

"""
This module implements invariant and its related computations.

:author: Gervaise B. Alina/UTK
:author: Mathieu Doucet/UTK
:author: Jae Cho/UTK

"""
import math
import numpy

# The minimum q-value to be used when extrapolating
Q_MINIMUM = 1e-5

# The maximum q-value to be used when extrapolating
Q_MAXIMUM = 10

# Number of steps in the extrapolation
INTEGRATION_NSTEPS = 1000

[docs]class Transform(object):
"""
Define interface that need to compute a function or an inverse
function given some x, y
"""

[docs]    def linearize_data(self, data):
"""
Linearize data so that a linear fit can be performed.
Filter out the data that can't be transformed.

"""
# Check that the vector lengths are equal
assert len(data.x) == len(data.y)
if data.dy is not None:
assert len(data.x) == len(data.dy)
dy = data.dy
else:
dy = numpy.ones(len(data.y))

# Transform the data
data_points = zip(data.x, data.y, dy)

output_points = [(self.linearize_q_value(p[0]),
math.log(p[1]),
p[2] / p[1]) for p in data_points if p[0] > 0 and \
p[1] > 0 and p[2] > 0]

x_out, y_out, dy_out = zip(*output_points)

# Create Data1D object
x_out = numpy.asarray(x_out)
y_out = numpy.asarray(y_out)
dy_out = numpy.asarray(dy_out)

return linear_data

[docs]    def get_allowed_bins(self, data):
"""
Goes through the data points and returns a list of boolean values
to indicate whether each points is allowed by the model or not.

:param data: Data1D object
"""
return [p[0] > 0 and p[1] > 0 and p[2] > 0 for p in zip(data.x, data.y,
data.dy)]

[docs]    def linearize_q_value(self, value):
"""
Transform the input q-value for linearization
"""
return NotImplemented

[docs]    def extract_model_parameters(self, constant, slope, dconstant=0, dslope=0):
"""
set private member
"""
return NotImplemented

[docs]    def evaluate_model(self, x):
"""
Returns an array f(x) values where f is the Transform function.
"""
return NotImplemented

[docs]    def evaluate_model_errors(self, x):
"""
Returns an array of I(q) errors
"""
return NotImplemented

[docs]class Guinier(Transform):
"""
class of type Transform that performs operations related to guinier
function
"""
Transform.__init__(self)
self.scale = scale
## Uncertainty of scale parameter
self.dscale = 0

[docs]    def linearize_q_value(self, value):
"""
Transform the input q-value for linearization

:param value: q-value

:return: q*q
"""
return value * value

[docs]    def extract_model_parameters(self, constant, slope, dconstant=0, dslope=0):
"""
assign new value to the scale and the radius
"""
self.scale = math.exp(constant)
if slope > 0:
slope = 0.0
# Errors
self.dscale = math.exp(constant) * dconstant
if slope == 0.0:
n_zero = -1.0e-24
self.dradius = -3.0 / 2.0 / math.sqrt(-3 * n_zero) * dslope
else:
self.dradius = -3.0 / 2.0 / math.sqrt(-3 * slope) * dslope

[docs]    def evaluate_model(self, x):
"""
"""
return self._guinier(x)

[docs]    def evaluate_model_errors(self, x):
"""
Returns the error on I(q) for the given array of q-values

:param x: array of q-values
"""
p1 = numpy.array([self.dscale * math.exp(-((self.radius * q) ** 2 / 3)) \
for q in x])
p2 = numpy.array([self.scale * math.exp(-((self.radius * q) ** 2 / 3))\
* (-(q ** 2 / 3)) * 2 * self.radius * self.dradius for q in x])
diq2 = p1 * p1 + p2 * p2
return numpy.array([math.sqrt(err) for err in diq2])

def _guinier(self, x):
"""
Retrieve the guinier function after apply an inverse guinier function
to x
Compute a F(x) = scale* e-((radius*x)**2/3).

:param x: a vector of q values
:param scale: the scale value

:return: F(x)
"""
# transform the radius of coming from the inverse guinier function to a
# a radius of a guinier function
msg = "Rg expected positive value, but got %s" % self.radius
raise ValueError(msg)
value = numpy.array([math.exp(-((self.radius * i) ** 2 / 3)) for i in x])
return self.scale * value

[docs]class PowerLaw(Transform):
"""
class of type transform that perform operation related to power_law
function
"""
def __init__(self, scale=1, power=4):
Transform.__init__(self)
self.scale = scale
self.power = power
self.dscale = 0.0
self.dpower = 0.0

[docs]    def linearize_q_value(self, value):
"""
Transform the input q-value for linearization

:param value: q-value

:return: log(q)
"""
return math.log(value)

[docs]    def extract_model_parameters(self, constant, slope, dconstant=0, dslope=0):
"""
Assign new value to the scale and the power
"""
self.power = -slope
self.scale = math.exp(constant)

# Errors
self.dscale = math.exp(constant) * dconstant
self.dpower = -dslope

return [self.power, self.scale], [self.dpower, self.dscale]

[docs]    def evaluate_model(self, x):
"""
given a scale and a radius transform x, y using a power_law
function
"""
return self._power_law(x)

[docs]    def evaluate_model_errors(self, x):
"""
Returns the error on I(q) for the given array of q-values
:param x: array of q-values
"""
p1 = numpy.array([self.dscale * math.pow(q, -self.power) for q in x])
p2 = numpy.array([self.scale * self.power * math.pow(q, -self.power - 1)\
* self.dpower for q in x])
diq2 = p1 * p1 + p2 * p2
return numpy.array([math.sqrt(err) for err in diq2])

def _power_law(self, x):
"""
F(x) = scale* (x)^(-power)
when power= 4. the model is porod
else power_law
The model has three parameters: ::
1. x: a vector of q values
2. power: power of the function
3. scale : scale factor value

:param x: array
:return: F(x)
"""
if self.power <= 0:
msg = "Power_law function expected positive power,"
msg += " but got %s" % self.power
raise ValueError(msg)
if self.scale <= 0:
msg = "scale expected positive value, but got %s" % self.scale
raise ValueError(msg)

value = numpy.array([math.pow(i, -self.power) for i in x])
return self.scale * value

[docs]class Extrapolator(object):
"""
Extrapolate I(q) distribution using a given model
"""
def __init__(self, data, model=None):
"""
Determine a and b given a linear equation y = ax + b

If a model is given, it will be used to linearize the data before
the extrapolation is performed. If None,
a simple linear fit will be done.

:param data: data containing x and y  such as  y = ax + b
:param model: optional Transform object
"""
self.data = data
self.model = model

# Set qmin as the lowest non-zero value
self.qmin = Q_MINIMUM
for q_value in self.data.x:
if q_value > 0:
self.qmin = q_value
break
self.qmax = max(self.data.x)

[docs]    def fit(self, power=None, qmin=None, qmax=None):
"""
Fit data for y = ax + b  return a and b

:param power: a fixed, otherwise None
:param qmin: Minimum Q-value
:param qmax: Maximum Q-value
"""
if qmin is None:
qmin = self.qmin
if qmax is None:
qmax = self.qmax

# Identify the bin range for the fit
idx = (self.data.x >= qmin) & (self.data.x <= qmax)

fx = numpy.zeros(len(self.data.x))

# Uncertainty
if type(self.data.dy) == numpy.ndarray and \
len(self.data.dy) == len(self.data.x) and \
numpy.all(self.data.dy > 0):
sigma = self.data.dy
else:
sigma = numpy.ones(len(self.data.x))

# Compute theory data f(x)
fx[idx] = self.data.y[idx]

# Linearize the data
if self.model is not None:
linearized_data = self.model.linearize_data(\
fx[idx],
dy=sigma[idx]))
else:
fx[idx],
dy=sigma[idx])

##power is given only for function = power_law
if power != None:
sigma2 = linearized_data.dy * linearized_data.dy
a = -(power)
b = (numpy.sum(linearized_data.y / sigma2) \
- a * numpy.sum(linearized_data.x / sigma2)) / numpy.sum(1.0 / sigma2)

deltas = linearized_data.x * a + \
numpy.ones(len(linearized_data.x)) * b - linearized_data.y
residuals = numpy.sum(deltas * deltas / sigma2)

err = math.fabs(residuals) / numpy.sum(1.0 / sigma2)
return [a, b], [0, math.sqrt(err)]
else:
A = numpy.vstack([linearized_data.x / linearized_data.dy, 1.0 / linearized_data.dy]).T
(p, residuals, _, _) = numpy.linalg.lstsq(A, linearized_data.y / linearized_data.dy)

# Get the covariance matrix, defined as inv_cov = a_transposed * a
err = numpy.zeros(2)
try:
inv_cov = numpy.dot(A.transpose(), A)
cov = numpy.linalg.pinv(inv_cov)
err_matrix = math.fabs(residuals) * cov
err = [math.sqrt(err_matrix[0][0]), math.sqrt(err_matrix[1][1])]
except:
err = [-1.0, -1.0]

return p, err

[docs]class InvariantCalculator(object):
"""
Compute invariant if data is given.
Can provide volume fraction and surface area if the user provides
Porod constant  and contrast values.

:precondition:  the user must send a data of type DataLoader.Data1D
the user provide background and scale values.

:note: Some computations depends on each others.
"""
def __init__(self, data, background=0, scale=1):
"""
Initialize variables.

:param data: data must be of type DataLoader.Data1D
:param background: Background value. The data will be corrected
before processing
:param scale: Scaling factor for I(q). The data will be corrected
before processing
"""
# Background and scale should be private data member if the only way to
# change them are by instantiating a new object.
self._background = background
self._scale = scale
# slit height for smeared data
self._smeared = None
# The data should be private
self._data = self._get_data(data)
# get the dxl if the data is smeared: This is done only once on init.
if self._data.dxl != None and self._data.dxl.all() > 0:
# assumes constant dxl
self._smeared = self._data.dxl[0]

# Since there are multiple variants of Q*, you should force the
# user to use the get method and keep Q* a private data member
self._qstar = None

# You should keep the error on Q* so you can reuse it without
# recomputing the whole thing.
self._qstar_err = 0

# Extrapolation parameters
self._low_extrapolation_npts = 4
self._low_extrapolation_function = Guinier()
self._low_extrapolation_power = None
self._low_extrapolation_power_fitted = None

self._high_extrapolation_npts = 4
self._high_extrapolation_function = PowerLaw()
self._high_extrapolation_power = None
self._high_extrapolation_power_fitted = None

# Extrapolation range
self._low_q_limit = Q_MINIMUM

def _get_data(self, data):
"""
:note: this function must be call before computing any type
of invariant

:return: new data = self._scale *data - self._background
"""
#Process only data that inherited from DataLoader.Data_info.Data1D
raise ValueError, "Data must be of type DataLoader.Data1D"
#from copy import deepcopy
new_data = (self._scale * data) - self._background

# Check that the vector lengths are equal
assert len(new_data.x) == len(new_data.y)

# Verify that the errors are set correctly
if new_data.dy is None or len(new_data.x) != len(new_data.dy) or \
(min(new_data.dy) == 0 and max(new_data.dy) == 0):
new_data.dy = numpy.ones(len(new_data.x))
return  new_data

def _fit(self, model, qmin=Q_MINIMUM, qmax=Q_MAXIMUM, power=None):
"""
fit data with function using
data = self._get_data()
fx = Functor(data , function)
y = data.y
slope, constant = linalg.lstsq(y,fx)

:param qmin: data first q value to consider during the fit
:param qmax: data last q value to consider during the fit
:param power : power value to consider for power-law
:param function: the function to use during the fit

:return a: the scale of the function
:return b: the other parameter of the function for guinier will be radius
for power_law will be the power value
"""
extrapolator = Extrapolator(data=self._data, model=model)
p, dp = extrapolator.fit(power=power, qmin=qmin, qmax=qmax)

return model.extract_model_parameters(constant=p[1], slope=p[0],
dconstant=dp[1], dslope=dp[0])

def _get_qstar(self, data):
"""
Compute invariant for pinhole data.
This invariant is given by: ::

q_star = x0**2 *y0 *dx0 +x1**2 *y1 *dx1
+ ..+ xn**2 *yn *dxn    for non smeared data

q_star = dxl0 *x0 *y0 *dx0 +dxl1 *x1 *y1 *dx1
+ ..+ dlxn *xn *yn *dxn    for smeared data

where n >= len(data.x)-1
dxl = slit height dQl
dxi = 1/2*(xi+1 - xi) + (xi - xi-1)
dx0 = (x1 - x0)/2
dxn = (xn - xn-1)/2

:param data: the data to use to compute invariant.

:return q_star: invariant value for pinhole data. q_star > 0
"""
if len(data.x) <= 1 or len(data.y) <= 1 or len(data.x) != len(data.y):
msg = "Length x and y must be equal"
msg += " and greater than 1; got x=%s, y=%s" % (len(data.x), len(data.y))
raise ValueError, msg
else:
# Take care of smeared data
if self._smeared is None:
gx = data.x * data.x
# assumes that len(x) == len(dxl).
else:
gx = data.dxl * data.x

n = len(data.x) - 1
#compute the first delta q
dx0 = (data.x[1] - data.x[0]) / 2
#compute the last delta q
dxn = (data.x[n] - data.x[n - 1]) / 2
total = 0
total += gx[0] * data.y[0] * dx0
total += gx[n] * data.y[n] * dxn

if len(data.x) == 2:
else:
#iterate between for element different
#from the first and the last
for i in xrange(1, n - 1):
dxi = (data.x[i + 1] - data.x[i - 1]) / 2
total += gx[i] * data.y[i] * dxi

def _get_qstar_uncertainty(self, data):
"""
Compute invariant uncertainty with with pinhole data.
This uncertainty is given as follow: ::

dq_star = math.sqrt[(x0**2*(dy0)*dx0)**2 +
(x1**2 *(dy1)*dx1)**2 + ..+ (xn**2 *(dyn)*dxn)**2 ]
where n >= len(data.x)-1
dxi = 1/2*(xi+1 - xi) + (xi - xi-1)
dx0 = (x1 - x0)/2
dxn = (xn - xn-1)/2
dyn: error on dy

:param data:
:note: if data doesn't contain dy assume dy= math.sqrt(data.y)
"""
if len(data.x) <= 1 or len(data.y) <= 1 or \
len(data.x) != len(data.y) or \
(data.dy is not None and (len(data.dy) != len(data.y))):
msg = "Length of data.x and data.y must be equal"
msg += " and greater than 1; got x=%s, y=%s" % (len(data.x), len(data.y))
raise ValueError, msg
else:
#Create error for data without dy error
if data.dy is None:
dy = math.sqrt(data.y)
else:
dy = data.dy
# Take care of smeared data
if self._smeared is None:
gx = data.x * data.x
# assumes that len(x) == len(dxl).
else:
gx = data.dxl * data.x

n = len(data.x) - 1
#compute the first delta
dx0 = (data.x[1] - data.x[0]) / 2
#compute the last delta
dxn = (data.x[n] - data.x[n - 1]) / 2
total = 0
total += (gx[0] * dy[0] * dx0) ** 2
total += (gx[n] * dy[n] * dxn) ** 2
if len(data.x) == 2:
return math.sqrt(total)
else:
#iterate between for element different
#from the first and the last
for i in xrange(1, n - 1):
dxi = (data.x[i + 1] - data.x[i - 1]) / 2
total += (gx[i] * dy[i] * dxi) ** 2
return math.sqrt(total)

def _get_extrapolated_data(self, model, npts=INTEGRATION_NSTEPS,
q_start=Q_MINIMUM, q_end=Q_MAXIMUM):
"""
:return: extrapolate data create from data
"""
#create new Data1D to compute the invariant
q = numpy.linspace(start=q_start,
stop=q_end,
num=npts,
endpoint=True)
iq = model.evaluate_model(q)
diq = model.evaluate_model_errors(q)

if self._smeared != None:
result_data.dxl = self._smeared * numpy.ones(len(q))
return result_data

[docs]    def get_data(self):
"""
:return: self._data
"""
return self._data

[docs]    def get_extrapolation_power(self, range='high'):
"""
:return: the fitted power for power law function for a given
extrapolation range
"""
if range == 'low':
return self._low_extrapolation_power_fitted
return self._high_extrapolation_power_fitted

[docs]    def get_qstar_low(self):
"""
Compute the invariant for extrapolated data at low q range.

Implementation:
data = self._get_extra_data_low()
return self._get_qstar()

:return q_star: the invariant for data extrapolated at low q.
"""
# Data boundaries for fitting
qmin = self._data.x[0]
qmax = self._data.x[self._low_extrapolation_npts - 1]

# Extrapolate the low-Q data
p, _ = self._fit(model=self._low_extrapolation_function,
qmin=qmin,
qmax=qmax,
power=self._low_extrapolation_power)
self._low_extrapolation_power_fitted = p[0]

# Distribution starting point
self._low_q_limit = Q_MINIMUM
if Q_MINIMUM >= qmin:
self._low_q_limit = qmin / 10

data = self._get_extrapolated_data(\
model=self._low_extrapolation_function,
npts=INTEGRATION_NSTEPS,
q_start=self._low_q_limit, q_end=qmin)

# Systematic error
# If we have smearing, the shape of the I(q) distribution at low Q will
# may not be a Guinier or simple power law. The following is
# a conservative estimation for the systematic error.
err = qmin * qmin * math.fabs((qmin - self._low_q_limit) * \
(data.y[0] - data.y[INTEGRATION_NSTEPS - 1]))
return self._get_qstar(data), self._get_qstar_uncertainty(data) + err

[docs]    def get_qstar_high(self):
"""
Compute the invariant for extrapolated data at high q range.

Implementation:
data = self._get_extra_data_high()
return self._get_qstar()

:return q_star: the invariant for data extrapolated at high q.
"""
# Data boundaries for fitting
x_len = len(self._data.x) - 1
qmin = self._data.x[x_len - (self._high_extrapolation_npts - 1)]
qmax = self._data.x[x_len]

# fit the data with a model to get the appropriate parameters
p, _ = self._fit(model=self._high_extrapolation_function,
qmin=qmin,
qmax=qmax,
power=self._high_extrapolation_power)
self._high_extrapolation_power_fitted = p[0]

#create new Data1D to compute the invariant
data = self._get_extrapolated_data(\
model=self._high_extrapolation_function,
npts=INTEGRATION_NSTEPS,
q_start=qmax, q_end=Q_MAXIMUM)

return self._get_qstar(data), self._get_qstar_uncertainty(data)

[docs]    def get_extra_data_low(self, npts_in=None, q_start=None, npts=20):
"""
Returns the extrapolated data used for the loew-Q invariant calculation.
By default, the distribution will cover the data points used for the
extrapolation. The number of overlap points is a parameter (npts_in).
By default, the maximum q-value of the distribution will be
the minimum q-value used when extrapolating for the purpose of the
invariant calculation.

:param npts_in: number of data points for which
the extrapolated data overlap
:param q_start: is the minimum value to uses for extrapolated data
:param npts: the number of points in the extrapolated distribution

"""
# Get extrapolation range
if q_start is None:
q_start = self._low_q_limit

if npts_in is None:
npts_in = self._low_extrapolation_npts
q_end = self._data.x[max(0, npts_in - 1)]

if q_start >= q_end:
return numpy.zeros(0), numpy.zeros(0)

return self._get_extrapolated_data(\
model=self._low_extrapolation_function,
npts=npts,
q_start=q_start, q_end=q_end)

[docs]    def get_extra_data_high(self, npts_in=None, q_end=Q_MAXIMUM, npts=20):
"""
Returns the extrapolated data used for the high-Q invariant calculation.
By default, the distribution will cover the data points used for the
extrapolation. The number of overlap points is a parameter (npts_in).
By default, the maximum q-value of the distribution will be Q_MAXIMUM,
the maximum q-value used when extrapolating for the purpose of the
invariant calculation.

:param npts_in: number of data points for which the
extrapolated data overlap
:param q_end: is the maximum value to uses for extrapolated data
:param npts: the number of points in the extrapolated distribution
"""
# Get extrapolation range
if npts_in is None:
npts_in = self._high_extrapolation_npts
_npts = len(self._data.x)
q_start = self._data.x[min(_npts, _npts - npts_in)]

if q_start >= q_end:
return numpy.zeros(0), numpy.zeros(0)

return self._get_extrapolated_data(\
model=self._high_extrapolation_function,
npts=npts,
q_start=q_start, q_end=q_end)

[docs]    def set_extrapolation(self, range, npts=4, function=None, power=None):
"""
Set the extrapolation parameters for the high or low Q-range.
Note that this does not turn extrapolation on or off.

:param range: a keyword set the type of extrapolation . type string
:param npts: the numbers of q points of data to consider
for extrapolation
:param function: a keyword to select the function to use
for extrapolation.
of type string.
:param power: an power to apply power_low function

"""
range = range.lower()
if range not in ['high', 'low']:
raise ValueError, "Extrapolation range should be 'high' or 'low'"
function = function.lower()
if function not in ['power_law', 'guinier']:
msg = "Extrapolation function should be 'guinier' or 'power_law'"
raise ValueError, msg

if range == 'high':
if function != 'power_law':
msg = "Extrapolation only allows a power law at high Q"
raise ValueError, msg
self._high_extrapolation_npts = npts
self._high_extrapolation_power = power
self._high_extrapolation_power_fitted = power
else:
if function == 'power_law':
self._low_extrapolation_function = PowerLaw()
else:
self._low_extrapolation_function = Guinier()
self._low_extrapolation_npts = npts
self._low_extrapolation_power = power
self._low_extrapolation_power_fitted = power

[docs]    def get_qstar(self, extrapolation=None):
"""
Compute the invariant of the local copy of data.

:param extrapolation: string to apply optional extrapolation

:return q_star: invariant of the data within data's q range

:warning: When using setting data to Data1D ,
the user is responsible of
checking that the scale and the background are
properly apply to the data

"""
self._qstar = self._get_qstar(self._data)
self._qstar_err = self._get_qstar_uncertainty(self._data)

if extrapolation is None:
return self._qstar

# Compute invariant plus invariant of extrapolated data
extrapolation = extrapolation.lower()
if extrapolation == "low":
qs_low, dqs_low = self.get_qstar_low()
qs_hi, dqs_hi = 0, 0

elif extrapolation == "high":
qs_low, dqs_low = 0, 0
qs_hi, dqs_hi = self.get_qstar_high()

elif extrapolation == "both":
qs_low, dqs_low = self.get_qstar_low()
qs_hi, dqs_hi = self.get_qstar_high()

self._qstar += qs_low + qs_hi
self._qstar_err = math.sqrt(self._qstar_err * self._qstar_err \
+ dqs_low * dqs_low + dqs_hi * dqs_hi)

return self._qstar

[docs]    def get_surface(self, contrast, porod_const, extrapolation=None):
"""
Compute the specific surface from the data.

Implementation::

V =  self.get_volume_fraction(contrast, extrapolation)

Compute the surface given by:
surface = (2*pi *V(1- V)*porod_const)/ q_star

:param contrast: contrast value to compute the volume
:param porod_const: Porod constant to compute the surface
:param extrapolation: string to apply optional extrapolation

:return: specific surface
"""
# Compute the volume
volume = self.get_volume_fraction(contrast, extrapolation)
return 2 * math.pi * volume * (1 - volume) * \
float(porod_const) / self._qstar

[docs]    def get_volume_fraction(self, contrast, extrapolation=None):
"""
Compute volume fraction is deduced as follow: ::

q_star = 2*(pi*contrast)**2* volume( 1- volume)
for k = 10^(-8)*q_star/(2*(pi*|contrast|)**2)
we get 2 values of volume:
with   1 - 4 * k >= 0
volume1 = (1- sqrt(1- 4*k))/2
volume2 = (1+ sqrt(1- 4*k))/2

q_star: the invariant value included extrapolation is applied
unit  1/A^(3)*1/cm
q_star = self.get_qstar()

the result returned will be 0 <= volume <= 1

:param contrast: contrast value provides by the user of type float.
contrast unit is 1/A^(2)= 10^(16)cm^(2)
:param extrapolation: string to apply optional extrapolation

:return: volume fraction

:note: volume fraction must have no unit
"""
if contrast <= 0:
raise ValueError, "The contrast parameter must be greater than zero"

# Make sure Q star is up to date
self.get_qstar(extrapolation)

if self._qstar <= 0:
msg = "Invalid invariant: Invariant Q* must be greater than zero"
raise RuntimeError, msg

# Compute intermediate constant
k = 1.e-8 * self._qstar / (2 * (math.pi * math.fabs(float(contrast))) ** 2)
# Check discriminant value
discrim = 1 - 4 * k

# Compute volume fraction
if discrim < 0:
msg = "Could not compute the volume fraction: negative discriminant"
raise RuntimeError, msg
elif discrim == 0:
return 1 / 2
else:
volume1 = 0.5 * (1 - math.sqrt(discrim))
volume2 = 0.5 * (1 + math.sqrt(discrim))

if 0 <= volume1 and volume1 <= 1:
return volume1
elif 0 <= volume2 and volume2 <= 1:
return volume2
msg = "Could not compute the volume fraction: inconsistent results"
raise RuntimeError, msg

[docs]    def get_qstar_with_error(self, extrapolation=None):
"""
Compute the invariant uncertainty.
This uncertainty computation depends on whether or not the data is
smeared.

:param extrapolation: string to apply optional extrapolation

:return: invariant, the invariant uncertainty
"""
self.get_qstar(extrapolation)
return self._qstar, self._qstar_err

[docs]    def get_volume_fraction_with_error(self, contrast, extrapolation=None):
"""
Compute uncertainty on volume value as well as the volume fraction
This uncertainty is given by the following equation: ::

dV = 0.5 * (4*k* dq_star) /(2* math.sqrt(1-k* q_star))

for k = 10^(-8)*q_star/(2*(pi*|contrast|)**2)

q_star: the invariant value including extrapolated value if existing
dq_star: the invariant uncertainty
dV: the volume uncertainty

The uncertainty will be set to -1 if it can't be computed.

:param contrast: contrast value
:param extrapolation: string to apply optional extrapolation

:return: V, dV = volume fraction, error on volume fraction
"""
volume = self.get_volume_fraction(contrast, extrapolation)

# Compute error
k = 1.e-8 * self._qstar / (2 * (math.pi * math.fabs(float(contrast))) ** 2)
# Check value inside the sqrt function
value = 1 - k * self._qstar
if (value) <= 0:
uncertainty = -1
# Compute uncertainty
uncertainty = math.fabs((0.5 * 4 * k * \
self._qstar_err) / (2 * math.sqrt(1 - k * self._qstar)))

return volume, uncertainty

[docs]    def get_surface_with_error(self, contrast, porod_const, extrapolation=None):
"""
Compute uncertainty of the surface value as well as the surface value.
The uncertainty is given as follow: ::

dS = porod_const *2*pi[( dV -2*V*dV)/q_star
+ dq_star(v-v**2)

q_star: the invariant value
dq_star: the invariant uncertainty
V: the volume fraction value
dV: the volume uncertainty

:param contrast: contrast value
:param porod_const: porod constant value
:param extrapolation: string to apply optional extrapolation

:return S, dS: the surface, with its uncertainty
"""
# We get the volume fraction, with error
#   get_volume_fraction_with_error calls get_volume_fraction
#   get_volume_fraction calls get_qstar
#   which computes Qstar and dQstar
v, dv = self.get_volume_fraction_with_error(contrast, extrapolation)

s = self.get_surface(contrast=contrast, porod_const=porod_const,
extrapolation=extrapolation)
ds = porod_const * 2 * math.pi * ((dv - 2 * v * dv) / self._qstar\
+ self._qstar_err * (v - v ** 2))

return s, ds