source: sasview/src/sas/sascalc/invariant/invariant.py @ 9cc1f49

# 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.
#See the license text in license.txt
#copyright 2010, University of Tennessee
######################################################################

"""
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 as np

from sas.sascalc.dataloader.data_info import Data1D as LoaderData1D

# 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

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

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

        :param data: LoadData1D instance

        """
        # 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 = np.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 = np.asarray(x_out)
        y_out = np.asarray(y_out)
        dy_out = np.asarray(dy_out)
        linear_data = LoaderData1D(x=x_out, y=y_out, dy=dy_out)

        return linear_data

    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)]

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

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

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

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

class Guinier(Transform):
    """
    class of type Transform that performs operations related to guinier
    function
    """
    def __init__(self, scale=1, radius=60):
        Transform.__init__(self)
        self.scale = scale
        self.radius = radius
        ## Uncertainty of scale parameter
        self.dscale = 0
        ## Unvertainty of radius parameter
        self.dradius = 0

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

        :param value: q-value

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

    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
        self.radius = math.sqrt(-3 * slope)
        # 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

        return [self.radius, self.scale], [self.dradius, self.dscale]

    def evaluate_model(self, x):
        """
        return F(x)= scale* e-((radius*x)**2/3)
        """
        return self._guinier(x)

    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 = np.array([self.dscale * math.exp(-((self.radius * q) ** 2 / 3)) \
                          for q in x])
        p2 = np.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 np.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
        :param radius: the guinier radius value

        :return: F(x)
        """
        # transform the radius of coming from the inverse guinier function to a
        # a radius of a guinier function
        if self.radius <= 0:
            msg = "Rg expected positive value, but got %s" % self.radius
            raise ValueError(msg)
        value = np.array([math.exp(-((self.radius * i) ** 2 / 3)) for i in x])
        return self.scale * value

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

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

        :param value: q-value

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

    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]

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

    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 = np.array([self.dscale * math.pow(q, -self.power) for q in x])
        p2 = np.array([self.scale * self.power * math.pow(q, -self.power - 1)\
                           * self.dpower for q in x])
        diq2 = p1 * p1 + p2 * p2
        return np.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 = np.array([math.pow(i, -self.power) for i in x])
        return self.scale * value

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)

    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 = np.zeros(len(self.data.x))

        # Uncertainty
        if type(self.data.dy) == np.ndarray and \
            len(self.data.dy) == len(self.data.x) and \
                np.all(self.data.dy > 0):
            sigma = self.data.dy
        else:
            sigma = np.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(\
                                            LoaderData1D(self.data.x[idx],
                                                         fx[idx],
                                                         dy=sigma[idx]))
        else:
            linearized_data = LoaderData1D(self.data.x[idx],
                                           fx[idx],
                                           dy=sigma[idx])

        ##power is given only for function = power_law
        if power is not None:
            sigma2 = linearized_data.dy * linearized_data.dy
            a = -(power)
            b = (np.sum(linearized_data.y / sigma2) \
                 - a * np.sum(linearized_data.x / sigma2)) / np.sum(1.0 / sigma2)


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

            err = math.fabs(residuals) / np.sum(1.0 / sigma2)
            return [a, b], [0, math.sqrt(err)]
        else:
            A = np.vstack([linearized_data.x / linearized_data.dy, 1.0 / linearized_data.dy]).T
            # CRUFT: numpy>=1.14.0 allows rcond=None for the following default
            rcond = np.finfo(float).eps * max(A.shape)
            p, residuals, _, _ = np.linalg.lstsq(A, linearized_data.y / linearized_data.dy,
                                                 rcond=rcond)

            # Get the covariance matrix, defined as inv_cov = a_transposed * a
            err = np.zeros(2)
            try:
                inv_cov = np.dot(A.transpose(), A)
                cov = np.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


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 is not 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
        """
        if not issubclass(data.__class__, LoaderData1D):
            #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 = np.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:
                return total
            else:
                #iterate between for element different
                #from the first and the last
                for i in range(1, n - 1):
                    dxi = (data.x[i + 1] - data.x[i - 1]) / 2
                    total += gx[i] * data.y[i] * dxi
                return total

    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 range(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 = np.linspace(start=q_start,
                           stop=q_end,
                           num=npts,
                           endpoint=True)
        iq = model.evaluate_model(q)
        diq = model.evaluate_model_errors(q)

        result_data = LoaderData1D(x=q, y=iq, dy=diq)
        if self._smeared is not None:
            result_data.dxl = self._smeared * np.ones(len(q))
        return result_data

    def get_data(self):
        """
        :return: self._data
        """
        return self._data

    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

    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[int(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

    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[int(x_len - (self._high_extrapolation_npts - 1))]
        qmax = self._data.x[int(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)

    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, int(npts_in - 1))]

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

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

    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 = int(self._high_extrapolation_npts)
        _npts = len(self._data.x)
        q_start = self._data.x[min(_npts, int(_npts - npts_in))]

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

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

    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

    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

    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

    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)

    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

    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

    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