""" 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 TODO: - intro / documentation """ import math import numpy from 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 = 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) 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) self.radius = math.sqrt(-3 * slope) # Errors self.dscale = math.exp(constant)*dconstant 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 = 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): """ Retrive 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: raise ValueError("Rg expected positive value, but got %s"%self.radius) value = numpy.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 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 = 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: @param x: a vector of q values @param power: power of the function @param scale : scale factor value @param F(x) """ if self.power <= 0: raise ValueError("Power_law function expected positive power, but got %s"%self.power) if self.scale <= 0: raise ValueError("scale expected positive value, but got %s"%self.scale) value = numpy.array([ math.pow(i, -self.power) for i in x ]) return self.scale * value class Extrapolator: """ 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 = numpy.zeros(len(self.data.x)) # Uncertainty if type(self.data.dy)==numpy.ndarray and len(self.data.dy)==len(self.data.x): 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(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 != 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, rank, s) = 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 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 # The data should be private self._data = self._get_data(data) # 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 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" 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 where n >= len(data.x)-1 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: 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 sum = 0 sum += data.x[0] * data.x[0] * data.y[0] * dx0 sum += data.x[n] * data.x[n] * data.y[n] * dxn if len(data.x) == 2: return sum 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 sum += data.x[i] * data.x[i] * data.y[i] * dxi return sum 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(y) else: dy = data.dy 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 sum = 0 sum += (data.x[0] * data.x[0] * dy[0] * dx0)**2 sum += (data.x[n] * data.x[n] * dy[n] * dxn)**2 if len(data.x) == 2: return math.sqrt(sum) 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 sum += (data.x[i] * data.x[i] * dy[i] * dxi)**2 return math.sqrt(sum) 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) result_data = LoaderData1D(x=q, y=iq, dy=diq) return result_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[self._low_extrapolation_npts - 1] # Extrapolate the low-Q data p, dp = 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[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, dp = 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, 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) 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) 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']: raise ValueError, "Extrapolation function should be 'guinier' or 'power_law'" if range == 'high': if function != 'power_law': raise ValueError, "Extrapolation only allows a power law at high Q" 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: raise RuntimeError, "Invalid invariant: Invariant Q* must be greater than zero" # 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: raise RuntimeError, "Could not compute the volume fraction: negative discriminant" 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 raise RuntimeError, "Could not compute the volume fraction: inconsistent results" 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