sas.sascalc.invariant package

Submodules

sas.sascalc.invariant.invariant module

This module implements invariant and its related computations.

author:

Gervaise B. Alina/UTK

author:

Mathieu Doucet/UTK

author:

Jae Cho/UTK

author:

Paul Butler/NIST/UD/UTK – refactor in 2020

class sas.sascalc.invariant.invariant.Extrapolator(data, model=None)

Bases: object

Extrapolate I(q) distribution using a given model

__dict__ = mappingproxy({'__module__': 'sas.sascalc.invariant.invariant', '__doc__': '\n    Extrapolate I(q) distribution using a given model\n    ', '__init__': <function Extrapolator.__init__>, 'fit': <function Extrapolator.fit>, '__dict__': <attribute '__dict__' of 'Extrapolator' objects>, '__weakref__': <attribute '__weakref__' of 'Extrapolator' objects>, '__annotations__': {}})

__doc__ = '\n    Extrapolate I(q) distribution using a given model\n    '

__init__(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.

Parameters:

• data – data containing x and y such as y = ax + b

• model – optional Transform object

__module__ = 'sas.sascalc.invariant.invariant'

__weakref__

list of weak references to the object (if defined)

fit(power=None, qmin=None, qmax=None)

Fit data for \(y = ax + b\) return \(a\) and \(b\)

Parameters:

• power – a fixed, otherwise None

• qmin – Minimum Q-value

• qmax – Maximum Q-value

class sas.sascalc.invariant.invariant.Guinier(scale=1, radius=60)

Bases: Transform

class of type Transform that performs operations related to guinier function

__doc__ = '\n    class of type Transform that performs operations related to guinier\n    function\n    '

__init__(scale=1, radius=60)

__module__ = 'sas.sascalc.invariant.invariant'

_guinier(x)

Retrieve the guinier function after apply an inverse guinier function to x Compute \(F(x) = s * \exp\left(-(r x)^{2/3}\right)\).

Parameters:

x – a vector of q values

Also uses:

• self.scale: \(s\), the scale value

• self.radius: \(r\), the guinier radius value

Returns:

F(x)

evaluate_model(x)

return calculated I(q) for the model

Calculates the Guinier expression \(F(x)= s * \exp\left(-(r x)^{2/3}\right)\)

evaluate_model_errors(x)

Returns the error on I(q) for the given array of q-values

Parameters:

x – array of q-values

extract_model_parameters(constant, slope, dconstant=0, dslope=0)

assign new value to the scale and the radius

linearize_q_value(value)

Transform the input q-value for linearization

Parameters:

value – q-value

Returns:

q*q

class sas.sascalc.invariant.invariant.InvariantCalculator(data, background=0, scale=1)

Bases: 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.

__dict__ = mappingproxy({'__module__': 'sas.sascalc.invariant.invariant', '__doc__': '\n    Compute invariant if data is given.\n    Can provide volume fraction and surface area if the user provides\n    Porod constant  and contrast values.\n\n    :precondition:  the user must send a data of type DataLoader.Data1D\n                    the user provide background and scale values.\n\n    :note: Some computations depends on each others.\n    ', '__init__': <function InvariantCalculator.__init__>, 'background': <property object>, 'scale': <property object>, 'set_data': <function InvariantCalculator.set_data>, '_get_data': <function InvariantCalculator._get_data>, '_fit': <function InvariantCalculator._fit>, '_get_qstar': <function InvariantCalculator._get_qstar>, '_get_qstar_uncertainty': <function InvariantCalculator._get_qstar_uncertainty>, '_get_extrapolated_data': <function InvariantCalculator._get_extrapolated_data>, 'get_data': <function InvariantCalculator.get_data>, 'get_extrapolation_power': <function InvariantCalculator.get_extrapolation_power>, 'get_qstar_low': <function InvariantCalculator.get_qstar_low>, 'get_qstar_high': <function InvariantCalculator.get_qstar_high>, 'get_extra_data_low': <function InvariantCalculator.get_extra_data_low>, 'get_extra_data_high': <function InvariantCalculator.get_extra_data_high>, 'set_extrapolation': <function InvariantCalculator.set_extrapolation>, 'get_qstar': <function InvariantCalculator.get_qstar>, 'get_surface': <function InvariantCalculator.get_surface>, 'get_volume_fraction': <function InvariantCalculator.get_volume_fraction>, 'get_qstar_with_error': <function InvariantCalculator.get_qstar_with_error>, 'get_volume_fraction_with_error': <function InvariantCalculator.get_volume_fraction_with_error>, 'get_surface_with_error': <function InvariantCalculator.get_surface_with_error>, '__dict__': <attribute '__dict__' of 'InvariantCalculator' objects>, '__weakref__': <attribute '__weakref__' of 'InvariantCalculator' objects>, '__annotations__': {}})

__doc__ = '\n    Compute invariant if data is given.\n    Can provide volume fraction and surface area if the user provides\n    Porod constant  and contrast values.\n\n    :precondition:  the user must send a data of type DataLoader.Data1D\n                    the user provide background and scale values.\n\n    :note: Some computations depends on each others.\n    '

__init__(data, background=0, scale=1)

Initialize variables.

Parameters:

• data – data must be of type DataLoader.Data1D

• background – Background value. The data will be corrected before processing

• scale – Scaling factor for I(q). The data will be corrected before processing

__module__ = 'sas.sascalc.invariant.invariant'

__weakref__

list of weak references to the object (if defined)

_fit(model, qmin=1e-05, qmax=10, power=None)

fit data with function using data = self._get_data() fx = Functor(data , function) y = data.y slope, constant = linalg.lstsq(y,fx)

Parameters:

• qmin – data first q value to consider during the fit

• 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

_get_data(data)

Note:

this function must be call before computing any type of invariant

Returns:

new data = self._scale *data - self._background

_get_extrapolated_data(model, npts=1000, q_start=1e-05, q_end=10)

Returns:

extrapolate data create from data

_get_qstar(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

Parameters:

data – the data to use to compute invariant.

Return q_star:

invariant value for pinhole data. q_star > 0

_get_qstar_uncertainty(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

Parameters:

data –

Note:

if data doesn’t contain dy return “None”

property background

get_data()

Returns:

self._data

get_extra_data_high(npts_in=None, q_end=10, 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.

Parameters:

• npts_in – number of data points for which the extrapolated data overlap

• q_end – is the maximum value to uses for extrapolated data

• npts – the number of points in the extrapolated distribution

get_extra_data_low(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.

Parameters:

• npts_in – number of data points for which the extrapolated data overlap

• q_start – is the minimum value to uses for extrapolated data

• npts – the number of points in the extrapolated distribution

get_extrapolation_power(range='high')

Returns:

the fitted power for power law function for a given extrapolation range

get_qstar(extrapolation=None)

Compute the invariant of the local copy of data.

Parameters:

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

get_qstar_high(high_q_limit=None)

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.

get_qstar_low(low_q_limit=None)

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.

get_qstar_with_error(extrapolation=None)

Compute the invariant uncertainty. This uncertainty computation depends on whether or not the data is smeared.

Parameters:

extrapolation – string to apply optional extrapolation

Returns:

invariant, the invariant uncertainty

get_surface(contrast, porod_const, extrapolation=None)

Compute the specific surface from the data.

Historically, Sv was computed with the invariant and the Porod constant so as not to have to know the contrast in order to get the Sv as:

surface = (pi * V * (1- V) * porod_const) / q_star

However, that turns out to be a pointless exercise since it also requires a knowledge of the volume fractions and from the volume fraction and the invariant the contrast can be calculated as:

contrast**2 = q_star / (2 * pi**2 * V * (1- V))

Thus either way, mathematically it is always identical to computing with only the contrast and the Porod Constant. up to and including SasView versions 4.2.2 and 5.0.1 the implementation used the traditional circular approach.

Implementation:

Given the above, as of SasView 4.3 and 5.0.2 we compute Sv simply
from the Porod Constant and the contrast between the two phases as:

surface = porod_const / (2 * pi contrast**2)

Parameters:

• contrast – contrast between the two phases

• porod_const – Porod constant

• extrapolation – string to apply optional extrapolation. This will only be needed if and when the contrast term is calculated from the invariant.

Returns:

specific surface

get_surface_with_error(contrast, porod_const, extrapolation=None)

As of SasView 4.3 and 5.0.3, the specific surface is computed directly from the contrast and porod_constant wich are currently user inputs with no option for any uncertainty so no uncertainty can be calculated. However we include the uncertainty computation for future use if and when these values get an uncertainty. This is given as:

ds = sqrt[(s'_cp)**2 * dcp**2 + (s'_contrast)**2 * dcontrast**2]

where s’_x is the partial derivative of S with respect to x

which gives (this should be checked before using in anger):

ds = sqrt((dporod_const**2 * contrast**2 + 4 * (porod_const *
              dcontrast)**2) / (4 * pi**2 * contrast**6))

We also assume some users will never enter a value for uncertainty so allow for None even when it is an option.

Parameters:

• contrast – contrast value eventually with the error

• porod_const – porod constant value eventually with the error

• extrapolation – string to apply optional extrapolation. This will only be needed if and when the contrast term is calculated from the invariant.

Return s, ds:

the surface, with its uncertainty

get_volume_fraction(contrast, extrapolation=None)

Compute volume fraction is deduced as follows:

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

Parameters:

• contrast – contrast value provides by the user of type float. contrast unit is 1/A^(2)= 10^(16)cm^(2)

• extrapolation – string to apply optional extrapolation

Returns:

volume fraction

Note:

volume fraction must have no unit

get_volume_fraction_with_error(contrast, extrapolation=None)

Compute uncertainty on volume value as well as the volume fraction This uncertainty is given by the following equation:

sigV = dV/dq_star * sigq_star

so that:

sigV = (k * sigq_star) /(q_star * math.sqrt(1 - 4 * k))

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

Notes:

• 10^(-8) converts from cm^-1 to A^-1

• q_star: the invariant, in cm^-1A^-3, including extrapolated values if they have been requested

• dq_star: the invariant uncertainty

• dV: the volume uncertainty

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

Parameters:

• contrast – contrast value

• extrapolation – string to apply optional extrapolation

Returns:

V, dV = volume fraction, error on volume fraction

property scale

set_data(data)

set_extrapolation(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.

Parameters:

• range – a keyword set the type of extrapolation . type string

• npts – the numbers of q points of data to consider for extrapolation

• function – a keyword to select the function to use for extrapolation. of type string.

• power – an power to apply power_low function

class sas.sascalc.invariant.invariant.PowerLaw(scale=1, power=4)

Bases: Transform

class of type transform that perform operation related to power_law function

__doc__ = '\n    class of type transform that perform operation related to power_law\n    function\n    '

__init__(scale=1, power=4)

__module__ = 'sas.sascalc.invariant.invariant'

_power_law(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

Parameters:

x – array

Returns:

F(x)

evaluate_model(x)

given a scale and a radius transform x, y using a power_law function

evaluate_model_errors(x)

Returns the error on I(q) for the given array of q-values :param x: array of q-values

extract_model_parameters(constant, slope, dconstant=0, dslope=0)

Assign new value to the scale and the power

linearize_q_value(value)

Transform the input q-value for linearization

Parameters:

value – q-value

Returns:

\(\log(q)\)

class sas.sascalc.invariant.invariant.Transform

Bases: object

Define interface that need to compute a function or an inverse function given some x, y

__dict__ = mappingproxy({'__module__': 'sas.sascalc.invariant.invariant', '__doc__': '\n    Define interface that need to compute a function or an inverse\n    function given some x, y\n    ', 'linearize_data': <function Transform.linearize_data>, 'get_allowed_bins': <function Transform.get_allowed_bins>, 'linearize_q_value': <function Transform.linearize_q_value>, 'extract_model_parameters': <function Transform.extract_model_parameters>, 'evaluate_model': <function Transform.evaluate_model>, 'evaluate_model_errors': <function Transform.evaluate_model_errors>, '__dict__': <attribute '__dict__' of 'Transform' objects>, '__weakref__': <attribute '__weakref__' of 'Transform' objects>, '__annotations__': {}})

__doc__ = '\n    Define interface that need to compute a function or an inverse\n    function given some x, y\n    '

__module__ = 'sas.sascalc.invariant.invariant'

__weakref__

list of weak references to the object (if defined)

evaluate_model(x)

Returns an array f(x) values where f is the Transform function.

evaluate_model_errors(x)

Returns an array of I(q) errors

extract_model_parameters(constant, slope, dconstant=0, dslope=0)

set private member

get_allowed_bins(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.

Parameters:

data – Data1D object

linearize_data(data)

Linearize data so that a linear fit can be performed. Filter out the data that can’t be transformed.

Parameters:

data – LoadData1D instance

linearize_q_value(value)

Transform the input q-value for linearization

sas.sascalc.invariant.invariant_mapper module

This module is a wrapper to a map function. It allows to loop through different invariant objects to call the same function

sas.sascalc.invariant.invariant_mapper.get_qstar(inv, extrapolation=None)

Get invariant value (Q*)

sas.sascalc.invariant.invariant_mapper.get_qstar_with_error(inv, extrapolation=None)

Get invariant value with uncertainty

sas.sascalc.invariant.invariant_mapper.get_surface(inv, contrast, porod_const, extrapolation=None)

Get surface with uncertainty

sas.sascalc.invariant.invariant_mapper.get_surface_with_error(inv, contrast, porod_const, extrapolation=None)

Get surface with uncertainty

sas.sascalc.invariant.invariant_mapper.get_volume_fraction(inv, contrast, extrapolation=None)

Get volume fraction

sas.sascalc.invariant.invariant_mapper.get_volume_fraction_with_error(inv, contrast, extrapolation=None)

Get volume fraction with uncertainty

Module contents