sas.sascalc.pr package

Submodules

sas.sascalc.pr.calc module

Converted invertor.c’s methods. Implements low level inversion functionality, with conditional Numba njit compilation.

sas.sascalc.pr.calc.dprdr(pars, d_max, r)

dP(r)/dr calculated from the expansion.

Parameters:

• pars – c-parameters.
• d_max – d_max.
• r – r-value.

Returns:

dP(r)/dr.

sas.sascalc.pr.calc.dprdr_calc(i, d_max, r)

sas.sascalc.pr.calc.int_pr(pars, d_max, nslice)

Integral of P(r).

Parameters:

• pars – c-parameters.
• d_max – d_max.
• nslice – nslice.

Returns:

Integral of P(r).

sas.sascalc.pr.calc.int_pr_square(pars, d_max, nslice)

Regularization term calculated from the expansion.

Parameters:

• pars – c-parameters.
• d_max – d_max.
• nslice – nslice.

Returns:

Regularization term calculated from the expansion.

sas.sascalc.pr.calc.iq(pars, d_max, q)

Scattering intensity calculated from the expansion.

Parameters:

• pars – c-parameters.
• d_max – d_max.
• q – q (vector).

Returns:

Scattering intensity from the expansion across all q.

sas.sascalc.pr.calc.iq_smeared(p, q, d_max, height, width, npts)

Scattering intensity calculated from the expansion, slit-smeared.

Parameters:

• p – c-parameters.
• q – q (vector).
• height – slit_height.
• width – slit_width.
• npts – npts.

Returns:

Scattering intensity from the expansion slit-smeared across all q.

sas.sascalc.pr.calc.njit(*args, **kw)

sas.sascalc.pr.calc.npeaks(pars, d_max, nslice)

Get the number of P(r) peaks.

Parameters:

• pars – c-parameters.
• d_max – d_max.
• nslice – nslice.

Returns:

Number of P(r) peaks.

sas.sascalc.pr.calc.ortho(d_max, n, r)

Orthogonal Functions: B(r) = 2r sin(pi*nr/d)

Parameters:

• d_max – d_max.
• n –

Returns:

B(r).

sas.sascalc.pr.calc.ortho_derived(d_max, n, r)

First derivative in of the orthogonal function dB(r)/dr.

Parameters:

• d_max – d_max.
• n –

Returns:

First derivative in dB(r)/dr.

sas.sascalc.pr.calc.ortho_transformed(q, d_max, n)

Fourier transform of the nth orthogonal function.

Parameters:

• q – q (vector).
• d_max – d_max.
• n –

Returns:

Fourier transform of nth orthogonal function across all q.

sas.sascalc.pr.calc.ortho_transformed_smeared(q, d_max, n, height, width, npts)

Slit-smeared Fourier transform of the nth orthogonal function. Smearing follows Lake, Acta Cryst. (1967) 23, 191.

Parameters:

• q – q (vector).
• d_max – d_max.
• n –
• height – slit_height.
• width – slit_width.
• npts – npts.

Returns:

Slit-smeared Fourier transform of nth orthogonal function across all q.

sas.sascalc.pr.calc.positive_errors(pars, err, d_max, nslice)

Get the fraction of the integral of P(r) over the whole range of r that is at least one sigma above 0.

Parameters:

• pars – c-parameters.
• err – error terms.
• d_max – d_max.
• nslice – nslice.

Returns:

The fraction of the integral of P(r) over the whole range of r that is at least one sigma above 0.

sas.sascalc.pr.calc.positive_integral(pars, d_max, nslice)

Get the fraction of the integral of P(r) over the whole range of r that is above 0. A valid P(r) is defined as being positive for all r.

Parameters:

• pars – c-parameters.
• d_max – d_max.
• nslice – nslice.

Returns:

The fraction of the integral of P(r) over the whole range of r that is above 0.

sas.sascalc.pr.calc.pr(pars, d_max, r)

P(r) calculated from the expansion

Parameters:

• pars – c-parameters.
• d_max – d_max.
• r – r-value to evaluate P(r).

Returns:

P(r).

sas.sascalc.pr.calc.pr_err(pars, err, d_max, r)

P(r) calculated from the expansion, with errors.

Parameters:

• pars – c-parameters.
• err – err.
• r – r-value.

Returns:

[P(r), dP(r)].

sas.sascalc.pr.calc.reg_term(pars, d_max, nslice)

Regularization term calculated from the expansion.

Parameters:

• pars – c-parameters.
• d_max – d_max.
• nslice – nslice.

Returns:

Regularization term calculated from the expansion.

sas.sascalc.pr.calc.rg(pars, d_max, nslice)

R_g radius of gyration calculation

R_g**2 = integral[r**2 * p(r) dr] / (2.0 * integral[p(r) dr])

Parameters:

• pars – c-parameters.
• d_max – d_max.
• nslice – nslice.

Returns:

R_g radius of gyration.

sas.sascalc.pr.distance_explorer module

Module to explore the P(r) inversion results for a range of D_max value. User picks a number of points and a range of distances, then get a series of outputs as a function of D_max over that range.

class sas.sascalc.pr.distance_explorer.DistExplorer(pr_state)

Bases: object

The explorer class

class sas.sascalc.pr.distance_explorer.Results

Bases: object

Class to hold the inversion output parameters as a function of D_max

sas.sascalc.pr.invertor module

Module to perform P(r) inversion. The module contains the Invertor class.

FIXME: The way the Invertor interacts with its C component should be cleaned up

class sas.sascalc.pr.invertor.Invertor

Bases: sas.sascalc.pr.p_invertor.Pinvertor

Invertor class to perform P(r) inversion

The problem is solved by posing the problem as Ax = b, where x is the set of coefficients we are looking for.

Npts is the number of points.

In the following i refers to the ith base function coefficient. The matrix has its entries j in its first Npts rows set to

A[j][i] = (Fourier transformed base function for point j)

We then choose a number of r-points, n_r, to evaluate the second derivative of P(r) at. This is used as our regularization term. For a vector r of length n_r, the following n_r rows are set to

A[j+Npts][i] = (2nd derivative of P(r), d**2(P(r))/d(r)**2,
evaluated at r[j])

The vector b has its first Npts entries set to

b[j] = (I(q) observed for point j)

The following n_r entries are set to zero.

The result is found by using scipy.linalg.basic.lstsq to invert the matrix and find the coefficients x.

Methods inherited from Cinvertor:

• get_peaks(pars): returns the number of P(r) peaks
• oscillations(pars): returns the oscillation parameters for the output P(r)
• get_positive(pars): returns the fraction of P(r) that is above zero
• get_pos_err(pars): returns the fraction of P(r) that is 1-sigma above zero

background = 0

chi2 = 0

clone()

Return a clone of this instance

cov = None

elapsed = 0

estimate_alpha(nfunc)

Returns a reasonable guess for the regularization constant alpha

Parameters:nfunc – number of terms to use in the expansion. Returns:alpha, message, elapsed

where alpha is the estimate for alpha, message is a message for the user, elapsed is the computation time

estimate_numterms(isquit_func=None)

Returns a reasonable guess for the number of terms

Parameters:isquit_func – reference to thread function to call to check whether the computation needs to be stopped. Returns:number of terms, alpha, message

from_file(path)

Load the state of the Invertor from a file, to be able to generate P(r) from a set of parameters.

Parameters:path – path of the file to load

info = {}

invert(nfunc=10, nr=20)

Perform inversion to P(r)

The problem is solved by posing the problem as Ax = b, where x is the set of coefficients we are looking for.

Npts is the number of points.

In the following i refers to the ith base function coefficient. The matrix has its entries j in its first Npts rows set to

A[i][j] = (Fourier transformed base function for point j)

We then choose a number of r-points, n_r, to evaluate the second derivative of P(r) at. This is used as our regularization term. For a vector r of length n_r, the following n_r rows are set to

A[i+Npts][j] = (2nd derivative of P(r), d**2(P(r))/d(r)**2, evaluated at r[j])

The vector b has its first Npts entries set to

b[j] = (I(q) observed for point j)

The following n_r entries are set to zero.

The result is found by using scipy.linalg.basic.lstsq to invert the matrix and find the coefficients x.

Parameters:

• nfunc – number of base functions to use.
• nr – number of r points to evaluate the 2nd derivative at for the reg. term.

Returns:

c_out, c_cov - the coefficients with covariance matrix

invert_optimize(nfunc=10, nr=20)

Slower version of the P(r) inversion that uses scipy.optimize.leastsq.

This probably produce more reliable results, but is much slower. The minimization function is set to sum_i[ (I_obs(q_i) - I_theo(q_i))/err**2 ] + alpha * reg_term, where the reg_term is given by Svergun: it is the integral of the square of the first derivative of P(r), d(P(r))/dr, integrated over the full range of r.

Parameters:

• nfunc – number of base functions to use.
• nr – number of r points to evaluate the 2nd derivative at for the reg. term.

Returns:

c_out, c_cov - the coefficients with covariance matrix

iq(out, q)

Function to call to evaluate the scattering intensity

Parameters:args – c-parameters, and q Returns:I(q)

lstsq(nfunc=5, nr=20)

The problem is solved by posing the problem as Ax = b, where x is the set of coefficients we are looking for.

Npts is the number of points.

In the following i refers to the ith base function coefficient. The matrix has its entries j in its first Npts rows set to

A[i][j] = (Fourier transformed base function for point j)

We then choose a number of r-points, n_r, to evaluate the second derivative of P(r) at. This is used as our regularization term. For a vector r of length n_r, the following n_r rows are set to

A[i+Npts][j] = (2nd derivative of P(r), d**2(P(r))/d(r)**2,
evaluated at r[j])

The vector b has its first Npts entries set to

b[j] = (I(q) observed for point j)

The following n_r entries are set to zero.

The result is found by using scipy.linalg.basic.lstsq to invert the matrix and find the coefficients x.

Parameters:

• nfunc – number of base functions to use.
• nr – number of r points to evaluate the 2nd derivative at for the reg. term.

If the result does not allow us to compute the covariance matrix, a matrix filled with zeros will be returned.

nfunc = 10

out = None

pr_err(c, c_cov, r)

Returns the value of P(r) for a given r, and base function coefficients, with error.

Parameters:

• c – base function coefficients
• c_cov – covariance matrice of the base function coefficients
• r – r-value to evaluate P(r) at

Returns:

P(r)

pr_fit(nfunc=5)

This is a direct fit to a given P(r). It assumes that the y data is set to some P(r) distribution that we are trying to reproduce with a set of base functions.

This method is provided as a test.

suggested_alpha = 0

to_file(path, npts=100)

Save the state to a file that will be readable by SliceView.

Parameters:

• path – path of the file to write
• npts – number of P(r) points to be written

sas.sascalc.pr.invertor.help()

Provide general online help text Future work: extend this function to allow topic selection

sas.sascalc.pr.num_term module

class sas.sascalc.pr.num_term.NTermEstimator(invertor)

Bases: object

compare_err()

get0_out()

is_odd(n)

ls_osc()

median_osc()

num_terms(isquit_func=None)

sort_osc()

sas.sascalc.pr.num_term.load(path)

sas.sascalc.pr.p_invertor module

Python implementation of the P(r) inversion Pinvertor is the base class for the Invertor class and provides the underlying computations.

class sas.sascalc.pr.p_invertor.Pinvertor

Bases: object

accept_q(q)

Check whether a q-value is within acceptable limits.

Returns:1 if accepted, 0 if rejected.

basefunc_ft(d_max, n, q)

Returns the value of the nth Fourier transformed base function.

Parameters:

• d_max – d_max.
• n –
• q – q, scalar or vector.

Returns:

nth Fourier transformed base function, evaluated at q.

check_for_zero(x)

err = array([], dtype=float64)

get_alpha()

Gets the alpha parameter.

Returns:alpha.

get_dmax()

Gets the maximum distance.

Returns:d_max.

get_err(data)

Function to get the err data.

Parameters:data – Array of doubles to place err into. Returns:npoints - number of entries found

get_est_bck()

Gets background flag.

Returns:est_bck.

get_iq_smeared(pars, q)

Function to call to evaluate the scattering intensity. The scattering intensity is slit-smeared.

Parameters:

• pars – c-parameters
• q – q, scalar or vector.

Returns:

I(q), either scalar or vector depending on q.

get_nerr()

Gets the number of error points.

Returns:nerr.

get_nx()

Gets the number of x points.

Returns:npoints.

get_ny()

Gets the number of y points.

Returns:ny.

get_peaks(pars)

Returns the number of peaks in the output P(r) distribution for the given set of coefficients.

Parameters:pars – c-parameters. Returns:number of P(r) peaks.

get_pos_err(pars, pars_err)

Returns the fraction of P(r) that is 1 standard deviation above zero over the full range of r for the given set of coefficients.

Parameters:

• pars – c-parameters.
• pars_err – pars_err.

Returns:

fraction of P(r) that is positive.

get_positive(pars)

Returns the fraction of P(r) that is positive over the full range of r for the given set of coefficients.

Parameters:pars – c-parameters. Returns:fraction of P(r) that is positive.

get_pr_err(pars, pars_err, r)

Function to call to evaluate P(r) with errors.

Parameters:

• pars – c-parameters.
• pars_err – pars_err.
• r – r-value.

Returns:

(P(r), dP(r))

get_qmax()

Gets the maximum q.

Returns:q_max.

get_qmin()

Gets the minimum q.

Returns:q_min.

get_slit_height()

Gets the slit height.

Returns:slit_height.

get_slit_width()

Gets the slit width.

Returns:slit_width.

get_x(data)

Function to get the x data.

Parameters:data – Array to place x into Returns:npoints - Number of entries found.

get_y(data)

Function to get the y data.

Parameters:data – Array of doubles to place y into. Returns:npoints - Number of entries found.

iq(pars, q)

Function to call to evaluate the scattering intensity.

Parameters:

• pars – c-parameters
• q – q, scalar or vector.

Returns:

I(q)

iq0(pars)

Returns the value of I(q=0).

Parameters:pars – c-parameters. Returns:I(q=0)

is_valid()

Check the validity of the stored data.

Returns:Returns the number of points if it’s all good, -1 otherwise.

nerr = 0

npoints = 0

ny = 0

oscillations(pars)

Returns the value of the oscillation figure of merit for the given set of coefficients. For a sphere, the oscillation figure of merit is 1.1.

Parameters:pars – c-parameters. Returns:oscillation figure of merit.

pr(pars, r)

Function to call to evaluate P(r).

Parameters:

• pars – c-parameters.
• r – r-value to evaluate P(r) at.

Returns:

P(r)

pr_residuals(pars)

Function to call to evaluate the residuals for P(r) minimization (for testing purposes).

Parameters:pars – input parameters. Returns:residuals - list of residuals.

residuals(pars)

Function to call to evaluate the residuals for P(r) inversion.

Parameters:pars – input parameters. Returns:residuals - list of residuals.

rg(pars)

Returns the value of the radius of gyration Rg.

Parameters:pars – c-parameters. Returns:Rg.

set_alpha(alpha)

Sets the alpha parameter.

Parameters:alpha – float to set alpha to. Returns:alpha.

set_dmax(d_max)

Sets the maximum distance.

Parameters:d_max – float to set d_max to. Returns:d_max

set_err(data)

Function to set the err data.

Parameters:data – Array of doubles to set err to. Returns:nerr - Number of entries found.

set_est_bck(est_bck)

Sets background flag.

Parameters:est_bck – int to set est_bck to. Returns:est_bck.

set_qmax(max_q)

Sets the maximum q.

Parameters:max_q – float to set q_max to. Returns:q_max.

set_qmin(min_q)

Sets the minimum q.

Parameters:min_q – float to set q_min to. Returns:q_min.

set_slit_height(slit_height)

Sets the slit height in units of q [A-1].

Parameters:slit_height – float to set slit-height to. Returns:slit_height.

set_slit_width(slit_width)

Sets the slit width in units of q [A-1].

Parameters:slit_width – float to set slit_width to. Returns:slit_width.

set_x(data)

Function to set the x data.

Parameters:data – Array of doubles to set x to. Returns:npoints - Number of entries found, the size of x.

set_y(data)

Function to set the y data.

Parameters:data – Array of doubles to set y to. Returns:ny - Number of entries found.

slit_height = 0.0

slit_width = 0.0

x = array([], dtype=float64)

y = array([], dtype=float64)

Module contents

P(r) inversion for SAS