Ubuntu Manpages

xrstools

XRStools Documentation Contents:

  • If you install from a Debian package you can skip the following points, install it , and then go directly to the code invocation section
  • Using Git, sources can be retrived with the following command

git clone https://gitlab.esrf.fr/ixstools/xrstools


  • for a local installation you can use

python setup.py install --prefix=~/packages


then to run the code you must do beforehand

export PYTHONPATH=/home/yourname/packages/lib/python2.7/site-packages
export PATH=/home/yourname/bin:$PATH


  • To install by creating a virtual environment

export MYPREFIX=/REPLACE/WITH/YOUR/TARGET
cd ${MYPREFIX}
python3 -m venv myenv
source ${MYPREFIX}/myenv/bin/activate
pip install pip --upgrade
pip install setuptools --upgrade
git clone https://gitlab.esrf.fr/ixstools/xrstools
cd ${MYPREFIX}/xrstools/
pip install -r requirements.txt
python setup.py install


  • Examples can be found in the nonregression directory.
  • For the roi selection tool you need a recent version of pymca installed on your sistem
  • Usage examples can be found in the non regression directory.



  • Some of the XRStools capabilities can be accessed by invocation of the XRS_swissknife script, providing as input a file in the yaml format.
  • To use the wizard the suggested instruction is

XRS_wizard  --wroot ~/software/XRStoolsSuperResolution/XRStools/WIZARD/methods/


the wroot argument tells where extra workflow can be found. In the above instruction we give workflows in the home source directory. This is practical because the wizard allows to edit them online and the modification will remain in the sources. or to access extra workflows that are not coming with the main disribution.

  • Depending on the details of your installation, you have the XRS_swissknife script sitting somewhere in a directory. Check the Installation page to see how to set PYTHONPATH and PATH in case of a local installation.

    The following documentation has been generated automatically from the comments found in the code.




embedded doc :

embedded doc :

embedded doc :

e_rois

  • A Tool to clean the spectra from Compton profile and absorption edge
  • A Tool to define ROI by using NNMF in spectral and spatial domain

XRStools.xrs_utilities Module

rotation around (1,0,0), pos sense

Returns the first order correction to filled 1s, 2s, and 2p Compton profiles.

Implementation after Holm and Ribberfors (citation ...).

  • pzprofile (np.array): Compton profile (e.g. tabulated from Biggs) to be corrected (2D matrix).
  • occupation (list): electron configuration.
  • q (float or np.array): momentum transfer in [a.u.].

asymmetry (np.array): asymmetries to be added to the raw profiles (normalized to the number of electrons on pz scale)


NNMFcost Returns cost and gradient for NNMF with constraints.


NNMFcost Returns cost and gradient for NNMF with constraints.

rotation around (0,0,1), pos sense

rotation around (0,1,0), neg sense

Rx Rotation matrix for vector rotations around the [1,0,0]-direction.
  • chi (float) : Angle of rotation.
  • degrees(bool) : Angle given in radians or degrees.

  • 3x3 rotation matrix.



Ry Rotation matrix for vector rotations around the [0,1,0]-direction.
  • phi (float) : Angle of rotation.
  • degrees(bool) : Angle given in radians or degrees.

  • 3x3 rotation matrix.



Rz Rotation matrix for vector rotations around the [0,0,1]-direction.
  • omega (float) : Angle of rotation.
  • degrees(bool) : Angle given in radians or degrees.

  • 3x3 rotation matrix.



TTsolver Solves the Takagi-Taupin equation for a bent crystal.

This function is based on a Matlab implementation by S. Huotari of M. Krisch's Fortran programs.

  • el_energy (float): Fixed nominal (working) energy in keV.
  • hkl (array): Reflection order vector, e.g. [6, 6, 0]
  • crystal (str): Crystal used (can be silicon 'Si' or 'Ge')
  • R (float): Crystal bending radius in m.
  • dev (np.array): Deviation parameter (in arc. seconds) for which the reflectivity curve should be calculated.
  • alpha (float): Crystal assymetry angle.

  • refl (np.array): Reflectivity curve.
  • e (np.array): Deviation from Bragg angle in meV.
  • dev (np.array): Deviation from Bragg angle in microrad.



absCorrection

Calculates absorption correction for given mu1 and mu2. Multiply the measured spectrum with this correction factor. This is a translation of Keijo Hamalainen's Matlab function (KH 30.05.96).

  • mu1 : np.array Absorption coefficient for the incident energy in [1/cm].
  • mu2 : np.array Absorption coefficient for the scattered energy in [1/cm].
  • alpha : float Incident angle relative to plane normal in [deg].
  • beta : float Exit angle relative to plane normal [deg].
  • samthick : float Sample thickness in [cm].
  • geometry : string, optional Key word for different sample geometries ('transmission', 'reflection', 'sphere'). If geometry is set to 'sphere', no angular dependence is assumed.

  • ac : np.array Absorption correction factor. Multiply this with your measured spectrum.



Calculates absorption correction for given mu1 and mu2. Multiply the measured spectrum with this correction factor.

This is a translation of Keijo Hamalainen's Matlab function (KH 30.05.96).

  • mu1 (np.array): absorption coefficient for the incident energy in [1/cm].
  • mu2 (np.array): absorption coefficient for the scattered energy in [1/cm].
  • alpha (float): incident angle relative to plane normal in [deg].
  • beta (float): exit angle relative to plane normal [deg] (for transmission geometry use beta < 0).
  • samthick (float): sample thickness in [cm].

  • ac (np.array): absorption correction factor. Multiply this with your measured spectrum.



# ADDCH Adds contents of given adjacent channels together # # [x2,y2] = addch(x,y,n,n0) # x = original x-scale (row or column vector) # y = original y-values (row or column vector) # n = number of channels to be summed up # n0 = offset for adding, default is 0 # x2 = new x-scale # y2 = new y-values # # KH 17.09.1990 # Modified 29.05.1995 to include offset

function [U,B,V]=bidiag_reduction(A) % [U B V]=bidiag_reduction(A) % Algorithm 6.5-1 in Golub & Van Loan, Matrix Computations % Johns Hopkins University Press % Finds an upper bidiagonal matrix B so that A=U*B*V' % with U,V orthogonal. A is an m x n matrix

bootstrapCNNMF Constrained non-negative matrix factorization with bootstrapping for error estimates.

bootstrapCNNMF Constrained non-negative matrix factorization with bootstrapping for error estimates.

% BRAGG Calculates Bragg angle for given reflection in RAD % output=bangle(hkl,e,xtal) % hkl can be a matrix i.e. hkl=[1,0,0 ; 1,1,1]; % e=energy in keV % xtal='Si', 'Ge', etc. (check dspace.m) or d0 (Si default) % % KH 28.09.93 %

Bases: object

Dynamical theory of diffraction.



Calculate polarization factor.




# BRAGGD Calculates Bragg angle for given reflection in deg # Call BRAGG.M # output=bangle(hkl,e,xtal) # hkl can be a matrix i.e. hkl=[1,0,0 ; 1,1,1]; # e=energy in keV # xtal='Si', 'Ge', etc. (check dspace.m) or d0 (Si default) # # KH 28.09.93


cixsUBfind




returns the Q0 given the detector position (tthv, tth) and th crystal orientation. This orientation is calculated considering :
this rotation is defined by psi which is a rotation around G






cixs_primo

cixs_secondo

cixs_terzo



cfactorizeOffDiaMatrix constrained version of factorizeOffDiaMatrix Returns main components from an off-diagonal Matrix (energy-loss x angular-departure).

constrained_nnmf Approximate singular value decomposition with constraints.

function [U, S, V] = constrained_svd(M,U_ini,S_ini,V_ini,U_up,max_iter=10000,verbose=False)


converts the old style EDF files (one image for horizontal and one image for vertical chambers) to the new style EDF (one single image).
converted.



Convolution with Gaussian x = x-vector y = y-vector fwhm = fulll width at half maximum of the gaussian with which y is convoluted

convtoprim converts diamond structure reciprocal lattice expressed in conventional lattice vectors to primitive one (Helsinki -> Palaiseau conversion) from S. Huotari

cshift Calculates Compton peak position.
  • w1 (float, array): Incident energy in [keV].
  • th (float): Scattering angle in [deg].

  • w2 (foat, array): Energy of Compton peak in [keV].


Funktion adapted from Keijo Hamalainen.


Calculates the Johann aberration of a spherical analyzer crystal.
E (float): Working energy in [eV]. A (float): Analyzer aperture [mm]. R (float): Radius of the Rowland circle [mm]. Theta (float): Analyzer Bragg angle [degree].
Johann abberation in [eV].


Calculates the intrinsic energy resolution of a diced crystal analyzer.
E (float): Working energy in [eV]. Dw (float): Darwin width of the used reflection [microRad]. Theta (float): Analyzer Bragg angle [degree].
Intrinsic energy resolution of a perfect analyzer crystal.


Calculates the off-Rowland contribution of a spherical analyzer crystal.
E (float): Working energy in [eV]. z (float): Off-Rowland distance [mm]. A (float): Analyzer aperture [mm]. R (float): Radius of the Rowland circle [mm]. Theta (float): Analyzer Bragg angle [degree].
Off-Rowland contribution in [eV] to the energy resolution.


Calculates the pixel size contribution to the resolution function of a diced analyzer crystal.
E (float): Working energy in [eV]. p (float): Pixel size in [mm]. R (float): Radius of the Rowland circle [mm]. Theta (float): Analyzer Bragg angle [degree].
Pixel size contribution in [eV] to the energy resolution for a diced analyzer crystal.


Calculates the source size contribution to the resolution function.
E (float): Working energy in [eV]. s (float): Source size in [mm]. R (float): Radius of the Rowland circle [mm]. Theta (float): Analyzer Bragg angle [degree].
Source size contribution in [eV] to the energy resolution.


Calculates the stress induced contribution to the resulution function of a spherically bent crystal analyzer.
E (float): Working energy in [eV]. t (float): Absorption length in the analyzer material [mm]. v (float): Poisson ratio of the analyzer material. R (float): Radius of the Rowland circle [mm]. Theta (float): Analyzer Bragg angle [degree].
Stress-induced contribution in [eV] to the energy resolution.


diode Calculates the number of photons incident for a Si PIPS diode.
  • current (float): Diode current in [pA].
  • energy (float): Photon energy in [keV].
  • thickness (float): Thickness of Si active layer in [cm].

  • flux (float): Number of photons per second.


Function adapted from Matlab function by S. Huotari.


% DSPACE Gives d-spacing for given xtal % d=dspace(hkl,xtal) % hkl can be a matrix i.e. hkl=[1,0,0 ; 1,1,1]; % xtal='Si','Ge','LiF','InSb','C','Dia','Li' (case insensitive) % if xtal is number this is user as a d0 % % KH 28.09.93 % SH 2005 %

Bases: object

class to hold all things dynamic theory of diffraction.






negative alpha -> more grazing incidence







Calculates the momentum scale and the relativistic Compton cross section correction according to P. Holm, PRA 37, 3706 (1988).

This function is translated from Keijo Hamalainen's Matlab implementation (KH 29.05.96).

  • w1 (float or np.array): incident energy in [keV]
  • w2 (float or np.array): scattered energy in [keV]
  • th (float): scattering angle two theta in [deg]

  • pz (float or np.array): momentum scale in [a.u.]
  • cf (float or np.array): cross section correction factor such that: J(pz) = cf * d^2(sigma)/d(w2)*d(Omega) [barn/atom/keV/srad]



reads edf-file with filename "filename" OUTPUT: data = 256x256 numpy array

reads edf-file with filename "filename" OUTPUT: data = 256x256 numpy array

here is how i opened the HH data: data = np.fromfile(f,np.int32) image = np.reshape(data,(dim,dim))


Converts atomic number into string of the element symbol and vice versa.

Returns atomic number of given element, if z is a string of the element symbol or string of element symbol of given atomic number z.

  • z (string or int): string of the element symbol or atomic number.

  • Z (string or int): string of the element symbol or atomic number.



% ENERGY Calculates energy corrresponing to Bragg angle for given d-spacing % function e=energy(dspace,bragg_angle) % % dspace for reflection % bragg_angle in DEG % % KH 28.09.93

% ENERGY Calculates energy corrresponing to Bragg angle for given d-spacing % function e=energy(dspace,bragg_angle) % % dspace for reflection (defaulf for Si(311) reflection) % bragg_angle in DEG % % KH 28.09.93 %

fermi Calculates the plasmon energy (in eV), Fermi energy (in eV), Fermi momentum (in a.u.), and critical plasmon cut-off vector (in a.u.).
  • rs (float): electron separation parameter

  • wp (float): plasmon energy (in eV)
  • ef (float): Fermi energy (in eV)
  • kf (float): Fermi momentum (in a.u.)
  • kc (float): critical plasmon cut-off vector (in a.u.)


Based on Matlab function from A. Soininen.


Returns the center of mass (first moment) for the given curve y(x)

find_diag_angles Finds the FOURC spectrometer and sample angles for a desired q.
  • q (array): Desired momentum transfer in Lab coordinates.
  • x0 (list): Guesses for the angles (tthv, tthh, chi, phi, omega).
  • U (array): 3x3 U-matrix Lab-to-sample transformation.
  • B (array): 3x3 B-matrix reciprocal lattice to absolute units transformation.
  • lambdai (float): Incident x-ray wavelength in Angstrom.
  • lambdao (float): Scattered x-ray wavelength in Angstrom.
  • tol (float): Toleranz for minimization (see scipy.optimize.minimize)
  • method (str): Method for minimization (see scipy.optimize.minimize)

  • ans (array): tthv, tthh, phi, chi, omega



finds full width at half maximum of the curve y vs. x returns f = FWHM x0 = position of the maximum


get_UB_Q Returns the momentum transfer and scattering vectors for given FOURC spectrometer and sample angles. U-, B-matrices and incident/scattered wavelength are passed as keyword-arguments.
  • tthv (float): Spectrometer vertical 2Theta angle.
  • tthh (float): Spectrometer horizontal 2Theta angle.
  • chi (float): Sample rotation around x-direction.
  • phi (float): Sample rotation around y-direction.
  • omega (float): Sample rotation around z-direction.
  • kwargs['U'] (array): 3x3 U-matrix Lab-to-sample transformation.
  • kwargs['B'] (array): 3x3 B-matrix reciprocal lattice to absolute units transformation.
  • kwargs['lambdai'] (float): Incident x-ray wavelength in Angstrom.
  • kwargs['lambdao'] (float): Scattered x-ray wavelength in Angstrom.



  • Q_sample (array): Momentum transfer in sample coordinates.
  • Ki_sample (array): Incident beam direction in sample coordinates.
  • Ko_sample (array): Scattered beam direction in sample coordinates.



get_gnuplot_rgb Prints out a progression of RGB hex-keys to use in Gnuplot.
  • start (array): RGB code to start from (must be numbers out of [0,1]).
  • end (array): RGB code to end at (must be numbers out of [0,1]).
  • length (int): How many colors to print out.



Calculates the number of steps in an MD simulation for a desired time (in ps) and given step size (in a.u.)
time_ps (float): Desired time span (ps). time_step (float): Chosen time step (a.u.).
The number of steps required to span the desired time span.


returns the penetration depth of a mixture of chemical formulas with certain concentrations and densities

returns the transmission through a sample composed of chemical formulas with certain densities mixed to certain concentrations, and a thickness


hlike_Rwfn Returns an array with the radial part of a hydrogen-like wave function.
  • n (integer): main quantum number n
  • l (integer): orbitalquantum number l
  • r (array): vector of radii on which the function should be evaluated
  • Z (float): effective nuclear charge



function H = householder(b, k) % H = householder(b, k) % Atkinson, Section 9.3, p. 611 % b is a column vector, k an index < length(b) % Constructs a matrix H that annihilates entries % in the product H*b below index k

% $Id: householder.m,v 1.1 2008-01-16 15:33:30 mike Exp $ % M. M. Sussman


Linear interpolation scheme after Martin Sundermann that conserves the absolute number of counts.

ONLY WORKS FOR EQUALLY/EVENLY SPACED XC, XI!

xc (np.array): The x-coordinates of the interpolated values. xi (np.array): The x-coordinates of the data points, must be increasing. yi (np.array): The y-coordinates of the data points, same length as xp. i0 (np.array): Normalization values for the data points, same length as xp.
ic (np.array): The interpolated and normalized data points.



from scipy.interpolate import Rbf x = arange(20) d = zeros(len(x)) d[10] = 1 xc = arange(0.5,19.5) rbfi = Rbf(x, d) di = rbfi(xc)


is_allowed_refl_fcc Check if given reflection is allowed for a FCC lattice.
  • H (array, list, tuple): H=[h,k,l]

  • boolean



lindhard_pol Calculates the Lindhard polarizability function (RPA) for certain q (a.u.), w (a.u.) and rs (a.u.).
  • q (float): momentum transfer (in a.u.)
  • w (float): energy (in a.u.)
  • rs (float): electron parameter
  • use_corr (boolean): if True, uses Bernardo's calculation for n(k) instead of the Fermi function.
  • lifetime (float): life time (default is 0.28 eV for Na).


Based on Matlab function by S. Huotari.


takes the profiles from 'makepzprofile()', converts them onto eloss scale and normalizes them to S(q,w) [1/eV] input: element = element symbol (e.g. 'Si', 'Al', etc.) filename = path and filename to tabulated profiles E0 = scattering energy [keV] tth = scattering angle [deg] returns: enscale = energy loss scale J = total CP C = only core contribution to CP V = only valence contribution to CP q = momentum transfer [a.u.]

returns the compton profile of a chemical compound with formula 'formula' input: formula = string of a chemical formula (e.g. 'SiO2', 'Ba8Si46', etc.) filename = path and filename to tabulated profiles E0 = scattering energy [keV] tth = scattering angle [deg] returns: eloss = energy loss scale J = total CP C = only core contribution to CP V = only valence contribution to CP q = momentum transfer [a.u.]

returns sum of compton profiles from a lost of chemical compounds weighted by the given concentration

constructs compton profiles of element 'element' on pz-scale (-100:100 a.u.) from the Biggs tables provided in 'filename'
  • element = element symbol (e.g. 'Si', 'Al', etc.)
  • filename = path and filename to tabulated profiles

  • pzprofile = numpy array of the CP: * 1. column: pz-scale * 2. ... n. columns: compton profile of nth shell * binden = binding energies of shells * occupation = number of electrons in the according shells





Bases: object

Class to store some useful values from the detectors used. To be used for arranging the ROIs.




Calculates the momentum transfer in atomic units input: e1 = incident energy [keV] e2 = scattered energy [keV] tth = scattering angle [deg] returns: q = momentum transfer [a.u.] (corresponding to sin(th)/lambda)

Calculates the momentum transfer in inverse angstrom input: e1 = incident energy [keV] e2 = scattered energy [keV] tth = scattering angle [deg] returns: q = momentum transfer [a.u.] (corresponding to sin(th)/lambda)

Calculates the photoelectric, elastic, and inelastic absorption of a chemical compound.

Calculates the photoelectric, elastic, and inelastic absorption of a chemical compound.

  • energy (np.array): energy scale in [keV].
  • compound (string): chemical sum formula (e.g. 'SiO2')

  • murho (np.array): absorption coefficient normalized by the density.
  • rho (float): density in UNITS?
  • m (float): atomic mass in UNITS?



Calculates the photoelectric, elastic, and inelastic absorption of a mix of compounds.

Returns the photoelectric absorption for a sum of different chemical compounds.

  • energy (np.array): energy scale in [keV].
  • formulas (list of strings): list of chemical sum formulas

  • murho (np.array): absorption coefficient normalized by the density.
  • rho (float): density in UNITS?
  • m (float): atomic mass in UNITS?



Calculates the photoelectric, elastic, and inelastic absorption of an element Z

Calculates the photelectric , elastic, and inelastic absorption of an element Z. Z can be atomic number or element symbol.

  • energy (np.array): energy scale in [keV].
  • Z (string or int): atomic number or string of element symbol.

  • murho (np.array): absorption coefficient normalized by the density.
  • rho (float): density in UNITS?
  • m (float): atomic mass in UNITS?




#% [T,Y] = ODE23(ODEFUN,TSPAN,Y0,OPTIONS,P1,P2,...) passes the additional #% parameters P1,P2,... to the ODE function as ODEFUN(T,Y,P1,P2...), and to #% all functions specified in OPTIONS. Use OPTIONS = [] as a place holder if #% no options are set.


Parses a chemical sum formula.

Parses the constituing elements and stoichiometries from a given chemical sum formula.

  • formula (string): string of a chemical formula (e.g. 'SiO2', 'Ba8Si46', etc.)

  • elements (list): list of strings of constituting elemental symbols.
  • stoichiometries (list): list of according stoichiometries in the same order as 'elements'.



opens a plot window of the penetration depth of a mixture of chemical formulas with certain concentrations and densities plotted along the given energy vector

opens a plot with the transmission plotted along the given energy vector

primtoconv converts diamond structure reciprocal lattice expressed in primitive basis to the conventional basis (Palaiseau -> Helsinki conversion) from S. Huotari

Calculates the incident energy for a specific scattered photon and momentum value.

Returns the incident energy for a given photon energy and scattering angle. This function is translated from Keijo Hamalainen's Matlab implementation (KH 29.05.96).

  • w2 (float): scattered photon energy in [keV]
  • pz (np.array): pz scale in [a.u.]
  • th (float): scattering angle two theta in [deg]

  • w1 (np.array): incident energy in [keV]



read_dft_wfn Parses radial parts of wavefunctions.
  • element (str): Element symbol.
  • n (int): Main quantum number.
  • l (int): Orbital quantum number.
  • spin (str): Which spin channel, default is average over up and down.
  • directory (str): Path to directory where the wavefunctions can be found.

  • r (np.array): radius
  • wfn (np.array):



Reads Hartree-Fock Profile of element 'element' from values tabulated by Biggs et al. (Atomic Data and Nuclear Data Tables 16, 201-309 (1975)) as provided by the DABAX library (http://ftp.esrf.eu/pub/scisoft/xop2.3/DabaxFiles/ComptonProfiles.dat). input: filename = path to the ComptonProfiles.dat file (the file should be distributed with this package) element = string of element name returns:
  • #UD Columns:
  • #UD col1: pz in atomic units
  • #UD col2: Total compton profile (sum over the atomic electrons
  • #UD col3,...coln: Compton profile for the individual sub-shells


  • occupation = occupation number of the according shells
  • bindingen = binding energies of the accorting shells
  • colnames = strings of column names as used in the file




if repnumber = 0: reads a spectra-file (name: prefix_scannumber.fio) if repnumber > 1: reads a spectra-file (name: prefix_scannumber_rrepnumber.fio)

reads a detector file from PetraIII beamline P01

reads a whole scan from PetraIII beamline P01 (experimental)

reads a whole scan with repititions from PetraIII beamline P01 (experimental)

Smooth (and optionally differentiate) data with a Savitzky-Golay filter. The Savitzky-Golay filter removes high frequency noise from data. It has the advantage of preserving the original shape and features of the signal better than other types of filtering approaches, such as moving averages techniques.
  • y : array_like, shape (N,) the values of the time history of the signal.
  • window_size : int the length of the window. Must be an odd integer number.
  • order : int the order of the polynomial used in the filtering. Must be less then window_size - 1.
  • deriv: int the order of the derivative to compute (default = 0 means only smoothing)

  • ys : ndarray, shape (N) the smoothed signal (or it's n-th derivative).

The Savitzky-Golay is a type of low-pass filter, particularly suited for smoothing noisy data. The main idea behind this approach is to make for each point a least-square fit with a polynomial of high order over a odd-sized window centered at the point.

Examples

t = np.linspace(-4, 4, 500)
y = np.exp( -t**2 ) + np.random.normal(0, 0.05, t.shape)
ysg = savitzky_golay(y, window_size=31, order=4)
import matplotlib.pyplot as plt
plt.plot(t, y, label='Noisy signal')
plt.plot(t, np.exp(-t**2), 'k', lw=1.5, label='Original signal')
plt.plot(t, ysg, 'r', label='Filtered signal')
plt.legend()
plt.show()


[1]
A. Savitzky, M. J. E. Golay, Smoothing and Differentiation of Data by Simplified Least Squares Procedures. Analytical Chemistry, 1964, 36 (8), pp 1627-1639.
[2]
Numerical Recipes 3rd Edition: The Art of Scientific Computing W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery Cambridge University Press ISBN-13: 9780521880688



sigmainc Calculates the Incoherent Scattering Cross Section in cm^2/g using Log-Log Fit.
  • z (int or string): Element number or elements symbol.
  • energy (float or array): Energy (can be number or vector)

  • tau (float or array): Photoelectric cross section in [cm**2/g]


Adapted from original Matlab function of Keijo Hamalainen.


reads scan "nscan" from SPEC-file "filename"
  • filename = string with the SPEC-file name
  • nscan = number (int) of desired scan

  • data =
  • motors =
  • counters = dictionary



Extrapolates the smaller and larger valuea as a constant


stiff_compl_matrix_Si Returns stiffnes and compliance tensor of Si for a given orientation.
  • e1 (np.array): unit vector normal to crystal surface
  • e2 (np.array): unit vector crystal surface
  • e3 (np.array): unit vector orthogonal to e2

  • S (np.array): compliance tensor in new coordinate system
  • C (np.array): stiffnes tensor in new coordinate system
  • E (np.array): Young's modulus in [GPa]
  • G (np.array): shear modulus in [GPa]
  • nu (np.array): Poisson ratio


Copied from S.I. of L. Zhang et al. "Anisotropic elasticity of silicon and its application to the modelling of X-ray optics." J. Synchrotron Rad. 21, no. 3 (2014): 507-517.


Short-hand command to sum over 1st dimension of a N-D matrix (N>2) and to squeeze it to N-1-D matrix.


% TAUPGEN Calculates the reflectivity curves of bent crystals % % function [refl,e,dev]=taupgen_new(e,hkl,crystals,R,dev,alpha); % % e = fixed nominal energy in keV % hkl = reflection order vector, e.g. [1 1 1] % crystals = crystal string, e.g. 'si' or 'ge' % R = bending radius in meters % dev = deviation parameter for which the % curve will be calculated (vector) (optional) % alpha = asymmetry angle % based on a FORTRAN program of Michael Krisch % Translitterated to Matlab by Simo Huotari 2006, 2007 % Is far away from being good matlab writing - mostly copy&paste from % the fortran routines. Frankly, my dear, I don't give a damn. % Complaints -> /dev/null

% TAUPGEN Calculates the reflectivity curves of bent crystals % % function [refl,e,dev]=taupgen_new(e,hkl,crystals,R,dev,alpha); % % e = fixed nominal energy in keV % hkl = reflection order vector, e.g. [1 1 1] % crystals = crystal string, e.g. 'si' or 'ge' % R = bending radius in meters % dev = deviation parameter for which the % curve will be calculated (vector) (optional) % alpha = asymmetry angle % based on a FORTRAN program of Michael Krisch % Translitterated to Matlab by Simo Huotari 2006, 2007 % Is far away from being good matlab writing - mostly copy&paste from % the fortran routines. Frankly, my dear, I don't give a damn. % Complaints -> /dev/null

tauphoto Calculates Photoelectric Cross Section in cm^2/g using Log-Log Fit.
  • z (int or string): Element number or elements symbol.
  • energy (float or array): Energy (can be number or vector)

  • tau (float or array): Photoelectric cross section in [cm**2/g]


Adapted from original Matlab function of Keijo Hamalainen.


unconstrained_mf Returns main components from an off-diagonal Matrix (energy-loss x angular-departure), using the power method iteratively on the different main components.

vangle Calculates the angle between two cartesian vectors v1 and v2 in degrees.
  • v1 (np.array): first vector.
  • v2 (np.array): second vector.

  • th (float): angle between first and second vector.


Function by S. Huotari, adopted for Python.



vrot Rotates a vector around a given axis.
  • v (np.array): vector to be rotated
  • vaxis (np.array): rotation axis
  • phi (float): angle [deg] respecting the right-hand rule

  • v2 (np.array): new rotated vector


Function by S. Huotari (2007) adopted to Python.


rotMatrix Rotate vector1 around vector2 by an angle.

xas_fluo_correct Fluorescence yield over-absorption correction as in Larch/Athena. see: https://www3.aps.anl.gov/haskel/FLUO/Fluo-manual.pdf
  • ene (np.array): energy axis in [keV]
  • mu (np.array): measured fluorescence spectrum
  • formula (str): chemical sum formulas (e.g. 'SiO2')
  • fluo_ene (float): energy in keV of main fluorescence line
  • edge_ene (float): edge energy in [keV]
  • angin (float): incidence angle (relative to sample normal) [deg.]
  • angout (float): exit angle (relative to sample normal) [deg.]

  • ene (np.array): energy axis in [keV]
  • mu_corr (np.array): corrected fluorescence spectrum



XRStools.xrs_calctools Module


axsfTrajParser

Draw samples from a Beta distribution.

The Beta distribution is a special case of the Dirichlet distribution, and is related to the Gamma distribution. It has the probability distribution function


f(x; a,b) = \frac{1}{B(\alpha, \beta)} x^{\alpha - 1}
(1 - x)^{\beta - 1},


where the normalization, B, is the beta function,


B(\alpha, \beta) = \int_0^1 t^{\alpha - 1}
(1 - t)^{\beta - 1} dt.


It is often seen in Bayesian inference and order statistics.

NOTE:

New code should use the beta method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Alpha, positive (>0).
float or array_like of floats Beta, positive (>0).
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a and b are both scalars. Otherwise, np.broadcast(a, b).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized beta distribution.

Generator.beta: which should be used for new code.


Draw samples from a binomial distribution.

Samples are drawn from a binomial distribution with specified parameters, n trials and p probability of success where n an integer >= 0 and p is in the interval [0,1]. (n may be input as a float, but it is truncated to an integer in use)

NOTE:

New code should use the binomial method of a default_rng() instance instead; please see the random-quick-start.


int or array_like of ints Parameter of the distribution, >= 0. Floats are also accepted, but they will be truncated to integers.
float or array_like of floats Parameter of the distribution, >= 0 and <=1.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if n and p are both scalars. Otherwise, np.broadcast(n, p).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized binomial distribution, where each sample is equal to the number of successes over the n trials.

probability density function, distribution or cumulative density function, etc.

Generator.binomial: which should be used for new code.

The probability density for the binomial distribution is


P(N) = \binom{n}{N}p^N(1-p)^{n-N},



where n is the number of trials, p is the probability of success, and N is the number of successes.

When estimating the standard error of a proportion in a population by using a random sample, the normal distribution works well unless the product p*n <=5, where p = population proportion estimate, and n = number of samples, in which case the binomial distribution is used instead. For example, a sample of 15 people shows 4 who are left handed, and 11 who are right handed. Then p = 4/15 = 27%. 0.27*15 = 4, so the binomial distribution should be used in this case.

[1]
Dalgaard, Peter, "Introductory Statistics with R", Springer-Verlag, 2002.
[2]
Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill, Fifth Edition, 2002.
[3]
Lentner, Marvin, "Elementary Applied Statistics", Bogden and Quigley, 1972.
[4]
Weisstein, Eric W. "Binomial Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BinomialDistribution.html
[5]
Wikipedia, "Binomial distribution", https://en.wikipedia.org/wiki/Binomial_distribution

Draw samples from the distribution:

>>> n, p = 10, .5  # number of trials, probability of each trial
>>> s = np.random.binomial(n, p, 1000)
# result of flipping a coin 10 times, tested 1000 times.

A real world example. A company drills 9 wild-cat oil exploration wells, each with an estimated probability of success of 0.1. All nine wells fail. What is the probability of that happening?

Let's do 20,000 trials of the model, and count the number that generate zero positive results.

>>> sum(np.random.binomial(9, 0.1, 20000) == 0)/20000.
# answer = 0.38885, or 38%.

parseXYZfile Reads an xyz-style file.

function [e2,s2] = broaden_diagram2(e,s,params,npoints)

% BROADEN_DIAGRAM2 Broaden a StoBe line diagram % % [ENE2,SQW2] = BROADEN_DIAGRAM2(ENE,SQW,PARAMS,NPOINTS) % % gives the broadened spectrum SQW2(ENE2) of the line-spectrum % SWQ(ENE). Each line is substituted with a Gaussian peak, % the FWHM of which is determined by PARAMS. ENE2 is a linear % scale of length NPOINTS (default 1000). % % PARAMS = [f_min f_max emin max] % % For ENE <= e_min, FWHM = f_min. % For ENE >= e_max, FWHM = f_min. % FWHM increases linearly from [f_min f_max] between [e_min e_max]. % % T Pylkkanen @ 2008-04-18 [17:37]


broadens a spectrum with a Gaussian of width params[0] below params[2] and width params[1] above params[3], width increases linear in between. returns two-column numpy array of length npoints with energy and the broadened spectrum

calculateCOMlist Calculates center of mass for a list of atoms.





Draw samples from a chi-square distribution.

When df independent random variables, each with standard normal distributions (mean 0, variance 1), are squared and summed, the resulting distribution is chi-square (see Notes). This distribution is often used in hypothesis testing.

NOTE:

New code should use the chisquare method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Number of degrees of freedom, must be > 0.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if df is a scalar. Otherwise, np.array(df).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized chi-square distribution.

When df <= 0 or when an inappropriate size (e.g. size=-1) is given.

Generator.chisquare: which should be used for new code.

The variable obtained by summing the squares of df independent, standard normally distributed random variables:


Q = \sum_{i=0}^{\mathtt{df}} X^2_i



is chi-square distributed, denoted


Q \sim \chi^2_k.



The probability density function of the chi-squared distribution is


p(x) = \frac{(1/2)^{k/2}}{\Gamma(k/2)}
x^{k/2 - 1} e^{-x/2},


where \Gamma is the gamma function,


\Gamma(x) = \int_0^{-\infty} t^{x - 1} e^{-t} dt.



[1]
NIST "Engineering Statistics Handbook" https://www.itl.nist.gov/div898/handbook/eda/section3/eda3666.htm 

>>> np.random.chisquare(2,4)
array([ 1.89920014,  9.00867716,  3.13710533,  5.62318272]) # random

Generates a random sample from a given 1-D array

New in version 1.7.0.

NOTE:

New code should use the choice method of a default_rng() instance instead; please see the random-quick-start.


1-D array-like or int If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated as if it were np.arange(a)
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
boolean, optional Whether the sample is with or without replacement. Default is True, meaning that a value of a can be selected multiple times.
1-D array-like, optional The probabilities associated with each entry in a. If not given, the sample assumes a uniform distribution over all entries in a.

single item or ndarray The generated random samples

If a is an int and less than zero, if a or p are not 1-dimensional, if a is an array-like of size 0, if p is not a vector of probabilities, if a and p have different lengths, or if replace=False and the sample size is greater than the population size

randint, shuffle, permutation Generator.choice: which should be used in new code

Setting user-specified probabilities through p uses a more general but less efficient sampler than the default. The general sampler produces a different sample than the optimized sampler even if each element of p is 1 / len(a).

Sampling random rows from a 2-D array is not possible with this function, but is possible with Generator.choice through its axis keyword.

Generate a uniform random sample from np.arange(5) of size 3:

>>> np.random.choice(5, 3)
array([0, 3, 4]) # random
>>> #This is equivalent to np.random.randint(0,5,3)

Generate a non-uniform random sample from np.arange(5) of size 3:

>>> np.random.choice(5, 3, p=[0.1, 0, 0.3, 0.6, 0])
array([3, 3, 0]) # random

Generate a uniform random sample from np.arange(5) of size 3 without replacement:

>>> np.random.choice(5, 3, replace=False)
array([3,1,0]) # random
>>> #This is equivalent to np.random.permutation(np.arange(5))[:3]

Generate a non-uniform random sample from np.arange(5) of size 3 without replacement:

>>> np.random.choice(5, 3, replace=False, p=[0.1, 0, 0.3, 0.6, 0])
array([2, 3, 0]) # random

Any of the above can be repeated with an arbitrary array-like instead of just integers. For instance:

>>> aa_milne_arr = ['pooh', 'rabbit', 'piglet', 'Christopher']
>>> np.random.choice(aa_milne_arr, 5, p=[0.5, 0.1, 0.1, 0.3])
array(['pooh', 'pooh', 'pooh', 'Christopher', 'piglet'], # random
      dtype='<U11')

Convolution with Gaussian







deletes lines of matrix with first column smaller than emin and larger than emax

Draw samples from the Dirichlet distribution.

Draw size samples of dimension k from a Dirichlet distribution. A Dirichlet-distributed random variable can be seen as a multivariate generalization of a Beta distribution. The Dirichlet distribution is a conjugate prior of a multinomial distribution in Bayesian inference.

NOTE:

New code should use the dirichlet method of a default_rng() instance instead; please see the random-quick-start.


sequence of floats, length k Parameter of the distribution (length k for sample of length k).
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n), then m * n * k samples are drawn. Default is None, in which case a vector of length k is returned.

ndarray, The drawn samples, of shape (size, k).

If any value in alpha is less than or equal to zero

Generator.dirichlet: which should be used for new code.

The Dirichlet distribution is a distribution over vectors x that fulfil the conditions x_i>0 and \sum_{i=1}^k x_i = 1.

The probability density function p of a Dirichlet-distributed random vector X is proportional to


p(x) \propto \prod_{i=1}^{k}{x^{\alpha_i-1}_i},



where \alpha is a vector containing the positive concentration parameters.

The method uses the following property for computation: let Y be a random vector which has components that follow a standard gamma distribution, then X = \frac{1}{\sum_{i=1}^k{Y_i}} Y is Dirichlet-distributed

[1]
David McKay, "Information Theory, Inference and Learning Algorithms," chapter 23, http://www.inference.org.uk/mackay/itila/
[2]
Wikipedia, "Dirichlet distribution", https://en.wikipedia.org/wiki/Dirichlet_distribution

Taking an example cited in Wikipedia, this distribution can be used if one wanted to cut strings (each of initial length 1.0) into K pieces with different lengths, where each piece had, on average, a designated average length, but allowing some variation in the relative sizes of the pieces.

>>> s = np.random.dirichlet((10, 5, 3), 20).transpose()
    

>>> import matplotlib.pyplot as plt
>>> plt.barh(range(20), s[0])
>>> plt.barh(range(20), s[1], left=s[0], color='g')
>>> plt.barh(range(20), s[2], left=s[0]+s[1], color='r')
>>> plt.title("Lengths of Strings")

Bases: object

class to analyze ERKALE XRS results.









Draw samples from an exponential distribution.

Its probability density function is


f(x; \frac{1}{\beta}) = \frac{1}{\beta} \exp(-\frac{x}{\beta}),



for x > 0 and 0 elsewhere. \beta is the scale parameter, which is the inverse of the rate parameter \lambda = 1/\beta. The rate parameter is an alternative, widely used parameterization of the exponential distribution

[3]_
.

The exponential distribution is a continuous analogue of the geometric distribution. It describes many common situations, such as the size of raindrops measured over many rainstorms

[1]_
, or the time between page requests to Wikipedia 
[2]_
.

NOTE:

New code should use the exponential method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats The scale parameter, \beta = 1/\lambda. Must be non-negative.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if scale is a scalar. Otherwise, np.array(scale).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized exponential distribution.

Generator.exponential: which should be used for new code.

[1]
Peyton Z. Peebles Jr., "Probability, Random Variables and Random Signal Principles", 4th ed, 2001, p. 57.
[2]
Wikipedia, "Poisson process", https://en.wikipedia.org/wiki/Poisson_process
[3]
Wikipedia, "Exponential distribution", https://en.wikipedia.org/wiki/Exponential_distribution

Draw samples from an F distribution.

Samples are drawn from an F distribution with specified parameters, dfnum (degrees of freedom in numerator) and dfden (degrees of freedom in denominator), where both parameters must be greater than zero.

The random variate of the F distribution (also known as the Fisher distribution) is a continuous probability distribution that arises in ANOVA tests, and is the ratio of two chi-square variates.

NOTE:

New code should use the f method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Degrees of freedom in numerator, must be > 0.
float or array_like of float Degrees of freedom in denominator, must be > 0.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if dfnum and dfden are both scalars. Otherwise, np.broadcast(dfnum, dfden).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized Fisher distribution.

probability density function, distribution or cumulative density function, etc.

Generator.f: which should be used for new code.

The F statistic is used to compare in-group variances to between-group variances. Calculating the distribution depends on the sampling, and so it is a function of the respective degrees of freedom in the problem. The variable dfnum is the number of samples minus one, the between-groups degrees of freedom, while dfden is the within-groups degrees of freedom, the sum of the number of samples in each group minus the number of groups.

[1]
Glantz, Stanton A. "Primer of Biostatistics.", McGraw-Hill, Fifth Edition, 2002.
[2]
Wikipedia, "F-distribution", https://en.wikipedia.org/wiki/F-distribution

An example from Glantz[1], pp 47-40:

Two groups, children of diabetics (25 people) and children from people without diabetes (25 controls). Fasting blood glucose was measured, case group had a mean value of 86.1, controls had a mean value of 82.2. Standard deviations were 2.09 and 2.49 respectively. Are these data consistent with the null hypothesis that the parents diabetic status does not affect their children's blood glucose levels? Calculating the F statistic from the data gives a value of 36.01.

Draw samples from the distribution:

>>> dfnum = 1. # between group degrees of freedom
>>> dfden = 48. # within groups degrees of freedom
>>> s = np.random.f(dfnum, dfden, 1000)

The lower bound for the top 1% of the samples is :

>>> np.sort(s)[-10]
7.61988120985 # random

So there is about a 1% chance that the F statistic will exceed 7.62, the measured value is 36, so the null hypothesis is rejected at the 1% level.








Draw samples from a Gamma distribution.

Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated "k") and scale (sometimes designated "theta"), where both parameters are > 0.

NOTE:

New code should use the gamma method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats The shape of the gamma distribution. Must be non-negative.
float or array_like of floats, optional The scale of the gamma distribution. Must be non-negative. Default is equal to 1.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if shape and scale are both scalars. Otherwise, np.broadcast(shape, scale).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized gamma distribution.

probability density function, distribution or cumulative density function, etc.

Generator.gamma: which should be used for new code.

The probability density for the Gamma distribution is


p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},



where k is the shape and \theta the scale, and \Gamma is the Gamma function.

The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant.

[1]
Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html
[2]
Wikipedia, "Gamma distribution", https://en.wikipedia.org/wiki/Gamma_distribution

Draw samples from the distribution:

>>> shape, scale = 2., 2.  # mean=4, std=2*sqrt(2)
>>> s = np.random.gamma(shape, scale, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps  
>>> count, bins, ignored = plt.hist(s, 50, density=True)
>>> y = bins**(shape-1)*(np.exp(-bins/scale) /  
...                      (sps.gamma(shape)*scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')  
>>> plt.show()


returns a gaussian with peak value normalized to unity a[0] = peak position a[1] = Full Width at Half Maximum

area-normalized gaussian

Draw samples from the geometric distribution.

Bernoulli trials are experiments with one of two outcomes: success or failure (an example of such an experiment is flipping a coin). The geometric distribution models the number of trials that must be run in order to achieve success. It is therefore supported on the positive integers, k = 1, 2, ....

The probability mass function of the geometric distribution is


f(k) = (1 - p)^{k - 1} p



where p is the probability of success of an individual trial.

NOTE:

New code should use the geometric method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats The probability of success of an individual trial.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if p is a scalar. Otherwise, np.array(p).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized geometric distribution.

Generator.geometric: which should be used for new code.

Draw ten thousand values from the geometric distribution, with the probability of an individual success equal to 0.35:

>>> z = np.random.geometric(p=0.35, size=10000)

How many trials succeeded after a single run?

>>> (z == 1).sum() / 10000.
0.34889999999999999 #random


getDistVectorPBC_arb

Calculates the distance vector between two atoms from an arbitrary simulation box using the minimum image convention.

atom1 (obj): Instance of the xzyAtom class. atom2 (obj): Instance of the xzyAtom class. lattice (np.array): Array with lattice vectors as columns. lattice_inv (np.array): Inverse of lattice.
The distance vector between the two atoms (np.array).




getDistancePBC_arb

Calculates the distance of two atoms from an arbitrary simulation box using the minimum image convention.

atom1 (obj): Instance of the xzyAtom class. atom2 (obj): Instance of the xzyAtom class. lattice (np.array): Array with lattice vectors as columns. lattice_inv (np.array): Inverse of lattice.
The distance between the two atoms.







according to NATURE, VOL 409, 18 JANUARY 2001

getTranslVec Returns the translation vector that brings atom2 closer to atom1 in case atom2 is further than boxLength away.

getTranslVec_geocen

Return a tuple representing the internal state of the generator.

For more details, see set_state.

bool, optional Flag indicating to return a legacy tuple state when the BitGenerator is MT19937, instead of a dict.

{tuple(str, ndarray of 624 uints, int, int, float), dict} The returned tuple has the following items:
1.
the string 'MT19937'.
2.
a 1-D array of 624 unsigned integer keys.
3.
an integer pos.
4.
an integer has_gauss.
5.
a float cached_gaussian.

If legacy is False, or the BitGenerator is not MT19937, then state is returned as a dictionary.


set_state

set_state and get_state are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing.


groBoxParser Parses an gromacs GRO-style file for the xyzBox class.

groTrajecParser Parses an gromacs GRO-style file for the xyzBox class.

Draw samples from a Gumbel distribution.

Draw samples from a Gumbel distribution with specified location and scale. For more information on the Gumbel distribution, see Notes and References below.

NOTE:

New code should use the gumbel method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats, optional The location of the mode of the distribution. Default is 0.
float or array_like of floats, optional The scale parameter of the distribution. Default is 1. Must be non- negative.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized Gumbel distribution.

scipy.stats.gumbel_l scipy.stats.gumbel_r scipy.stats.genextreme weibull Generator.gumbel: which should be used for new code.

The Gumbel (or Smallest Extreme Value (SEV) or the Smallest Extreme Value Type I) distribution is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. The Gumbel is a special case of the Extreme Value Type I distribution for maximums from distributions with "exponential-like" tails.

The probability density for the Gumbel distribution is


p(x) = \frac{e^{-(x - \mu)/ \beta}}{\beta} e^{ -e^{-(x - \mu)/
\beta}},


where \mu is the mode, a location parameter, and \beta is the scale parameter.

The Gumbel (named for German mathematician Emil Julius Gumbel) was used very early in the hydrology literature, for modeling the occurrence of flood events. It is also used for modeling maximum wind speed and rainfall rates. It is a "fat-tailed" distribution - the probability of an event in the tail of the distribution is larger than if one used a Gaussian, hence the surprisingly frequent occurrence of 100-year floods. Floods were initially modeled as a Gaussian process, which underestimated the frequency of extreme events.

It is one of a class of extreme value distributions, the Generalized Extreme Value (GEV) distributions, which also includes the Weibull and Frechet.

The function has a mean of \mu + 0.57721\beta and a variance of \frac{\pi^2}{6}\beta^2.

[1]
Gumbel, E. J., "Statistics of Extremes," New York: Columbia University Press, 1958.
[2]
Reiss, R.-D. and Thomas, M., "Statistical Analysis of Extreme Values from Insurance, Finance, Hydrology and Other Fields," Basel: Birkhauser Verlag, 2001.

Draw samples from the distribution:

>>> mu, beta = 0, 0.1 # location and scale
>>> s = np.random.gumbel(mu, beta, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
...          * np.exp( -np.exp( -(bins - mu) /beta) ),
...          linewidth=2, color='r')
>>> plt.show()

Show how an extreme value distribution can arise from a Gaussian process and compare to a Gaussian:

>>> means = []
>>> maxima = []
>>> for i in range(0,1000) :
...    a = np.random.normal(mu, beta, 1000)
...    means.append(a.mean())
...    maxima.append(a.max())
>>> count, bins, ignored = plt.hist(maxima, 30, density=True)
>>> beta = np.std(maxima) * np.sqrt(6) / np.pi
>>> mu = np.mean(maxima) - 0.57721*beta
>>> plt.plot(bins, (1/beta)*np.exp(-(bins - mu)/beta)
...          * np.exp(-np.exp(-(bins - mu)/beta)),
...          linewidth=2, color='r')
>>> plt.plot(bins, 1/(beta * np.sqrt(2 * np.pi))
...          * np.exp(-(bins - mu)**2 / (2 * beta**2)),
...          linewidth=2, color='g')
>>> plt.show()

Draw samples from a Hypergeometric distribution.

Samples are drawn from a hypergeometric distribution with specified parameters, ngood (ways to make a good selection), nbad (ways to make a bad selection), and nsample (number of items sampled, which is less than or equal to the sum ngood + nbad).

NOTE:

New code should use the hypergeometric method of a default_rng() instance instead; please see the random-quick-start.


int or array_like of ints Number of ways to make a good selection. Must be nonnegative.
int or array_like of ints Number of ways to make a bad selection. Must be nonnegative.
int or array_like of ints Number of items sampled. Must be at least 1 and at most ngood + nbad.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if ngood, nbad, and nsample are all scalars. Otherwise, np.broadcast(ngood, nbad, nsample).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized hypergeometric distribution. Each sample is the number of good items within a randomly selected subset of size nsample taken from a set of ngood good items and nbad bad items.

probability density function, distribution or cumulative density function, etc.

Generator.hypergeometric: which should be used for new code.

The probability density for the Hypergeometric distribution is


P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}},



where 0 \le x \le n and n-b \le x \le g

for P(x) the probability of x good results in the drawn sample, g = ngood, b = nbad, and n = nsample.

Consider an urn with black and white marbles in it, ngood of them are black and nbad are white. If you draw nsample balls without replacement, then the hypergeometric distribution describes the distribution of black balls in the drawn sample.

Note that this distribution is very similar to the binomial distribution, except that in this case, samples are drawn without replacement, whereas in the Binomial case samples are drawn with replacement (or the sample space is infinite). As the sample space becomes large, this distribution approaches the binomial.

[1]
Lentner, Marvin, "Elementary Applied Statistics", Bogden and Quigley, 1972.
[2]
Weisstein, Eric W. "Hypergeometric Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/HypergeometricDistribution.html
[3]
Wikipedia, "Hypergeometric distribution", https://en.wikipedia.org/wiki/Hypergeometric_distribution

Draw samples from the distribution:

>>> ngood, nbad, nsamp = 100, 2, 10
# number of good, number of bad, and number of samples
>>> s = np.random.hypergeometric(ngood, nbad, nsamp, 1000)
>>> from matplotlib.pyplot import hist
>>> hist(s)
#   note that it is very unlikely to grab both bad items

Suppose you have an urn with 15 white and 15 black marbles. If you pull 15 marbles at random, how likely is it that 12 or more of them are one color?

>>> s = np.random.hypergeometric(15, 15, 15, 100000)
>>> sum(s>=12)/100000. + sum(s<=3)/100000.
#   answer = 0.003 ... pretty unlikely!

keithBoxParser

Reads structure files from Keith's SiO2 simulations.


Draw samples from the Laplace or double exponential distribution with specified location (or mean) and scale (decay).

The Laplace distribution is similar to the Gaussian/normal distribution, but is sharper at the peak and has fatter tails. It represents the difference between two independent, identically distributed exponential random variables.

NOTE:

New code should use the laplace method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats, optional The position, \mu, of the distribution peak. Default is 0.
float or array_like of floats, optional \lambda, the exponential decay. Default is 1. Must be non- negative.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized Laplace distribution.

Generator.laplace: which should be used for new code.

It has the probability density function


f(x; \mu, \lambda) = \frac{1}{2\lambda}
\exp\left(-\frac{|x - \mu|}{\lambda}\right).


The first law of Laplace, from 1774, states that the frequency of an error can be expressed as an exponential function of the absolute magnitude of the error, which leads to the Laplace distribution. For many problems in economics and health sciences, this distribution seems to model the data better than the standard Gaussian distribution.

[1]
Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972.
[2]
Kotz, Samuel, et. al. "The Laplace Distribution and Generalizations, " Birkhauser, 2001.
[3]
Weisstein, Eric W. "Laplace Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LaplaceDistribution.html
[4]
Wikipedia, "Laplace distribution", https://en.wikipedia.org/wiki/Laplace_distribution

Draw samples from the distribution

>>> loc, scale = 0., 1.
>>> s = np.random.laplace(loc, scale, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> x = np.arange(-8., 8., .01)
>>> pdf = np.exp(-abs(x-loc)/scale)/(2.*scale)
>>> plt.plot(x, pdf)

Plot Gaussian for comparison:

>>> g = (1/(scale * np.sqrt(2 * np.pi)) *
...      np.exp(-(x - loc)**2 / (2 * scale**2)))
>>> plt.plot(x,g)

returns an erkale spectrum

returns a list of erkale spectra

load a bunch of StoBe calculations, which filenames are made up of the prefix, postfix, and the counter in the between the prefix and postfix runs from 'fromnumber' to 'tonumber' in steps of 'step' (number of digits is 'stepformat')

Draw samples from a logistic distribution.

Samples are drawn from a logistic distribution with specified parameters, loc (location or mean, also median), and scale (>0).

NOTE:

New code should use the logistic method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats, optional Parameter of the distribution. Default is 0.
float or array_like of floats, optional Parameter of the distribution. Must be non-negative. Default is 1.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized logistic distribution.

probability density function, distribution or cumulative density function, etc.

Generator.logistic: which should be used for new code.

The probability density for the Logistic distribution is


P(x) = P(x) = \frac{e^{-(x-\mu)/s}}{s(1+e^{-(x-\mu)/s})^2},



where \mu = location and s = scale.

The Logistic distribution is used in Extreme Value problems where it can act as a mixture of Gumbel distributions, in Epidemiology, and by the World Chess Federation (FIDE) where it is used in the Elo ranking system, assuming the performance of each player is a logistically distributed random variable.

[1]
Reiss, R.-D. and Thomas M. (2001), "Statistical Analysis of Extreme Values, from Insurance, Finance, Hydrology and Other Fields," Birkhauser Verlag, Basel, pp 132-133.
[2]
Weisstein, Eric W. "Logistic Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticDistribution.html
[3]
Wikipedia, "Logistic-distribution", https://en.wikipedia.org/wiki/Logistic_distribution

Draw samples from the distribution:

>>> loc, scale = 10, 1
>>> s = np.random.logistic(loc, scale, 10000)
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, bins=50)

# plot against distribution

>>> def logist(x, loc, scale):
...     return np.exp((loc-x)/scale)/(scale*(1+np.exp((loc-x)/scale))**2)
>>> lgst_val = logist(bins, loc, scale)
>>> plt.plot(bins, lgst_val * count.max() / lgst_val.max())
>>> plt.show()

Draw samples from a log-normal distribution.

Draw samples from a log-normal distribution with specified mean, standard deviation, and array shape. Note that the mean and standard deviation are not the values for the distribution itself, but of the underlying normal distribution it is derived from.

NOTE:

New code should use the lognormal method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats, optional Mean value of the underlying normal distribution. Default is 0.
float or array_like of floats, optional Standard deviation of the underlying normal distribution. Must be non-negative. Default is 1.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if mean and sigma are both scalars. Otherwise, np.broadcast(mean, sigma).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized log-normal distribution.

probability density function, distribution, cumulative density function, etc.

Generator.lognormal: which should be used for new code.

A variable x has a log-normal distribution if log(x) is normally distributed. The probability density function for the log-normal distribution is:


p(x) = \frac{1}{\sigma x \sqrt{2\pi}}
e^{(-\frac{(ln(x)-\mu)^2}{2\sigma^2})}


where \mu is the mean and \sigma is the standard deviation of the normally distributed logarithm of the variable. A log-normal distribution results if a random variable is the product of a large number of independent, identically-distributed variables in the same way that a normal distribution results if the variable is the sum of a large number of independent, identically-distributed variables.

[1]
Limpert, E., Stahel, W. A., and Abbt, M., "Log-normal Distributions across the Sciences: Keys and Clues," BioScience, Vol. 51, No. 5, May, 2001. https://stat.ethz.ch/~stahel/lognormal/bioscience.pdf
[2]
Reiss, R.D. and Thomas, M., "Statistical Analysis of Extreme Values," Basel: Birkhauser Verlag, 2001, pp. 31-32.

Draw samples from the distribution:

>>> mu, sigma = 3., 1. # mean and standard deviation
>>> s = np.random.lognormal(mu, sigma, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 100, density=True, align='mid')
    

>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
...        / (x * sigma * np.sqrt(2 * np.pi)))
    

>>> plt.plot(x, pdf, linewidth=2, color='r')
>>> plt.axis('tight')
>>> plt.show()

Demonstrate that taking the products of random samples from a uniform distribution can be fit well by a log-normal probability density function.

>>> # Generate a thousand samples: each is the product of 100 random
>>> # values, drawn from a normal distribution.
>>> b = []
>>> for i in range(1000):
...    a = 10. + np.random.standard_normal(100)
...    b.append(np.product(a))
    

>>> b = np.array(b) / np.min(b) # scale values to be positive
>>> count, bins, ignored = plt.hist(b, 100, density=True, align='mid')
>>> sigma = np.std(np.log(b))
>>> mu = np.mean(np.log(b))
    

>>> x = np.linspace(min(bins), max(bins), 10000)
>>> pdf = (np.exp(-(np.log(x) - mu)**2 / (2 * sigma**2))
...        / (x * sigma * np.sqrt(2 * np.pi)))
    

>>> plt.plot(x, pdf, color='r', linewidth=2)
>>> plt.show()

Draw samples from a logarithmic series distribution.

Samples are drawn from a log series distribution with specified shape parameter, 0 < p < 1.

NOTE:

New code should use the logseries method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Shape parameter for the distribution. Must be in the range (0, 1).
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if p is a scalar. Otherwise, np.array(p).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized logarithmic series distribution.

probability density function, distribution or cumulative density function, etc.

Generator.logseries: which should be used for new code.

The probability density for the Log Series distribution is


P(k) = \frac{-p^k}{k \ln(1-p)},



where p = probability.

The log series distribution is frequently used to represent species richness and occurrence, first proposed by Fisher, Corbet, and Williams in 1943 [2]. It may also be used to model the numbers of occupants seen in cars [3].

[1]
Buzas, Martin A.; Culver, Stephen J., Understanding regional species diversity through the log series distribution of occurrences: BIODIVERSITY RESEARCH Diversity & Distributions, Volume 5, Number 5, September 1999 , pp. 187-195(9).
[2]
Fisher, R.A,, A.S. Corbet, and C.B. Williams. 1943. The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12:42-58.
[3]
D. J. Hand, F. Daly, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, CRC Press, 1994.
[4]
Wikipedia, "Logarithmic distribution", https://en.wikipedia.org/wiki/Logarithmic_distribution

Draw samples from the distribution:

>>> a = .6
>>> s = np.random.logseries(a, 10000)
>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s)

# plot against distribution

>>> def logseries(k, p):
...     return -p**k/(k*np.log(1-p))
>>> plt.plot(bins, logseries(bins, a)*count.max()/
...          logseries(bins, a).max(), 'r')
>>> plt.show()

Draw samples from a multinomial distribution.

The multinomial distribution is a multivariate generalization of the binomial distribution. Take an experiment with one of p possible outcomes. An example of such an experiment is throwing a dice, where the outcome can be 1 through 6. Each sample drawn from the distribution represents n such experiments. Its values, X_i = [X_0, X_1, ..., X_p], represent the number of times the outcome was i.

NOTE:

New code should use the multinomial method of a default_rng() instance instead; please see the random-quick-start.


int Number of experiments.
sequence of floats, length p Probabilities of each of the p different outcomes. These must sum to 1 (however, the last element is always assumed to account for the remaining probability, as long as sum(pvals[:-1]) <= 1).
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.

ndarray The drawn samples, of shape size, if that was provided. If not, the shape is (N,).

In other words, each entry out[i,j,...,:] is an N-dimensional value drawn from the distribution.


Generator.multinomial: which should be used for new code.

Throw a dice 20 times:

>>> np.random.multinomial(20, [1/6.]*6, size=1)
array([[4, 1, 7, 5, 2, 1]]) # random

It landed 4 times on 1, once on 2, etc.

Now, throw the dice 20 times, and 20 times again:

>>> np.random.multinomial(20, [1/6.]*6, size=2)
array([[3, 4, 3, 3, 4, 3], # random
       [2, 4, 3, 4, 0, 7]])

For the first run, we threw 3 times 1, 4 times 2, etc. For the second, we threw 2 times 1, 4 times 2, etc.

A loaded die is more likely to land on number 6:

>>> np.random.multinomial(100, [1/7.]*5 + [2/7.])
array([11, 16, 14, 17, 16, 26]) # random

The probability inputs should be normalized. As an implementation detail, the value of the last entry is ignored and assumed to take up any leftover probability mass, but this should not be relied on. A biased coin which has twice as much weight on one side as on the other should be sampled like so:

>>> np.random.multinomial(100, [1.0 / 3, 2.0 / 3])  # RIGHT
array([38, 62]) # random

not like:

>>> np.random.multinomial(100, [1.0, 2.0])  # WRONG
Traceback (most recent call last):
ValueError: pvals < 0, pvals > 1 or pvals contains NaNs

Draw random samples from a multivariate normal distribution.

The multivariate normal, multinormal or Gaussian distribution is a generalization of the one-dimensional normal distribution to higher dimensions. Such a distribution is specified by its mean and covariance matrix. These parameters are analogous to the mean (average or "center") and variance (standard deviation, or "width," squared) of the one-dimensional normal distribution.

NOTE:

New code should use the multivariate_normal method of a default_rng() instance instead; please see the random-quick-start.


1-D array_like, of length N Mean of the N-dimensional distribution.
2-D array_like, of shape (N, N) Covariance matrix of the distribution. It must be symmetric and positive-semidefinite for proper sampling.
int or tuple of ints, optional Given a shape of, for example, (m,n,k), m*n*k samples are generated, and packed in an m-by-n-by-k arrangement. Because each sample is N-dimensional, the output shape is (m,n,k,N). If no shape is specified, a single (N-D) sample is returned.
{ 'warn', 'raise', 'ignore' }, optional Behavior when the covariance matrix is not positive semidefinite.
float, optional Tolerance when checking the singular values in covariance matrix. cov is cast to double before the check.

ndarray The drawn samples, of shape size, if that was provided. If not, the shape is (N,).

In other words, each entry out[i,j,...,:] is an N-dimensional value drawn from the distribution.


Generator.multivariate_normal: which should be used for new code.

The mean is a coordinate in N-dimensional space, which represents the location where samples are most likely to be generated. This is analogous to the peak of the bell curve for the one-dimensional or univariate normal distribution.

Covariance indicates the level to which two variables vary together. From the multivariate normal distribution, we draw N-dimensional samples, X = [x_1, x_2, ... x_N]. The covariance matrix element C_{ij} is the covariance of x_i and x_j. The element C_{ii} is the variance of x_i (i.e. its "spread").

Instead of specifying the full covariance matrix, popular approximations include:

  • Spherical covariance (cov is a multiple of the identity matrix)
  • Diagonal covariance (cov has non-negative elements, and only on the diagonal)



This geometrical property can be seen in two dimensions by plotting generated data-points:

>>> mean = [0, 0]
>>> cov = [[1, 0], [0, 100]]  # diagonal covariance

Diagonal covariance means that points are oriented along x or y-axis:

>>> import matplotlib.pyplot as plt
>>> x, y = np.random.multivariate_normal(mean, cov, 5000).T
>>> plt.plot(x, y, 'x')
>>> plt.axis('equal')
>>> plt.show()

Note that the covariance matrix must be positive semidefinite (a.k.a. nonnegative-definite). Otherwise, the behavior of this method is undefined and backwards compatibility is not guaranteed.

[1]
Papoulis, A., "Probability, Random Variables, and Stochastic Processes," 3rd ed., New York: McGraw-Hill, 1991.
[2]
Duda, R. O., Hart, P. E., and Stork, D. G., "Pattern Classification," 2nd ed., New York: Wiley, 2001. 

>>> mean = (1, 2)
>>> cov = [[1, 0], [0, 1]]
>>> x = np.random.multivariate_normal(mean, cov, (3, 3))
>>> x.shape
(3, 3, 2)

The following is probably true, given that 0.6 is roughly twice the standard deviation:

>>> list((x[0,0,:] - mean) < 0.6)
[True, True] # random

Draw samples from a negative binomial distribution.

Samples are drawn from a negative binomial distribution with specified parameters, n successes and p probability of success where n is > 0 and p is in the interval [0, 1].

NOTE:

New code should use the negative_binomial method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Parameter of the distribution, > 0.
float or array_like of floats Parameter of the distribution, >= 0 and <=1.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if n and p are both scalars. Otherwise, np.broadcast(n, p).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized negative binomial distribution, where each sample is equal to N, the number of failures that occurred before a total of n successes was reached.

Generator.negative_binomial: which should be used for new code.

The probability mass function of the negative binomial distribution is


P(N;n,p) = \frac{\Gamma(N+n)}{N!\Gamma(n)}p^{n}(1-p)^{N},



where n is the number of successes, p is the probability of success, N+n is the number of trials, and \Gamma is the gamma function. When n is an integer, \frac{\Gamma(N+n)}{N!\Gamma(n)} = \binom{N+n-1}{N}, which is the more common form of this term in the the pmf. The negative binomial distribution gives the probability of N failures given n successes, with a success on the last trial.

If one throws a die repeatedly until the third time a "1" appears, then the probability distribution of the number of non-"1"s that appear before the third "1" is a negative binomial distribution.

[1]
Weisstein, Eric W. "Negative Binomial Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NegativeBinomialDistribution.html
[2]
Wikipedia, "Negative binomial distribution", https://en.wikipedia.org/wiki/Negative_binomial_distribution

Draw samples from the distribution:

A real world example. A company drills wild-cat oil exploration wells, each with an estimated probability of success of 0.1. What is the probability of having one success for each successive well, that is what is the probability of a single success after drilling 5 wells, after 6 wells, etc.?

>>> s = np.random.negative_binomial(1, 0.1, 100000)
>>> for i in range(1, 11): 
...    probability = sum(s<i) / 100000.
...    print(i, "wells drilled, probability of one success =", probability)

Draw samples from a noncentral chi-square distribution.

The noncentral \chi^2 distribution is a generalization of the \chi^2 distribution.

NOTE:

New code should use the noncentral_chisquare method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Degrees of freedom, must be > 0.

Changed in version 1.10.0: Earlier NumPy versions required dfnum > 1.

float or array_like of floats Non-centrality, must be non-negative.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if df and nonc are both scalars. Otherwise, np.broadcast(df, nonc).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized noncentral chi-square distribution.

Generator.noncentral_chisquare: which should be used for new code.

The probability density function for the noncentral Chi-square distribution is


P(x;df,nonc) = \sum^{\infty}_{i=0}
\frac{e^{-nonc/2}(nonc/2)^{i}}{i!} P_{Y_{df+2i}}(x),


where Y_{q} is the Chi-square with q degrees of freedom.

[1]
Wikipedia, "Noncentral chi-squared distribution" https://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution

Draw values from the distribution and plot the histogram

>>> import matplotlib.pyplot as plt
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
...                   bins=200, density=True)
>>> plt.show()

Draw values from a noncentral chisquare with very small noncentrality, and compare to a chisquare.

>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, .0000001, 100000),
...                   bins=np.arange(0., 25, .1), density=True)
>>> values2 = plt.hist(np.random.chisquare(3, 100000),
...                    bins=np.arange(0., 25, .1), density=True)
>>> plt.plot(values[1][0:-1], values[0]-values2[0], 'ob')
>>> plt.show()

Demonstrate how large values of non-centrality lead to a more symmetric distribution.

>>> plt.figure()
>>> values = plt.hist(np.random.noncentral_chisquare(3, 20, 100000),
...                   bins=200, density=True)
>>> plt.show()

Draw samples from the noncentral F distribution.

Samples are drawn from an F distribution with specified parameters, dfnum (degrees of freedom in numerator) and dfden (degrees of freedom in denominator), where both parameters > 1. nonc is the non-centrality parameter.

NOTE:

New code should use the noncentral_f method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Numerator degrees of freedom, must be > 0.

Changed in version 1.14.0: Earlier NumPy versions required dfnum > 1.

float or array_like of floats Denominator degrees of freedom, must be > 0.
float or array_like of floats Non-centrality parameter, the sum of the squares of the numerator means, must be >= 0.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if dfnum, dfden, and nonc are all scalars. Otherwise, np.broadcast(dfnum, dfden, nonc).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized noncentral Fisher distribution.

Generator.noncentral_f: which should be used for new code.

When calculating the power of an experiment (power = probability of rejecting the null hypothesis when a specific alternative is true) the non-central F statistic becomes important. When the null hypothesis is true, the F statistic follows a central F distribution. When the null hypothesis is not true, then it follows a non-central F statistic.

[1]
Weisstein, Eric W. "Noncentral F-Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NoncentralF-Distribution.html
[2]
Wikipedia, "Noncentral F-distribution", https://en.wikipedia.org/wiki/Noncentral_F-distribution

In a study, testing for a specific alternative to the null hypothesis requires use of the Noncentral F distribution. We need to calculate the area in the tail of the distribution that exceeds the value of the F distribution for the null hypothesis. We'll plot the two probability distributions for comparison.

>>> dfnum = 3 # between group deg of freedom
>>> dfden = 20 # within groups degrees of freedom
>>> nonc = 3.0
>>> nc_vals = np.random.noncentral_f(dfnum, dfden, nonc, 1000000)
>>> NF = np.histogram(nc_vals, bins=50, density=True)
>>> c_vals = np.random.f(dfnum, dfden, 1000000)
>>> F = np.histogram(c_vals, bins=50, density=True)
>>> import matplotlib.pyplot as plt
>>> plt.plot(F[1][1:], F[0])
>>> plt.plot(NF[1][1:], NF[0])
>>> plt.show()

Draw random samples from a normal (Gaussian) distribution.

The probability density function of the normal distribution, first derived by De Moivre and 200 years later by both Gauss and Laplace independently

[2]_
, is often called the bell curve because of its characteristic shape (see the example below).

The normal distributions occurs often in nature. For example, it describes the commonly occurring distribution of samples influenced by a large number of tiny, random disturbances, each with its own unique distribution

[2]_
.

NOTE:

New code should use the normal method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Mean ("centre") of the distribution.
float or array_like of floats Standard deviation (spread or "width") of the distribution. Must be non-negative.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if loc and scale are both scalars. Otherwise, np.broadcast(loc, scale).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized normal distribution.

probability density function, distribution or cumulative density function, etc.

Generator.normal: which should be used for new code.

The probability density for the Gaussian distribution is


p(x) = \frac{1}{\sqrt{ 2 \pi \sigma^2 }}
e^{ - \frac{ (x - \mu)^2 } {2 \sigma^2} },


where \mu is the mean and \sigma the standard deviation. The square of the standard deviation, \sigma^2, is called the variance.

The function has its peak at the mean, and its "spread" increases with the standard deviation (the function reaches 0.607 times its maximum at x + \sigma and x - \sigma

[2]_
). This implies that normal is more likely to return samples lying close to the mean, rather than those far away.
[1]
Wikipedia, "Normal distribution", https://en.wikipedia.org/wiki/Normal_distribution
[2]
P. R. Peebles Jr., "Central Limit Theorem" in "Probability, Random Variables and Random Signal Principles", 4th ed., 2001, pp. 51, 51, 125.

Draw samples from the distribution:

>>> mu, sigma = 0, 0.1 # mean and standard deviation
>>> s = np.random.normal(mu, sigma, 1000)

Verify the mean and the variance:

>>> abs(mu - np.mean(s))
0.0  # may vary
    

>>> abs(sigma - np.std(s, ddof=1))
0.1  # may vary

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 30, density=True)
>>> plt.plot(bins, 1/(sigma * np.sqrt(2 * np.pi)) *
...                np.exp( - (bins - mu)**2 / (2 * sigma**2) ),
...          linewidth=2, color='r')
>>> plt.show()

Two-by-four array of samples from N(3, 6.25):

>>> np.random.normal(3, 2.5, size=(2, 4))
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random

Draw samples from a Pareto II or Lomax distribution with specified shape.

The Lomax or Pareto II distribution is a shifted Pareto distribution. The classical Pareto distribution can be obtained from the Lomax distribution by adding 1 and multiplying by the scale parameter m (see Notes). The smallest value of the Lomax distribution is zero while for the classical Pareto distribution it is mu, where the standard Pareto distribution has location mu = 1. Lomax can also be considered as a simplified version of the Generalized Pareto distribution (available in SciPy), with the scale set to one and the location set to zero.

The Pareto distribution must be greater than zero, and is unbounded above. It is also known as the "80-20 rule". In this distribution, 80 percent of the weights are in the lowest 20 percent of the range, while the other 20 percent fill the remaining 80 percent of the range.

NOTE:

New code should use the pareto method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Shape of the distribution. Must be positive.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized Pareto distribution.

probability density function, distribution or cumulative density function, etc.
probability density function, distribution or cumulative density function, etc.

Generator.pareto: which should be used for new code.

The probability density for the Pareto distribution is


p(x) = \frac{am^a}{x^{a+1}}



where a is the shape and m the scale.

The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution useful in many real world problems. Outside the field of economics it is generally referred to as the Bradford distribution. Pareto developed the distribution to describe the distribution of wealth in an economy. It has also found use in insurance, web page access statistics, oil field sizes, and many other problems, including the download frequency for projects in Sourceforge

[1]_
. It is one of the so-called "fat-tailed" distributions.
[1]
Francis Hunt and Paul Johnson, On the Pareto Distribution of Sourceforge projects.
[2]
Pareto, V. (1896). Course of Political Economy. Lausanne.
[3]
Reiss, R.D., Thomas, M.(2001), Statistical Analysis of Extreme Values, Birkhauser Verlag, Basel, pp 23-30.
[4]
Wikipedia, "Pareto distribution", https://en.wikipedia.org/wiki/Pareto_distribution

Draw samples from the distribution:

>>> a, m = 3., 2.  # shape and mode
>>> s = (np.random.pareto(a, 1000) + 1) * m

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, _ = plt.hist(s, 100, density=True)
>>> fit = a*m**a / bins**(a+1)
>>> plt.plot(bins, max(count)*fit/max(fit), linewidth=2, color='r')
>>> plt.show()

parseOCEANinputFile

Parses an OCEAN input file and returns lattice vectors, atom names, and relative atom positions.

  • fname (str): Absolute filename of OCEAN input file.
  • atoms (list): List of elemental symbols in the same order as they appear in the input file.

  • lattice (np.array): Array of lattice vectors.
  • rel_coords (np.array): Array of relative atomic coordinates.
  • oceaatoms (list): List of atomic names.



parsePwscfFile

Parses a PWSCF file and returns a xyzBox object.

fname (str): Absolute filename of OCEAN input file.
xyzBox object


parseVaspFile

Parses a VASPS file and returns a xyzBox object.

fname (str): Absolute filename of VASP file.
xyzBox object


parseXYZfile Reads an xyz-style file.

Randomly permute a sequence, or return a permuted range.

If x is a multi-dimensional array, it is only shuffled along its first index.

NOTE:

New code should use the permutation method of a default_rng() instance instead; please see the random-quick-start.


int or array_like If x is an integer, randomly permute np.arange(x). If x is an array, make a copy and shuffle the elements randomly.

ndarray Permuted sequence or array range.

Generator.permutation: which should be used for new code.

>>> np.random.permutation(10)
array([1, 7, 4, 3, 0, 9, 2, 5, 8, 6]) # random


>>> np.random.permutation([1, 4, 9, 12, 15])
array([15,  1,  9,  4, 12]) # random


>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.permutation(arr)
array([[6, 7, 8], # random
       [0, 1, 2],
       [3, 4, 5]])

Draw samples from a Poisson distribution.

The Poisson distribution is the limit of the binomial distribution for large N.

NOTE:

New code should use the poisson method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Expected number of events occurring in a fixed-time interval, must be >= 0. A sequence must be broadcastable over the requested size.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if lam is a scalar. Otherwise, np.array(lam).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized Poisson distribution.

Generator.poisson: which should be used for new code.

The Poisson distribution


f(k; \lambda)=\frac{\lambda^k e^{-\lambda}}{k!}



For events with an expected separation \lambda the Poisson distribution f(k; \lambda) describes the probability of k events occurring within the observed interval \lambda.

Because the output is limited to the range of the C int64 type, a ValueError is raised when lam is within 10 sigma of the maximum representable value.

[1]
Weisstein, Eric W. "Poisson Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PoissonDistribution.html
[2]
Wikipedia, "Poisson distribution", https://en.wikipedia.org/wiki/Poisson_distribution

Draw samples from the distribution:

>>> import numpy as np
>>> s = np.random.poisson(5, 10000)

Display histogram of the sample:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 14, density=True)
>>> plt.show()

Draw each 100 values for lambda 100 and 500:

>>> s = np.random.poisson(lam=(100., 500.), size=(100, 2))

Draws samples in [0, 1] from a power distribution with positive exponent a - 1.

Also known as the power function distribution.

NOTE:

New code should use the power method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Parameter of the distribution. Must be non-negative.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized power distribution.

If a < 1.

Generator.power: which should be used for new code.

The probability density function is


P(x; a) = ax^{a-1}, 0 \le x \le 1, a>0.



The power function distribution is just the inverse of the Pareto distribution. It may also be seen as a special case of the Beta distribution.

It is used, for example, in modeling the over-reporting of insurance claims.

[1]
Christian Kleiber, Samuel Kotz, "Statistical size distributions in economics and actuarial sciences", Wiley, 2003.
[2]
Heckert, N. A. and Filliben, James J. "NIST Handbook 148: Dataplot Reference Manual, Volume 2: Let Subcommands and Library Functions", National Institute of Standards and Technology Handbook Series, June 2003. https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/powpdf.pdf

Draw samples from the distribution:

>>> a = 5. # shape
>>> samples = 1000
>>> s = np.random.power(a, samples)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, bins=30)
>>> x = np.linspace(0, 1, 100)
>>> y = a*x**(a-1.)
>>> normed_y = samples*np.diff(bins)[0]*y
>>> plt.plot(x, normed_y)
>>> plt.show()

Compare the power function distribution to the inverse of the Pareto.

>>> from scipy import stats 
>>> rvs = np.random.power(5, 1000000)
>>> rvsp = np.random.pareto(5, 1000000)
>>> xx = np.linspace(0,1,100)
>>> powpdf = stats.powerlaw.pdf(xx,5)  
    

>>> plt.figure()
>>> plt.hist(rvs, bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  
>>> plt.title('np.random.power(5)')
    

>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  
>>> plt.title('inverse of 1 + np.random.pareto(5)')
    

>>> plt.figure()
>>> plt.hist(1./(1.+rvsp), bins=50, density=True)
>>> plt.plot(xx,powpdf,'r-')  
>>> plt.title('inverse of stats.pareto(5)')

Random values in a given shape.

NOTE:

This is a convenience function for users porting code from Matlab, and wraps random_sample. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.


Create an array of the given shape and populate it with random samples from a uniform distribution over [0, 1).

int, optional The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.

ndarray, shape (d0, d1, ..., dn) Random values.

random

>>> np.random.rand(3,2)
array([[ 0.14022471,  0.96360618],  #random
       [ 0.37601032,  0.25528411],  #random
       [ 0.49313049,  0.94909878]]) #random

Return random integers from low (inclusive) to high (exclusive).

Return random integers from the "discrete uniform" distribution of the specified dtype in the "half-open" interval [low, high). If high is None (the default), then results are from [0, low).

NOTE:

New code should use the integers method of a default_rng() instance instead; please see the random-quick-start.


int or array-like of ints Lowest (signed) integers to be drawn from the distribution (unless high=None, in which case this parameter is one above the highest such integer).
int or array-like of ints, optional If provided, one above the largest (signed) integer to be drawn from the distribution (see above for behavior if high=None). If array-like, must contain integer values
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.
dtype, optional Desired dtype of the result. Byteorder must be native. The default value is int.

New in version 1.11.0.


int or ndarray of ints size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.

similar to randint, only for the closed interval [low, high], and 1 is the lowest value if high is omitted.

Generator.integers: which should be used for new code.

>>> np.random.randint(2, size=10)
array([1, 0, 0, 0, 1, 1, 0, 0, 1, 0]) # random
>>> np.random.randint(1, size=10)
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0])

Generate a 2 x 4 array of ints between 0 and 4, inclusive:

>>> np.random.randint(5, size=(2, 4))
array([[4, 0, 2, 1], # random
       [3, 2, 2, 0]])

Generate a 1 x 3 array with 3 different upper bounds

>>> np.random.randint(1, [3, 5, 10])
array([2, 2, 9]) # random

Generate a 1 by 3 array with 3 different lower bounds

>>> np.random.randint([1, 5, 7], 10)
array([9, 8, 7]) # random

Generate a 2 by 4 array using broadcasting with dtype of uint8

>>> np.random.randint([1, 3, 5, 7], [[10], [20]], dtype=np.uint8)
array([[ 8,  6,  9,  7], # random
       [ 1, 16,  9, 12]], dtype=uint8)

Return a sample (or samples) from the "standard normal" distribution.

NOTE:

This is a convenience function for users porting code from Matlab, and wraps standard_normal. That function takes a tuple to specify the size of the output, which is consistent with other NumPy functions like numpy.zeros and numpy.ones.


NOTE:

New code should use the standard_normal method of a default_rng() instance instead; please see the random-quick-start.


If positive int_like arguments are provided, randn generates an array of shape (d0, d1, ..., dn), filled with random floats sampled from a univariate "normal" (Gaussian) distribution of mean 0 and variance 1. A single float randomly sampled from the distribution is returned if no argument is provided.

int, optional The dimensions of the returned array, must be non-negative. If no argument is given a single Python float is returned.

ndarray or float A (d0, d1, ..., dn)-shaped array of floating-point samples from the standard normal distribution, or a single such float if no parameters were supplied.

standard_normal : Similar, but takes a tuple as its argument. normal : Also accepts mu and sigma arguments. Generator.standard_normal: which should be used for new code.

For random samples from N(\mu, \sigma^2), use:

sigma * np.random.randn(...) + mu

>>> np.random.randn()
2.1923875335537315  # random

Two-by-four array of samples from N(3, 6.25):

>>> 3 + 2.5 * np.random.randn(2, 4)
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random

Return random floats in the half-open interval [0.0, 1.0). Alias for random_sample to ease forward-porting to the new random API.

Random integers of type np.int_ between low and high, inclusive.

Return random integers of type np.int_ from the "discrete uniform" distribution in the closed interval [low, high]. If high is None (the default), then results are from [1, low]. The np.int_ type translates to the C long integer type and its precision is platform dependent.

This function has been deprecated. Use randint instead.

Deprecated since version 1.11.0.

int Lowest (signed) integer to be drawn from the distribution (unless high=None, in which case this parameter is the highest such integer).
int, optional If provided, the largest (signed) integer to be drawn from the distribution (see above for behavior if high=None).
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.

int or ndarray of ints size-shaped array of random integers from the appropriate distribution, or a single such random int if size not provided.

Similar to random_integers, only for the half-open interval [low, high), and 0 is the lowest value if high is omitted.

To sample from N evenly spaced floating-point numbers between a and b, use:

a + (b - a) * (np.random.random_integers(N) - 1) / (N - 1.)



>>> np.random.random_integers(5)
4 # random
>>> type(np.random.random_integers(5))
<class 'numpy.int64'>
>>> np.random.random_integers(5, size=(3,2))
array([[5, 4], # random
       [3, 3],
       [4, 5]])

Choose five random numbers from the set of five evenly-spaced numbers between 0 and 2.5, inclusive (i.e., from the set {0, 5/8, 10/8, 15/8, 20/8}):

>>> 2.5 * (np.random.random_integers(5, size=(5,)) - 1) / 4.
array([ 0.625,  1.25 ,  0.625,  0.625,  2.5  ]) # random

Roll two six sided dice 1000 times and sum the results:

>>> d1 = np.random.random_integers(1, 6, 1000)
>>> d2 = np.random.random_integers(1, 6, 1000)
>>> dsums = d1 + d2

Display results as a histogram:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(dsums, 11, density=True)
>>> plt.show()

Return random floats in the half-open interval [0.0, 1.0).

Results are from the "continuous uniform" distribution over the stated interval. To sample Unif[a, b), b > a multiply the output of random_sample by (b-a) and add a:

(b - a) * random_sample() + a


NOTE:

New code should use the random method of a default_rng() instance instead; please see the random-quick-start.


int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.

float or ndarray of floats Array of random floats of shape size (unless size=None, in which case a single float is returned).

Generator.random: which should be used for new code.

>>> np.random.random_sample()
0.47108547995356098 # random
>>> type(np.random.random_sample())
<class 'float'>
>>> np.random.random_sample((5,))
array([ 0.30220482,  0.86820401,  0.1654503 ,  0.11659149,  0.54323428]) # random

Three-by-two array of random numbers from [-5, 0):

>>> 5 * np.random.random_sample((3, 2)) - 5
array([[-3.99149989, -0.52338984], # random
       [-2.99091858, -0.79479508],
       [-1.23204345, -1.75224494]])

Draw samples from a Rayleigh distribution.

The \chi and Weibull distributions are generalizations of the Rayleigh.

NOTE:

New code should use the rayleigh method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats, optional Scale, also equals the mode. Must be non-negative. Default is 1.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if scale is a scalar. Otherwise, np.array(scale).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized Rayleigh distribution.

Generator.rayleigh: which should be used for new code.

The probability density function for the Rayleigh distribution is


P(x;scale) = \frac{x}{scale^2}e^{\frac{-x^2}{2 \cdotp scale^2}}



The Rayleigh distribution would arise, for example, if the East and North components of the wind velocity had identical zero-mean Gaussian distributions. Then the wind speed would have a Rayleigh distribution.

[1]
Brighton Webs Ltd., "Rayleigh Distribution," https://web.archive.org/web/20090514091424/http://brighton-webs.co.uk:80/distributions/rayleigh.asp
[2]
Wikipedia, "Rayleigh distribution" https://en.wikipedia.org/wiki/Rayleigh_distribution

Draw values from the distribution and plot the histogram

>>> from matplotlib.pyplot import hist
>>> values = hist(np.random.rayleigh(3, 100000), bins=200, density=True)

Wave heights tend to follow a Rayleigh distribution. If the mean wave height is 1 meter, what fraction of waves are likely to be larger than 3 meters?

>>> meanvalue = 1
>>> modevalue = np.sqrt(2 / np.pi) * meanvalue
>>> s = np.random.rayleigh(modevalue, 1000000)

The percentage of waves larger than 3 meters is:

>>> 100.*sum(s>3)/1000000.
0.087300000000000003 # random

function output = readxas(filename)%[e,p,s,px,py,pz] = readxas(filename)

% READSTF Load StoBe fort.11 (XAS output) data % % [E,P,S,PX,PY,PZ] = READXAS(FILENAME) % % E energy transfer [eV] % P dipole transition intensity % S r^2 transition intensity % PX dipole transition intensity along x % PY dipole transition intensity along y % PZ dipole transition intensity along z % % as line diagrams. % % T Pylkkanen @ 2011-10-17



Reseed a legacy MT19937 BitGenerator

This is a convenience, legacy function.

The best practice is to not reseed a BitGenerator, rather to recreate a new one. This method is here for legacy reasons. This example demonstrates best practice.

>>> from numpy.random import MT19937
>>> from numpy.random import RandomState, SeedSequence
>>> rs = RandomState(MT19937(SeedSequence(123456789)))
# Later, you want to restart the stream
>>> rs = RandomState(MT19937(SeedSequence(987654321)))

Set the internal state of the generator from a tuple.

For use if one has reason to manually (re-)set the internal state of the bit generator used by the RandomState instance. By default, RandomState uses the "Mersenne Twister"

[1]_

pseudo-random number generating algorithm.
{tuple(str, ndarray of 624 uints, int, int, float), dict} The state tuple has the following items:
1.
the string 'MT19937', specifying the Mersenne Twister algorithm.
2.
a 1-D array of 624 unsigned integers keys.
3.
an integer pos.
4.
an integer has_gauss.
5.
a float cached_gaussian.

If state is a dictionary, it is directly set using the BitGenerators state property.


None Returns 'None' on success.

get_state

set_state and get_state are not needed to work with any of the random distributions in NumPy. If the internal state is manually altered, the user should know exactly what he/she is doing.

For backwards compatibility, the form (str, array of 624 uints, int) is also accepted although it is missing some information about the cached Gaussian value: state = ('MT19937', keys, pos).

[1]
M. Matsumoto and T. Nishimura, "Mersenne Twister: A 623-dimensionally equidistributed uniform pseudorandom number generator," ACM Trans. on Modeling and Computer Simulation, Vol. 8, No. 1, pp. 3-30, Jan. 1998.

Modify a sequence in-place by shuffling its contents.

This function only shuffles the array along the first axis of a multi-dimensional array. The order of sub-arrays is changed but their contents remains the same.

NOTE:

New code should use the shuffle method of a default_rng() instance instead; please see the random-quick-start.


ndarray or MutableSequence The array, list or mutable sequence to be shuffled.

None

Generator.shuffle: which should be used for new code.

>>> arr = np.arange(10)
>>> np.random.shuffle(arr)
>>> arr
[1 7 5 2 9 4 3 6 0 8] # random

Multi-dimensional arrays are only shuffled along the first axis:

>>> arr = np.arange(9).reshape((3, 3))
>>> np.random.shuffle(arr)
>>> arr
array([[3, 4, 5], # random
       [6, 7, 8],
       [0, 1, 2]])


Extrapolates the smaller and larger valuea as a constant

Draw samples from a standard Cauchy distribution with mode = 0.

Also known as the Lorentz distribution.

NOTE:

New code should use the standard_cauchy method of a default_rng() instance instead; please see the random-quick-start.


int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.

ndarray or scalar The drawn samples.

Generator.standard_cauchy: which should be used for new code.

The probability density function for the full Cauchy distribution is


P(x; x_0, \gamma) = \frac{1}{\pi \gamma \bigl[ 1+
(\frac{x-x_0}{\gamma})^2 \bigr] }


and the Standard Cauchy distribution just sets x_0=0 and \gamma=1

The Cauchy distribution arises in the solution to the driven harmonic oscillator problem, and also describes spectral line broadening. It also describes the distribution of values at which a line tilted at a random angle will cut the x axis.

When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of their sensitivity to a heavy-tailed distribution, since the Cauchy looks very much like a Gaussian distribution, but with heavier tails.

[1]
NIST/SEMATECH e-Handbook of Statistical Methods, "Cauchy Distribution", https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm
[2]
Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CauchyDistribution.html
[3]
Wikipedia, "Cauchy distribution" https://en.wikipedia.org/wiki/Cauchy_distribution

Draw samples and plot the distribution:

>>> import matplotlib.pyplot as plt
>>> s = np.random.standard_cauchy(1000000)
>>> s = s[(s>-25) & (s<25)]  # truncate distribution so it plots well
>>> plt.hist(s, bins=100)
>>> plt.show()

Draw samples from the standard exponential distribution.

standard_exponential is identical to the exponential distribution with a scale parameter of 1.

NOTE:

New code should use the standard_exponential method of a default_rng() instance instead; please see the random-quick-start.


int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.

float or ndarray Drawn samples.

Generator.standard_exponential: which should be used for new code.

Output a 3x8000 array:

>>> n = np.random.standard_exponential((3, 8000))

Draw samples from a standard Gamma distribution.

Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated "k") and scale=1.

NOTE:

New code should use the standard_gamma method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Parameter, must be non-negative.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if shape is a scalar. Otherwise, np.array(shape).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized standard gamma distribution.

probability density function, distribution or cumulative density function, etc.

Generator.standard_gamma: which should be used for new code.

The probability density for the Gamma distribution is


p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},



where k is the shape and \theta the scale, and \Gamma is the Gamma function.

The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant.

[1]
Weisstein, Eric W. "Gamma Distribution." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html
[2]
Wikipedia, "Gamma distribution", https://en.wikipedia.org/wiki/Gamma_distribution

Draw samples from the distribution:

>>> shape, scale = 2., 1. # mean and width
>>> s = np.random.standard_gamma(shape, 1000000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps  
>>> count, bins, ignored = plt.hist(s, 50, density=True)
>>> y = bins**(shape-1) * ((np.exp(-bins/scale))/  
...                       (sps.gamma(shape) * scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')  
>>> plt.show()

Draw samples from a standard Normal distribution (mean=0, stdev=1).

NOTE:

New code should use the standard_normal method of a default_rng() instance instead; please see the random-quick-start.


int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.

float or ndarray A floating-point array of shape size of drawn samples, or a single sample if size was not specified.

Equivalent function with additional loc and scale arguments for setting the mean and standard deviation.

Generator.standard_normal: which should be used for new code.

For random samples from N(\mu, \sigma^2), use one of:

mu + sigma * np.random.standard_normal(size=...)
np.random.normal(mu, sigma, size=...)



>>> np.random.standard_normal()
2.1923875335537315 #random


>>> s = np.random.standard_normal(8000)
>>> s
array([ 0.6888893 ,  0.78096262, -0.89086505, ...,  0.49876311,  # random
       -0.38672696, -0.4685006 ])                                # random
>>> s.shape
(8000,)
>>> s = np.random.standard_normal(size=(3, 4, 2))
>>> s.shape
(3, 4, 2)

Two-by-four array of samples from N(3, 6.25):

>>> 3 + 2.5 * np.random.standard_normal(size=(2, 4))
array([[-4.49401501,  4.00950034, -1.81814867,  7.29718677],   # random
       [ 0.39924804,  4.68456316,  4.99394529,  4.84057254]])  # random

Draw samples from a standard Student's t distribution with df degrees of freedom.

A special case of the hyperbolic distribution. As df gets large, the result resembles that of the standard normal distribution (standard_normal).

NOTE:

New code should use the standard_t method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Degrees of freedom, must be > 0.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if df is a scalar. Otherwise, np.array(df).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized standard Student's t distribution.

Generator.standard_t: which should be used for new code.

The probability density function for the t distribution is


P(x, df) = \frac{\Gamma(\frac{df+1}{2})}{\sqrt{\pi df}
\Gamma(\frac{df}{2})}\Bigl( 1+\frac{x^2}{df} \Bigr)^{-(df+1)/2}


The t test is based on an assumption that the data come from a Normal distribution. The t test provides a way to test whether the sample mean (that is the mean calculated from the data) is a good estimate of the true mean.

The derivation of the t-distribution was first published in 1908 by William Gosset while working for the Guinness Brewery in Dublin. Due to proprietary issues, he had to publish under a pseudonym, and so he used the name Student.

[1]
Dalgaard, Peter, "Introductory Statistics With R", Springer, 2002.
[2]
Wikipedia, "Student's t-distribution" https://en.wikipedia.org/wiki/Student's_t-distribution

From Dalgaard page 83

[1]_
, suppose the daily energy intake for 11 women in kilojoules (kJ) is: 

>>> intake = np.array([5260., 5470, 5640, 6180, 6390, 6515, 6805, 7515, \
...                    7515, 8230, 8770])

Does their energy intake deviate systematically from the recommended value of 7725 kJ? Our null hypothesis will be the absence of deviation, and the alternate hypothesis will be the presence of an effect that could be either positive or negative, hence making our test 2-tailed.

Because we are estimating the mean and we have N=11 values in our sample, we have N-1=10 degrees of freedom. We set our significance level to 95% and compute the t statistic using the empirical mean and empirical standard deviation of our intake. We use a ddof of 1 to base the computation of our empirical standard deviation on an unbiased estimate of the variance (note: the final estimate is not unbiased due to the concave nature of the square root).

>>> np.mean(intake)
6753.636363636364
>>> intake.std(ddof=1)
1142.1232221373727
>>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake)))
>>> t
-2.8207540608310198

We draw 1000000 samples from Student's t distribution with the adequate degrees of freedom.

>>> import matplotlib.pyplot as plt
>>> s = np.random.standard_t(10, size=1000000)
>>> h = plt.hist(s, bins=100, density=True)

Does our t statistic land in one of the two critical regions found at both tails of the distribution?

>>> np.sum(np.abs(t) < np.abs(s)) / float(len(s))
0.018318  #random < 0.05, statistic is in critical region

The probability value for this 2-tailed test is about 1.83%, which is lower than the 5% pre-determined significance threshold.

Therefore, the probability of observing values as extreme as our intake conditionally on the null hypothesis being true is too low, and we reject the null hypothesis of no deviation.


Bases: object

class to analyze StoBe results







Draw samples from the triangular distribution over the interval [left, right].

The triangular distribution is a continuous probability distribution with lower limit left, peak at mode, and upper limit right. Unlike the other distributions, these parameters directly define the shape of the pdf.

NOTE:

New code should use the triangular method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Lower limit.
float or array_like of floats The value where the peak of the distribution occurs. The value must fulfill the condition left <= mode <= right.
float or array_like of floats Upper limit, must be larger than left.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if left, mode, and right are all scalars. Otherwise, np.broadcast(left, mode, right).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized triangular distribution.

Generator.triangular: which should be used for new code.

The probability density function for the triangular distribution is


P(x;l, m, r) = \begin{cases}
\frac{2(x-l)}{(r-l)(m-l)}& \text{for $l \leq x \leq m$},\\ \frac{2(r-x)}{(r-l)(r-m)}& \text{for $m \leq x \leq r$},\\ 0& \text{otherwise}. \end{cases}


The triangular distribution is often used in ill-defined problems where the underlying distribution is not known, but some knowledge of the limits and mode exists. Often it is used in simulations.

[1]
Wikipedia, "Triangular distribution" https://en.wikipedia.org/wiki/Triangular_distribution

Draw values from the distribution and plot the histogram:

>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.triangular(-3, 0, 8, 100000), bins=200,
...              density=True)
>>> plt.show()

Draw samples from a uniform distribution.

Samples are uniformly distributed over the half-open interval [low, high) (includes low, but excludes high). In other words, any value within the given interval is equally likely to be drawn by uniform.

NOTE:

New code should use the uniform method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats, optional Lower boundary of the output interval. All values generated will be greater than or equal to low. The default value is 0.
float or array_like of floats Upper boundary of the output interval. All values generated will be less than or equal to high. The default value is 1.0.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if low and high are both scalars. Otherwise, np.broadcast(low, high).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized uniform distribution.

randint : Discrete uniform distribution, yielding integers. random_integers : Discrete uniform distribution over the closed

interval [low, high].


random_sample : Floats uniformly distributed over [0, 1). random : Alias for random_sample. rand : Convenience function that accepts dimensions as input, e.g.,

rand(2,2) would generate a 2-by-2 array of floats, uniformly distributed over [0, 1).


Generator.uniform: which should be used for new code.

The probability density function of the uniform distribution is


p(x) = \frac{1}{b - a}



anywhere within the interval [a, b), and zero elsewhere.

When high == low, values of low will be returned. If high < low, the results are officially undefined and may eventually raise an error, i.e. do not rely on this function to behave when passed arguments satisfying that inequality condition. The high limit may be included in the returned array of floats due to floating-point rounding in the equation low + (high-low) * random_sample(). For example:

>>> x = np.float32(5*0.99999999)
>>> x
5.0

Draw samples from the distribution:

>>> s = np.random.uniform(-1,0,1000)

All values are within the given interval:

>>> np.all(s >= -1)
True
>>> np.all(s < 0)
True

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> count, bins, ignored = plt.hist(s, 15, density=True)
>>> plt.plot(bins, np.ones_like(bins), linewidth=2, color='r')
>>> plt.show()

groTrajecParser Parses an gromacs GRO-style file for the xyzBox class.

groTrajecParser Parses an gromacs GRO-style file for the xyzBox class.

Draw samples from a von Mises distribution.

Samples are drawn from a von Mises distribution with specified mode (mu) and dispersion (kappa), on the interval [-pi, pi].

The von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the unit circle. It may be thought of as the circular analogue of the normal distribution.

NOTE:

New code should use the vonmises method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Mode ("center") of the distribution.
float or array_like of floats Dispersion of the distribution, has to be >=0.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if mu and kappa are both scalars. Otherwise, np.broadcast(mu, kappa).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized von Mises distribution.

probability density function, distribution, or cumulative density function, etc.

Generator.vonmises: which should be used for new code.

The probability density for the von Mises distribution is


p(x) = \frac{e^{\kappa cos(x-\mu)}}{2\pi I_0(\kappa)},



where \mu is the mode and \kappa the dispersion, and I_0(\kappa) is the modified Bessel function of order 0.

The von Mises is named for Richard Edler von Mises, who was born in Austria-Hungary, in what is now the Ukraine. He fled to the United States in 1939 and became a professor at Harvard. He worked in probability theory, aerodynamics, fluid mechanics, and philosophy of science.

[1]
Abramowitz, M. and Stegun, I. A. (Eds.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing," New York: Dover, 1972.
[2]
von Mises, R., "Mathematical Theory of Probability and Statistics", New York: Academic Press, 1964.

Draw samples from the distribution:

>>> mu, kappa = 0.0, 4.0 # mean and dispersion
>>> s = np.random.vonmises(mu, kappa, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> from scipy.special import i0  
>>> plt.hist(s, 50, density=True)
>>> x = np.linspace(-np.pi, np.pi, num=51)
>>> y = np.exp(kappa*np.cos(x-mu))/(2*np.pi*i0(kappa))  
>>> plt.plot(x, y, linewidth=2, color='r')  
>>> plt.show()

Draw samples from a Wald, or inverse Gaussian, distribution.

As the scale approaches infinity, the distribution becomes more like a Gaussian. Some references claim that the Wald is an inverse Gaussian with mean equal to 1, but this is by no means universal.

The inverse Gaussian distribution was first studied in relationship to Brownian motion. In 1956 M.C.K. Tweedie used the name inverse Gaussian because there is an inverse relationship between the time to cover a unit distance and distance covered in unit time.

NOTE:

New code should use the wald method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Distribution mean, must be > 0.
float or array_like of floats Scale parameter, must be > 0.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if mean and scale are both scalars. Otherwise, np.broadcast(mean, scale).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized Wald distribution.

Generator.wald: which should be used for new code.

The probability density function for the Wald distribution is


P(x;mean,scale) = \sqrt{\frac{scale}{2\pi x^3}}e^
\frac{-scale(x-mean)^2}{2\cdotp mean^2x}


As noted above the inverse Gaussian distribution first arise from attempts to model Brownian motion. It is also a competitor to the Weibull for use in reliability modeling and modeling stock returns and interest rate processes.

[1]
Brighton Webs Ltd., Wald Distribution, https://web.archive.org/web/20090423014010/http://www.brighton-webs.co.uk:80/distributions/wald.asp
[2]
Chhikara, Raj S., and Folks, J. Leroy, "The Inverse Gaussian Distribution: Theory : Methodology, and Applications", CRC Press, 1988.
[3]
Wikipedia, "Inverse Gaussian distribution" https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution

Draw values from the distribution and plot the histogram:

>>> import matplotlib.pyplot as plt
>>> h = plt.hist(np.random.wald(3, 2, 100000), bins=200, density=True)
>>> plt.show()

Draw samples from a Weibull distribution.

Draw samples from a 1-parameter Weibull distribution with the given shape parameter a.


X = (-ln(U))^{1/a}



Here, U is drawn from the uniform distribution over (0,1].

The more common 2-parameter Weibull, including a scale parameter \lambda is just X = \lambda(-ln(U))^{1/a}.

NOTE:

New code should use the weibull method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Shape parameter of the distribution. Must be nonnegative.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized Weibull distribution.

scipy.stats.weibull_max scipy.stats.weibull_min scipy.stats.genextreme gumbel Generator.weibull: which should be used for new code.

The Weibull (or Type III asymptotic extreme value distribution for smallest values, SEV Type III, or Rosin-Rammler distribution) is one of a class of Generalized Extreme Value (GEV) distributions used in modeling extreme value problems. This class includes the Gumbel and Frechet distributions.

The probability density for the Weibull distribution is


p(x) = \frac{a}
{\lambda}(\frac{x}{\lambda})^{a-1}e^{-(x/\lambda)^a},


where a is the shape and \lambda the scale.

The function has its peak (the mode) at \lambda(\frac{a-1}{a})^{1/a}.

When a = 1, the Weibull distribution reduces to the exponential distribution.

[1]
Waloddi Weibull, Royal Technical University, Stockholm, 1939 "A Statistical Theory Of The Strength Of Materials", Ingeniorsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Forlag, Stockholm.
[2]
Waloddi Weibull, "A Statistical Distribution Function of Wide Applicability", Journal Of Applied Mechanics ASME Paper 1951.
[3]
Wikipedia, "Weibull distribution", https://en.wikipedia.org/wiki/Weibull_distribution

Draw samples from the distribution:

>>> a = 5. # shape
>>> s = np.random.weibull(a, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> x = np.arange(1,100.)/50.
>>> def weib(x,n,a):
...     return (a / n) * (x / n)**(a - 1) * np.exp(-(x / n)**a)
    

>>> count, bins, ignored = plt.hist(np.random.weibull(5.,1000))
>>> x = np.arange(1,100.)/50.
>>> scale = count.max()/weib(x, 1., 5.).max()
>>> plt.plot(x, weib(x, 1., 5.)*scale)
>>> plt.show()

writeFDMNESinput_file Writes an input file to be used for FDMNES.

writeFEFFinput_arb

writeWFN1input Writes an input for cp.x by Quantum espresso for electronic wave function minimization.

writeOCEAN_XESInput Writes an input for ONEAN XES calculation for 17 molecule water boxes.

writeOCEANinput

writeOCEANinput

Writes a complete OCEAN input file.
  • fname (str): Filename for the input file to be written.
  • xyzBox (xyzBox): Instance of the xyzBox class to be converted into an OCEAN input file.
  • exatom (str): Atomic symbol for the excited atom.
  • edge (int): Integer defining which shell to excite (e.g. 0 for K-shell, 1 for L, etc.).
  • subshell (int): Integer defining which sub-shell to excite ( e.g. 0 for s, 1 for p, etc.).



writeOCEANinput

writePWinuptFile


writeWFN1input Writes an input for cp.x by Quantum espresso for electronic wave function minimization.



Bases: object

xyzAtom

Class to hold information about and manipulate a single atom in xyz-style format.

  • name (str): Atomic symbol.
  • coordinates (np.array): Array of xyz-coordinates.
  • number (int): Integer, e.g. number of atom in a cluster.


get_angle Return angle between the three given atoms (as seen from atom2).











Bases: object

xyzBox

Class to hold information about and manipulate a xyz-periodic cubic box.

  • xyzAtoms (list): List of instances of the xyzAtoms class that make up the molecule.
  • boxLength (float): Box length.


changeOHBondlength Changes all OH covalent bond lengths inside the box by a fraction.


count_hbonds Counts the number of hydrogen bonds around all oxygen atoms and sets that number as attribute to the accorting xyzAtom.

count_neighbors

Counts number of neighbors (of name2) around atom of name1.

  • name1 (str): Name of first type of atom.
  • name2 (str): Name of second type of atom.
  • cutoff_low (float): Lower cutoff (Angstrom).
  • cutoff_high (float): Upper cutoff (Angstrom).
  • counter_name (str): Attribute namer under which the result should be saved.



deleteTip4pCOM Deletes the ficticious atoms used in the TIP4P water model.

CH3OH

CH3OH

find_hydroniums Returns a list of hydronium molecules.

find_hydroxides Returns a list of hydroxide molecules.

find_tmao_molecules Returns a list of TMAO molecules.

find_urea_molecules Returns a list of Urea molecules.

getCoordinates Return coordinates of all atoms in the cluster.

getDistVectorPBC_arb

Calculates the distance vector between two atoms from an arbitrary simulation box using the minimum image convention.

atom1 (obj): Instance of the xzyAtom class. atom2 (obj): Instance of the xzyAtom class.
The distance vector between the two atoms (np.array).


getDistancePBC_arb Calculates the distance of two atoms from an arbitrary simulation box using the minimum image convention.
atom1 (obj): Instance of the xzyAtom class. atom2 (obj): Instance of the xzyAtom class.
The distance between the two atoms.


getTetraParameter Returns a list of tetrahedrality paprameters, according to NATURE, VOL 409, 18 JANUARY (2001).

UNTESTED!!!


get_OO_neighbors Returns list of numbers of nearest oxygen neighbors within readius 'Roocut'.

get_OO_neighbors_pbc Returns a list of numbers of nearest oxygen atoms, uses periodic boundary conditions.

get_angle Return angle between the three given atoms (as seen from atom2).

get_angle Return angle between the three given atoms (as seen from atom2).

get_atoms_by_name Return a list of all xyzAtoms of a given name 'name'.

get_atoms_from_molecules Parses all atoms inside self.xyzMolecules into self.xyzAtoms (useful for turning an xyzMolecule into an xyzBox).

get_h2o_molecules Finds all water molecules inside the box and collects them inside the self.xyzMolecules attribute.


get_hbonds Counts the hydrogen bonds inside the box, returns the number of H-bond donors and H-bond acceptors.

multiplyBoxPBC Applies the periodic boundary conditions and multiplies the box in shells around the original.

multiplyBoxPBC_arb Applies the periodic boundary conditions and multiplies the box in shells around the original. Works with arbitrary lattices.



scatterPlot Opens a plot window with a scatter-plot of all coordinates of the box.

setBoxLength Set the box length.

translateAtomsMinimumImage

Brings back all atoms into the original box using periodic boundary conditions and minimal image convention.


writeBox Creates an xyz-style text file with all coordinates of the box.

writeXYZclusters Write water clusters into files.

writeXYZclusters Write water clusters into files.

writeFDMNESinput Creates an input file to be used for q-dependent calculations with FDMNES.

writeXYZclusters Write water clusters into files.

writeMoleculeCluster Careful, this works only for a single molecule in water.

writeOCEANinput Creates an OCEAN input file based on the headerfile.

writeRelBox Writes all relative atom coordinates into a text file (useful as OCEAN input).


Bases: object

xyzMolecule

Class to hold information about and manipulate an xyz-style molecule.

  • xyzAtoms (list): List of instances of the xyzAtoms class that make up the molecule.


appendAtom Add an xzyAtom to the molecule.

getCoordinates Return coordinates of all atoms in the cluster.

getCoordinates_name Return coordintes of all atoms with 'name'.

getGeometricCenter Return the geometric center of the xyz-molecule.


get_atoms_by_name Return a list of all xyzAtoms of a given name 'name'.

popAtom Delete an xyzAtom from the molecule.

scatterPlot Opens a plot window with a scatter-plot of all coordinates of the molecule.

translateAtomsMinimumImage

Brings back all atoms into the original box using periodic boundary conditions and minimal image convention.


translateSelf Translate all atoms of the molecule by a vector 'vector'.

writeXYZfile Creates an xyz-style text file with all coordinates of the molecule.


Parses a Trajectory of xyz-files.
filename (str): Filename of the xyz Trajectory file.
A list of xzyBoxes.


Bases: object







Draw samples from a Zipf distribution.

Samples are drawn from a Zipf distribution with specified parameter a > 1.

The Zipf distribution (also known as the zeta distribution) is a continuous probability distribution that satisfies Zipf's law: the frequency of an item is inversely proportional to its rank in a frequency table.

NOTE:

New code should use the zipf method of a default_rng() instance instead; please see the random-quick-start.


float or array_like of floats Distribution parameter. Must be greater than 1.
int or tuple of ints, optional Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if a is a scalar. Otherwise, np.array(a).size samples are drawn.

ndarray or scalar Drawn samples from the parameterized Zipf distribution.

probability density function, distribution, or cumulative density function, etc.

Generator.zipf: which should be used for new code.

The probability density for the Zipf distribution is


p(x) = \frac{x^{-a}}{\zeta(a)},



where \zeta is the Riemann Zeta function.

It is named for the American linguist George Kingsley Zipf, who noted that the frequency of any word in a sample of a language is inversely proportional to its rank in the frequency table.

[1]
Zipf, G. K., "Selected Studies of the Principle of Relative Frequency in Language," Cambridge, MA: Harvard Univ. Press, 1932.

Draw samples from the distribution:

>>> a = 2. # parameter
>>> s = np.random.zipf(a, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> from scipy import special

Truncate s values at 50 so plot is interesting:

>>> count, bins, ignored = plt.hist(s[s<50], 50, density=True)
>>> x = np.arange(1., 50.)
>>> y = x**(-a) / special.zetac(a)  
>>> plt.plot(x, y/max(y), linewidth=2, color='r')  
>>> plt.show()

XRStools.xrs_extraction Module

Bases: object

dataset A class to hold all information from HF Compton profiles necessary to subtract background from the experiment.





Bases: object

edge_extraction Class to destill core edge spectra from x-ray Raman scattering experiments.

analyzerAverage Averages signals from several crystals before background subtraction.

Args:

list, str list of ROI numbers to average over of keyword for analyzer chamber (e.g. 'VD','VU','VB','HR','HL','HB')

boolean (True by default) keyword if error weighing should be used for the averaging or not





removeCorePearsonAv

guess (list): [position, FWHM, shape, intensity, ax, b, scale ]


removeCorePearsonAv_new

removePearsonAv

removePolyCoreAv Subtract a polynomial from averaged data guided by the HF core Compton profile.

Args

  • element : str String (e.g. 'Si') for the element you want to work on.
  • edge: str String (e.g. 'K' or 'L23') for the edge to extract.
  • range1 : list List with start and end value for fit-region 1.
  • range2 : list List with start and end value for fit-region 2.
  • weigths : list of ints List with weights for the respective fit-regions 1 and 2. Default is [1,1].
  • guess : list List of starting values for the fit. Default is [1.0,0.0,0.0] (i.e. a quadratic function. Change the number of guess values to get other degrees of polynomials (i.e. [1.0, 0.0] for a constant, [1.0,0.0,0.0,0.0] for a cubic, etc.). The first guess value passed is for scaling of the experimental data to the HF core Compton profile.
  • ewindow: float Width of energy window used in the plot. Default is 100.0.




save_average_Sqw Save the S(q,w) into a ascii file (energy loss, S(q,w), Poisson errors).
  • filename : str Filename for the ascii file.
  • emin : float Use this to save only part of the spectrum.
  • emax : float Use this to save only part of the spectrum.
  • normrange : list of floats E_start and E_end for possible area-normalization before saving.




Bases: object


map_chamber_names Maps names of chambers to range of ROI numbers.

Bases: object

valence_CP Class to organize information about extracted experimental valence Compton profiles.




XRStools.xrs_ComptonProfiles Module

Bases: object

AtomProfile

Class to construct and handle Hartree-Fock atomic Compton Profile of a single atoms.

  • filename : string Path and filename to the HF profile table.
  • element : string Element symbol as in the periodic table.
  • elementNr : int Number of the element as in the periodic table.
  • shells : list of strings Names of the shells.
  • edges : list List of edge onsets (eV).
  • C_total : np.array Total core Compton profile.
  • J_total : np.array Total Compton profile.
  • V_total : np.array Total valence Compton profile.
  • CperShell : dict. of np.arrays Core Compton profile per electron shell.
  • JperShell : dict. of np.arrays Total Compton profile per electron shell.
  • VperShell : dict. of np.arrays Valence Compton profile per electron shell.
  • stoichiometry : float, optional Stoichiometric weight (default is 1.0).
  • atomic_weight : float Atomic weight.
  • atomic_density : float Density (g/cm**3).
  • twotheta : float Scattering angle 2Th (degrees).
  • alpha : float Incident angle (degrees).
  • beta : float Exit angle (degrees).
  • thickness : float Sample thickness (cm).


absorptionCorrectProfiles

Apply absorption correction to the Compton profiles on energy loss scale.

  • alpha :float Angle of incidence (degrees).
  • beta : float Exit angle for the scattered x-rays (degrees). If 'beta' is negative, transmission geometry is assumed, if 'beta' is positive, reflection geometry.
  • thickness : float Sample thickness.



get_elossProfiles Convert the HF Compton profile on to energy loss scale.

Args: E0 : float

Analyzer energy, enery of the scattered r-rays.


float or list of floats Scattering angle 2Th.
float, optional Scaling factor to be multiplied to the asymmetry.
float, optional Energy cut off as to what is considered the boundary between core and valence.




Bases: object

Class for multiple HF Compton profiles.

This class should hold one or more instances of the ComptonProfile class and have methods to return profiles from single atoms, single shells, all atoms. It should be able to apply corrections etc. on those...

  • element (string): Element symbol as in the periodic table.
  • elementNr (int) : Number of the element as in the periodic table.
  • shells (list) :
  • edges (list) :
  • C (np.array) :
  • J (np.array) :
  • V (np.array) :
  • CperShell (dict. of np.arrays):
  • JperShell (dict. of np.arrays):
  • VperShell (dict. of np.arrays):



Bases: object

FormulaProfile

Class to construct and handle Hartree-Fock atomic Compton Profile of a single chemical compound.

  • filename : string Path and filename to Biggs database.
  • formula : string Chemical sum formula for the compound of interest (e.g. 'SiO2' or 'H2O').
  • elements : list of strings List of atomic symbols that make up the chemical sum formula.
  • stoichiometries : list of integers List of the stoichimetric weights for each of the elements in the list elements.
  • element_Nrs : list of integers List of atomic numbers for each element in the elements list.
  • AtomProfiles : list of AtomProfiles List of instances of the AtomProfiles class for each element in the list.
  • eloss : np.ndarray Energy loss scale for the Compton profiles.
  • C_total : np.ndarray Core HF Compton profile (one column per 2Th).
  • J_total : np.ndarray Total HF Compton profile (one column per 2Th).
  • V_total :np.ndarray Valence HF Compton profile (one column per 2Th).
  • E0 : float Analyzer energy (keV).
  • twotheta : float, list, or np.ndarray Value or list/np.ndarray of the scattering angle.






Bases: object

HFProfile

Class to construct and handle Hartree-Fock atomic Compton Profile of sample composed of several chemical compounds.

Attributes



Returns the first order correction to filled 1s, 2s, and 2p Compton profiles.

Implementation after Holm and Ribberfors (citation ...).

  • pzprofile (np.array): Compton profile (e.g. tabulated from Biggs) to be corrected (2D matrix).
  • occupation (list): electron configuration.
  • q (float or np.array): momentum transfer in [a.u.].

  • asymmetry (np.array): asymmetries to be added to the raw profiles (normalized to the number of electrons on pz scale)



Returnes tabulated HF Compton profiles.

Reads in tabulated HF Compton profiles from the Biggs paper, interpolates them, and normalizes them to the # of electrons in the shell.

  • element (string): element symbol (e.g. 'Si', 'Al', etc.)
  • filename (string): absolute path and filename to tabulated profiles

  • CP_profile (np.array): Matrix of the Compton profile * 1. column: pz-scale * 2. ... n. columns: Compton profile of nth shell
  • binding_energy (list): binding energies of shells
  • occupation_num (list): number of electrons in the according shells



Bases: object

Class to build a S(q,w) prediction based on HF Compton Profiles.

Attributes:

  • sampleStr (list of strings): one string per compound (e.g. ['C','SiO2'])
  • concentrations (list of floats): relative compositional weight for each compound




Returns HF Compton profiles on energy loss scale.

Uses the PzProfile function to read read in Biggs HF profiles and converts them onto energy loss scale. The profiles are cut at the respective electron binding energies and are normalized to the f-sum rule (i.e. S(q,w) is in units of [1/eV]).

  • element (string): element symbol.
  • filename (string): absolute path and filename to tabulated Compton profiles.
  • E0 (float): analyzer energy in [keV].
  • tth (float): scattering angle two theta in [deg].
  • correctasym (np.array): vector of scaling factors to be applied.
  • valence_cutoff (float): energy value below which edges are considered as valence

  • enScale (np.array): energy loss scale in [eV]
  • J_total (np.array): total S(q,w) in [1/eV]
  • C_total (np.array): core contribution to S(q,w) in [1/eV]
  • V_total (np.array): valence contribution to S(q,w) in [1/eV], the valence is defined by valence_cutoff
  • q (np.array): momentum transfer in [a.u]
  • J_shell (dict of np.arrays): dictionary of contributions for each shell, the key are defines as in Biggs table.
  • C_shell (dict of np.arrays): same as J_shell for core contribution
  • V_shell (dict of np.arrays): same as J_shell for valence contribution



Returns the atomic density.

Returns the atomic weight.


mapShellNames

Translates to and from spectroscopic edge notation and the convention of the Biggs database.

  • shell_str : string Spectroscopic symbol to be converted to Biggs database convention.
  • atomicNumber : int Z for the atom in question.





XRStools.xrs_fileIO Module


Returns the EDF-data using FabIO.

Returns np.zeros of the shape of the detector.

Returns np.zeros for old data (horizontal and vertical Maxipix images in different files).

Returns the EDF-data using PyMCA.

Returns data, counter-names, and EDF-files using PyMCA.

Returns data, counter-names, and EDF-files using PyMCA.

Reads a series of EDF-images and returs them in a 3D Numpy array (horizontal and vertical Maxipix images in different files).

Reads a series of EDF-images and returs them in a 3D Numpy array (horizontal and vertical Maxipix images in different files).

Reads a series of EDF-images and returs them in a 3D Numpy array (horizontal and vertical Maxipix images in different files).

Reads a series of EDF-images and returs them in a 3D Numpy array (horizontal and vertical Maxipix images in different files).


Returns a scan stored in a Numpy archive.

Returns data, motors, counter-names, and labels using Silx.

Parses a SPEC file and returns a specified scan.
  • filename (string): SPEC file name (inlc. path)
  • nscan (int): Number of the desired scan.

  • data (np.array): array of the data from the specified scan.
  • motors (list): list of all motor positions from the header of the specified scan.
  • counters (dict): all counters in a dictionary with the counter names as keys.



Writes a scan into a Numpy archive.


Returns EDF-data, if PyMCA is not installed (this is slow).

Reads Hartree-Fock Profile of element 'element' from values tabulated by Biggs et al. (Atomic Data and Nuclear Data Tables 16, 201-309 (1975)) as provided by the DABAX library (http://ftp.esrf.eu/pub/scisoft/xop2.3/DabaxFiles/ComptonProfiles.dat). input: filename = path to the ComptonProfiles.dat file (the file should be distributed with this package) element = string of element name returns: data = the data for the according element as in the file:
#UD Columns: #UD col1: pz in atomic units #UD col2: Total compton profile (sum over the atomic electrons #UD col3,...coln: Compton profile for the individual sub-shells


occupation = occupation number of the according shells bindingen = binding energies of the accorting shells colnames = strings of column names as used in the file


XRStools.xrs_utilities Module

rotation around (1,0,0), pos sense

Returns the first order correction to filled 1s, 2s, and 2p Compton profiles.

Implementation after Holm and Ribberfors (citation ...).

  • pzprofile (np.array): Compton profile (e.g. tabulated from Biggs) to be corrected (2D matrix).
  • occupation (list): electron configuration.
  • q (float or np.array): momentum transfer in [a.u.].

asymmetry (np.array): asymmetries to be added to the raw profiles (normalized to the number of electrons on pz scale)


NNMFcost Returns cost and gradient for NNMF with constraints.


NNMFcost Returns cost and gradient for NNMF with constraints.

rotation around (0,0,1), pos sense

rotation around (0,1,0), neg sense

Rx Rotation matrix for vector rotations around the [1,0,0]-direction.
  • chi (float) : Angle of rotation.
  • degrees(bool) : Angle given in radians or degrees.

  • 3x3 rotation matrix.



Ry Rotation matrix for vector rotations around the [0,1,0]-direction.
  • phi (float) : Angle of rotation.
  • degrees(bool) : Angle given in radians or degrees.

  • 3x3 rotation matrix.



Rz Rotation matrix for vector rotations around the [0,0,1]-direction.
  • omega (float) : Angle of rotation.
  • degrees(bool) : Angle given in radians or degrees.

  • 3x3 rotation matrix.



TTsolver Solves the Takagi-Taupin equation for a bent crystal.

This function is based on a Matlab implementation by S. Huotari of M. Krisch's Fortran programs.

  • el_energy (float): Fixed nominal (working) energy in keV.
  • hkl (array): Reflection order vector, e.g. [6, 6, 0]
  • crystal (str): Crystal used (can be silicon 'Si' or 'Ge')
  • R (float): Crystal bending radius in m.
  • dev (np.array): Deviation parameter (in arc. seconds) for which the reflectivity curve should be calculated.
  • alpha (float): Crystal assymetry angle.

  • refl (np.array): Reflectivity curve.
  • e (np.array): Deviation from Bragg angle in meV.
  • dev (np.array): Deviation from Bragg angle in microrad.



absCorrection

Calculates absorption correction for given mu1 and mu2. Multiply the measured spectrum with this correction factor. This is a translation of Keijo Hamalainen's Matlab function (KH 30.05.96).

  • mu1 : np.array Absorption coefficient for the incident energy in [1/cm].
  • mu2 : np.array Absorption coefficient for the scattered energy in [1/cm].
  • alpha : float Incident angle relative to plane normal in [deg].
  • beta : float Exit angle relative to plane normal [deg].
  • samthick : float Sample thickness in [cm].
  • geometry : string, optional Key word for different sample geometries ('transmission', 'reflection', 'sphere'). If geometry is set to 'sphere', no angular dependence is assumed.

  • ac : np.array Absorption correction factor. Multiply this with your measured spectrum.



Calculates absorption correction for given mu1 and mu2. Multiply the measured spectrum with this correction factor.

This is a translation of Keijo Hamalainen's Matlab function (KH 30.05.96).

  • mu1 (np.array): absorption coefficient for the incident energy in [1/cm].
  • mu2 (np.array): absorption coefficient for the scattered energy in [1/cm].
  • alpha (float): incident angle relative to plane normal in [deg].
  • beta (float): exit angle relative to plane normal [deg] (for transmission geometry use beta < 0).
  • samthick (float): sample thickness in [cm].

  • ac (np.array): absorption correction factor. Multiply this with your measured spectrum.



# ADDCH Adds contents of given adjacent channels together # # [x2,y2] = addch(x,y,n,n0) # x = original x-scale (row or column vector) # y = original y-values (row or column vector) # n = number of channels to be summed up # n0 = offset for adding, default is 0 # x2 = new x-scale # y2 = new y-values # # KH 17.09.1990 # Modified 29.05.1995 to include offset

function [U,B,V]=bidiag_reduction(A) % [U B V]=bidiag_reduction(A) % Algorithm 6.5-1 in Golub & Van Loan, Matrix Computations % Johns Hopkins University Press % Finds an upper bidiagonal matrix B so that A=U*B*V' % with U,V orthogonal. A is an m x n matrix

bootstrapCNNMF Constrained non-negative matrix factorization with bootstrapping for error estimates.

bootstrapCNNMF Constrained non-negative matrix factorization with bootstrapping for error estimates.

% BRAGG Calculates Bragg angle for given reflection in RAD % output=bangle(hkl,e,xtal) % hkl can be a matrix i.e. hkl=[1,0,0 ; 1,1,1]; % e=energy in keV % xtal='Si', 'Ge', etc. (check dspace.m) or d0 (Si default) % % KH 28.09.93 %

Bases: object

Dynamical theory of diffraction.



Calculate polarization factor.




# BRAGGD Calculates Bragg angle for given reflection in deg # Call BRAGG.M # output=bangle(hkl,e,xtal) # hkl can be a matrix i.e. hkl=[1,0,0 ; 1,1,1]; # e=energy in keV # xtal='Si', 'Ge', etc. (check dspace.m) or d0 (Si default) # # KH 28.09.93


cixsUBfind




returns the Q0 given the detector position (tthv, tth) and th crystal orientation. This orientation is calculated considering :
this rotation is defined by psi which is a rotation around G






cixs_primo

cixs_secondo

cixs_terzo



cfactorizeOffDiaMatrix constrained version of factorizeOffDiaMatrix Returns main components from an off-diagonal Matrix (energy-loss x angular-departure).

constrained_nnmf Approximate singular value decomposition with constraints.

function [U, S, V] = constrained_svd(M,U_ini,S_ini,V_ini,U_up,max_iter=10000,verbose=False)


converts the old style EDF files (one image for horizontal and one image for vertical chambers) to the new style EDF (one single image).
converted.



Convolution with Gaussian x = x-vector y = y-vector fwhm = fulll width at half maximum of the gaussian with which y is convoluted

convtoprim converts diamond structure reciprocal lattice expressed in conventional lattice vectors to primitive one (Helsinki -> Palaiseau conversion) from S. Huotari

cshift Calculates Compton peak position.
  • w1 (float, array): Incident energy in [keV].
  • th (float): Scattering angle in [deg].

  • w2 (foat, array): Energy of Compton peak in [keV].


Funktion adapted from Keijo Hamalainen.


Calculates the Johann aberration of a spherical analyzer crystal.
E (float): Working energy in [eV]. A (float): Analyzer aperture [mm]. R (float): Radius of the Rowland circle [mm]. Theta (float): Analyzer Bragg angle [degree].
Johann abberation in [eV].


Calculates the intrinsic energy resolution of a diced crystal analyzer.
E (float): Working energy in [eV]. Dw (float): Darwin width of the used reflection [microRad]. Theta (float): Analyzer Bragg angle [degree].
Intrinsic energy resolution of a perfect analyzer crystal.


Calculates the off-Rowland contribution of a spherical analyzer crystal.
E (float): Working energy in [eV]. z (float): Off-Rowland distance [mm]. A (float): Analyzer aperture [mm]. R (float): Radius of the Rowland circle [mm]. Theta (float): Analyzer Bragg angle [degree].
Off-Rowland contribution in [eV] to the energy resolution.


Calculates the pixel size contribution to the resolution function of a diced analyzer crystal.
E (float): Working energy in [eV]. p (float): Pixel size in [mm]. R (float): Radius of the Rowland circle [mm]. Theta (float): Analyzer Bragg angle [degree].
Pixel size contribution in [eV] to the energy resolution for a diced analyzer crystal.


Calculates the source size contribution to the resolution function.
E (float): Working energy in [eV]. s (float): Source size in [mm]. R (float): Radius of the Rowland circle [mm]. Theta (float): Analyzer Bragg angle [degree].
Source size contribution in [eV] to the energy resolution.


Calculates the stress induced contribution to the resulution function of a spherically bent crystal analyzer.
E (float): Working energy in [eV]. t (float): Absorption length in the analyzer material [mm]. v (float): Poisson ratio of the analyzer material. R (float): Radius of the Rowland circle [mm]. Theta (float): Analyzer Bragg angle [degree].
Stress-induced contribution in [eV] to the energy resolution.


diode Calculates the number of photons incident for a Si PIPS diode.
  • current (float): Diode current in [pA].
  • energy (float): Photon energy in [keV].
  • thickness (float): Thickness of Si active layer in [cm].

  • flux (float): Number of photons per second.


Function adapted from Matlab function by S. Huotari.


% DSPACE Gives d-spacing for given xtal % d=dspace(hkl,xtal) % hkl can be a matrix i.e. hkl=[1,0,0 ; 1,1,1]; % xtal='Si','Ge','LiF','InSb','C','Dia','Li' (case insensitive) % if xtal is number this is user as a d0 % % KH 28.09.93 % SH 2005 %

Bases: object

class to hold all things dynamic theory of diffraction.






negative alpha -> more grazing incidence







Calculates the momentum scale and the relativistic Compton cross section correction according to P. Holm, PRA 37, 3706 (1988).

This function is translated from Keijo Hamalainen's Matlab implementation (KH 29.05.96).

  • w1 (float or np.array): incident energy in [keV]
  • w2 (float or np.array): scattered energy in [keV]
  • th (float): scattering angle two theta in [deg]

  • pz (float or np.array): momentum scale in [a.u.]
  • cf (float or np.array): cross section correction factor such that: J(pz) = cf * d^2(sigma)/d(w2)*d(Omega) [barn/atom/keV/srad]



reads edf-file with filename "filename" OUTPUT: data = 256x256 numpy array

reads edf-file with filename "filename" OUTPUT: data = 256x256 numpy array

here is how i opened the HH data: data = np.fromfile(f,np.int32) image = np.reshape(data,(dim,dim))


Converts atomic number into string of the element symbol and vice versa.

Returns atomic number of given element, if z is a string of the element symbol or string of element symbol of given atomic number z.

  • z (string or int): string of the element symbol or atomic number.

  • Z (string or int): string of the element symbol or atomic number.



% ENERGY Calculates energy corrresponing to Bragg angle for given d-spacing % function e=energy(dspace,bragg_angle) % % dspace for reflection % bragg_angle in DEG % % KH 28.09.93

% ENERGY Calculates energy corrresponing to Bragg angle for given d-spacing % function e=energy(dspace,bragg_angle) % % dspace for reflection (defaulf for Si(311) reflection) % bragg_angle in DEG % % KH 28.09.93 %

fermi Calculates the plasmon energy (in eV), Fermi energy (in eV), Fermi momentum (in a.u.), and critical plasmon cut-off vector (in a.u.).
  • rs (float): electron separation parameter

  • wp (float): plasmon energy (in eV)
  • ef (float): Fermi energy (in eV)
  • kf (float): Fermi momentum (in a.u.)
  • kc (float): critical plasmon cut-off vector (in a.u.)


Based on Matlab function from A. Soininen.


Returns the center of mass (first moment) for the given curve y(x)

find_diag_angles Finds the FOURC spectrometer and sample angles for a desired q.
  • q (array): Desired momentum transfer in Lab coordinates.
  • x0 (list): Guesses for the angles (tthv, tthh, chi, phi, omega).
  • U (array): 3x3 U-matrix Lab-to-sample transformation.
  • B (array): 3x3 B-matrix reciprocal lattice to absolute units transformation.
  • lambdai (float): Incident x-ray wavelength in Angstrom.
  • lambdao (float): Scattered x-ray wavelength in Angstrom.
  • tol (float): Toleranz for minimization (see scipy.optimize.minimize)
  • method (str): Method for minimization (see scipy.optimize.minimize)

  • ans (array): tthv, tthh, phi, chi, omega



finds full width at half maximum of the curve y vs. x returns f = FWHM x0 = position of the maximum


get_UB_Q Returns the momentum transfer and scattering vectors for given FOURC spectrometer and sample angles. U-, B-matrices and incident/scattered wavelength are passed as keyword-arguments.
  • tthv (float): Spectrometer vertical 2Theta angle.
  • tthh (float): Spectrometer horizontal 2Theta angle.
  • chi (float): Sample rotation around x-direction.
  • phi (float): Sample rotation around y-direction.
  • omega (float): Sample rotation around z-direction.
  • kwargs['U'] (array): 3x3 U-matrix Lab-to-sample transformation.
  • kwargs['B'] (array): 3x3 B-matrix reciprocal lattice to absolute units transformation.
  • kwargs['lambdai'] (float): Incident x-ray wavelength in Angstrom.
  • kwargs['lambdao'] (float): Scattered x-ray wavelength in Angstrom.



  • Q_sample (array): Momentum transfer in sample coordinates.
  • Ki_sample (array): Incident beam direction in sample coordinates.
  • Ko_sample (array): Scattered beam direction in sample coordinates.



get_gnuplot_rgb Prints out a progression of RGB hex-keys to use in Gnuplot.
  • start (array): RGB code to start from (must be numbers out of [0,1]).
  • end (array): RGB code to end at (must be numbers out of [0,1]).
  • length (int): How many colors to print out.



Calculates the number of steps in an MD simulation for a desired time (in ps) and given step size (in a.u.)
time_ps (float): Desired time span (ps). time_step (float): Chosen time step (a.u.).
The number of steps required to span the desired time span.


returns the penetration depth of a mixture of chemical formulas with certain concentrations and densities

returns the transmission through a sample composed of chemical formulas with certain densities mixed to certain concentrations, and a thickness


hlike_Rwfn Returns an array with the radial part of a hydrogen-like wave function.
  • n (integer): main quantum number n
  • l (integer): orbitalquantum number l
  • r (array): vector of radii on which the function should be evaluated
  • Z (float): effective nuclear charge



function H = householder(b, k) % H = householder(b, k) % Atkinson, Section 9.3, p. 611 % b is a column vector, k an index < length(b) % Constructs a matrix H that annihilates entries % in the product H*b below index k

% $Id: householder.m,v 1.1 2008-01-16 15:33:30 mike Exp $ % M. M. Sussman


Linear interpolation scheme after Martin Sundermann that conserves the absolute number of counts.

ONLY WORKS FOR EQUALLY/EVENLY SPACED XC, XI!

xc (np.array): The x-coordinates of the interpolated values. xi (np.array): The x-coordinates of the data points, must be increasing. yi (np.array): The y-coordinates of the data points, same length as xp. i0 (np.array): Normalization values for the data points, same length as xp.
ic (np.array): The interpolated and normalized data points.



from scipy.interpolate import Rbf x = arange(20) d = zeros(len(x)) d[10] = 1 xc = arange(0.5,19.5) rbfi = Rbf(x, d) di = rbfi(xc)


is_allowed_refl_fcc Check if given reflection is allowed for a FCC lattice.
  • H (array, list, tuple): H=[h,k,l]

  • boolean



lindhard_pol Calculates the Lindhard polarizability function (RPA) for certain q (a.u.), w (a.u.) and rs (a.u.).
  • q (float): momentum transfer (in a.u.)
  • w (float): energy (in a.u.)
  • rs (float): electron parameter
  • use_corr (boolean): if True, uses Bernardo's calculation for n(k) instead of the Fermi function.
  • lifetime (float): life time (default is 0.28 eV for Na).


Based on Matlab function by S. Huotari.


takes the profiles from 'makepzprofile()', converts them onto eloss scale and normalizes them to S(q,w) [1/eV] input: element = element symbol (e.g. 'Si', 'Al', etc.) filename = path and filename to tabulated profiles E0 = scattering energy [keV] tth = scattering angle [deg] returns: enscale = energy loss scale J = total CP C = only core contribution to CP V = only valence contribution to CP q = momentum transfer [a.u.]

returns the compton profile of a chemical compound with formula 'formula' input: formula = string of a chemical formula (e.g. 'SiO2', 'Ba8Si46', etc.) filename = path and filename to tabulated profiles E0 = scattering energy [keV] tth = scattering angle [deg] returns: eloss = energy loss scale J = total CP C = only core contribution to CP V = only valence contribution to CP q = momentum transfer [a.u.]

returns sum of compton profiles from a lost of chemical compounds weighted by the given concentration

constructs compton profiles of element 'element' on pz-scale (-100:100 a.u.) from the Biggs tables provided in 'filename'
  • element = element symbol (e.g. 'Si', 'Al', etc.)
  • filename = path and filename to tabulated profiles

  • pzprofile = numpy array of the CP: * 1. column: pz-scale * 2. ... n. columns: compton profile of nth shell * binden = binding energies of shells * occupation = number of electrons in the according shells





Bases: object

Class to store some useful values from the detectors used. To be used for arranging the ROIs.




Calculates the momentum transfer in atomic units input: e1 = incident energy [keV] e2 = scattered energy [keV] tth = scattering angle [deg] returns: q = momentum transfer [a.u.] (corresponding to sin(th)/lambda)

Calculates the momentum transfer in inverse angstrom input: e1 = incident energy [keV] e2 = scattered energy [keV] tth = scattering angle [deg] returns: q = momentum transfer [a.u.] (corresponding to sin(th)/lambda)

Calculates the photoelectric, elastic, and inelastic absorption of a chemical compound.

Calculates the photoelectric, elastic, and inelastic absorption of a chemical compound.

  • energy (np.array): energy scale in [keV].
  • compound (string): chemical sum formula (e.g. 'SiO2')

  • murho (np.array): absorption coefficient normalized by the density.
  • rho (float): density in UNITS?
  • m (float): atomic mass in UNITS?



Calculates the photoelectric, elastic, and inelastic absorption of a mix of compounds.

Returns the photoelectric absorption for a sum of different chemical compounds.

  • energy (np.array): energy scale in [keV].
  • formulas (list of strings): list of chemical sum formulas

  • murho (np.array): absorption coefficient normalized by the density.
  • rho (float): density in UNITS?
  • m (float): atomic mass in UNITS?



Calculates the photoelectric, elastic, and inelastic absorption of an element Z

Calculates the photelectric , elastic, and inelastic absorption of an element Z. Z can be atomic number or element symbol.

  • energy (np.array): energy scale in [keV].
  • Z (string or int): atomic number or string of element symbol.

  • murho (np.array): absorption coefficient normalized by the density.
  • rho (float): density in UNITS?
  • m (float): atomic mass in UNITS?




#% [T,Y] = ODE23(ODEFUN,TSPAN,Y0,OPTIONS,P1,P2,...) passes the additional #% parameters P1,P2,... to the ODE function as ODEFUN(T,Y,P1,P2...), and to #% all functions specified in OPTIONS. Use OPTIONS = [] as a place holder if #% no options are set.


Parses a chemical sum formula.

Parses the constituing elements and stoichiometries from a given chemical sum formula.

  • formula (string): string of a chemical formula (e.g. 'SiO2', 'Ba8Si46', etc.)

  • elements (list): list of strings of constituting elemental symbols.
  • stoichiometries (list): list of according stoichiometries in the same order as 'elements'.



opens a plot window of the penetration depth of a mixture of chemical formulas with certain concentrations and densities plotted along the given energy vector

opens a plot with the transmission plotted along the given energy vector

primtoconv converts diamond structure reciprocal lattice expressed in primitive basis to the conventional basis (Palaiseau -> Helsinki conversion) from S. Huotari

Calculates the incident energy for a specific scattered photon and momentum value.

Returns the incident energy for a given photon energy and scattering angle. This function is translated from Keijo Hamalainen's Matlab implementation (KH 29.05.96).

  • w2 (float): scattered photon energy in [keV]
  • pz (np.array): pz scale in [a.u.]
  • th (float): scattering angle two theta in [deg]

  • w1 (np.array): incident energy in [keV]



read_dft_wfn Parses radial parts of wavefunctions.
  • element (str): Element symbol.
  • n (int): Main quantum number.
  • l (int): Orbital quantum number.
  • spin (str): Which spin channel, default is average over up and down.
  • directory (str): Path to directory where the wavefunctions can be found.

  • r (np.array): radius
  • wfn (np.array):



Reads Hartree-Fock Profile of element 'element' from values tabulated by Biggs et al. (Atomic Data and Nuclear Data Tables 16, 201-309 (1975)) as provided by the DABAX library (http://ftp.esrf.eu/pub/scisoft/xop2.3/DabaxFiles/ComptonProfiles.dat). input: filename = path to the ComptonProfiles.dat file (the file should be distributed with this package) element = string of element name returns:
  • #UD Columns:
  • #UD col1: pz in atomic units
  • #UD col2: Total compton profile (sum over the atomic electrons
  • #UD col3,...coln: Compton profile for the individual sub-shells


  • occupation = occupation number of the according shells
  • bindingen = binding energies of the accorting shells
  • colnames = strings of column names as used in the file




if repnumber = 0: reads a spectra-file (name: prefix_scannumber.fio) if repnumber > 1: reads a spectra-file (name: prefix_scannumber_rrepnumber.fio)

reads a detector file from PetraIII beamline P01

reads a whole scan from PetraIII beamline P01 (experimental)

reads a whole scan with repititions from PetraIII beamline P01 (experimental)

Smooth (and optionally differentiate) data with a Savitzky-Golay filter. The Savitzky-Golay filter removes high frequency noise from data. It has the advantage of preserving the original shape and features of the signal better than other types of filtering approaches, such as moving averages techniques.
  • y : array_like, shape (N,) the values of the time history of the signal.
  • window_size : int the length of the window. Must be an odd integer number.
  • order : int the order of the polynomial used in the filtering. Must be less then window_size - 1.
  • deriv: int the order of the derivative to compute (default = 0 means only smoothing)

  • ys : ndarray, shape (N) the smoothed signal (or it's n-th derivative).

The Savitzky-Golay is a type of low-pass filter, particularly suited for smoothing noisy data. The main idea behind this approach is to make for each point a least-square fit with a polynomial of high order over a odd-sized window centered at the point.

Examples

t = np.linspace(-4, 4, 500)
y = np.exp( -t**2 ) + np.random.normal(0, 0.05, t.shape)
ysg = savitzky_golay(y, window_size=31, order=4)
import matplotlib.pyplot as plt
plt.plot(t, y, label='Noisy signal')
plt.plot(t, np.exp(-t**2), 'k', lw=1.5, label='Original signal')
plt.plot(t, ysg, 'r', label='Filtered signal')
plt.legend()
plt.show()


[1]
A. Savitzky, M. J. E. Golay, Smoothing and Differentiation of Data by Simplified Least Squares Procedures. Analytical Chemistry, 1964, 36 (8), pp 1627-1639.
[2]
Numerical Recipes 3rd Edition: The Art of Scientific Computing W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery Cambridge University Press ISBN-13: 9780521880688



sigmainc Calculates the Incoherent Scattering Cross Section in cm^2/g using Log-Log Fit.
  • z (int or string): Element number or elements symbol.
  • energy (float or array): Energy (can be number or vector)

  • tau (float or array): Photoelectric cross section in [cm**2/g]


Adapted from original Matlab function of Keijo Hamalainen.


reads scan "nscan" from SPEC-file "filename"
  • filename = string with the SPEC-file name
  • nscan = number (int) of desired scan

  • data =
  • motors =
  • counters = dictionary



Extrapolates the smaller and larger valuea as a constant


stiff_compl_matrix_Si Returns stiffnes and compliance tensor of Si for a given orientation.
  • e1 (np.array): unit vector normal to crystal surface
  • e2 (np.array): unit vector crystal surface
  • e3 (np.array): unit vector orthogonal to e2

  • S (np.array): compliance tensor in new coordinate system
  • C (np.array): stiffnes tensor in new coordinate system
  • E (np.array): Young's modulus in [GPa]
  • G (np.array): shear modulus in [GPa]
  • nu (np.array): Poisson ratio


Copied from S.I. of L. Zhang et al. "Anisotropic elasticity of silicon and its application to the modelling of X-ray optics." J. Synchrotron Rad. 21, no. 3 (2014): 507-517.


Short-hand command to sum over 1st dimension of a N-D matrix (N>2) and to squeeze it to N-1-D matrix.


% TAUPGEN Calculates the reflectivity curves of bent crystals % % function [refl,e,dev]=taupgen_new(e,hkl,crystals,R,dev,alpha); % % e = fixed nominal energy in keV % hkl = reflection order vector, e.g. [1 1 1] % crystals = crystal string, e.g. 'si' or 'ge' % R = bending radius in meters % dev = deviation parameter for which the % curve will be calculated (vector) (optional) % alpha = asymmetry angle % based on a FORTRAN program of Michael Krisch % Translitterated to Matlab by Simo Huotari 2006, 2007 % Is far away from being good matlab writing - mostly copy&paste from % the fortran routines. Frankly, my dear, I don't give a damn. % Complaints -> /dev/null

% TAUPGEN Calculates the reflectivity curves of bent crystals % % function [refl,e,dev]=taupgen_new(e,hkl,crystals,R,dev,alpha); % % e = fixed nominal energy in keV % hkl = reflection order vector, e.g. [1 1 1] % crystals = crystal string, e.g. 'si' or 'ge' % R = bending radius in meters % dev = deviation parameter for which the % curve will be calculated (vector) (optional) % alpha = asymmetry angle % based on a FORTRAN program of Michael Krisch % Translitterated to Matlab by Simo Huotari 2006, 2007 % Is far away from being good matlab writing - mostly copy&paste from % the fortran routines. Frankly, my dear, I don't give a damn. % Complaints -> /dev/null

tauphoto Calculates Photoelectric Cross Section in cm^2/g using Log-Log Fit.
  • z (int or string): Element number or elements symbol.
  • energy (float or array): Energy (can be number or vector)

  • tau (float or array): Photoelectric cross section in [cm**2/g]


Adapted from original Matlab function of Keijo Hamalainen.


unconstrained_mf Returns main components from an off-diagonal Matrix (energy-loss x angular-departure), using the power method iteratively on the different main components.

vangle Calculates the angle between two cartesian vectors v1 and v2 in degrees.
  • v1 (np.array): first vector.
  • v2 (np.array): second vector.

  • th (float): angle between first and second vector.


Function by S. Huotari, adopted for Python.



vrot Rotates a vector around a given axis.
  • v (np.array): vector to be rotated
  • vaxis (np.array): rotation axis
  • phi (float): angle [deg] respecting the right-hand rule

  • v2 (np.array): new rotated vector


Function by S. Huotari (2007) adopted to Python.


rotMatrix Rotate vector1 around vector2 by an angle.

xas_fluo_correct Fluorescence yield over-absorption correction as in Larch/Athena. see: https://www3.aps.anl.gov/haskel/FLUO/Fluo-manual.pdf
  • ene (np.array): energy axis in [keV]
  • mu (np.array): measured fluorescence spectrum
  • formula (str): chemical sum formulas (e.g. 'SiO2')
  • fluo_ene (float): energy in keV of main fluorescence line
  • edge_ene (float): edge energy in [keV]
  • angin (float): incidence angle (relative to sample normal) [deg.]
  • angout (float): exit angle (relative to sample normal) [deg.]

  • ene (np.array): energy axis in [keV]
  • mu_corr (np.array): corrected fluorescence spectrum



XRStools.roifinder_and_gui Module

  • genindex
  • modindex
  • search

Christoph Sahle, Alessandro Mirone

2022, Christoph Sahle, Alessandro Mirone